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	diff --git a/admin/site/content/submission.md b/admin/site/content/submission.md
	--- a/admin/site/content/submission.md
	+++ b/admin/site/content/submission.md
	@@ -1,65 +1,66 @@
	---
	title: Entry Submission
	menu:
	main:
	name: "Submission"
	weight: 5
	rss: false
	---

	## Submission Guidelines

	The submission must follow the following Isabelle style rules. For additional guidelines on Isabelle proofs, also see the this [guide](https://proofcraft.org/blog/isabelle-style.html) (feel free to follow all of these; only the below are mandatory). Technical details about the submission process and the format of the submission are explained on the submission site.

	* No use of the commands `sorry` or `back`.
	* Instantiations must not use Isabelle-generated names such as `xa` — use Isar, the `subgoal` command or `rename_tac` to avoid such names.
	* No use of the command `smt_oracle`.
	* If your theories contain calls to `nitpick`, `quickcheck`, or `nunchaku` those calls must include the `expect` parameter. Alternatively the `expect` parameter must be set globally via, e.g. `nitpick_params`.
	* `apply` scripts should be indented by subgoal as in the Isabelle distribution. If an `apply` command is applied to a state with `n+1` subgoals, it must be indented by `n` spaces relative to the first `apply` in the sequence.
	* Only named lemmas should carry attributes such as `[simp]`.
	* We prefer structured Isar proofs over apply style, but do not mandate them.
	* If there are proof steps that take significant time, i.e. longer than roughly 1 min, please add a short comment to that step, so maintainers will know what to expect.
	* The entry must contain a ROOT file with one session that has the name of the entry. We strongly encourage precisely one session per entry, but exceptions can be made. All sessions must be in chapter AFP, and all theory files of the submission must be contained in at least one session. See also the example [ROOT](https://foss.heptapod.net/isa-afp/afp-2020/-/blob/branch/default/thys/Example-Submission/ROOT) file in the [Example submission](/entries/Example-Submission.html).
	* The entry should cite all sources that the theories are based on, for example textbooks or research articles containing informal versions of the proofs.
	+* All sessions must have a timeout after which they are assumed to be in a non-terminating state. Timeouts may be no finer than five-minute increments, i.e. must be divisible by 300.

	Your submission must contain an abstract to be displayed on the web site – usually this will be the same as the abstract of your proof document in the root.tex file. You can use LaTeX formulae in this web site abstract, either inline formulae in the form $a+b$ or \$a+b\$ or display formulae in the form $$a + b$$ or \\\[a + b\\\]. Other occurrences of these characters must be escaped (e.g. \\$ or \\\\(). Note that LaTeX in the title of an entry is _not_ allowed. Most basic LaTeX functionality should be supported. For details on what parts of LaTeX are supported, see the [MathJax documentation.](https://docs.mathjax.org/en/v2.7-latest/tex.html)

	It is possible and encouraged to build on other archive entries in your submission. There is a standardised way to [refer to other AFP entries](/help) in your theories.

	## Submission Form

	[Submission form](/webapp/submit)

	Your submission will be refereed and you will receive notification as soon as possible. If accepted, you must agree to maintain your archive entry or nominate someone else to maintain it. The Isabelle development team will assist with maintenance, but it does not have the resources to fully maintain the complete archive.

	If you have questions regarding your submission, please email [afp-submit@in.tum.de](mailto:afp-submit@in.tum.de). If you need help with Isabelle, please use the [isabelle-users@cl.cam.ac.uk](mailto:isabelle-users@cl.cam.ac.uk) mailing list. It is always a good idea to [subscribe](https://lists.cam.ac.uk/mailman/listinfo/cl-isabelle-users).

	# Updating Entries

	## Change

	The Archive of Formal Proofs is an online resource and therefore more dynamic than a normal scientific journal. Existing entries can and do evolve and can also be updated significantly by their authors.

	This conflicts with the purpose of archiving and preserving entries as they have been submitted and with the purpose of providing a clear and simple interface to readers.

	The AFP deals with this by synchronizing such updates with Isabelle releases:

	* The entries released and visible on the main site are always working with the most recent stable Isabelle version and do not change.
	* In the background, the archive maintainers evolve all entries to be up to date with the current Isabelle development version. Authors can contribute changes to this version which is available as a [Heptapod mercurial repository](https://foss.heptapod.net/isa-afp/afp-devel/) or as tar.gz package on the [download page](/download).
	* When a new Isabelle version is released, the above mentioned development version of AFP is frozen and turns into the main version displayed on the front page. Older versions (including the original submission) of all entries are archived and remain accessible.

	Significant changes of an entry should be recorded in the metadata of the entry using the keyword "extra-history". The resulting web page should look [something like this](https://www.isa-afp.org/entries/JinjaThreads.html).

	## Monotonicity

	Updating an entry should be mostly monotone: you add new material, but you do not modify existing material in a major way. Ideally, entries (by other people) that build on yours should not be affected. Otherwise you have to liaise with them first. If you intend to carry out major non-monotone changes, you will need to submit a completely new entry (with a description of how it relates to the old one). This should be required only very rarely: AFP entries should be mature enough not to require major changes to their interface (i.e. the main functions and theorems provided).

	Major monotone changes, e.g. adding a new concept rather than more results on existing concepts, may also call for a new entry, but one that builds on the existing one. This depends on how you would like to organize your entries.

	## If you are an Author

	The above means that if you are an author and would like to provide a new, better version of your AFP entry, you can do so.

	To achieve this, you should base your changes on the [mercurial development version](https:/foss.heptapod.net/isa-afp/afp-devel/) of your AFP entry and test it against the current [Isabelle development version](https://isabelle.in.tum.de/devel/).

	If you would like to get write access to your entry in the mercurial repository or if you need assistance, please contact the [editors](/about#editors).
	diff --git a/admin/site/themes/afp/static/js/theory.js b/admin/site/themes/afp/static/js/theory.js
	--- a/admin/site/themes/afp/static/js/theory.js
	+++ b/admin/site/themes/afp/static/js/theory.js
	@@ -1,279 +1,283 @@
	/* constants */

	const ID_THEORY_LIST = 'theories'
	const CLASS_LOADER = 'loader'
	const CLASS_ANIMATION = 'animation'
	const ATTRIBUTE_THEORY_SRC = 'theory-src'
	const CLASS_NAVBAR_TYPE = 'theory-navbar-type'
	const CLASS_THY_NAV = 'thy-nav'
	const PARAM_NAVBAR_TYPE = 'theory-navbar-type'
	const ID_NAVBAR_TYPE_SELECTOR = 'navbar-type-selector'
	const ID_NAVBAR = 'theory-navbar'
	const NAVBAR_TYPES = ['fact', 'type', 'const']


	/* routing */

	function target(base_href, rel_href) {
	const href_parts = rel_href.split('/')

	- if (href_parts.length === 1) return `#${href_parts[0]}`
	- else if (href_parts.length === 3 && href_parts[0] === '..' && href_parts[1] !== '..') {
	- return `../${href_parts[1].toLowerCase()}/#${href_parts[2]}`
	+ if (href_parts.length === 1) {
	+ const ref_parts = href_parts[0].split('.html')
	+ const file_parts = ref_parts[0].split('.')
	+ if (file_parts.length == 1) return `#${href_parts[0]}`
	+ else {
	+ return `../${file_parts[0].toLowerCase()}/#${file_parts.slice(1).join('.')}.html${ref_parts[1]}`
	+ }
	}
	else return `${base_href}/../${rel_href}`
	}

	function to_id(thy_name, ref) {
	if (ref) return `${thy_name}.html#${ref}`
	else return `${thy_name}.html`
	}

	const to_fresh_id = (id) => `${id}#`
	const to_svg_id = (id) => `${id}#svg`
	const to_container_id = (id) => `${id}#container`
	const to_collapsible_id = (id) => `${id}#collapsible`
	const to_spinner_id = (id) => `${id}#spinner`
	const to_nav_id = (id) => `${id}#nav`
	const to_ul_id = (id) => `${id}#ul`
	const of_ul_id = (id) => id.split('#').slice(0, -1).join('#')
	const to_a_id = (id) => `${id}#a`

	/* document translation */

	function translate(base_href, thy_name, thy_body) {
	const thy_refs = [...thy_body.getElementsByTagName('span')].map((span) => {
	let ref = span.getAttribute('id')
	if (ref) {
	span.setAttribute('id', to_fresh_id(to_id(thy_name, ref)))
	}
	return ref
	}).filter(e => e)
	for (const link of thy_body.getElementsByTagName('a')) {
	const rel_href = link.getAttribute('href')
	link.setAttribute('href', target(base_href, rel_href))
	}
	return thy_refs
	}


	/* theory lazy-loading */

	async function fetch_theory_body(href) {
	const html_str = await fetch(href).then((http_res) => {
	if (http_res.status !== 200) return Promise.resolve(`<body>${http_res.statusText}</body>`)
	else return http_res.text()
	}).catch((_) => {
	console.log(`Could not load theory at '${href}'. Redirecting...`)
	window.location.replace(href)
	})

	const parser = new DOMParser()
	const html = parser.parseFromString(html_str, 'text/html')
	return html.getElementsByTagName('body')[0]
	}

	async function load_theory(thy_name, href) {
	const thy_body = await fetch_theory_body(href)
	const refs = translate(href, thy_name, thy_body)

	const collapse = document.getElementById(to_collapsible_id(thy_name))
	collapse.append(...Array(...thy_body.children).slice(1))
	return refs
	}

	async function open_theory(thy_name) {
	const container = document.getElementById(to_container_id(thy_name))

	if (container) {
	if (document.getElementById(to_collapsible_id(thy_name))) open(container)
	else {
	const collapsible = parse_elem(`
	<div id="${to_collapsible_id(thy_name)}" class="${CLASS_COLLAPSIBLE}">
	<div id="${to_spinner_id(thy_name)}" class=${CLASS_LOADER}><div class=${CLASS_ANIMATION}></div></div>
	</div>`)
	container.appendChild(collapsible)
	open(container)
	let refs = await load_theory(thy_name, container.getAttribute(ATTRIBUTE_THEORY_SRC))
	await load_theory_nav(thy_name, refs)
	const spinner = document.getElementById(to_spinner_id(thy_name))
	spinner.parentNode.removeChild(spinner)
	}
	}
	}

	function nav_tree_rec(thy_name, path, key, ref_parts, type) {
	const rec_ref = ref_parts.filter(e => e.length > 0)
	const ref = `${path.join('.')}.${key}\|${type}`
	const id = to_id(thy_name, ref)
	let res
	if (rec_ref.length < ref_parts.length) {
	res = `<a id="${to_a_id(id)}" class="${CLASS_SPY_LINK}" href="#${id}">${escape_html(key)}</a>`
	} else {
	const head_id = to_id(thy_name, `${[...path, key, ...ref_parts[0]].join('.')}\|${type}`)
	res = `<a id="${to_a_id(id)}" class="${CLASS_SPY_LINK}" href="#${head_id}">${escape_html(key)}</a>`
	}

	if (rec_ref.length > 1) {
	const by_key = group_by(rec_ref)
	const children = Object.keys(by_key).map((key1) => `
	<li>${nav_tree_rec(thy_name, [...path, key], key1, by_key[key1], type)}</li>`)
	return `
	${res}
	<ul id="${to_ul_id(id)}" class="${CLASS_COLLAPSIBLE} ${CLASS_COLLAPSED}">
	${children.join('')}
	</ul>`
	} else return res
	}

	function nav_tree(thy_name, refs, type) {
	let trees = Object.entries(group_by(refs \|\| [])).map(([key, parts]) =>
	`<li>${nav_tree_rec(thy_name, [thy_name], key, parts, type)}</li>`)

	return parse_elem(`
	<ul id="${to_ul_id(thy_name)}" class="${CLASS_NAVBAR_TYPE} ${CLASS_COLLAPSIBLE} ${CLASS_COLLAPSED}">
	${trees.join('')}
	</ul>`)
	}


	const cached_refs = Object.fromEntries(NAVBAR_TYPES.map(t => [t, {}]))
	const load_theory_nav = (thy_name, refs) => {
	let selected = get_query(PARAM_NAVBAR_TYPE) \|\| NAVBAR_TYPES[0]

	let by_type = group_by(refs.filter(ref => ref.includes('\|')).map((id) => id.split('\|').reverse()))
	let type_selector = document.getElementById(ID_NAVBAR_TYPE_SELECTOR)
	let options = [...type_selector.options].map(e => e.value)

	for (let [type, elems] of Object.entries(by_type)) {
	if (NAVBAR_TYPES.includes(type) && !options.includes(type)) {
	type_selector.appendChild(parse_elem(`<option value=${type}>${type}</option>`))
	}

	let parts_by_thy = group_by(elems.map((s) => s[0].split('.')))
	if (NAVBAR_TYPES.includes(type)) cached_refs[type][thy_name] = parts_by_thy[thy_name]
	}

	let tree = nav_tree(thy_name, cached_refs[selected][thy_name], selected)
	document.getElementById(to_nav_id(thy_name)).appendChild(tree)

	ScrollSpy.instance.refresh()
	}

	/* state */

	let navbar_last_opened = []


	/* controls */

	const follow_theory_hash = async () => {
	let hash = window.location.hash

	if (hash.length > 1) {
	const id = hash.slice(1)

	const thy_name = strip_suffix(id.split('#')[0], '.html')
	await open_theory(thy_name)

	ScrollSpy.instance.scroll_to(id)
	}
	}

	const toggle_theory = async (thy_name) => {
	const hash = `#${to_id(thy_name)}`
	const collapsible = document.getElementById(to_container_id(thy_name))
	if (collapsible) {
	if (!collapse(collapsible)) {
	if (window.location.hash === hash) open(collapsible)
	else window.location.hash = hash
	}
	} else window.location.hash = hash
	}

	const change_selector = (type) => {
	let old_type = get_query(PARAM_NAVBAR_TYPE)
	if (!old_type \|\| old_type !== type) {
	set_query(PARAM_NAVBAR_TYPE, type)

	for (const elem of document.getElementsByClassName(CLASS_NAVBAR_TYPE)) {
	let thy_name = of_ul_id(elem.id)
	elem.replaceWith(nav_tree(thy_name, cached_refs[type][thy_name], type))
	}

	ScrollSpy.instance.refresh()
	}
	}

	const open_tree = (elem) => {
	if (elem.classList.contains(CLASS_COLLAPSIBLE)) {
	if (open(elem)) navbar_last_opened.push(elem)
	}
	if (elem.parentElement) open_tree(elem.parentElement)
	}

	const sync_navbar = (link) => {
	for (const elem of navbar_last_opened){
	collapse(elem)
	}

	open_tree(link.parentElement)

	link.scrollIntoView({block: "center"})
	}


	/* setup */

	const init = async () => {
	const theory_list = document.getElementById(ID_THEORY_LIST)
	const navbar = document.getElementById(ID_NAVBAR)

	if (theory_list && navbar) {
	const thy_names = []

	for (const theory of theory_list.children) {
	thy_names.push(theory.id)

	const href = theory.getAttribute('href')
	const thy_name = theory.id

	const thy_collapsible = parse_elem(`
	<div id="${to_container_id(thy_name)}" theory-src="${href}" style="min-width: fit-content" class="${CLASS_COLLAPSE_CONTAINER} ${CLASS_COLLAPSED}">
	<h2 id="${to_id(thy_name)}" style="cursor: pointer" onclick="toggle_theory('${thy_name}')">
	${thy_name}
	<svg id="${to_svg_id(thy_name)}" viewBox="0 0 10 10" aria-hidden="true" focusable="false"
	class="${CLASS_INVERTIBLE}">
	<path d="m1.6953 6.7407 3.3047-3.3929 3.3047 3.3927" fill="none" stroke-linecap="round"
	stroke-linejoin="round" stroke-width="2"/>
	</svg>
	</h2>
	</div>`)
	theory.replaceWith(thy_collapsible)
	}

	const type = get_query(PARAM_NAVBAR_TYPE) ? get_query(PARAM_NAVBAR_TYPE) : NAVBAR_TYPES[0]
	navbar.appendChild(parse_elem(`
	<li>
	<select id=${ID_NAVBAR_TYPE_SELECTOR} onchange="change_selector(this.options[this.selectedIndex].value)">
	<option value=${type} selected="selected">${type}</option>
	</select>
	</li>`))
	navbar.append(...thy_names.map((thy_name) => parse_elem(`
	<li id="${to_nav_id(thy_name)}">
	<a id="${to_a_id(thy_name)}" class="${CLASS_SPY_LINK} ${CLASS_THY_NAV}" href="#${to_id(thy_name)}">
	${thy_name}
	</a>
	</li>`)))
	navbar.insertAdjacentElement('afterend', document.createElement('hr'));

	window.onhashchange = follow_theory_hash
	window.addEventListener(EVENT_SPY_ACTIVATE, (e) => sync_navbar(e.relatedTarget))

	new ScrollSpy(document.body, ID_NAVBAR, "#")

	await follow_theory_hash()
	}
	}

	document.addEventListener('DOMContentLoaded', init)
	diff --git a/metadata/authors.toml b/metadata/authors.toml
	--- a/metadata/authors.toml
	+++ b/metadata/authors.toml
	@@ -1,7427 +1,7468 @@

	[abdulaziz]
	name = "Mohammad Abdulaziz"

	[abdulaziz.emails]

	[abdulaziz.emails.abdulaziz_email]
	user = [
	"mohammad",
	"abdulaziz",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[abdulaziz.emails.abdulaziz_email1]
	user = [
	"mohammad",
	"abdulaziz8",
	]
	host = [
	"gmail",
	"com",
	]

	[abdulaziz.homepages]
	abdulaziz_homepage = "http://home.in.tum.de/~mansour/"

	[adelsberger]
	name = "Stephan Adelsberger"

	[adelsberger.emails]

	[adelsberger.emails.adelsberger_email]
	user = [
	"stvienna",
	]
	host = [
	"gmail",
	"com",
	]

	[adelsberger.homepages]
	adelsberger_homepage = "http://nm.wu.ac.at/nm/sadelsbe"

	[aehlig]
	name = "Klaus Aehlig"

	[aehlig.emails]

	[aehlig.homepages]
	aehlig_homepage = "http://www.linta.de/~aehlig/"

	[aissat]
	name = "Romain Aissat"

	[aissat.emails]

	[aissat.homepages]

	[amani]
	name = "Sidney Amani"

	[amani.emails]

	[amani.emails.amani_email]
	user = [
	"sidney",
	"amani",
	]
	host = [
	"data61",
	"csiro",
	"au",
	]

	[amani.homepages]

	[ammer]
	name = "Thomas Ammer"

	[ammer.emails]

	[ammer.emails.ammer_email]
	user = [
	"thomas",
	"ammer",
	]
	host = [
	"tum",
	"de",
	]

	[ammer.homepages]

	[andreka]
	name = "Hajnal Andreka"

	[andreka.emails]

	[andreka.homepages]
	andreka_homepage = "https://renyi.hu/en/researchers/hajnal-andreka"

	[andronick]
	name = "June Andronick"

	[andronick.emails]

	[andronick.homepages]

	[aransay]
	name = "Jesús Aransay"

	[aransay.emails]

	[aransay.emails.aransay_email]
	user = [
	"jesus-maria",
	"aransay",
	]
	host = [
	"unirioja",
	"es",
	]

	[aransay.homepages]
	aransay_homepage = "https://www.unirioja.es/cu/jearansa"

	[argyraki]
	name = "Angeliki Koutsoukou-Argyraki"

	[argyraki.emails]

	[argyraki.emails.argyraki_email]
	user = [
	"ak2110",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[argyraki.homepages]
	argyraki_homepage = "https://www.cl.cam.ac.uk/~ak2110/"
	argyraki_homepage2 = "https://www.cst.cam.ac.uk/people/ak2110"

	[armstrong]
	name = "Alasdair Armstrong"

	[armstrong.emails]

	[armstrong.homepages]

	[aspinall]
	name = "David Aspinall"

	[aspinall.emails]

	[aspinall.homepages]
	aspinall_homepage = "http://homepages.inf.ed.ac.uk/da/"

	[ausaf]
	name = "Fahad Ausaf"

	[ausaf.emails]

	[ausaf.homepages]
	ausaf_homepage = "http://kcl.academia.edu/FahadAusaf"

	[avigad]
	name = "Jeremy Avigad"

	[avigad.emails]

	[avigad.emails.avigad_email]
	user = [
	"avigad",
	]
	host = [
	"cmu",
	"edu",
	]

	[avigad.homepages]
	avigad_homepage = "http://www.andrew.cmu.edu/user/avigad/"

	[back]
	name = "Ralph-Johan Back"

	[back.emails]

	[back.homepages]
	back_homepage = "http://users.abo.fi/Ralph-Johan.Back/"

	[baksys]
	name = "Mantas Bakšys"

	[baksys.emails]

	[baksys.emails.baksys_email]
	user = [
	"mb2412",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[baksys.homepages]
	baksys_homepage = "https://github.com/MantasBaksys"

	[balbach]
	name = "Frank J. Balbach"

	[balbach.emails]

	[balbach.emails.balbach_email]
	user = [
	"frank-balbach",
	]
	host = [
	"gmx",
	"de",
	]

	[balbach.homepages]

	[ballarin]
	name = "Clemens Ballarin"

	[ballarin.emails]

	[ballarin.emails.ballarin_email]
	user = [
	"ballarin",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[ballarin.homepages]
	ballarin_homepage = "http://www21.in.tum.de/~ballarin/"

	[barsotti]
	name = "Damián Barsotti"

	[barsotti.emails]

	[barsotti.homepages]
	barsotti_homepage = "http://www.cs.famaf.unc.edu.ar/~damian/"

	[bauer]
	name = "Gertrud Bauer"

	[bauer.emails]

	[bauer.homepages]

	[bauereiss]
	name = "Thomas Bauereiss"

	[bauereiss.emails]

	[bauereiss.emails.bauereiss_email]
	user = [
	"thomas",
	]
	host = [
	"bauereiss",
	"name",
	]

	[bauereiss.homepages]

	[bayer]
	name = "Jonas Bayer"

	[bayer.emails]

	[bayer.emails.bayer_email]
	user = [
	"jonas",
	"bayer999",
	]
	host = [
	"gmail",
	"com",
	]

	[bayer.homepages]

	[becker]
	name = "Heiko Becker"

	[becker.emails]

	[becker.emails.becker_email]
	user = [
	"hbecker",
	]
	host = [
	"mpi-sws",
	"org",
	]

	[becker.homepages]

	[beeren]
	name = "Joel Beeren"

	[beeren.emails]

	[beeren.homepages]

	[bella]
	name = "Giampaolo Bella"

	[bella.emails]

	[bella.emails.bella_email]
	user = [
	"giamp",
	]
	host = [
	"dmi",
	"unict",
	"it",
	]

	[bella.homepages]
	bella_homepage = "http://www.dmi.unict.it/~giamp/"

	[bengtson]
	name = "Jesper Bengtson"

	[bengtson.emails]

	[bengtson.homepages]
	bengtson_homepage = "http://www.itu.dk/people/jebe"

	[bentkamp]
	name = "Alexander Bentkamp"

	[bentkamp.emails]

	[bentkamp.emails.bentkamp_email]
	user = [
	"bentkamp",
	]
	host = [
	"gmail",
	"com",
	]

	[bentkamp.emails.bentkamp_email1]
	user = [
	"a",
	"bentkamp",
	]
	host = [
	"vu",
	"nl",
	]

	[bentkamp.homepages]
	bentkamp_homepage = "https://www.cs.vu.nl/~abp290/"

	[benzmueller]
	name = "Christoph Benzmüller"

	[benzmueller.emails]

	[benzmueller.emails.benzmueller_email]
	user = [
	"c",
	"benzmueller",
	]
	host = [
	"gmail",
	"com",
	]

	[benzmueller.emails.benzmueller_email1]
	user = [
	"c",
	"benzmueller",
	]
	host = [
	"fu-berlin",
	"de",
	]

	[benzmueller.homepages]
	benzmueller_homepage = "http://christoph-benzmueller.de"
	benzmueller_homepage1 = "http://page.mi.fu-berlin.de/cbenzmueller/"

	[beresford]
	name = "Alastair R. Beresford"

	[beresford.emails]

	[beresford.emails.beresford_email]
	user = [
	"arb33",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[beresford.homepages]

	[berghofer]
	name = "Stefan Berghofer"

	[berghofer.emails]

	[berghofer.emails.berghofer_email]
	user = [
	"berghofe",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[berghofer.homepages]
	berghofer_homepage = "http://www.in.tum.de/~berghofe"

	[beringer]
	name = "Lennart Beringer"

	[beringer.emails]

	[beringer.emails.beringer_email]
	user = [
	"lennart",
	"beringer",
	]
	host = [
	"ifi",
	"lmu",
	"de",
	]

	[beringer.homepages]

	[bharadwaj]
	name = "Abhijith Bharadwaj"

	[bharadwaj.emails]

	[bharadwaj.homepages]

	[bhatt]
	name = "Bhargav Bhatt"

	[bhatt.emails]

	[bhatt.emails.bhatt_email]
	user = [
	"bhargav",
	"bhatt",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[bhatt.homepages]

	[biendarra]
	name = "Julian Biendarra"

	[biendarra.emails]

	[biendarra.homepages]

	[bisping]
	name = "Benjamin Bisping"

	[bisping.emails]

	[bisping.emails.bisping_email]
	user = [
	"benjamin",
	"bisping",
	]
	host = [
	"campus",
	"tu-berlin",
	"de",
	]

	[bisping.homepages]

	[blanchette]
	name = "Jasmin Christian Blanchette"

	[blanchette.emails]

	[blanchette.emails.blanchette_email]
	user = [
	"jasmin",
	"blanchette",
	]
	host = [
	"gmail",
	"com",
	]

	[blanchette.emails.blanchette_email1]
	user = [
	"j",
	"c",
	"blanchette",
	]
	host = [
	"vu",
	"nl",
	]

	[blanchette.homepages]
	blanchette_homepage = "http://www21.in.tum.de/~blanchet"
	blanchette_homepage1 = "https://www.cs.vu.nl/~jbe248/"

	[blasum]
	name = "Holger Blasum"

	[blasum.emails]

	[blasum.emails.blasum_email]
	user = [
	"holger",
	"blasum",
	]
	host = [
	"sysgo",
	"com",
	]

	[blasum.homepages]

	[blumson]
	name = "Ben Blumson"

	[blumson.emails]

	[blumson.emails.blumson_email]
	user = [
	"benblumson",
	]
	host = [
	"gmail",
	"com",
	]

	[blumson.homepages]
	blumson_homepage = "https://philpeople.org/profiles/ben-blumson"

	[bockenek]
	name = "Joshua Bockenek"

	[bockenek.emails]

	[bockenek.homepages]

	[boehme]
	name = "Sascha Böhme"

	[boehme.emails]

	[boehme.emails.boehme_email]
	user = [
	"boehmes",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[boehme.homepages]
	boehme_homepage = "http://www21.in.tum.de/~boehmes/"

	[bohrer]
	name = "Rose Bohrer"

	[bohrer.emails]

	[bohrer.emails.bohrer_email]
	user = [
	"rose",
	"bohrer",
	"cs",
	]
	host = [
	"gmail",
	"com",
	]

	[bohrer.homepages]

	[bordg]
	name = "Anthony Bordg"

	[bordg.emails]

	[bordg.emails.bordg_email]
	user = [
	"apdb3",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[bordg.homepages]
	bordg_homepage = "https://sites.google.com/site/anthonybordg/"

	[borgstroem]
	name = "Johannes Borgström"

	[borgstroem.emails]

	[borgstroem.emails.borgstroem_email]
	user = [
	"johannes",
	"borgstrom",
	]
	host = [
	"it",
	"uu",
	"se",
	]

	[borgstroem.homepages]

	[bortin]
	name = "Maksym Bortin"

	[bortin.emails]

	[bortin.emails.bortin_email]
	user = [
	"maksym",
	"bortin",
	]
	host = [
	"nicta",
	"com",
	"au",
	]

	[bortin.emails.bortin_email1]
	user = [
	"mbortin",
	]
	host = [
	"gmail",
	"com",
	]

	[bortin.homepages]

	[bottesch]
	name = "Ralph Bottesch"

	[bottesch.emails]

	[bottesch.emails.bottesch_email]
	user = [
	"ralph",
	"bottesch",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[bottesch.homepages]
	bottesch_homepage = "http://cl-informatik.uibk.ac.at/users/bottesch/"

	[boulanger]
	name = "Frédéric Boulanger"

	[boulanger.emails]

	[boulanger.emails.boulanger_email]
	user = [
	"frederic",
	"boulanger",
	]
	host = [
	"centralesupelec",
	"fr",
	]

	[boulanger.homepages]

	[bourke]
	name = "Timothy Bourke"

	[bourke.emails]

	[bourke.emails.bourke_email]
	user = [
	"tim",
	]
	host = [
	"tbrk",
	"org",
	]

	[bourke.homepages]
	bourke_homepage = "http://www.tbrk.org"

	[boutry]
	name = "Pierre Boutry"

	[boutry.emails]

	[boutry.emails.boutry_email]
	user = [
	"boutry",
	]
	host = [
	"unistra",
	"fr",
	]

	[boutry.homepages]

	[boyton]
	name = "Andrew Boyton"

	[boyton.emails]

	[boyton.emails.boyton_email]
	user = [
	"andrew",
	"boyton",
	]
	host = [
	"nicta",
	"com",
	"au",
	]

	[boyton.homepages]

	[bracevac]
	name = "Oliver Bračevac"

	[bracevac.emails]

	[bracevac.emails.bracevac_email]
	user = [
	"bracevac",
	]
	host = [
	"st",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[bracevac.homepages]

	[brandt]
	name = "Felix Brandt"

	[brandt.emails]

	[brandt.homepages]
	brandt_homepage = "http://dss.in.tum.de/staff/brandt.html"

	[breitner]
	name = "Joachim Breitner"

	[breitner.emails]

	[breitner.emails.breitner_email]
	user = [
	"mail",
	]
	host = [
	"joachim-breitner",
	"de",
	]

	[breitner.emails.breitner_email1]
	user = [
	"joachim",
	]
	host = [
	"cis",
	"upenn",
	"edu",
	]

	[breitner.homepages]
	breitner_homepage = "http://pp.ipd.kit.edu/~breitner"

	[brien]
	name = "Nicolas Robinson-O'Brien"

	[brien.emails]

	[brien.homepages]

	[brinkop]
	name = "Hauke Brinkop"

	[brinkop.emails]

	[brinkop.emails.brinkop_email]
	user = [
	"hauke",
	"brinkop",
	]
	host = [
	"googlemail",
	"com",
	]

	[brinkop.homepages]

	[brodmann]
	name = "Paul-David Brodmann"

	[brodmann.emails]

	[brodmann.emails.brodmann_email]
	user = [
	"p",
	"brodmann",
	]
	host = [
	"tu-berlin",
	"de",
	]

	[brodmann.homepages]

	[brucker]
	name = "Achim D. Brucker"

	[brucker.emails]

	[brucker.emails.brucker_email]
	user = [
	"a",
	"brucker",
	]
	host = [
	"exeter",
	"ac",
	"uk",
	]

	[brucker.emails.brucker_email1]
	user = [
	"brucker",
	]
	host = [
	"spamfence",
	"net",
	]

	[brucker.emails.brucker_email2]
	user = [
	"adbrucker",
	]
	host = [
	"0x5f",
	"org",
	]

	[brucker.homepages]
	brucker_homepage = "https://www.brucker.ch/"

	[bruegger]
	name = "Lukas Brügger"

	[bruegger.emails]

	[bruegger.emails.bruegger_email]
	user = [
	"lukas",
	"a",
	"bruegger",
	]
	host = [
	"gmail",
	"com",
	]

	[bruegger.homepages]

	[brun]
	name = "Matthias Brun"

	[brun.emails]

	[brun.emails.brun_email]
	user = [
	"matthias",
	"brun",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[brun.homepages]

	[brunner]
	name = "Julian Brunner"

	[brunner.emails]

	[brunner.emails.brunner_email]
	user = [
	"brunnerj",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[brunner.homepages]
	brunner_homepage = "http://www21.in.tum.de/~brunnerj/"

	[bulwahn]
	name = "Lukas Bulwahn"

	[bulwahn.emails]

	[bulwahn.emails.bulwahn_email]
	user = [
	"lukas",
	"bulwahn",
	]
	host = [
	"gmail",
	"com",
	]

	[bulwahn.homepages]

	[butler]
	name = "David Butler"

	[butler.emails]

	[butler.emails.butler_email]
	user = [
	"dbutler",
	]
	host = [
	"turing",
	"ac",
	"uk",
	]

	[butler.homepages]
	butler_homepage = "https://www.turing.ac.uk/people/doctoral-students/david-butler"

	[buyse]
	name = "Maxime Buyse"

	[buyse.emails]

	[buyse.emails.buyse_email]
	user = [
	"maxime",
	"buyse",
	]
	host = [
	"polytechnique",
	"edu",
	]

	[buyse.homepages]

	[caballero]
	name = "José Manuel Rodríguez Caballero"

	[caballero.emails]

	[caballero.emails.caballero_email]
	user = [
	"jose",
	"manuel",
	"rodriguez",
	"caballero",
	]
	host = [
	"ut",
	"ee",
	]

	[caballero.homepages]
	caballero_homepage = "https://josephcmac.github.io/"

	+[calk]
	+name = "Cameron Calk"
	+
	+[calk.emails]
	+
	+[calk.homepages]
	+
	[caminati]
	name = "Marco B. Caminati"

	[caminati.emails]

	[caminati.homepages]

	[campo]
	name = "Alejandro del Campo"

	[campo.emails]

	[campo.emails.campo_email]
	user = [
	"alejandro",
	"del-campo",
	]
	host = [
	"alum",
	"unirioja",
	"es",
	]

	[campo.homepages]

	[chaieb]
	name = "Amine Chaieb"

	[chaieb.emails]

	[chaieb.homepages]

	[chapman]
	name = "Peter Chapman"

	[chapman.emails]

	[chapman.emails.chapman_email]
	user = [
	"pc",
	]
	host = [
	"cs",
	"st-andrews",
	"ac",
	"uk",
	]

	[chapman.homepages]

	[chen]
	name = "L. Chen"

	[chen.emails]

	[chen.homepages]

	[chevalier]
	name = "Loïc Chevalier"

	[chevalier.emails]

	[chevalier.homepages]

	[christfort]
	name = "Axel Christfort"

	[christfort.emails]

	[christfort.emails.christfort_email]
	user = [
	"axel",
	]
	host = [
	"di",
	"ku",
	"dk",
	]

	[christfort.homepages]

	[clouston]
	name = "Ranald Clouston"

	[clouston.emails]

	[clouston.emails.clouston_email]
	user = [
	"ranald",
	"clouston",
	]
	host = [
	"cs",
	"au",
	"dk",
	]

	[clouston.homepages]

	[cock]
	name = "David Cock"

	[cock.emails]

	[cock.emails.cock_email]
	user = [
	"david",
	"cock",
	]
	host = [
	"nicta",
	"com",
	"au",
	]

	[cock.homepages]

	[coghetto]
	name = "Roland Coghetto"

	[coghetto.emails]

	[coghetto.emails.coghetto_email]
	user = [
	"roland_coghetto",
	]
	host = [
	"hotmail",
	"com",
	]

	[coghetto.homepages]

	[coglio]
	name = "Alessandro Coglio"

	[coglio.emails]

	[coglio.emails.coglio_email]
	user = [
	"coglio",
	]
	host = [
	"kestrel",
	"edu",
	]

	[coglio.homepages]
	coglio_homepage = "http://www.kestrel.edu/~coglio"

	[cohen]
	name = "Ernie Cohen"

	[cohen.emails]

	[cohen.emails.cohen_email]
	user = [
	"ecohen",
	]
	host = [
	"amazon",
	"com",
	]

	[cohen.homepages]

	[cordwell]
	name = "Katherine Kosaian"

	[cordwell.emails]

	[cordwell.emails.cordwell_email]
	user = [
	"kcordwel",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[cordwell.homepages]
	cordwell_homepage = "https://www.cs.cmu.edu/~kcordwel/"

	[cousin]
	name = "Marie Cousin"

	[cousin.emails]

	[cousin.emails.cousin_email]
	user = [
	"marie",
	"cousin",
	]
	host = [
	"grenoble-inp",
	"org",
	]

	[cousin.homepages]

	[cremer]
	name = "Nils Cremer"

	[cremer.emails]

	[cremer.emails.cremer_email]
	user = [
	"nils",
	"cremer",
	]
	host = [
	"tum",
	"de",
	]

	[cremer.homepages]

	[crighton]
	name = "Aaron Crighton"

	[crighton.emails]

	[crighton.emails.crighton_email]
	user = [
	"crightoa",
	]
	host = [
	"mcmaster",
	"ca",
	]

	[crighton.homepages]

	[dalvit]
	name = "Christian Dalvit"

	[dalvit.emails]

	[dalvit.emails.dalvit_email]
	user = [
	"chris",
	"dalvit",
	]
	host = [
	"gmail",
	"com",
	]

	[dalvit.homepages]

	[danilkin]
	name = "Anton Danilkin"

	[danilkin.emails]

	[danilkin.emails.danilkin_email]
	user = [
	"anton",
	"danilkin",
	]
	host = [
	"ens",
	"psl",
	"eu",
	]

	[danilkin.homepages]

	[dardinier]
	name = "Thibault Dardinier"

	[dardinier.emails]

	[dardinier.emails.dardinier_email]
	user = [
	"thibault",
	"dardinier",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[dardinier.homepages]
	dardinier_homepage = "https://dardinier.me/"

	[david]
	name = "Marco David"

	[david.emails]

	[david.emails.david_email]
	user = [
	"marco",
	"david",
	]
	host = [
	"hotmail",
	"de",
	]

	[david.homepages]

	[debois]
	name = "Søren Debois"

	[debois.emails]

	[debois.homepages]

	[debrat]
	name = "Henri Debrat"

	[debrat.emails]

	[debrat.emails.debrat_email]
	user = [
	"henri",
	"debrat",
	]
	host = [
	"loria",
	"fr",
	]

	[debrat.homepages]

	[decova]
	name = "Sára Decova"

	[decova.emails]

	[decova.homepages]

	[delemazure]
	name = "Théo Delemazure"

	[delemazure.emails]

	[delemazure.homepages]
	delemazure_homepage = "https://theo.delemazure.fr/"

	[demeulemeester]
	name = "Tom Demeulemeester"

	[demeulemeester.emails]

	[demeulemeester.homepages]
	demeulemeester_homepage = "https://www.kuleuven.be/wieiswie/en/person/00131528"

	[derrick]
	name = "John Derrick"

	[derrick.emails]

	[derrick.emails.derrick_email]
	user = [
	"j",
	"derrick",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[derrick.homepages]

	[desharnais]
	name = "Martin Desharnais"

	[desharnais.emails]

	[desharnais.emails.desharnais_email]
	user = [
	"martin",
	"desharnais",
	]
	host = [
	"unibw",
	"de",
	]

	[desharnais.emails.desharnais_email1]
	user = [
	"desharnais",
	]
	host = [
	"mpi-inf",
	"mpg",
	"de",
	]

	[desharnais.homepages]
	desharnais_homepage = "https://martin.desharnais.me"

	[diaz]
	name = "Javier Díaz"

	[diaz.emails]

	[diaz.emails.diaz_email]
	user = [
	"javier",
	"diaz",
	"manzi",
	]
	host = [
	"gmail",
	"com",
	]

	[diaz.homepages]

	[diekmann]
	name = "Cornelius Diekmann"

	[diekmann.emails]

	[diekmann.emails.diekmann_email]
	user = [
	"diekmann",
	]
	host = [
	"net",
	"in",
	"tum",
	"de",
	]

	[diekmann.homepages]
	diekmann_homepage = "http://net.in.tum.de/~diekmann"

	[dirix]
	name = "Stefan Dirix"

	[dirix.emails]

	[dirix.homepages]

	[dittmann]
	name = "Christoph Dittmann"

	[dittmann.emails]

	[dittmann.emails.dittmann_email]
	user = [
	"isabelle",
	]
	host = [
	"christoph-d",
	"de",
	]

	[dittmann.homepages]
	dittmann_homepage = "http://logic.las.tu-berlin.de/Members/Dittmann/"

	[divason]
	name = "Jose Divasón"

	[divason.emails]

	[divason.emails.divason_email]
	user = [
	"jose",
	"divason",
	]
	host = [
	"unirioja",
	"es",
	]

	[divason.homepages]
	divason_homepage = "https://www.unirioja.es/cu/jodivaso/"

	[doczkal]
	name = "Christian Doczkal"

	[doczkal.emails]

	[doczkal.emails.doczkal_email]
	user = [
	"doczkal",
	]
	host = [
	"ps",
	"uni-saarland",
	"de",
	]

	[doczkal.homepages]

	[dongol]
	name = "Brijesh Dongol"

	[dongol.emails]

	[dongol.emails.dongol_email]
	user = [
	"brijesh",
	"dongol",
	]
	host = [
	"brunel",
	"ac",
	"uk",
	]

	[dongol.homepages]

	[doty]
	name = "Matthew Doty"

	[doty.emails]

	[doty.emails.doty_email]
	user = [
	"matt",
	]
	host = [
	"w-d",
	"org",
	]

	[doty.homepages]

	[dubut]
	name = "Jérémy Dubut"

	[dubut.emails]

	[dubut.emails.dubut_email]
	user = [
	"dubut",
	]
	host = [
	"nii",
	"ac",
	"jp",
	]

	[dubut.emails.dubut_email1]
	user = [
	"jeremy",
	"dubut",
	]
	host = [
	"aist",
	"go",
	"jp",
	]

	[dubut.homepages]
	dubut_homepage = "http://group-mmm.org/~dubut/"

	[dunaev]
	name = "Georgy Dunaev"

	[dunaev.emails]

	[dunaev.emails.dunaev_email]
	user = [
	"georgedunaev",
	]
	host = [
	"gmail",
	"com",
	]

	[dunaev.homepages]

	[dyckhoff]
	name = "Roy Dyckhoff"

	[dyckhoff.emails]

	[dyckhoff.homepages]
	dyckhoff_homepage = "https://rd.host.cs.st-andrews.ac.uk"

	[eberl]
	name = "Manuel Eberl"
	orcid = "0000-0002-4263-6571"

	[eberl.emails]

	[eberl.emails.eberl_email]
	user = [
	"manuel",
	]
	host = [
	"pruvisto",
	"org",
	]

	[eberl.emails.eberl_email1]
	user = [
	"manuel",
	"eberl",
	]
	host = [
	"tum",
	"de",
	]

	[eberl.emails.eberl_email2]
	user = [
	"manuel",
	"eberl",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[eberl.homepages]
	eberl_homepage = "https://pruvisto.org/"
	eberl_homepage2 = "https://www.in.tum.de/~eberlm"

	[echenim]
	name = "Mnacho Echenim"

	[echenim.emails]

	[echenim.emails.echenim_email]
	user = [
	"mnacho",
	"echenim",
	]
	host = [
	"univ-grenoble-alpes",
	"fr",
	]

	[echenim.homepages]
	echenim_homepage = "https://lig-membres.imag.fr/mechenim/"

	[edmonds]
	name = "Chelsea Edmonds"

	[edmonds.emails]

	[edmonds.emails.edmonds_email]
	user = [
	"cle47",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[edmonds.homepages]
	edmonds_homepage = "https://www.cst.cam.ac.uk/people/cle47"

	[engelhardt]
	name = "Kai Engelhardt"

	[engelhardt.emails]

	[engelhardt.homepages]

	[eriksson]
	name = "Lars-Henrik Eriksson"

	[eriksson.emails]

	[eriksson.emails.eriksson_email]
	user = [
	"lhe",
	]
	host = [
	"it",
	"uu",
	"se",
	]

	[eriksson.homepages]

	[esparza]
	name = "Javier Esparza"

	[esparza.emails]

	[esparza.homepages]
	esparza_homepage = "https://www7.in.tum.de/~esparza/"

	[essmann]
	name = "Robin Eßmann"

	[essmann.emails]

	[essmann.emails.essmann_email]
	user = [
	"robin",
	"essmann",
	]
	host = [
	"tum",
	"de",
	]

	[essmann.homepages]

	[felgenhauer]
	name = "Bertram Felgenhauer"

	[felgenhauer.emails]

	[felgenhauer.emails.felgenhauer_email]
	user = [
	"bertram",
	"felgenhauer",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[felgenhauer.emails.felgenhauer_email1]
	user = [
	"int-e",
	]
	host = [
	"gmx",
	"de",
	]

	[felgenhauer.homepages]

	[feliachi]
	name = "Abderrahmane Feliachi"

	[feliachi.emails]

	[feliachi.emails.feliachi_email]
	user = [
	"abderrahmane",
	"feliachi",
	]
	host = [
	"lri",
	"fr",
	]

	[feliachi.homepages]

	[fell]
	name = "Julian Fell"

	[fell.emails]

	[fell.emails.fell_email]
	user = [
	"julian",
	"fell",
	]
	host = [
	"uq",
	"net",
	"au",
	]

	[fell.homepages]

	[fernandez]
	name = "Matthew Fernandez"

	[fernandez.emails]

	[fernandez.homepages]

	[fiedler]
	name = "Ben Fiedler"

	[fiedler.emails]

	[fiedler.emails.fiedler_email]
	user = [
	"ben",
	"fiedler",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[fiedler.homepages]

	[fleuriot]
	name = "Jacques D. Fleuriot"

	[fleuriot.emails]

	[fleuriot.emails.fleuriot_email]
	user = [
	"Jacques",
	"Fleuriot",
	]
	host = [
	"ed",
	"ac",
	"uk",
	]

	[fleuriot.emails.fleuriot_email1]
	user = [
	"jdf",
	]
	host = [
	"ed",
	"ac",
	"uk",
	]

	[fleuriot.homepages]
	fleuriot_homepage = "https://www.inf.ed.ac.uk/people/staff/Jacques_Fleuriot.html"

	[fleury]
	name = "Mathias Fleury"

	[fleury.emails]

	[fleury.emails.fleury_email]
	user = [
	"fleury",
	]
	host = [
	"mpi-inf",
	"mpg",
	"de",
	]

	[fleury.emails.fleury_email1]
	user = [
	"mathias",
	"fleury",
	]
	host = [
	"jku",
	"at",
	]

	[fleury.homepages]
	fleury_homepage = "http://fmv.jku.at/fleury"

	[foster]
	name = "Michael Foster"

	[foster.emails]

	[foster.emails.foster_email]
	user = [
	"m",
	"foster",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[foster.homepages]

	[fosterj]
	name = "J. Nathan Foster"

	[fosterj.emails]

	[fosterj.homepages]
	fosterj_homepage = "http://www.cs.cornell.edu/~jnfoster/"

	[fosters]
	name = "Simon Foster"

	[fosters.emails]

	[fosters.emails.fosters_email]
	user = [
	"simon",
	"foster",
	]
	host = [
	"york",
	"ac",
	"uk",
	]

	[fosters.homepages]
	fosters_homepage = "https://www-users.cs.york.ac.uk/~simonf/"

	[fouillard]
	name = "Valentin Fouillard"

	[fouillard.emails]

	[fouillard.emails.fouillard_email]
	user = [
	"valentin",
	"fouillard",
	]
	host = [
	"limsi",
	"fr",
	]

	[fouillard.homepages]

	[friedrich]
	name = "Stefan Friedrich"

	[friedrich.emails]

	[friedrich.homepages]

	[from]
	name = "Asta Halkjær From"

	[from.emails]

	[from.emails.from_email]
	user = [
	"ahfrom",
	]
	host = [
	"dtu",
	"dk",
	]

	[from.homepages]
	from_homepage = "https://people.compute.dtu.dk/ahfrom/"

	[fuenmayor]
	name = "David Fuenmayor"

	[fuenmayor.emails]

	[fuenmayor.emails.fuenmayor_email]
	user = [
	"davfuenmayor",
	]
	host = [
	"gmail",
	"com",
	]

	[fuenmayor.homepages]

	[furusawa]
	name = "Hitoshi Furusawa"

	[furusawa.emails]

	[furusawa.homepages]
	furusawa_homepage = "http://www.sci.kagoshima-u.ac.jp/~furusawa/"

	[gammie]
	name = "Peter Gammie"

	[gammie.emails]

	[gammie.emails.gammie_email]
	user = [
	"peteg42",
	]
	host = [
	"gmail",
	"com",
	]

	[gammie.homepages]
	gammie_homepage = "http://peteg.org"

	[gao]
	name = "Xin Gao"

	[gao.emails]

	[gao.homepages]

	[gaudel]
	name = "Marie-Claude Gaudel"

	[gaudel.emails]

	[gaudel.emails.gaudel_email]
	user = [
	"mcg",
	]
	host = [
	"lri",
	"fr",
	]

	[gaudel.homepages]

	[gay]
	name = "Richard Gay"

	[gay.emails]

	[gay.emails.gay_email]
	user = [
	"gay",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[gay.homepages]

	[georgescu]
	name = "George Georgescu"

	[georgescu.emails]

	[georgescu.homepages]

	[gheri]
	name = "Lorenzo Gheri"

	[gheri.emails]

	[gheri.emails.gheri_email]
	user = [
	"lor",
	"gheri",
	]
	host = [
	"gmail",
	"com",
	]

	[gheri.homepages]

	[ghourabi]
	name = "Fadoua Ghourabi"

	[ghourabi.emails]

	[ghourabi.emails.ghourabi_email]
	user = [
	"fadouaghourabi",
	]
	host = [
	"gmail",
	"com",
	]

	[ghourabi.homepages]

	[gioiosa]
	name = "Gianpaolo Gioiosa"

	[gioiosa.emails]

	[gioiosa.homepages]

	[glabbeek]
	name = "Rob van Glabbeek"

	[glabbeek.emails]

	[glabbeek.homepages]
	glabbeek_homepage = "http://theory.stanford.edu/~rvg/"

	[gomes]
	name = "Victor B. F. Gomes"

	[gomes.emails]

	[gomes.emails.gomes_email]
	user = [
	"victor",
	"gomes",
	]
	host = [
	"cl",
	"cam",
	"ac",
	"uk",
	]

	[gomes.emails.gomes_email2]
	user = [
	"victorborgesfg",
	]
	host = [
	"gmail",
	"com",
	]

	[gomes.emails.gomes_email4]
	user = [
	"vborgesferreiragomes1",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[gomes.homepages]
	gomes_homepage = "http://www.dcs.shef.ac.uk/~victor"

	[gonzalez]
	name = "Edgar Gonzàlez"
	orcid = "0000-0002-9169-0769"

	[gonzalez.emails]

	[gonzalez.emails.gonzalez_email]
	user = [
	"edgargip",
	]
	host = [
	"google",
	"com",
	]

	[gonzalez.homepages]

	[gore]
	name = "Rajeev Gore"

	[gore.emails]

	[gore.emails.gore_email]
	user = [
	"rajeev",
	"gore",
	]
	host = [
	"anu",
	"edu",
	"au",
	]

	[gore.homepages]

	[gouezel]
	name = "Sebastien Gouezel"

	[gouezel.emails]

	[gouezel.emails.gouezel_email]
	user = [
	"sebastien",
	"gouezel",
	]
	host = [
	"univ-rennes1",
	"fr",
	]

	[gouezel.homepages]
	gouezel_homepage = "http://www.math.sciences.univ-nantes.fr/~gouezel/"

	[grechuk]
	name = "Bogdan Grechuk"

	[grechuk.emails]

	[grechuk.emails.grechuk_email]
	user = [
	"grechukbogdan",
	]
	host = [
	"yandex",
	"ru",
	]

	[grechuk.homepages]

	[grewe]
	name = "Sylvia Grewe"

	[grewe.emails]

	[grewe.emails.grewe_email]
	user = [
	"grewe",
	]
	host = [
	"cs",
	"tu-darmstadt",
	"de",
	]

	[grewe.homepages]

	[griebel]
	name = "Simon Griebel"

	[griebel.emails]

	[griebel.emails.griebel_email]
	user = [
	"s",
	"griebel",
	]
	host = [
	"tum",
	"de",
	]

	[griebel.homepages]

	[grov]
	name = "Gudmund Grov"

	[grov.emails]

	[grov.emails.grov_email]
	user = [
	"ggrov",
	]
	host = [
	"inf",
	"ed",
	"ac",
	"uk",
	]

	[grov.homepages]
	grov_homepage = "http://homepages.inf.ed.ac.uk/ggrov"

	[guerraoui]
	name = "Rachid Guerraoui"

	[guerraoui.emails]

	[guerraoui.emails.guerraoui_email]
	user = [
	"rachid",
	"guerraoui",
	]
	host = [
	"epfl",
	"ch",
	]

	[guerraoui.homepages]

	[guiol]
	name = "Hervé Guiol"

	[guiol.emails]

	[guiol.emails.guiol_email]
	user = [
	"herve",
	"guiol",
	]
	host = [
	"univ-grenoble-alpes",
	"fr",
	]

	[guiol.homepages]

	[gunther]
	name = "Emmanuel Gunther"

	[gunther.emails]

	[gunther.emails.gunther_email]
	user = [
	"gunther",
	]
	host = [
	"famaf",
	"unc",
	"edu",
	"ar",
	]

	[gunther.homepages]

	[gutkovas]
	name = "Ramunas Gutkovas"

	[gutkovas.emails]

	[gutkovas.emails.gutkovas_email]
	user = [
	"ramunas",
	"gutkovas",
	]
	host = [
	"it",
	"uu",
	"se",
	]

	[gutkovas.homepages]

	[guttmann]
	name = "Walter Guttmann"

	[guttmann.emails]

	[guttmann.emails.guttmann_email]
	user = [
	"walter",
	"guttmann",
	]
	host = [
	"canterbury",
	"ac",
	"nz",
	]

	[guttmann.homepages]
	guttmann_homepage = "https://www.cosc.canterbury.ac.nz/walter.guttmann/"

	[guzman]
	name = "Laura P. Gamboa Guzman"

	[guzman.emails]

	[guzman.emails.guzman_email]
	user = [
	"lpgamboa",
	]
	host = [
	"iastate",
	"edu",
	]

	[guzman.homepages]
	guzman_homepage = "https://sites.google.com/view/lpgamboa/home"

	[haftmann]
	name = "Florian Haftmann"

	[haftmann.emails]

	[haftmann.emails.haftmann_email]
	user = [
	"florian",
	"haftmann",
	]
	host = [
	"informatik",
	"tu-muenchen",
	"de",
	]

	[haftmann.homepages]
	haftmann_homepage = "http://isabelle.in.tum.de/~haftmann"

	[haslbeck]
	name = "Max W. Haslbeck"

	[haslbeck.emails]

	[haslbeck.emails.haslbeck_email]
	user = [
	"maximilian",
	"haslbeck",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[haslbeck.emails.haslbeck_email1]
	user = [
	"haslbecm",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[haslbeck.emails.haslbeck_email2]
	user = [
	"max",
	"haslbeck",
	]
	host = [
	"gmx",
	"de",
	]

	[haslbeck.homepages]
	haslbeck_homepage = "http://cl-informatik.uibk.ac.at/users/mhaslbeck/"

	[haslbeckm]
	name = "Maximilian P. L. Haslbeck"

	[haslbeckm.emails]

	[haslbeckm.emails.haslbeckm_email]
	user = [
	"haslbema",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[haslbeckm.homepages]
	haslbeckm_homepage = "http://in.tum.de/~haslbema/"

	[havle]
	name = "Oto Havle"

	[havle.emails]

	[havle.emails.havle_email]
	user = [
	"oha",
	]
	host = [
	"sysgo",
	"com",
	]

	[havle.homepages]

	[hayes]
	name = "Ian J. Hayes"

	[hayes.emails]

	[hayes.emails.hayes_email]
	user = [
	"ian",
	"hayes",
	]
	host = [
	"itee",
	"uq",
	"edu",
	"au",
	]

	[hayes.homepages]

	[he]
	name = "Yijun He"

	[he.emails]

	[he.emails.he_email]
	user = [
	"yh403",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[he.homepages]

	[heimes]
	name = "Lukas Heimes"

	[heimes.emails]

	[heimes.emails.heimes_email]
	user = [
	"heimesl",
	]
	host = [
	"student",
	"ethz",
	"ch",
	]

	[heimes.homepages]

	[helke]
	name = "Steffen Helke"

	[helke.emails]

	[helke.emails.helke_email]
	user = [
	"helke",
	]
	host = [
	"cs",
	"tu-berlin",
	"de",
	]

	[helke.homepages]

	[hellauer]
	name = "Fabian Hellauer"

	[hellauer.emails]

	[hellauer.emails.hellauer_email]
	user = [
	"hellauer",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[hellauer.homepages]

	[heller]
	name = "Armin Heller"

	[heller.emails]

	[heller.homepages]

	[henrio]
	name = "Ludovic Henrio"

	[henrio.emails]

	[henrio.emails.henrio_email]
	user = [
	"Ludovic",
	"Henrio",
	]
	host = [
	"sophia",
	"inria",
	"fr",
	]

	[henrio.homepages]

	[herzberg]
	name = "Michael Herzberg"

	[herzberg.emails]

	[herzberg.emails.herzberg_email]
	user = [
	"mail",
	]
	host = [
	"michael-herzberg",
	"de",
	]

	[herzberg.homepages]
	herzberg_homepage = "http://www.dcs.shef.ac.uk/cgi-bin/makeperson?M.Herzberg"

	[hess]
	name = "Andreas V. Hess"

	[hess.emails]

	[hess.emails.hess_email]
	user = [
	"avhe",
	]
	host = [
	"dtu",
	"dk",
	]

	[hess.emails.hess_email1]
	user = [
	"andreasvhess",
	]
	host = [
	"gmail",
	"com",
	]

	[hess.homepages]

	[hetzl]
	name = "Stefan Hetzl"

	[hetzl.emails]

	[hetzl.emails.hetzl_email]
	user = [
	"hetzl",
	]
	host = [
	"logic",
	"at",
	]

	[hetzl.homepages]
	hetzl_homepage = "http://www.logic.at/people/hetzl/"

	[hibon]
	name = "Quentin Hibon"

	[hibon.emails]

	[hibon.emails.hibon_email]
	user = [
	"qh225",
	]
	host = [
	"cl",
	"cam",
	"ac",
	"uk",
	]

	[hibon.homepages]

	[higgins]
	name = "Edward Higgins"

	[higgins.emails]

	[higgins.homepages]

	[hirata]
	name = "Michikazu Hirata"

	[hirata.emails]

	[hirata.emails.hirata_email]
	user = [
	"hirata",
	"m",
	"ac",
	]
	host = [
	"m",
	"titech",
	"ac",
	"jp",
	]

	[hirata.homepages]

	[hoefner]
	name = "Peter Höfner"

	[hoefner.emails]

	[hoefner.emails.hoefner_email]
	user = [
	"peter",
	]
	host = [
	"hoefner-online",
	"de",
	]

	[hoefner.homepages]
	hoefner_homepage = "http://www.hoefner-online.de/"

	[hoelzl]
	name = "Johannes Hölzl"

	[hoelzl.emails]

	[hoelzl.emails.hoelzl_email]
	user = [
	"hoelzl",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[hoelzl.homepages]
	hoelzl_homepage = "http://home.in.tum.de/~hoelzl"

	[hofmann]
	name = "Martin Hofmann"

	[hofmann.emails]

	[hofmann.homepages]
	hofmann_homepage = "http://www.tcs.informatik.uni-muenchen.de/~mhofmann"

	[hofmeier]
	name = "Paul Hofmeier"

	[hofmeier.emails]

	[hofmeier.emails.hofmeier_email]
	user = [
	"paul",
	"hofmeier",
	]
	host = [
	"tum",
	"de",
	]

	[hofmeier.homepages]

	[holub]
	name = "Štěpán Holub"

	[holub.emails]

	[holub.emails.holub_email]
	user = [
	"holub",
	]
	host = [
	"karlin",
	"mff",
	"cuni",
	"cz",
	]

	[holub.homepages]
	holub_homepage = "https://www2.karlin.mff.cuni.cz/~holub/"

	[hosking]
	name = "Tony Hosking"

	[hosking.emails]

	[hosking.homepages]
	hosking_homepage = "https://www.cs.purdue.edu/homes/hosking/"

	[hou]
	name = "Zhe Hou"

	[hou.emails]

	[hou.emails.hou_email]
	user = [
	"zhe",
	"hou",
	]
	host = [
	"ntu",
	"edu",
	"sg",
	]

	[hou.homepages]

	[hu]
	name = "Shuwei Hu"

	[hu.emails]

	[hu.emails.hu_email]
	user = [
	"shuwei",
	"hu",
	]
	host = [
	"tum",
	"de",
	]

	[hu.homepages]

	[huffman]
	name = "Brian Huffman"

	[huffman.emails]

	[huffman.emails.huffman_email]
	user = [
	"huffman",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[huffman.emails.huffman_email1]
	user = [
	"brianh",
	]
	host = [
	"cs",
	"pdx",
	"edu",
	]

	[huffman.homepages]
	huffman_homepage = "http://cs.pdx.edu/~brianh/"

	[hupel]
	name = "Lars Hupel"

	[hupel.emails]

	[hupel.emails.hupel_email]
	user = [
	"lars",
	]
	host = [
	"hupel",
	"info",
	]

	[hupel.homepages]
	hupel_homepage = "https://lars.hupel.info/"

	[ijbema]
	name = "Mark Ijbema"

	[ijbema.emails]

	[ijbema.emails.ijbema_email]
	user = [
	"ijbema",
	]
	host = [
	"fmf",
	"nl",
	]

	[ijbema.homepages]

	[immler]
	name = "Fabian Immler"

	[immler.emails]

	[immler.emails.immler_email]
	user = [
	"immler",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[immler.emails.immler_email1]
	user = [
	"fimmler",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[immler.homepages]
	immler_homepage = "https://home.in.tum.de/~immler/"

	[israel]
	name = "Jonas Israel"

	[israel.emails]

	[israel.homepages]
	israel_homepage = "https://www.algo.tu-berlin.de/menue/people/jonas_israel/"

	[ito]
	name = "Yosuke Ito"

	[ito.emails]

	[ito.emails.ito_email]
	user = [
	"glacier345",
	]
	host = [
	"gmail",
	"com",
	]

	[ito.homepages]

	[iwama]
	name = "Fumiya Iwama"

	[iwama.emails]

	[iwama.emails.iwama_email]
	user = [
	"d1623001",
	]
	host = [
	"s",
	"konan-u",
	"ac",
	"jp",
	]

	[iwama.homepages]

	[jacobsen]
	name = "Frederik Krogsdal Jacobsen"

	[jacobsen.emails]

	[jacobsen.emails.jacobsen_email]
	user = [
	"fkjac",
	]
	host = [
	"dtu",
	"dk",
	]

	[jacobsen.homepages]
	jacobsen_homepage = "http://people.compute.dtu.dk/fkjac/"

	[jaskelioff]
	name = "Mauro Jaskelioff"

	[jaskelioff.emails]

	[jaskelioff.homepages]
	jaskelioff_homepage = "http://www.fceia.unr.edu.ar/~mauro/"

	[jaskolka]
	name = "Jason Jaskolka"

	[jaskolka.emails]

	[jaskolka.emails.jaskolka_email]
	user = [
	"jason",
	"jaskolka",
	]
	host = [
	"carleton",
	"ca",
	]

	[jaskolka.homepages]
	jaskolka_homepage = "https://carleton.ca/jaskolka/"

	[jensen]
	name = "Alexander Birch Jensen"

	[jensen.emails]

	[jensen.emails.jensen_email]
	user = [
	"aleje",
	]
	host = [
	"dtu",
	"dk",
	]

	[jensen.homepages]
	jensen_homepage = "https://people.compute.dtu.dk/aleje/"

	[jiang]
	name = "Nan Jiang"

	[jiang.emails]

	[jiang.emails.jiang_email]
	user = [
	"nanjiang",
	]
	host = [
	"whu",
	"edu",
	"cn",
	]

	[jiang.homepages]

	[jiangd]
	name = "Dongchen Jiang"

	[jiangd.emails]

	[jiangd.emails.jiangd_email]
	user = [
	"dongchenjiang",
	]
	host = [
	"googlemail",
	"com",
	]

	[jiangd.homepages]

	[joosten]
	name = "Sebastiaan J. C. Joosten"

	[joosten.emails]

	[joosten.emails.joosten_email]
	user = [
	"sebastiaan",
	"joosten",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[joosten.emails.joosten_email1]
	user = [
	"sjcjoosten",
	]
	host = [
	"gmail",
	"com",
	]

	[joosten.emails.joosten_email2]
	user = [
	"s",
	"j",
	"c",
	"joosten",
	]
	host = [
	"utwente",
	"nl",
	]

	[joosten.homepages]
	joosten_homepage = "https://sjcjoosten.nl/"

	[jungnickel]
	name = "Tim Jungnickel"

	[jungnickel.emails]

	[jungnickel.emails.jungnickel_email]
	user = [
	"tim",
	"jungnickel",
	]
	host = [
	"tu-berlin",
	"de",
	]

	[jungnickel.homepages]

	[kadzioka]
	name = "Maya Kądziołka"

	[kadzioka.emails]

	[kadzioka.emails.kadzioka_email]
	user = [
	"afp",
	]
	host = [
	"compilercrim",
	"es",
	]

	[kadzioka.homepages]

	[kaliszyk]
	name = "Cezary Kaliszyk"

	[kaliszyk.emails]

	[kaliszyk.emails.kaliszyk_email]
	user = [
	"cezary",
	"kaliszyk",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[kaliszyk.homepages]
	kaliszyk_homepage = "http://cl-informatik.uibk.ac.at/users/cek/"

	[kammueller]
	name = "Florian Kammüller"

	[kammueller.emails]

	[kammueller.emails.kammueller_email]
	user = [
	"flokam",
	]
	host = [
	"cs",
	"tu-berlin",
	"de",
	]

	[kammueller.emails.kammueller_email1]
	user = [
	"florian",
	"kammuller",
	]
	host = [
	"gmail",
	"com",
	]

	[kammueller.homepages]
	kammueller_homepage = "http://www.cs.mdx.ac.uk/people/florian-kammueller/"

	[kappelmann]
	name = "Kevin Kappelmann"

	[kappelmann.emails]

	[kappelmann.emails.kappelmann_email]
	user = [
	"kevin",
	"kappelmann",
	]
	host = [
	"tum",
	"de",
	]

	[kappelmann.homepages]
	kappelmann_homepage = "https://www21.in.tum.de/team/kappelmk/"

	[karayel]
	name = "Emin Karayel"
	orcid = "0000-0003-3290-5034"

	[karayel.emails]

	[karayel.emails.karayel_email]
	user = [
	"me",
	]
	host = [
	"eminkarayel",
	"de",
	]

	[karayel.homepages]
	karayel_homepage = "https://orcid.org/0000-0003-3290-5034"

	[kastermans]
	name = "Bart Kastermans"

	[kastermans.emails]

	[kastermans.homepages]
	kastermans_homepage = "http://kasterma.net"

	[katovsky]
	name = "Alexander Katovsky"

	[katovsky.emails]

	[katovsky.emails.katovsky_email]
	user = [
	"apk32",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[katovsky.emails.katovsky_email1]
	user = [
	"alexander",
	"katovsky",
	]
	host = [
	"cantab",
	"net",
	]

	[katovsky.homepages]

	[kaufmann]
	name = "Daniela Kaufmann"

	[kaufmann.emails]

	[kaufmann.homepages]
	kaufmann_homepage = "http://fmv.jku.at/kaufmann"

	[keefe]
	name = "Greg O'Keefe"

	[keefe.emails]

	[keefe.homepages]
	keefe_homepage = "http://users.rsise.anu.edu.au/~okeefe/"

	[keinholz]
	name = "Jonas Keinholz"

	[keinholz.emails]

	[keinholz.homepages]

	[kerber]
	name = "Manfred Kerber"

	[kerber.emails]

	[kerber.emails.kerber_email]
	user = [
	"mnfrd",
	"krbr",
	]
	host = [
	"gmail",
	"com",
	]

	[kerber.homepages]
	kerber_homepage = "http://www.cs.bham.ac.uk/~mmk"

	[keskin]
	name = "Ata Keskin"
	orcid = "0000-0002-8296-1766"

	[keskin.emails]

	[keskin.emails.keskin_email]
	user = [
	"ata",
	"keskin",
	]
	host = [
	"tum",
	"de",
	]

	[keskin.homepages]

	[ketland]
	name = "Jeffrey Ketland"

	[ketland.emails]

	[ketland.emails.ketland_email]
	user = [
	"jeffreyketland",
	]
	host = [
	"gmail",
	"com",
	]

	[ketland.homepages]

	[kim]
	name = "Sunpill Kim"
	orcid = "0000-0002-7767-4084"

	[kim.emails]

	[kim.homepages]

	[kirchner]
	name = "Daniel Kirchner"

	[kirchner.emails]

	[kirchner.emails.kirchner_email]
	user = [
	"daniel",
	]
	host = [
	"ekpyron",
	"org",
	]

	[kirchner.homepages]

	[klein]
	name = "Gerwin Klein"

	[klein.emails]

	[klein.emails.klein_email]
	user = [
	"kleing",
	]
	host = [
	"unsw",
	"edu",
	"au",
	]

	[klein.homepages]
	klein_homepage = "http://www.cse.unsw.edu.au/~kleing/"

	[klenze]
	name = "Tobias Klenze"

	[klenze.emails]

	[klenze.emails.klenze_email]
	user = [
	"tobias",
	"klenze",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[klenze.homepages]

	[kleppmann]
	name = "Martin Kleppmann"

	[kleppmann.emails]

	[kleppmann.emails.kleppmann_email]
	user = [
	"martin",
	"kleppmann",
	]
	host = [
	"cl",
	"cam",
	"ac",
	"uk",
	]

	[kleppmann.homepages]

	[kobayashi]
	name = "Hidetsune Kobayashi"

	[kobayashi.emails]

	[kobayashi.homepages]

	[koerner]
	name = "Stefan Körner"

	[koerner.emails]

	[koerner.emails.koerner_email]
	user = [
	"s_koer03",
	]
	host = [
	"uni-muenster",
	"de",
	]

	[koerner.homepages]

	[kolanski]
	name = "Rafal Kolanski"

	[kolanski.emails]

	[kolanski.emails.kolanski_email]
	user = [
	"rafal",
	"kolanski",
	]
	host = [
	"nicta",
	"com",
	"au",
	]

	[kolanski.homepages]

	[koller]
	name = "Lukas Koller"

	[koller.emails]

	[koller.emails.koller_email]
	user = [
	"lukas",
	"koller",
	]
	host = [
	"tum",
	"de",
	]

	[koller.homepages]

	[krauss]
	name = "Alexander Krauss"

	[krauss.emails]

	[krauss.emails.krauss_email]
	user = [
	"krauss",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[krauss.homepages]
	krauss_homepage = "http://www.in.tum.de/~krauss"

	[kreuzer]
	name = "Katharina Kreuzer"

	[kreuzer.emails]

	[kreuzer.emails.kreuzer_email]
	user = [
	"kreuzerk",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[kreuzer.emails.kreuzer_email1]
	user = [
	"k",
	"kreuzer",
	]
	host = [
	"tum",
	"de",
	]

	[kreuzer.homepages]
	kreuzer_homepage = "https://www21.in.tum.de/team/kreuzer/"

	[kuncak]
	name = "Viktor Kuncak"

	[kuncak.emails]

	[kuncak.homepages]
	kuncak_homepage = "http://lara.epfl.ch/~kuncak/"

	[kuncar]
	name = "Ondřej Kunčar"

	[kuncar.emails]

	[kuncar.homepages]
	kuncar_homepage = "http://www21.in.tum.de/~kuncar/"

	[kurz]
	name = "Friedrich Kurz"

	[kurz.emails]

	[kurz.emails.kurz_email]
	user = [
	"friedrich",
	"kurz",
	]
	host = [
	"tum",
	"de",
	]

	[kurz.homepages]

	[lachnitt]
	name = "Hanna Lachnitt"

	[lachnitt.emails]

	[lachnitt.emails.lachnitt_email]
	user = [
	"lachnitt",
	]
	host = [
	"stanford",
	"edu",
	]

	[lachnitt.homepages]

	[lallemand]
	name = "Joseph Lallemand"

	[lallemand.emails]

	[lallemand.emails.lallemand_email]
	user = [
	"joseph",
	"lallemand",
	]
	host = [
	"loria",
	"fr",
	]

	[lallemand.homepages]

	[lammich]
	name = "Peter Lammich"

	[lammich.emails]

	[lammich.emails.lammich_email]
	user = [
	"lammich",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[lammich.emails.lammich_email1]
	user = [
	"peter",
	"lammich",
	]
	host = [
	"uni-muenster",
	"de",
	]

	[lammich.homepages]
	lammich_homepage = "http://www21.in.tum.de/~lammich"

	[lange]
	name = "Christoph Lange"

	[lange.emails]

	[lange.emails.lange_email]
	user = [
	"math",
	"semantic",
	"web",
	]
	host = [
	"gmail",
	"com",
	]

	[lange.homepages]

	[langenstein]
	name = "Bruno Langenstein"

	[langenstein.emails]

	[langenstein.emails.langenstein_email]
	user = [
	"langenstein",
	]
	host = [
	"dfki",
	"de",
	]

	[langenstein.homepages]

	[lattuada]
	name = "Andrea Lattuada"

	[lattuada.emails]

	[lattuada.homepages]
	lattuada_homepage = "https://andrea.lattuada.me"

	[lauermann]
	name = "Nils Lauermann"

	[lauermann.emails]

	[lauermann.emails.lauermann_email]
	user = [
	"Nils",
	"Lauermann",
	]
	host = [
	"cl",
	"cam",
	"ac",
	"uk",
	]

	[lauermann.homepages]

	[laursen]
	name = "Christian Pardillo-Laursen"

	[laursen.emails]

	[laursen.emails.laursen_email]
	user = [
	"christian",
	"laursen",
	]
	host = [
	"york",
	"ac",
	"uk",
	]

	[laursen.homepages]

	[lederer]
	name = "Patrick Lederer"

	[lederer.emails]

	[lederer.homepages]
	lederer_homepage = "https://www.cs.cit.tum.de/en/dss/members/patrick-lederer/"

	[lee]
	name = "Holden Lee"

	[lee.emails]

	[lee.emails.lee_email]
	user = [
	"holdenl",
	]
	host = [
	"princeton",
	"edu",
	]

	[lee.homepages]

	+[leek]
	+name = "Kevin Lee"
	+
	+[leek.emails]
	+
	+[leek.emails.leek_email]
	+user = [
	+ "kfkl2",
	+]
	+host = [
	+ "cam",
	+ "ac",
	+ "uk",
	+]
	+
	+[leek.homepages]
	+
	[leustean]
	name = "Laurentiu Leustean"

	[leustean.emails]

	[leustean.homepages]

	[lewis]
	name = "Corey Lewis"

	[lewis.emails]

	[lewis.emails.lewis_email]
	user = [
	"corey",
	"lewis",
	]
	host = [
	"data61",
	"csiro",
	"au",
	]

	[lewis.homepages]

	[li]
	name = "Wenda Li"

	[li.emails]

	[li.emails.li_email]
	user = [
	"wl302",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[li.emails.li_email1]
	user = [
	"liwenda1990",
	]
	host = [
	"hotmail",
	"com",
	]

	[li.homepages]
	li_homepage = "https://www.cl.cam.ac.uk/~wl302/"

	[lim]
	name = "Japheth Lim"

	[lim.emails]

	[lim.homepages]

	[lindenberg]
	name = "Christina Lindenberg"

	[lindenberg.emails]

	[lindenberg.homepages]

	[linker]
	name = "Sven Linker"

	[linker.emails]

	[linker.emails.linker_email]
	user = [
	"s",
	"linker",
	]
	host = [
	"liverpool",
	"ac",
	"uk",
	]

	[linker.homepages]

	[liu]
	name = "Junyi Liu"

	[liu.emails]

	[liu.homepages]

	[liut]
	name = "Tao Liu"

	[liut.emails]

	[liut.homepages]

	[liuy]
	name = "Yang Liu"

	[liuy.emails]

	[liuy.emails.liuy_email]
	user = [
	"yangliu",
	]
	host = [
	"ntu",
	"edu",
	"sg",
	]

	[liuy.homepages]

	[liy]
	name = "Yangjia Li"

	[liy.emails]

	[liy.homepages]

	[lochbihler]
	name = "Andreas Lochbihler"

	[lochbihler.emails]

	[lochbihler.emails.lochbihler_email]
	user = [
	"andreas",
	"lochbihler",
	]
	host = [
	"digitalasset",
	"com",
	]

	[lochbihler.emails.lochbihler_email1]
	user = [
	"mail",
	]
	host = [
	"andreas-lochbihler",
	"de",
	]

	[lochbihler.homepages]
	lochbihler_homepage = "http://www.andreas-lochbihler.de/"

	[lochmann]
	name = "Alexander Lochmann"

	[lochmann.emails]

	[lochmann.emails.lochmann_email]
	user = [
	"alexander",
	"lochmann",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[lochmann.homepages]

	[lohner]
	name = "Denis Lohner"

	[lohner.emails]

	[lohner.emails.lohner_email]
	user = [
	"denis",
	"lohner",
	]
	host = [
	"kit",
	"edu",
	]

	[lohner.homepages]
	lohner_homepage = "http://pp.ipd.kit.edu/person.php?id=88"

	[loibl]
	name = "Matthias Loibl"

	[loibl.emails]

	[loibl.homepages]

	[londono]
	name = "Alejandro Gómez-Londoño"

	[londono.emails]

	[londono.emails.londono_email]
	user = [
	"alejandro",
	"gomez",
	]
	host = [
	"chalmers",
	"se",
	]

	[londono.homepages]

	[losa]
	name = "Giuliano Losa"

	[losa.emails]

	[losa.emails.losa_email]
	user = [
	"giuliano",
	"losa",
	]
	host = [
	"epfl",
	"ch",
	]

	[losa.emails.losa_email1]
	user = [
	"giuliano",
	]
	host = [
	"galois",
	"com",
	]

	[losa.emails.losa_email2]
	user = [
	"giuliano",
	]
	host = [
	"losa",
	"fr",
	]

	[losa.homepages]

	[lutz]
	name = "Bianca Lutz"

	[lutz.emails]

	[lutz.emails.lutz_email]
	user = [
	"sowilo",
	]
	host = [
	"cs",
	"tu-berlin",
	"de",
	]

	[lutz.homepages]

	[lux]
	name = "Alexander Lux"

	[lux.emails]

	[lux.emails.lux_email]
	user = [
	"lux",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[lux.homepages]

	[madarasz]
	name = "Judit Madarasz"

	[madarasz.emails]

	[madarasz.homepages]
	madarasz_homepage = "https://users.renyi.hu/~madarasz/"

	[makarios]
	name = "T. J. M. Makarios"

	[makarios.emails]

	[makarios.emails.makarios_email]
	user = [
	"tjm1983",
	]
	host = [
	"gmail",
	"com",
	]

	[makarios.homepages]

	[maletzky]
	name = "Alexander Maletzky"

	[maletzky.emails]

	[maletzky.emails.maletzky_email]
	user = [
	"alexander",
	"maletzky",
	]
	host = [
	"risc",
	"jku",
	"at",
	]

	[maletzky.emails.maletzky_email1]
	user = [
	"alexander",
	"maletzky",
	]
	host = [
	"risc-software",
	"at",
	]

	[maletzky.homepages]
	maletzky_homepage = "https://risc.jku.at/m/alexander-maletzky/"

	[mansky]
	name = "Susannah Mansky"

	[mansky.emails]

	[mansky.emails.mansky_email]
	user = [
	"sjohnsn2",
	]
	host = [
	"illinois",
	"edu",
	]

	[mansky.emails.mansky_email1]
	user = [
	"susannahej",
	]
	host = [
	"gmail",
	"com",
	]

	[mansky.homepages]

	[mantel]
	name = "Heiko Mantel"

	[mantel.emails]

	[mantel.emails.mantel_email]
	user = [
	"mantel",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[mantel.homepages]

	[margetson]
	name = "James Margetson"

	[margetson.emails]

	[margetson.homepages]

	[maric]
	name = "Ognjen Marić"

	[maric.emails]

	[maric.emails.maric_email]
	user = [
	"ogi",
	"afp",
	]
	host = [
	"mynosefroze",
	"com",
	]

	[maric.homepages]

	[maricf]
	name = "Filip Marić"

	[maricf.emails]

	[maricf.emails.maricf_email]
	user = [
	"filip",
	]
	host = [
	"matf",
	"bg",
	"ac",
	"rs",
	]

	[maricf.homepages]
	maricf_homepage = "http://www.matf.bg.ac.rs/~filip"

	[marmsoler]
	name = "Diego Marmsoler"

	[marmsoler.emails]

	[marmsoler.emails.marmsoler_email]
	user = [
	"diego",
	"marmsoler",
	]
	host = [
	"tum",
	"de",
	]

	[marmsoler.emails.marmsoler_email1]
	user = [
	"d",
	"marmsoler",
	]
	host = [
	"exeter",
	"ac",
	"uk",
	]

	[marmsoler.homepages]
	marmsoler_homepage = "http://marmsoler.com"

	[matache]
	name = "Cristina Matache"

	[matache.emails]

	[matache.emails.matache_email]
	user = [
	"cris",
	"matache",
	]
	host = [
	"gmail",
	"com",
	]

	[matache.homepages]

	[mateo]
	name = "Adrián Doña Mateo"

	[mateo.emails]

	[mateo.emails.mateo_email]
	user = [
	"adrian",
	"dona",
	]
	host = [
	"ed",
	"ac",
	"uk",
	]

	[mateo.homepages]

	[matichuk]
	name = "Daniel Matichuk"

	[matichuk.emails]

	[matichuk.homepages]

	[matiyasevich]
	name = "Yuri Matiyasevich"

	[matiyasevich.emails]

	[matiyasevich.homepages]

	[maximova]
	name = "Alexandra Maximova"

	[maximova.emails]

	[maximova.emails.maximova_email]
	user = [
	"amaximov",
	]
	host = [
	"student",
	"ethz",
	"ch",
	]

	[maximova.homepages]

	[meis]
	name = "Rene Meis"

	[meis.emails]

	[meis.emails.meis_email]
	user = [
	"rene",
	"meis",
	]
	host = [
	"uni-muenster",
	"de",
	]

	[meis.emails.meis_email1]
	user = [
	"rene",
	"meis",
	]
	host = [
	"uni-due",
	"de",
	]

	[meis.homepages]

	[merz]
	name = "Stephan Merz"

	[merz.emails]

	[merz.emails.merz_email]
	user = [
	"Stephan",
	"Merz",
	]
	host = [
	"loria",
	"fr",
	]

	[merz.homepages]
	merz_homepage = "http://www.loria.fr/~merz"

	[messner]
	name = "Florian Messner"

	[messner.emails]

	[messner.emails.messner_email]
	user = [
	"florian",
	"g",
	"messner",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[messner.homepages]

	[mhalla]
	name = "Mehdi Mhalla"

	[mhalla.emails]

	[mhalla.emails.mhalla_email]
	user = [
	"mhallam",
	]
	host = [
	"univ-grenoble-alpes",
	"fr",
	]

	[mhalla.homepages]

	[michaelis]
	name = "Julius Michaelis"

	[michaelis.emails]

	[michaelis.emails.michaelis_email]
	user = [
	"isabelleopenflow",
	]
	host = [
	"liftm",
	"de",
	]

	[michaelis.emails.michaelis_email1]
	user = [
	"maintainafpppt",
	]
	host = [
	"liftm",
	"de",
	]

	[michaelis.emails.michaelis_email2]
	user = [
	"bdd",
	]
	host = [
	"liftm",
	"de",
	]

	[michaelis.emails.michaelis_email3]
	user = [
	"afp",
	]
	host = [
	"liftm",
	"de",
	]

	[michaelis.homepages]
	michaelis_homepage = "http://liftm.de/"

	[milehins]
	name = "Mihails Milehins"

	[milehins.emails]

	[milehins.emails.milehins_email]
	user = [
	"mihailsmilehins",
	]
	host = [
	"gmail",
	"com",
	]

	[milehins.homepages]

	[minamide]
	name = "Yasuhiko Minamide"

	[minamide.emails]

	[minamide.emails.minamide_email]
	user = [
	"minamide",
	]
	host = [
	"is",
	"titech",
	"ac",
	"jp",
	]

	[minamide.homepages]
	minamide_homepage = "https://sv.c.titech.ac.jp/minamide/index.en.html"

	[mitchell]
	name = "Neil Mitchell"

	[mitchell.emails]

	[mitchell.homepages]

	[mitsch]
	name = "Stefan Mitsch"

	[mitsch.emails]

	[mitsch.emails.mitsch_email]
	user = [
	"smitsch",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[mitsch.homepages]

	[moedersheim]
	name = "Sebastian Mödersheim"

	[moedersheim.emails]

	[moedersheim.emails.moedersheim_email]
	user = [
	"samo",
	]
	host = [
	"dtu",
	"dk",
	]

	[moedersheim.homepages]
	moedersheim_homepage = "https://people.compute.dtu.dk/samo/"

	[moeller]
	name = "Bernhard Möller"

	[moeller.emails]

	[moeller.homepages]
	moeller_homepage = "https://www.informatik.uni-augsburg.de/en/chairs/dbis/pmi/staff/moeller/"

	[mori]
	name = "Coraline Mori"

	[mori.emails]

	[mori.emails.mori_email]
	user = [
	"coraline",
	"mori",
	]
	host = [
	"grenoble-inp",
	"org",
	]

	[mori.homepages]

	[muendler]
	name = "Niels Mündler"

	[muendler.emails]

	[muendler.emails.muendler_email]
	user = [
	"n",
	"muendler",
	]
	host = [
	"tum",
	"de",
	]

	[muendler.homepages]

	[mulligan]
	name = "Dominic P. Mulligan"

	[mulligan.emails]

	[mulligan.emails.mulligan_email]
	user = [
	"dominic",
	"p",
	"mulligan",
	]
	host = [
	"googlemail",
	"com",
	]

	[mulligan.emails.mulligan_email1]
	user = [
	"Dominic",
	"Mulligan",
	]
	host = [
	"arm",
	"com",
	]

	[mulligan.homepages]

	[munive]
	name = "Jonathan Julian Huerta y Munive"

	[munive.emails]

	[munive.emails.munive_email]
	user = [
	"jjhuertaymunive1",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[munive.emails.munive_email1]
	user = [
	"jonjulian23",
	]
	host = [
	"gmail",
	"com",
	]

	[munive.homepages]

	[murao]
	name = "H. Murao"

	[murao.emails]

	[murao.homepages]

	[murray]
	name = "Toby Murray"

	[murray.emails]

	[murray.emails.murray_email]
	user = [
	"toby",
	"murray",
	]
	host = [
	"unimelb",
	"edu",
	"au",
	]

	[murray.homepages]
	murray_homepage = "https://people.eng.unimelb.edu.au/tobym/"

	[myreen]
	name = "Magnus O. Myreen"
	orcid = "0000-0002-9504-4107"

	[myreen.emails]

	[myreen.emails.myreen_email]
	user = [
	"myreen",
	]
	host = [
	"chalmers",
	"se",
	]

	[myreen.homepages]

	[nagashima]
	name = "Yutaka Nagashima"

	[nagashima.emails]

	[nagashima.emails.nagashima_email]
	user = [
	"Yutaka",
	"Nagashima",
	]
	host = [
	"data61",
	"csiro",
	"au",
	]

	[nagashima.homepages]

	[nagele]
	name = "Julian Nagele"

	[nagele.emails]

	[nagele.emails.nagele_email]
	user = [
	"julian",
	"nagele",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[nagele.homepages]

	[naraschewski]
	name = "Wolfgang Naraschewski"

	[naraschewski.emails]

	[naraschewski.homepages]

	[nedzelsky]
	name = "Michael Nedzelsky"

	[nedzelsky.emails]

	[nedzelsky.emails.nedzelsky_email]
	user = [
	"MichaelNedzelsky",
	]
	host = [
	"yandex",
	"ru",
	]

	[nedzelsky.homepages]

	[nemeti]
	name = "István Németi"

	[nemeti.emails]

	[nemeti.homepages]
	nemeti_homepage = "http://www.renyi.hu/~nemeti/"

	[nemouchi]
	name = "Yakoub Nemouchi"

	[nemouchi.emails]

	[nemouchi.emails.nemouchi_email]
	user = [
	"nemouchi",
	]
	host = [
	"lri",
	"fr",
	]

	[nemouchi.emails.nemouchi_email1]
	user = [
	"yakoub",
	"nemouchi",
	]
	host = [
	"york",
	"ac",
	"uk",
	]

	[nemouchi.homepages]

	[nestmann]
	name = "Uwe Nestmann"

	[nestmann.emails]

	[nestmann.homepages]
	nestmann_homepage = "https://www.mtv.tu-berlin.de/nestmann/"

	[neumann]
	name = "René Neumann"

	[neumann.emails]

	[neumann.emails.neumann_email]
	user = [
	"rene",
	"neumann",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[neumann.homepages]

	[nielsen]
	name = "Finn Nielsen"

	[nielsen.emails]

	[nielsen.emails.nielsen_email]
	user = [
	"finn",
	"nielsen",
	]
	host = [
	"uni-muenster",
	"de",
	]

	[nielsen.homepages]

	[nikiforov]
	name = "Denis Nikiforov"

	[nikiforov.emails]

	[nikiforov.emails.nikiforov_email]
	user = [
	"denis",
	"nikif",
	]
	host = [
	"gmail",
	"com",
	]

	[nikiforov.homepages]

	[nipkow]
	name = "Tobias Nipkow"
	orcid = "0000-0003-0730-515X"

	[nipkow.emails]

	[nipkow.emails.nipkow_email]
	user = [
	"nipkow",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[nipkow.homepages]
	nipkow_homepage = "https://www.in.tum.de/~nipkow/"

	[nishihara]
	name = "Toshiaki Nishihara"

	[nishihara.emails]

	[nishihara.homepages]

	[noce]
	name = "Pasquale Noce"

	[noce.emails]

	[noce.emails.noce_email]
	user = [
	"pasquale",
	"noce",
	"lavoro",
	]
	host = [
	"gmail",
	"com",
	]

	[noce.homepages]

	[nordhoff]
	name = "Benedikt Nordhoff"

	[nordhoff.emails]

	[nordhoff.emails.nordhoff_email]
	user = [
	"b",
	"n",
	]
	host = [
	"wwu",
	"de",
	]

	[nordhoff.emails.nordhoff_email1]
	user = [
	"b_nord01",
	]
	host = [
	"uni-muenster",
	"de",
	]

	[nordhoff.homepages]

	[noschinski]
	name = "Lars Noschinski"

	[noschinski.emails]

	[noschinski.emails.noschinski_email]
	user = [
	"noschinl",
	]
	host = [
	"gmail",
	"com",
	]

	[noschinski.homepages]
	noschinski_homepage = "http://www21.in.tum.de/~noschinl/"

	[obua]
	name = "Steven Obua"

	[obua.emails]

	[obua.emails.obua_email]
	user = [
	"steven",
	]
	host = [
	"recursivemind",
	"com",
	]

	[obua.homepages]

	[ogawa]
	name = "Mizuhito Ogawa"

	[ogawa.emails]

	[ogawa.homepages]

	[oldenburg]
	name = "Lennart Oldenburg"

	[oldenburg.emails]

	[oldenburg.homepages]

	[olm]
	name = "Markus Müller-Olm"

	[olm.emails]

	[olm.homepages]
	olm_homepage = "http://cs.uni-muenster.de/u/mmo/"

	[oosterhuis]
	name = "Roelof Oosterhuis"

	[oosterhuis.emails]

	[oosterhuis.emails.oosterhuis_email]
	user = [
	"roelofoosterhuis",
	]
	host = [
	"gmail",
	"com",
	]

	[oosterhuis.homepages]

	[oostrom]
	name = "Vincent van Oostrom"

	[oostrom.emails]

	[oostrom.homepages]

	[ortner]
	name = "Veronika Ortner"

	[ortner.emails]

	[ortner.homepages]

	[overbeek]
	name = "Roy Overbeek"

	[overbeek.emails]

	[overbeek.emails.overbeek_email]
	user = [
	"Roy",
	"Overbeek",
	]
	host = [
	"cwi",
	"nl",
	]

	[overbeek.homepages]

	[pagano]
	name = "Miguel Pagano"

	[pagano.emails]

	[pagano.emails.pagano_email]
	user = [
	"miguel",
	"pagano",
	]
	host = [
	"unc",
	"edu",
	"ar",
	]

	[pagano.homepages]
	pagano_homepage = "https://cs.famaf.unc.edu.ar/~mpagano/"

	[pal]
	name = "Abhik Pal"

	[pal.emails]

	[pal.homepages]

	[paleo]
	name = "Bruno Woltzenlogel Paleo"

	[paleo.emails]

	[paleo.homepages]
	paleo_homepage = "http://www.logic.at/staff/bruno/"

	[palmer]
	name = "Jake Palmer"

	[palmer.emails]

	[palmer.emails.palmer_email]
	user = [
	"jake",
	"palmer",
	]
	host = [
	"ed",
	"ac",
	"uk",
	]

	[palmer.homepages]

	[park]
	name = "Seung Hoon Park"
	orcid = "0000-0001-7165-6857"

	[park.emails]

	[park.emails.park_email]
	user = [
	"seunghoon",
	"park",
	]
	host = [
	"cs",
	"ox",
	"ac",
	"uk",
	]

	[park.homepages]
	park_homepage = "https://www.cs.ox.ac.uk/people/simon.park/"

	[parkinson]
	name = "Matthew Parkinson"

	[parkinson.emails]

	[parkinson.homepages]
	parkinson_homepage = "http://research.microsoft.com/people/mattpark/"

	[parrow]
	name = "Joachim Parrow"

	[parrow.emails]

	[parrow.emails.parrow_email]
	user = [
	"joachim",
	"parrow",
	]
	host = [
	"it",
	"uu",
	"se",
	]

	[parrow.homepages]

	[parsert]
	name = "Julian Parsert"

	[parsert.emails]

	[parsert.emails.parsert_email]
	user = [
	"julian",
	"parsert",
	]
	host = [
	"gmail",
	"com",
	]

	[parsert.emails.parsert_email1]
	user = [
	"julian",
	"parsert",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[parsert.homepages]
	parsert_homepage = "http://www.parsert.com/"

	[paulson]
	name = "Lawrence C. Paulson"

	[paulson.emails]

	[paulson.emails.paulson_email]
	user = [
	"lp15",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[paulson.homepages]
	paulson_homepage = "https://www.cl.cam.ac.uk/~lp15/"

	[peltier]
	name = "Nicolas Peltier"

	[peltier.emails]

	[peltier.emails.peltier_email]
	user = [
	"Nicolas",
	"Peltier",
	]
	host = [
	"imag",
	"fr",
	]

	[peltier.homepages]
	peltier_homepage = "http://membres-lig.imag.fr/peltier/"

	[peters]
	name = "Kirstin Peters"

	[peters.emails]

	[peters.emails.peters_email]
	user = [
	"kirstin",
	"peters",
	]
	host = [
	"tu-berlin",
	"de",
	]

	[peters.homepages]

	[petrovic]
	name = "Danijela Petrovic"

	[petrovic.emails]

	[petrovic.homepages]
	petrovic_homepage = "http://www.matf.bg.ac.rs/~danijela"

	[pierzchalski]
	name = "Edward Pierzchalski"

	[pierzchalski.emails]

	[pierzchalski.homepages]

	[platzer]
	name = "André Platzer"

	[platzer.emails]

	[platzer.emails.platzer_email]
	user = [
	"aplatzer",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[platzer.homepages]
	platzer_homepage = "https://www.cs.cmu.edu/~aplatzer/"

	[pohjola]
	name = "Johannes Åman Pohjola"

	[pohjola.emails]

	[pohjola.emails.pohjola_email]
	user = [
	"j",
	"amanpohjola",
	]
	host = [
	"unsw",
	"edu",
	"au",
	]

	[pohjola.homepages]

	[pollak]
	name = "Florian Pollak"

	[pollak.emails]

	[pollak.emails.pollak_email]
	user = [
	"florian",
	"pollak",
	]
	host = [
	"gmail",
	"com",
	]

	[pollak.homepages]

	[popescu]
	name = "Andrei Popescu"

	[popescu.emails]

	[popescu.emails.popescu_email]
	user = [
	"a",
	"popescu",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[popescu.emails.popescu_email1]
	user = [
	"uuomul",
	]
	host = [
	"yahoo",
	"com",
	]

	[popescu.emails.popescu_email2]
	user = [
	"a",
	"popescu",
	]
	host = [
	"mdx",
	"ac",
	"uk",
	]

	[popescu.homepages]
	popescu_homepage = "https://www.andreipopescu.uk"

	[porter]
	name = "Benjamin Porter"

	[porter.emails]

	[porter.homepages]

	[prathamesh]
	name = "T.V.H. Prathamesh"

	[prathamesh.emails]

	[prathamesh.emails.prathamesh_email]
	user = [
	"prathamesh",
	]
	host = [
	"imsc",
	"res",
	"in",
	]

	[prathamesh.homepages]

	[preoteasa]
	name = "Viorel Preoteasa"

	[preoteasa.emails]

	[preoteasa.emails.preoteasa_email]
	user = [
	"viorel",
	"preoteasa",
	]
	host = [
	"aalto",
	"fi",
	]

	[preoteasa.homepages]
	preoteasa_homepage = "http://users.abo.fi/vpreotea/"

	[pusch]
	name = "Cornelia Pusch"

	[pusch.emails]

	[pusch.homepages]

	[qiu]
	name = "Qi Qiu"

	[qiu.emails]

	[qiu.emails.qiu_email]
	user = [
	"qi",
	"qiu",
	]
	host = [
	"univ-lyon1",
	"fr",
	]

	[qiu.homepages]

	[rabe]
	name = "Markus N. Rabe"

	[rabe.emails]

	[rabe.homepages]
	rabe_homepage = "http://www.react.uni-saarland.de/people/rabe.html"

	[raedle]
	name = "Jonas Rädle"

	[raedle.emails]

	[raedle.emails.raedle_email]
	user = [
	"jonas",
	"raedle",
	]
	host = [
	"gmail",
	"com",
	]

	[raedle.emails.raedle_email1]
	user = [
	"jonas",
	"raedle",
	]
	host = [
	"tum",
	"de",
	]

	[raedle.homepages]

	[raska]
	name = "Martin Raška"

	[raska.emails]

	[raska.homepages]

	[raszyk]
	name = "Martin Raszyk"

	[raszyk.emails]

	[raszyk.emails.raszyk_email]
	user = [
	"martin",
	"raszyk",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[raszyk.emails.raszyk_email1]
	user = [
	"m",
	"raszyk",
	]
	host = [
	"gmail",
	"com",
	]

	[raszyk.homepages]

	[rau]
	name = "Martin Rau"

	[rau.emails]

	[rau.emails.rau_email]
	user = [
	"martin",
	"rau",
	]
	host = [
	"tum",
	"de",
	]

	[rau.emails.rau_email1]
	user = [
	"mrtnrau",
	]
	host = [
	"googlemail",
	"com",
	]

	[rau.homepages]

	[rauch]
	name = "Nicole Rauch"

	[rauch.emails]

	[rauch.emails.rauch_email]
	user = [
	"rauch",
	]
	host = [
	"informatik",
	"uni-kl",
	"de",
	]

	[rauch.homepages]

	[raumer]
	name = "Jakob von Raumer"

	[raumer.emails]

	[raumer.emails.raumer_email]
	user = [
	"psxjv4",
	]
	host = [
	"nottingham",
	"ac",
	"uk",
	]

	[raumer.homepages]

	[ravindran]
	name = "Binoy Ravindran"

	[ravindran.emails]

	[ravindran.homepages]

	[rawson]
	name = "Michael Rawson"

	[rawson.emails]

	[rawson.emails.rawson_email]
	user = [
	"michaelrawson76",
	]
	host = [
	"gmail",
	"com",
	]

	[rawson.emails.rawson_email1]
	user = [
	"mr644",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[rawson.homepages]

	[raya]
	name = "Rodrigo Raya"

	[raya.emails]

	[raya.emails.raya_email]
	user = [
	"rodrigo",
	"raya",
	]
	host = [
	"epfl",
	"ch",
	]

	[raya.homepages]
	raya_homepage = "https://people.epfl.ch/rodrigo.raya"

	[regensburger]
	name = "Franz Regensburger"

	[regensburger.emails]

	[regensburger.emails.regensburger_email]
	user = [
	"Franz",
	"Regensburger",
	]
	host = [
	"thi",
	"de",
	]

	[regensburger.homepages]
	regensburger_homepage = "https://www.thi.de/suche/mitarbeiter/prof-dr-rer-nat-franz-regensburger"

	[reiche]
	name = "Sebastian Reiche"

	[reiche.emails]

	[reiche.homepages]
	reiche_homepage = "https://www.linkedin.com/in/sebastian-reiche-0b2093178"

	[reiter]
	name = "Markus Reiter"

	[reiter.emails]

	[reiter.homepages]

	[reynaud]
	name = "Alban Reynaud"

	[reynaud.emails]

	[reynaud.homepages]

	[ribeiro]
	name = "Pedro Ribeiro"

	[ribeiro.emails]

	[ribeiro.homepages]

	[richter]
	name = "Stefan Richter"

	[richter.emails]

	[richter.emails.richter_email]
	user = [
	"richter",
	]
	host = [
	"informatik",
	"rwth-aachen",
	"de",
	]

	[richter.homepages]
	richter_homepage = "http://www-lti.informatik.rwth-aachen.de/~richter/"

	[rickmann]
	name = "Christina Rickmann"

	[rickmann.emails]

	[rickmann.emails.rickmann_email]
	user = [
	"c",
	"rickmann",
	]
	host = [
	"tu-berlin",
	"de",
	]

	[rickmann.homepages]

	[ridge]
	name = "Tom Ridge"

	[ridge.emails]

	[ridge.homepages]

	[rizaldi]
	name = "Albert Rizaldi"

	[rizaldi.emails]

	[rizaldi.emails.rizaldi_email]
	user = [
	"albert",
	"rizaldi",
	]
	host = [
	"ntu",
	"edu",
	"sg",
	]

	[rizaldi.homepages]

	[rizkallah]
	name = "Christine Rizkallah"

	[rizkallah.emails]

	[rizkallah.homepages]
	rizkallah_homepage = "https://www.mpi-inf.mpg.de/~crizkall/"

	[robillard]
	name = "Simon Robillard"

	[robillard.emails]

	[robillard.homepages]
	robillard_homepage = "https://simon-robillard.net/"

	[roessle]
	name = "Ian Roessle"

	[roessle.emails]

	[roessle.homepages]

	[romanos]
	name = "Ralph Romanos"

	[romanos.emails]

	[romanos.emails.romanos_email]
	user = [
	"ralph",
	"romanos",
	]
	host = [
	"student",
	"ecp",
	"fr",
	]

	[romanos.homepages]

	[rosskopf]
	name = "Simon Roßkopf"

	[rosskopf.emails]

	[rosskopf.emails.rosskopf_email]
	user = [
	"rosskops",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[rosskopf.homepages]
	rosskopf_homepage = "http://www21.in.tum.de/~rosskops"

	[rowat]
	name = "Colin Rowat"

	[rowat.emails]

	[rowat.emails.rowat_email]
	user = [
	"c",
	"rowat",
	]
	host = [
	"bham",
	"ac",
	"uk",
	]

	[rowat.homepages]

	[sabouret]
	name = "Nicolas Sabouret"

	[sabouret.emails]

	[sabouret.homepages]

	[sachtleben]
	name = "Robert Sachtleben"

	[sachtleben.emails]

	[sachtleben.emails.sachtleben_email]
	user = [
	"rob_sac",
	]
	host = [
	"uni-bremen",
	"de",
	]

	[sachtleben.homepages]

	[saile]
	name = "Christian Saile"

	[saile.emails]

	[saile.homepages]
	saile_homepage = "http://dss.in.tum.de/staff/christian-saile.html"

	[sanan]
	name = "David Sanan"

	[sanan.emails]

	[sanan.emails.sanan_email]
	user = [
	"sanan",
	]
	host = [
	"ntu",
	"edu",
	"sg",
	]

	[sanan.homepages]

	[sato]
	name = "Tetsuya Sato"

	[sato.emails]

	[sato.emails.sato_email]
	user = [
	"tsato",
	]
	host = [
	"c",
	"titech",
	"ac",
	"jp",
	]

	[sato.homepages]
	sato_homepage = "https://sites.google.com/view/tetsuyasato/"

	[sauer]
	name = "Jens Sauer"

	[sauer.emails]

	[sauer.emails.sauer_email]
	user = [
	"sauer",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[sauer.homepages]

	[schaeffeler]
	name = "Maximilian Schäffeler"

	[schaeffeler.emails]

	[schaeffeler.emails.schaeffeler_email]
	user = [
	"schaeffm",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[schaeffeler.homepages]

	[scharager]
	name = "Matias Scharager"

	[scharager.emails]

	[scharager.emails.scharager_email]
	user = [
	"mscharag",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[scharager.homepages]

	[schimpf]
	name = "Alexander Schimpf"

	[schimpf.emails]

	[schimpf.emails.schimpf_email]
	user = [
	"schimpfa",
	]
	host = [
	"informatik",
	"uni-freiburg",
	"de",
	]

	[schimpf.homepages]

	[schirmer]
	name = "Norbert Schirmer"

	[schirmer.emails]

	[schirmer.emails.schirmer_email]
	user = [
	"norbert",
	"schirmer",
	]
	host = [
	"web",
	"de",
	]

	[schirmer.homepages]

	[schleicher]
	name = "Dierk Schleicher"

	[schleicher.emails]

	[schleicher.homepages]

	[schlichtkrull]
	name = "Anders Schlichtkrull"

	[schlichtkrull.emails]

	[schlichtkrull.emails.schlichtkrull_email]
	user = [
	"andschl",
	]
	host = [
	"dtu",
	"dk",
	]

	[schlichtkrull.homepages]
	schlichtkrull_homepage = "https://people.compute.dtu.dk/andschl/"

	[schmaltz]
	name = "Julien Schmaltz"

	[schmaltz.emails]

	[schmaltz.emails.schmaltz_email]
	user = [
	"Julien",
	"Schmaltz",
	]
	host = [
	"ou",
	"nl",
	]

	[schmaltz.homepages]

	[schmidinger]
	name = "Lukas Schmidinger"

	[schmidinger.emails]

	[schmidinger.homepages]

	[schmoetten]
	name = "Richard Schmoetten"

	[schmoetten.emails]

	[schmoetten.emails.schmoetten_email]
	user = [
	"s1311325",
	]
	host = [
	"sms",
	"ed",
	"ac",
	"uk",
	]

	[schmoetten.homepages]

	[schneider]
	name = "Joshua Schneider"

	[schneider.emails]

	[schneider.emails.schneider_email]
	user = [
	"joshua",
	"schneider",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[schneider.homepages]

	[schoepe]
	name = "Daniel Schoepe"

	[schoepe.emails]

	[schoepe.emails.schoepe_email]
	user = [
	"daniel",
	]
	host = [
	"schoepe",
	"org",
	]

	[schoepe.homepages]

	[schoepf]
	name = "Jonas Schöpf"

	[schoepf.emails]

	[schoepf.emails.schoepf_email]
	user = [
	"jonas",
	"schoepf",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[schoepf.homepages]

	[scott]
	name = "Dana Scott"

	[scott.emails]

	[scott.homepages]
	scott_homepage = "http://www.cs.cmu.edu/~scott/"

	[sefidgar]
	name = "S. Reza Sefidgar"

	[sefidgar.emails]

	[sefidgar.emails.sefidgar_email]
	user = [
	"reza",
	"sefidgar",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[sefidgar.homepages]

	[seidl]
	name = "Benedikt Seidl"

	[seidl.emails]

	[seidl.emails.seidl_email]
	user = [
	"benedikt",
	"seidl",
	]
	host = [
	"tum",
	"de",
	]

	[seidl.homepages]

	[seidler]
	name = "Henning Seidler"

	[seidler.emails]

	[seidler.emails.seidler_email]
	user = [
	"henning",
	"seidler",
	]
	host = [
	"mailbox",
	"tu-berlin",
	"de",
	]

	[seidler.homepages]

	[sewell]
	name = "Thomas Sewell"

	[sewell.emails]

	[sewell.homepages]

	[sickert]
	name = "Salomon Sickert"

	[sickert.emails]

	[sickert.emails.sickert_email]
	user = [
	"s",
	"sickert",
	]
	host = [
	"tum",
	"de",
	]

	[sickert.homepages]
	sickert_homepage = "https://www7.in.tum.de/~sickert"

	[siek]
	name = "Jeremy Siek"

	[siek.emails]

	[siek.emails.siek_email]
	user = [
	"jsiek",
	]
	host = [
	"indiana",
	"edu",
	]

	[siek.homepages]
	siek_homepage = "http://homes.soic.indiana.edu/jsiek/"

	[simic]
	name = "Danijela Simić"

	[simic.emails]

	[simic.emails.simic_email]
	user = [
	"danijela",
	]
	host = [
	"matf",
	"bg",
	"ac",
	"rs",
	]

	[simic.homepages]
	simic_homepage = "http://poincare.matf.bg.ac.rs/~danijela"

	[sison]
	name = "Robert Sison"

	[sison.emails]

	[sison.homepages]

	[smaus]
	name = "Jan-Georg Smaus"

	[smaus.emails]

	[smaus.homepages]
	smaus_homepage = "http://www.irit.fr/~Jan-Georg.Smaus"

	[smola]
	name = "Filip Smola"

	[smola.emails]

	[smola.emails.smola_email]
	user = [
	"f",
	"smola",
	]
	host = [
	"sms",
	"ed",
	"ac",
	"uk",
	]

	[smola.homepages]

	[snelting]
	name = "Gregor Snelting"

	[snelting.emails]

	[snelting.homepages]
	snelting_homepage = "http://pp.info.uni-karlsruhe.de/personhp/gregor_snelting.php"

	[somaini]
	name = "Ivano Somaini"

	[somaini.emails]

	[somaini.homepages]

	[somogyi]
	name = "Dániel Somogyi"

	[somogyi.emails]

	[somogyi.homepages]

	[spasic]
	name = "Mirko Spasić"

	[spasic.emails]

	[spasic.emails.spasic_email]
	user = [
	"mirko",
	]
	host = [
	"matf",
	"bg",
	"ac",
	"rs",
	]

	[spasic.homepages]

	[spichkova]
	name = "Maria Spichkova"

	[spichkova.emails]

	[spichkova.emails.spichkova_email]
	user = [
	"maria",
	"spichkova",
	]
	host = [
	"rmit",
	"edu",
	"au",
	]

	[spichkova.homepages]

	[spitz]
	name = "Maximilian Spitz"

	[spitz.emails]

	[spitz.emails.spitz_email]
	user = [
	"maximilian",
	"spitz",
	]
	host = [
	"tum",
	"de",
	]

	[spitz.homepages]

	[sprenger]
	name = "Christoph Sprenger"

	[sprenger.emails]

	[sprenger.emails.sprenger_email]
	user = [
	"sprenger",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[sprenger.homepages]


	[springsklee]
	name = "Valentin Springsklee"

	[springsklee.emails]

	[springsklee.emails.springsklee_email]
	user = [
	"uidpn>",
	]
	host = [
	"student",
	"kit",
	"edu",
	]

	[springsklee.homepages]



	[staats]
	name = "Charles Staats"

	[staats.emails]

	[staats.emails.staats_email]
	user = [
	"cstaats",
	]
	host = [
	"google",
	"com",
	]

	[staats.emails.staats_email1]
	user = [
	"charles",
	"staats",
	"iii",
	]
	host = [
	"gmail",
	"com",
	]

	[staats.homepages]

	[stannett]
	name = "Mike Stannett"

	[stannett.emails]

	[stannett.emails.stannett_email]
	user = [
	"m",
	"stannett",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[stannett.homepages]

	[stark]
	name = "Eugene W. Stark"

	[stark.emails]

	[stark.emails.stark_email]
	user = [
	"stark",
	]
	host = [
	"cs",
	"stonybrook",
	"edu",
	]

	[stark.homepages]

	[starosta]
	name = "Štěpán Starosta"

	[starosta.emails]

	[starosta.emails.starosta_email]
	user = [
	"stepan",
	"starosta",
	]
	host = [
	"fit",
	"cvut",
	"cz",
	]

	[starosta.homepages]
	starosta_homepage = "https://users.fit.cvut.cz/~staroste/"

	[steen]
	name = "Alexander Steen"

	[steen.emails]

	[steen.homepages]

	[steinberg]
	name = "Matías Steinberg"

	[steinberg.emails]

	[steinberg.emails.steinberg_email]
	user = [
	"matias",
	"steinberg",
	]
	host = [
	"mi",
	"unc",
	"edu",
	"ar",
	]

	[steinberg.homepages]

	[stephan]
	name = "Werner Stephan"

	[stephan.emails]

	[stephan.emails.stephan_email]
	user = [
	"stephan",
	]
	host = [
	"dfki",
	"de",
	]

	[stephan.homepages]

	[sternagel]
	name = "Christian Sternagel"

	[sternagel.emails]

	[sternagel.emails.sternagel_email]
	user = [
	"c",
	"sternagel",
	]
	host = [
	"gmail",
	"com",
	]

	[sternagel.emails.sternagel_email1]
	user = [
	"christian",
	"sternagel",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[sternagel.homepages]
	sternagel_homepage = "http://cl-informatik.uibk.ac.at/users/griff/"

	[sternagelt]
	name = "Thomas Sternagel"

	[sternagelt.emails]

	[sternagelt.homepages]

	[stevens]
	name = "Lukas Stevens"

	[stevens.emails]

	[stevens.emails.stevens_email]
	user = [
	"lukas",
	"stevens",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[stevens.homepages]
	stevens_homepage = "https://www21.in.tum.de/team/stevensl"

	[stock]
	name = "Benedikt Stock"

	[stock.emails]

	[stock.emails.stock_email]
	user = [
	"benedikt1999",
	]
	host = [
	"freenet",
	"de",
	]

	[stock.homepages]

	[stoeckl]
	name = "Bernhard Stöckl"

	[stoeckl.emails]

	[stoeckl.emails.stoeckl_email]
	user = [
	"stoeckl",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[stoeckl.homepages]

	[stricker]
	name = "Christian Stricker"

	[stricker.emails]

	[stricker.homepages]
	stricker_homepage = "http://dss.in.tum.de/staff/christian-stricker.html"

	[strnisa]
	name = "Rok Strniša"

	[strnisa.emails]

	[strnisa.emails.strnisa_email]
	user = [
	"rok",
	]
	host = [
	"strnisa",
	"com",
	]

	[strnisa.homepages]
	strnisa_homepage = "http://rok.strnisa.com/lj/"

	[struth]
	name = "Georg Struth"

	[struth.emails]

	[struth.emails.struth_email]
	user = [
	"g",
	"struth",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[struth.homepages]
	struth_homepage = "http://staffwww.dcs.shef.ac.uk/people/G.Struth/"

	[stueber]
	name = "Anke Stüber"

	[stueber.emails]

	[stueber.emails.stueber_email]
	user = [
	"anke",
	"stueber",
	]
	host = [
	"campus",
	"tu-berlin",
	"de",
	]

	[stueber.homepages]

	[stuewe]
	name = "Daniel Stüwe"

	[stuewe.emails]

	[stuewe.homepages]

	[sudbrock]
	name = "Henning Sudbrock"

	[sudbrock.emails]

	[sudbrock.emails.sudbrock_email]
	user = [
	"sudbrock",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[sudbrock.homepages]

	[sudhof]
	name = "Henry Sudhof"

	[sudhof.emails]

	[sudhof.emails.sudhof_email]
	user = [
	"hsudhof",
	]
	host = [
	"cs",
	"tu-berlin",
	"de",
	]

	[sudhof.homepages]

	[sulejmani]
	name = "Ujkan Sulejmani"

	[sulejmani.emails]

	[sulejmani.emails.sulejmani_email]
	user = [
	"ujkan",
	"sulejmani",
	]
	host = [
	"tum",
	"de",
	]

	[sulejmani.emails.sulejmani_email1]
	user = [
	"ujkan99",
	]
	host = [
	"gmail",
	"com",
	]

	[sulejmani.homepages]

	[sutcliffe]
	name = "Geoff Sutcliffe"

	[sutcliffe.emails]

	[sutcliffe.homepages]

	[sylvestre]
	name = "Jeremy Sylvestre"

	[sylvestre.emails]

	[sylvestre.emails.sylvestre_email]
	user = [
	"jeremy",
	"sylvestre",
	]
	host = [
	"ualberta",
	"ca",
	]

	[sylvestre.emails.sylvestre_email1]
	user = [
	"jsylvest",
	]
	host = [
	"ualberta",
	"ca",
	]

	[sylvestre.homepages]
	sylvestre_homepage = "http://ualberta.ca/~jsylvest/"

	[szekely]
	name = "Gergely Szekely"

	[szekely.emails]

	[szekely.homepages]
	szekely_homepage = "https://users.renyi.hu/~turms/"

	[taha]
	name = "Safouan Taha"

	[taha.emails]

	[taha.emails.taha_email]
	user = [
	"safouan",
	"taha",
	]
	host = [
	"lri",
	"fr",
	]

	[taha.homepages]

	[tan]
	name = "Yong Kiam Tan"

	[tan.emails]

	[tan.emails.tan_email]
	user = [
	"yongkiat",
	]
	host = [
	"cs",
	"cmu",
	"edu",
	]

	[tan.homepages]
	tan_homepage = "https://www.cs.cmu.edu/~yongkiat/"

	[tanaka]
	name = "Miki Tanaka"

	[tanaka.emails]

	[tanaka.emails.tanaka_email]
	user = [
	"miki",
	"tanaka",
	]
	host = [
	"unsw",
	"edu",
	"au",
	]

	[tanaka.homepages]

	[tasch]
	name = "Markus Tasch"

	[tasch.emails]

	[tasch.emails.tasch_email]
	user = [
	"tasch",
	]
	host = [
	"mais",
	"informatik",
	"tu-darmstadt",
	"de",
	]

	[tasch.homepages]

	[taylor]
	name = "Ramsay G. Taylor"

	[taylor.emails]

	[taylor.emails.taylor_email]
	user = [
	"r",
	"g",
	"taylor",
	]
	host = [
	"sheffield",
	"ac",
	"uk",
	]

	[taylor.homepages]

	[terraf]
	name = "Pedro Sánchez Terraf"

	[terraf.emails]

	[terraf.emails.terraf_email]
	user = [
	"psterraf",
	]
	host = [
	"unc",
	"edu",
	"ar",
	]

	[terraf.homepages]
	terraf_homepage = "https://cs.famaf.unc.edu.ar/~pedro/home_en.html"

	[thiemann]
	name = "René Thiemann"

	[thiemann.emails]

	[thiemann.emails.thiemann_email]
	user = [
	"rene",
	"thiemann",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[thiemann.homepages]
	thiemann_homepage = "http://cl-informatik.uibk.ac.at/users/thiemann/"

	[thommes]
	name = "Joseph Thommes"

	[thommes.emails]

	[thommes.emails.thommes_email]
	user = [
	"joseph-thommes",
	]
	host = [
	"gmx",
	"de",
	]

	[thommes.homepages]

	[thomson]
	name = "Fox Thomson"

	[thomson.emails]

	[thomson.emails.thomson_email]
	user = [
	"foxthomson0",
	]
	host = [
	"gmail",
	"com",
	]

	[thomson.homepages]

	[tiu]
	name = "Alwen Tiu"

	[tiu.emails]

	[tiu.emails.tiu_email]
	user = [
	"ATiu",
	]
	host = [
	"ntu",
	"edu",
	"sg",
	]

	[tiu.homepages]
	tiu_homepage = "http://users.cecs.anu.edu.au/~tiu/"

	[toth]
	name = "Balazs Toth"

	[toth.emails]

	[toth.emails.toth_email]
	user = [
	"balazs",
	"toth",
	]
	host = [
	"tum",
	"de",
	]

	[toth.homepages]

	[tourret]
	name = "Sophie Tourret"

	[tourret.emails]

	[tourret.emails.tourret_email]
	user = [
	"stourret",
	]
	host = [
	"mpi-inf",
	"mpg",
	"de",
	]

	[tourret.homepages]
	tourret_homepage = "https://www.mpi-inf.mpg.de/departments/automation-of-logic/people/sophie-tourret/"

	[trachtenherz]
	name = "David Trachtenherz"

	[trachtenherz.emails]

	[trachtenherz.homepages]

	[traut]
	name = "Christoph Traut"

	[traut.emails]

	[traut.homepages]

	[traytel]
	name = "Dmitriy Traytel"

	[traytel.emails]

	[traytel.emails.traytel_email]
	user = [
	"traytel",
	]
	host = [
	"in",
	"tum",
	"de",
	]

	[traytel.emails.traytel_email1]
	user = [
	"traytel",
	]
	host = [
	"inf",
	"ethz",
	"ch",
	]

	[traytel.emails.traytel_email2]
	user = [
	"traytel",
	]
	host = [
	"di",
	"ku",
	"dk",
	]

	[traytel.homepages]
	traytel_homepage = "https://traytel.bitbucket.io/"

	[trelat]
	name = "Vincent Trélat"

	[trelat.emails]

	[trelat.emails.trelat_email]
	user = [
	"vincent",
	"trelat",
	]
	host = [
	"depinfonancy",
	"net",
	]

	[trelat.homepages]

	[tuerk]
	name = "Thomas Tuerk"

	[tuerk.emails]

	[tuerk.homepages]

	[tuong]
	name = "Frédéric Tuong"

	[tuong.emails]

	[tuong.emails.tuong_email]
	user = [
	"tuong",
	]
	host = [
	"users",
	"gforge",
	"inria",
	"fr",
	]

	[tuong.emails.tuong_email1]
	user = [
	"ftuong",
	]
	host = [
	"lri",
	"fr",
	]

	[tuong.homepages]
	tuong_homepage = "https://www.lri.fr/~ftuong/"

	[tuongj]
	name = "Joseph Tuong"

	[tuongj.emails]

	[tuongj.homepages]

	[tverdyshev]
	name = "Sergey Tverdyshev"

	[tverdyshev.emails]

	[tverdyshev.emails.tverdyshev_email]
	user = [
	"stv",
	]
	host = [
	"sysgo",
	"com",
	]

	[tverdyshev.homepages]

	[ullrich]
	name = "Sebastian Ullrich"

	[ullrich.emails]

	[ullrich.emails.ullrich_email]
	user = [
	"sebasti",
	]
	host = [
	"nullri",
	"ch",
	]

	[ullrich.homepages]

	[unruh]
	name = "Dominique Unruh"

	[unruh.emails]

	[unruh.emails.unruh_email]
	user = [
	"unruh",
	]
	host = [
	"ut",
	"ee",
	]

	[unruh.homepages]
	unruh_homepage = "https://www.ut.ee/~unruh/"

	[urban]
	name = "Christian Urban"

	[urban.emails]

	[urban.emails.urban_email]
	user = [
	"christian",
	"urban",
	]
	host = [
	"kcl",
	"ac",
	"uk",
	]

	[urban.homepages]
	urban_homepage = "https://nms.kcl.ac.uk/christian.urban/"

	[van]
	name = "Hai Nguyen Van"

	[van.emails]

	[van.emails.van_email]
	user = [
	"hai",
	"nguyenvan",
	"phie",
	]
	host = [
	"gmail",
	"com",
	]

	[van.homepages]

	[velykis]
	name = "Andrius Velykis"

	[velykis.emails]

	[velykis.homepages]
	velykis_homepage = "http://andrius.velykis.lt"

	[verbeek]
	name = "Freek Verbeek"

	[verbeek.emails]

	[verbeek.emails.verbeek_email]
	user = [
	"Freek",
	"Verbeek",
	]
	host = [
	"ou",
	"nl",
	]

	[verbeek.emails.verbeek_email1]
	user = [
	"freek",
	]
	host = [
	"vt",
	"edu",
	]

	[verbeek.homepages]

	[villadsen]
	name = "Jørgen Villadsen"

	[villadsen.emails]

	[villadsen.emails.villadsen_email]
	user = [
	"jovi",
	]
	host = [
	"dtu",
	"dk",
	]

	[villadsen.homepages]
	villadsen_homepage = "https://people.compute.dtu.dk/jovi/"

	[voisin]
	name = "Frederic Voisin"

	[voisin.emails]

	[voisin.homepages]

	[vytiniotis]
	name = "Dimitrios Vytiniotis"

	[vytiniotis.emails]

	[vytiniotis.homepages]
	vytiniotis_homepage = "http://research.microsoft.com/en-us/people/dimitris/"

	[wagner]
	name = "Max Wagner"

	[wagner.emails]

	[wagner.emails.wagner_email]
	user = [
	"max",
	]
	host = [
	"trollbu",
	"de",
	]

	[wagner.homepages]

	[waldmann]
	name = "Uwe Waldmann"

	[waldmann.emails]

	[waldmann.emails.waldmann_email]
	user = [
	"waldmann",
	]
	host = [
	"mpi-inf",
	"mpg",
	"de",
	]

	[waldmann.homepages]

	[wand]
	name = "Daniel Wand"

	[wand.emails]

	[wand.emails.wand_email]
	user = [
	"dwand",
	]
	host = [
	"mpi-inf",
	"mpg",
	"de",
	]

	[wand.homepages]

	[wang]
	name = "Shuling Wang"

	[wang.emails]

	[wang.homepages]

	[wassell]
	name = "Mark Wassell"

	[wassell.emails]

	[wassell.emails.wassell_email]
	user = [
	"mpwassell",
	]
	host = [
	"gmail",
	"com",
	]

	[wassell.homepages]

	[wasserrab]
	name = "Daniel Wasserrab"

	[wasserrab.emails]

	[wasserrab.homepages]
	wasserrab_homepage = "http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php"

	[watt]
	name = "Conrad Watt"

	[watt.emails]

	[watt.emails.watt_email]
	user = [
	"caw77",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[watt.homepages]
	watt_homepage = "http://www.cl.cam.ac.uk/~caw77/"

	[weber]
	name = "Tjark Weber"

	[weber.emails]

	[weber.emails.weber_email]
	user = [
	"tjark",
	"weber",
	]
	host = [
	"it",
	"uu",
	"se",
	]

	[weber.homepages]
	weber_homepage = "http://user.it.uu.se/~tjawe125/"

	[weerwag]
	name = "Timmy Weerwag"

	[weerwag.emails]

	[weerwag.homepages]

	[weidner]
	name = "Arno Wilhelm-Weidner"

	[weidner.emails]

	[weidner.emails.weidner_email]
	user = [
	"arno",
	"wilhelm-weidner",
	]
	host = [
	"tu-berlin",
	"de",
	]

	[weidner.homepages]

	[wenninger]
	name = "Elias Wenninger"

	[wenninger.emails]

	[wenninger.homepages]

	[wenzel]
	name = "Makarius Wenzel"

	[wenzel.emails]

	[wenzel.emails.wenzel_email]
	user = [
	"makarius",
	]
	host = [
	"sketis",
	"net",
	]

	[wenzel.homepages]
	wenzel_homepage = "https://sketis.net"

	[whitley]
	name = "A Whitley"

	[whitley.emails]

	[whitley.emails.whitley_email]
	user = [
	"awhitley",
	]
	host = [
	"gmail",
	"com",
	]

	[whitley.homepages]

	[wickerson]
	name = "John Wickerson"

	[wickerson.emails]

	[wickerson.homepages]
	wickerson_homepage = "http://www.doc.ic.ac.uk/~jpw48"

	[willenbrink]
	name = "Sebastian Willenbrink"

	[willenbrink.emails]

	[willenbrink.emails.willenbrink_email]
	user = [
	"sebastian",
	"willenbrink",
	]
	host = [
	"tum",
	"de",
	]

	[willenbrink.homepages]

	[wimmer]
	name = "Simon Wimmer"

	[wimmer.emails]

	[wimmer.emails.wimmer_email]
	user = [
	"simon",
	"wimmer",
	]
	host = [
	"tum",
	"de",
	]

	[wimmer.homepages]
	wimmer_homepage = "http://home.in.tum.de/~wimmers/"

	[wirt]
	name = "Kai Wirt"

	[wirt.emails]

	[wirt.homepages]

	[wolff]
	name = "Burkhart Wolff"

	[wolff.emails]

	[wolff.emails.wolff_email]
	user = [
	"burkhart",
	"wolff",
	]
	host = [
	"lri",
	"fr",
	]

	[wolff.homepages]
	wolff_homepage = "https://www.lri.fr/~wolff/"

	[wu]
	name = "Chunhan Wu"

	[wu.emails]

	[wu.homepages]

	[xu]
	name = "Jian Xu"

	[xu.emails]

	[xu.homepages]

	[yamada]
	name = "Akihisa Yamada"

	[yamada.emails]

	[yamada.emails.yamada_email]
	user = [
	"akihisa",
	"yamada",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[yamada.emails.yamada_email1]
	user = [
	"ayamada",
	]
	host = [
	"trs",
	"cm",
	"is",
	"nagoya-u",
	"ac",
	"jp",
	]

	[yamada.emails.yamada_email2]
	user = [
	"akihisa",
	"yamada",
	]
	host = [
	"aist",
	"go",
	"jp",
	]

	[yamada.emails.yamada_email3]
	user = [
	"akihisayamada",
	]
	host = [
	"nii",
	"ac",
	"jp",
	]

	[yamada.homepages]
	yamada_homepage = "http://group-mmm.org/~ayamada/"

	[ye]
	name = "Lina Ye"

	[ye.emails]

	[ye.emails.ye_email]
	user = [
	"lina",
	"ye",
	]
	host = [
	"lri",
	"fr",
	]

	[ye.homepages]

	+[yez]
	+name = "Zhengkun Ye"
	+
	+[yez.emails]
	+
	+[yez.emails.yez_email]
	+user = [
	+ "zy307",
	+]
	+host = [
	+ "cam",
	+ "ac",
	+ "uk",
	+]
	+
	+[yez.homepages]
	+
	[ying]
	name = "Shenggang Ying"

	[ying.emails]

	[ying.homepages]

	[yingm]
	name = "Mingsheng Ying"

	[yingm.emails]

	[yingm.homepages]

	[yu]
	name = "Lei Yu"

	[yu.emails]

	[yu.emails.yu_email]
	user = [
	"ly271",
	]
	host = [
	"cam",
	"ac",
	"uk",
	]

	[yu.homepages]

	[zankl]
	name = "Harald Zankl"

	[zankl.emails]

	[zankl.emails.zankl_email]
	user = [
	"Harald",
	"Zankl",
	]
	host = [
	"uibk",
	"ac",
	"at",
	]

	[zankl.homepages]
	zankl_homepage = "http://cl-informatik.uibk.ac.at/users/hzankl"

	[zee]
	name = "Karen Zee"

	[zee.emails]

	[zee.emails.zee_email]
	user = [
	"kkz",
	]
	host = [
	"mit",
	"edu",
	]

	[zee.homepages]
	zee_homepage = "http://www.mit.edu/~kkz/"

	[zeller]
	name = "Peter Zeller"

	[zeller.emails]

	[zeller.emails.zeller_email]
	user = [
	"p_zeller",
	]
	host = [
	"cs",
	"uni-kl",
	"de",
	]

	[zeller.homepages]

	[zeyda]
	name = "Frank Zeyda"

	[zeyda.emails]

	[zeyda.emails.zeyda_email]
	user = [
	"frank",
	"zeyda",
	]
	host = [
	"york",
	"ac",
	"uk",
	]

	[zeyda.homepages]

	[zhan]
	name = "Bohua Zhan"

	[zhan.emails]

	[zhan.emails.zhan_email]
	user = [
	"bzhan",
	]
	host = [
	"ios",
	"ac",
	"cn",
	]

	[zhan.homepages]
	zhan_homepage = "http://lcs.ios.ac.cn/~bzhan/"

	[zhang]
	name = "Yu Zhang"

	[zhang.emails]

	[zhang.homepages]

	[zhangx]
	name = "Xingyuan Zhang"

	[zhangx.emails]

	[zhangx.homepages]

	[zhann]
	name = "Naijun Zhan"

	[zhann.emails]

	[zhann.homepages]
	diff --git a/metadata/entries/Earley_Parser.toml b/metadata/entries/Earley_Parser.toml
	new file mode 100644
	--- /dev/null
	+++ b/metadata/entries/Earley_Parser.toml
	@@ -0,0 +1,34 @@
	+title = "Earley Parser"
	+date = 2023-07-16
	+topics = [
	+ "Computer science/Algorithms",
	+]
	+abstract = """
	+In 1968, Earley introduced his parsing algorithm, capable of parsing all context-free grammars in cubic
	+space and time. This entry contains a formalization of an executable Earley parser. We base our development on Jones'
	+extensive paper proof of Earley's recognizer and Obua's formalization of context-free grammars
	+and derivations. We implement and prove correct a functional recognizer modeling Earley's
	+original imperative implementation and extend it with the necessary data structures to enable the construction
	+of parse trees, following the work of Scott. We then develop a functional algorithm that
	+builds a single parse tree, and we prove its correctness. Finally, we generalize this approach to an algorithm
	+for a complete parse forest and prove soundness."""
	+license = "bsd"
	+note = ""
	+
	+[authors]
	+
	+[authors.rau]
	+email = "rau_email"
	+
	+[contributors]
	+
	+[notify]
	+rau = "rau_email"
	+
	+[history]
	+
	+[extra]
	+
	+[related]
	+dois = []
	+pubs = []
	diff --git a/metadata/entries/Polygonal_Number_Theorem.toml b/metadata/entries/Polygonal_Number_Theorem.toml
	new file mode 100644
	--- /dev/null
	+++ b/metadata/entries/Polygonal_Number_Theorem.toml
	@@ -0,0 +1,46 @@
	+title = "Polygonal Number Theorem"
	+date = 2023-08-10
	+topics = [
	+ "Mathematics/Number theory",
	+ "Mathematics/Combinatorics",
	+]
	+abstract = """
	+<p>We formalize the proofs of Cauchy's and Legendre's Polygonal Number Theorems given in Melvyn B. Nathanson's book \"Additive Number Theory: The Classical Bases\".</p>
	+
	+<p>For $m \\geq 1$, the $k$-th polygonal number of order $m+2$ is defined to be $p_m(k)=\\frac{mk(k-1)}{2}+k$. The theorems state that:</p>
	+
	+<p>1. If $m \\ge 4$ and $N \\geq 108m$, then $N$ can be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four of which are different from $0$ or $1$. If $N \\geq 324$, then $N$ can be written as the sum of five pentagonal numbers, at least one of which is $0$ or $1$.</p>
	+
	+<p>2. Let $m \\geq 3$ and $N \\geq 28m^3$. If $m$ is odd, then $N$ is the sum of four polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.</p>
	+
	+<p>We also formalize the proof of Gauss's theorem which states that every non-negative integer is the sum of three triangular numbers.</p>"""
	+license = "bsd"
	+note = ""
	+
	+[authors]
	+
	+[authors.leek]
	+email = "leek_email"
	+
	+[authors.yez]
	+email = "yez_email"
	+
	+[authors.argyraki]
	+email = "argyraki_email"
	+
	+[contributors]
	+
	+[notify]
	+leek = "leek_email"
	+yez = "yez_email"
	+argyraki = "argyraki_email"
	+
	+[history]
	+
	+[extra]
	+
	+[related]
	+dois = [
	+ "10.1007/978-1-4757-3845-2",
	+]
	+pubs = []
	diff --git a/metadata/entries/Quantales_Converse.toml b/metadata/entries/Quantales_Converse.toml
	new file mode 100644
	--- /dev/null
	+++ b/metadata/entries/Quantales_Converse.toml
	@@ -0,0 +1,42 @@
	+title = "Modal quantales, involutive quantales, Dedekind Quantales"
	+date = 2023-07-25
	+topics = [
	+ "Computer science/Semantics and reasoning",
	+ "Mathematics/Algebra",
	+ "Mathematics/Order",
	+]
	+abstract = """
	+This AFP entry provides mathematical components for modal quantales,
	+ involutive quantales and Dedekind quantales. Modal quantales are
	+ simple extensions of modal Kleene algebras useful for the
	+ verification of recursive programs. Involutive quantales appear in
	+ the study of C*-algebras. Dedekind quantales are relatives of
	+ Tarski's relation algebras, hence relevant to program verification
	+ and beyond that to higher rewriting. We also provide
	+ components for weaker variants such as Kleene algebras with converse
	+ and modal Kleene algebras with converse."""
	+license = "bsd"
	+note = ""
	+
	+[authors]
	+
	+[authors.struth]
	+email = "struth_email"
	+
	+[authors.calk]
	+
	+[contributors]
	+
	+[notify]
	+struth = "struth_email"
	+
	+[history]
	+
	+[extra]
	+
	+[related]
	+dois = [
	+ "10.1007/s00012-023-00805-9",
	+ "10.48550/arXiv.2307.09253",
	+]
	+pubs = []
	diff --git a/thys/Earley_Parser/CFG.thy b/thys/Earley_Parser/CFG.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/CFG.thy
	@@ -0,0 +1,67 @@
	+theory CFG
	+ imports Main
	+begin
	+
	+section \<open>Adjusted content from AFP/LocalLexing\<close>
	+
	+type_synonym 'a rule = "'a \<times> 'a list"
	+
	+type_synonym 'a rules = "'a rule list"
	+
	+type_synonym 'a sentence = "'a list"
	+
	+datatype 'a cfg =
	+ CFG (\<NN> : "'a list") (\<TT> : "'a list") (\<RR> : "'a rules") (\<SS> : "'a")
	+
	+definition disjunct_symbols :: "'a cfg \<Rightarrow> bool" where
	+ "disjunct_symbols \<G> \<equiv> set (\<NN> \<G>) \<inter> set (\<TT> \<G>) = {}"
	+
	+definition valid_startsymbol :: "'a cfg \<Rightarrow> bool" where
	+ "valid_startsymbol \<G> \<equiv> \<SS> \<G> \<in> set (\<NN> \<G>)"
	+
	+definition valid_rules :: "'a cfg \<Rightarrow> bool" where
	+ "valid_rules \<G> \<equiv> \<forall>(N, \<alpha>) \<in> set (\<RR> \<G>). N \<in> set (\<NN> \<G>) \<and> (\<forall>s \<in> set \<alpha>. s \<in> set (\<NN> \<G>) \<union> set (\<TT> \<G>))"
	+
	+definition distinct_rules :: "'a cfg \<Rightarrow> bool" where
	+ "distinct_rules \<G> \<equiv> distinct (\<RR> \<G>)"
	+
	+definition wf_\<G> :: "'a cfg \<Rightarrow> bool" where
	+ "wf_\<G> \<G> \<equiv> disjunct_symbols \<G> \<and> valid_startsymbol \<G> \<and> valid_rules \<G> \<and> distinct_rules \<G>"
	+
	+lemmas wf_\<G>_defs = wf_\<G>_def valid_rules_def valid_startsymbol_def disjunct_symbols_def distinct_rules_def
	+
	+definition is_terminal :: "'a cfg \<Rightarrow> 'a \<Rightarrow> bool" where
	+ "is_terminal \<G> x \<equiv> x \<in> set (\<TT> \<G>)"
	+
	+definition is_nonterminal :: "'a cfg \<Rightarrow> 'a \<Rightarrow> bool" where
	+ "is_nonterminal \<G> x \<equiv> x \<in> set (\<NN> \<G>)"
	+
	+definition is_symbol :: "'a cfg \<Rightarrow> 'a \<Rightarrow> bool" where
	+ "is_symbol \<G> x \<equiv> is_terminal \<G> x \<or> is_nonterminal \<G> x"
	+
	+definition wf_sentence :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "wf_sentence \<G> \<omega> \<equiv> \<forall>x \<in> set \<omega>. is_symbol \<G> x"
	+
	+lemma is_nonterminal_startsymbol:
	+ "wf_\<G> \<G> \<Longrightarrow> is_nonterminal \<G> (\<SS> \<G>)"
	+ by (simp add: is_nonterminal_def wf_\<G>_defs)
	+
	+definition is_word :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "is_word \<G> \<omega> \<equiv> \<forall>x \<in> set \<omega>. is_terminal \<G> x"
	+
	+definition derives1 :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "derives1 \<G> u v \<equiv> \<exists> x y N \<alpha>.
	+ u = x @ [N] @ y \<and>
	+ v = x @ \<alpha> @ y \<and>
	+ (N, \<alpha>) \<in> set (\<RR> \<G>)"
	+
	+definition derivations1 :: "'a cfg \<Rightarrow> ('a sentence \<times> 'a sentence) set" where
	+ "derivations1 \<G> \<equiv> { (u,v) \| u v. derives1 \<G> u v }"
	+
	+definition derivations :: "'a cfg \<Rightarrow> ('a sentence \<times> 'a sentence) set" where
	+ "derivations \<G> \<equiv> (derivations1 \<G>)^*"
	+
	+definition derives :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "derives \<G> u v \<equiv> ((u, v) \<in> derivations \<G>)"
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/Derivations.thy b/thys/Earley_Parser/Derivations.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Derivations.thy
	@@ -0,0 +1,255 @@
	+theory Derivations
	+ imports
	+ CFG
	+begin
	+
	+section \<open>Adjusted content from AFP/LocalLexing\<close>
	+
	+type_synonym 'a derivation = "(nat \<times> 'a rule) list"
	+
	+lemma is_word_empty: "is_word \<G> []" by (auto simp add: is_word_def)
	+
	+lemma derives1_implies_derives[simp]:
	+ "derives1 \<G> a b \<Longrightarrow> derives \<G> a b"
	+ by (auto simp add: derives_def derivations_def derivations1_def)
	+
	+lemma derives_trans:
	+ "derives \<G> a b \<Longrightarrow> derives \<G> b c \<Longrightarrow> derives \<G> a c"
	+ by (auto simp add: derives_def derivations_def)
	+
	+lemma derives1_eq_derivations1:
	+ "derives1 \<G> x y = ((x, y) \<in> derivations1 \<G>)"
	+ by (simp add: derivations1_def)
	+
	+lemma derives_induct[consumes 1, case_names Base Step]:
	+ assumes derives: "derives \<G> a b"
	+ assumes Pa: "P a"
	+ assumes induct: "\<And>y z. derives \<G> a y \<Longrightarrow> derives1 \<G> y z \<Longrightarrow> P y \<Longrightarrow> P z"
	+ shows "P b"
	+proof -
	+ note rtrancl_lemma = rtrancl_induct[where a = a and b = b and r = "derivations1 \<G>" and P=P]
	+ from derives Pa induct rtrancl_lemma show "P b"
	+ by (metis derives_def derivations_def derives1_eq_derivations1)
	+qed
	+
	+definition Derives1 :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> nat \<Rightarrow> 'a rule \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "Derives1 \<G> u i r v \<equiv> \<exists> x y N \<alpha>.
	+ u = x @ [N] @ y \<and>
	+ v = x @ \<alpha> @ y \<and>
	+ (N, \<alpha>) \<in> set (\<RR> \<G>) \<and> r = (N, \<alpha>) \<and> i = length x"
	+
	+lemma Derives1_split:
	+ "Derives1 \<G> u i r v \<Longrightarrow> \<exists> x y. u = x @ [fst r] @ y \<and> v = x @ (snd r) @ y \<and> length x = i"
	+ by (metis Derives1_def fst_conv snd_conv)
	+
	+lemma Derives1_implies_derives1: "Derives1 \<G> u i r v \<Longrightarrow> derives1 \<G> u v"
	+ by (auto simp add: Derives1_def derives1_def)
	+
	+lemma derives1_implies_Derives1: "derives1 \<G> u v \<Longrightarrow> \<exists> i r. Derives1 \<G> u i r v"
	+ by (auto simp add: Derives1_def derives1_def)
	+
	+fun Derivation :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a derivation \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "Derivation _ a [] b = (a = b)"
	+\| "Derivation \<G> a (d#D) b = (\<exists> x. Derives1 \<G> a (fst d) (snd d) x \<and> Derivation \<G> x D b)"
	+
	+lemma Derivation_implies_derives: "Derivation \<G> a D b \<Longrightarrow> derives \<G> a b"
	+proof (induct D arbitrary: a b)
	+ case Nil thus ?case
	+ by (simp add: derives_def derivations_def)
	+next
	+ case (Cons d D)
	+ note ihyps = this
	+ from ihyps have "\<exists> x. Derives1 \<G> a (fst d) (snd d) x \<and> Derivation \<G> x D b" by auto
	+ then obtain x where "Derives1 \<G> a (fst d) (snd d) x" and xb: "Derivation \<G> x D b" by blast
	+ with Derives1_implies_derives1 have d1: "derives \<G> a x" by fastforce
	+ from ihyps xb have d2:"derives \<G> x b" by simp
	+ show "derives \<G> a b" by (rule derives_trans[OF d1 d2])
	+qed
	+
	+lemma Derivation_Derives1: "Derivation \<G> a S y \<Longrightarrow> Derives1 \<G> y i r z \<Longrightarrow> Derivation \<G> a (S@[(i,r)]) z"
	+proof (induct S arbitrary: a y z i r)
	+ case Nil thus ?case by simp
	+next
	+ case (Cons s S) thus ?case
	+ by (metis Derivation.simps(2) append_Cons)
	+qed
	+
	+lemma derives_implies_Derivation: "derives \<G> a b \<Longrightarrow> \<exists> D. Derivation \<G> a D b"
	+proof (induct rule: derives_induct)
	+ case Base
	+ show ?case by (rule exI[where x="[]"], simp)
	+next
	+ case (Step y z)
	+ note ihyps = this
	+ from ihyps obtain D where ay: "Derivation \<G> a D y" by blast
	+ from ihyps derives1_implies_Derives1 obtain i r where yz: "Derives1 \<G> y i r z" by blast
	+ from Derivation_Derives1[OF ay yz] show ?case by auto
	+qed
	+
	+lemma rule_nonterminal_type[simp]: "wf_\<G> \<G> \<Longrightarrow> (N, \<alpha>) \<in> set (\<RR> \<G>) \<Longrightarrow> is_nonterminal \<G> N"
	+ by (auto simp add: is_nonterminal_def wf_\<G>_defs)
	+
	+lemma Derives1_rule [elim]: "Derives1 \<G> a i r b \<Longrightarrow> r \<in> set (\<RR> \<G>)"
	+ using Derives1_def by metis
	+
	+lemma is_terminal_nonterminal: "wf_\<G> \<G> \<Longrightarrow> is_terminal \<G> x \<Longrightarrow> is_nonterminal \<G> x \<Longrightarrow> False"
	+ by (auto simp: wf_\<G>_defs disjoint_iff is_nonterminal_def is_terminal_def)
	+
	+lemma is_word_is_terminal: "i < length u \<Longrightarrow> is_word \<G> u \<Longrightarrow> is_terminal \<G> (u ! i)"
	+ using is_word_def by force
	+
	+lemma Derivation_append: "Derivation \<G> a (D@E) c = (\<exists> b. Derivation \<G> a D b \<and> Derivation \<G> b E c)"
	+ by (induct D arbitrary: a c E) auto
	+
	+lemma Derivation_implies_append:
	+ "Derivation \<G> a D b \<Longrightarrow> Derivation \<G> b E c \<Longrightarrow> Derivation \<G> a (D@E) c"
	+ using Derivation_append by blast
	+
	+
	+section \<open>Additional derivation lemmas\<close>
	+
	+lemma Derives1_prepend:
	+ assumes "Derives1 \<G> u i r v"
	+ shows "Derives1 \<G> (w@u) (i + length w) r (w@v)"
	+proof -
	+ obtain x y N \<alpha> where *:
	+ "u = x @ [N] @ y" "v = x @ \<alpha> @ y"
	+ "(N, \<alpha>) \<in> set (\<RR> \<G>)" "r = (N, \<alpha>)" "i = length x"
	+ using assms Derives1_def by (smt (verit))
	+ hence "w@u = w @ x @ [N] @ y" "w@v = w @ x @ \<alpha> @ y"
	+ by auto
	+ thus ?thesis
	+ unfolding Derives1_def using *
	+ apply (rule_tac exI[where x="w@x"])
	+ apply (rule_tac exI[where x="y"])
	+ by simp
	+qed
	+
	+lemma Derivation_prepend:
	+ "Derivation \<G> b D b' \<Longrightarrow> Derivation \<G> (a@b) (map (\<lambda>(i, r). (i + length a, r)) D) (a@b')"
	+ using Derives1_prepend by (induction D arbitrary: b b') (auto, fast)
	+
	+lemma Derives1_append:
	+ assumes "Derives1 \<G> u i r v"
	+ shows "Derives1 \<G> (u@w) i r (v@w)"
	+proof -
	+ obtain x y N \<alpha> where *:
	+ "u = x @ [N] @ y" "v = x @ \<alpha> @ y"
	+ "(N, \<alpha>) \<in> set (\<RR> \<G>)" "r = (N, \<alpha>)" "i = length x"
	+ using assms Derives1_def by (smt (verit))
	+ hence "u@w = x @ [N] @ y @ w" "v@w = x @ \<alpha> @ y @ w"
	+ by auto
	+ thus ?thesis
	+ unfolding Derives1_def using *
	+ apply (rule_tac exI[where x="x"])
	+ apply (rule_tac exI[where x="y@w"])
	+ by blast
	+qed
	+
	+lemma Derivation_append':
	+ "Derivation \<G> a D a' \<Longrightarrow> Derivation \<G> (a@b) D (a'@b)"
	+ using Derives1_append by (induction D arbitrary: a a') (auto, fast)
	+
	+lemma Derivation_append_rewrite:
	+ assumes "Derivation \<G> a D (b @ c @ d) " "Derivation \<G> c E c'"
	+ shows "\<exists>F. Derivation \<G> a F (b @ c' @ d)"
	+ using assms Derivation_append' Derivation_prepend Derivation_implies_append by fast
	+
	+lemma derives1_if_valid_rule:
	+ "(N, \<alpha>) \<in> set (\<RR> \<G>) \<Longrightarrow> derives1 \<G> [N] \<alpha>"
	+ unfolding derives1_def
	+ apply (rule_tac exI[where x="[]"])
	+ apply (rule_tac exI[where x="[]"])
	+ by simp
	+
	+lemma derives_if_valid_rule:
	+ "(N, \<alpha>) \<in> set (\<RR> \<G>) \<Longrightarrow> derives \<G> [N] \<alpha>"
	+ using derives1_if_valid_rule by fastforce
	+
	+lemma Derivation_from_empty:
	+ "Derivation \<G> [] D a \<Longrightarrow> a = []"
	+ by (cases D) (auto simp: Derives1_def)
	+
	+lemma Derivation_concat_split:
	+ "Derivation \<G> (a@b) D c \<Longrightarrow> \<exists>E F a' b'. Derivation \<G> a E a' \<and> Derivation \<G> b F b' \<and>
	+ c = a' @ b' \<and> length E \<le> length D \<and> length F \<le> length D"
	+proof (induction D arbitrary: a b)
	+ case Nil
	+ thus ?case
	+ by (metis Derivation.simps(1) order_refl)
	+next
	+ case (Cons d D)
	+ then obtain ab where *: "Derives1 \<G> (a@b) (fst d) (snd d) ab" "Derivation \<G> ab D c"
	+ by auto
	+ then obtain x y N \<alpha> where #:
	+ "a@b = x @ [N] @ y" "ab = x @ \<alpha> @ y" "(N,\<alpha>) \<in> set (\<RR> \<G>)" "snd d = (N,\<alpha>)" "fst d = length x"
	+ using * unfolding Derives1_def by blast
	+ show ?case
	+ proof (cases "length a \<le> length x")
	+ case True
	+ hence ab_def:
	+ "a = take (length a) x"
	+ "b = drop (length a) x @ [N] @ y"
	+ "ab = take (length a) x @ drop (length a) x @ \<alpha> @ y"
	+ using #(1,2) True by (metis append_eq_append_conv_if)+
	+ then obtain E F a' b' where IH:
	+ "Derivation \<G> (take (length a) x) E a'"
	+ "Derivation \<G> (drop (length a) x @ \<alpha> @ y) F b'"
	+ "c = a' @ b'"
	+ "length E \<le> length D"
	+ "length F \<le> length D"
	+ using Cons *(2) by blast
	+ have "Derives1 \<G> b (fst d - length a) (snd d) (drop (length a) x @ \<alpha> @ y)"
	+ unfolding Derives1_def using *(1) #(3-5) ab_def(2) by (metis length_drop)
	+ hence "Derivation \<G> b ((fst d - length a, snd d) # F) b'"
	+ using IH(2) by force
	+ moreover have "Derivation \<G> a E a'"
	+ using IH(1) ab_def(1) by fastforce
	+ ultimately show ?thesis
	+ using IH(3-5) by fastforce
	+ next
	+ case False
	+ hence a_def: "a = x @ [N] @ take (length a - length x - 1) y"
	+ using #(1) append_eq_conv_conj[of a b "x @ [N] @ y"] take_all_iff take_append
	+ by (metis append_Cons append_Nil diff_is_0_eq le_cases take_Cons')
	+ hence b_def: "b = drop (length a - length x - 1) y"
	+ using #(1) by (metis List.append.assoc append_take_drop_id same_append_eq)
	+ have "ab = x @ \<alpha> @ take (length a - length x - 1) y @ drop (length a - length x - 1) y"
	+ using #(2) by force
	+ then obtain E F a' b' where IH:
	+ "Derivation \<G> (x @ \<alpha> @ take (length a - length x - 1) y) E a'"
	+ "Derivation \<G> (drop (length a - length x - 1) y) F b'"
	+ "c = a' @ b'"
	+ "length E \<le> length D"
	+ "length F \<le> length D"
	+ using Cons.IH[of "x @ \<alpha> @ take (length a - length x - 1) y" "drop (length a - length x - 1) y"] *(2) by auto
	+ have "Derives1 \<G> a (fst d) (snd d) (x @ \<alpha> @ take (length a - length x - 1) y)"
	+ unfolding Derives1_def using #(3-5) a_def by blast
	+ hence "Derivation \<G> a ((fst d, snd d) # E) a'"
	+ using IH(1) by fastforce
	+ moreover have "Derivation \<G> b F b'"
	+ using b_def IH(2) by blast
	+ ultimately show ?thesis
	+ using IH(3-5) by fastforce
	+ qed
	+qed
	+
	+lemma Derivation_\<SS>1:
	+ assumes "Derivation \<G> [\<SS> \<G>] D \<omega>" "is_word \<G> \<omega>" "wf_\<G> \<G>"
	+ shows "\<exists>\<alpha> E. Derivation \<G> \<alpha> E \<omega> \<and> (\<SS> \<G>,\<alpha>) \<in> set (\<RR> \<G>)"
	+proof (cases D)
	+ case Nil
	+ thus ?thesis
	+ using assms is_nonterminal_startsymbol is_terminal_nonterminal by (metis Derivation.simps(1) is_word_def list.set_intros(1))
	+next
	+ case (Cons d D)
	+ then obtain \<alpha> where "Derives1 \<G> [\<SS> \<G>] (fst d) (snd d) \<alpha>" "Derivation \<G> \<alpha> D \<omega>"
	+ using assms by auto
	+ hence "(\<SS> \<G>, \<alpha>) \<in> set (\<RR> \<G>)"
	+ unfolding Derives1_def
	+ by (metis List.append.right_neutral List.list.discI append_eq_Cons_conv append_is_Nil_conv nth_Cons_0 self_append_conv2)
	+ thus ?thesis
	+ using \<open>Derivation \<G> \<alpha> D \<omega>\<close> by auto
	+qed
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/Earley.thy b/thys/Earley_Parser/Earley.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Earley.thy
	@@ -0,0 +1,493 @@
	+theory Earley
	+ imports
	+ Derivations
	+begin
	+
	+section \<open>Slices\<close>
	+
	+fun slice :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
	+ "slice _ _ [] = []"
	+\| "slice _ 0 (x#xs) = []"
	+\| "slice 0 (Suc b) (x#xs) = x # slice 0 b xs"
	+\| "slice (Suc a) (Suc b) (x#xs) = slice a b xs"
	+
	+lemma slice_drop_take:
	+ "slice a b xs = drop a (take b xs)"
	+ by (induction a b xs rule: slice.induct) auto
	+
	+lemma slice_append_aux:
	+ "Suc b \<le> c \<Longrightarrow> slice (Suc b) c (x # xs) = slice b (c-1) xs"
	+ using Suc_le_D by fastforce
	+
	+lemma slice_concat:
	+ "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> slice a b xs @ slice b c xs = slice a c xs"
	+proof (induction a b xs arbitrary: c rule: slice.induct)
	+ case (3 b x xs)
	+ then show ?case
	+ using Suc_le_D by(fastforce simp: slice_append_aux)
	+qed (auto simp: slice_append_aux)
	+
	+lemma slice_concat_Ex:
	+ "a \<le> c \<Longrightarrow> slice a c xs = ys @ zs \<Longrightarrow> \<exists>b. ys = slice a b xs \<and> zs = slice b c xs \<and> a \<le> b \<and> b \<le> c"
	+proof (induction a c xs arbitrary: ys zs rule: slice.induct)
	+ case (3 b x xs)
	+ show ?case
	+ proof (cases ys)
	+ case Nil
	+ then obtain zs' where "x # slice 0 b xs = x # zs'" "x # zs' = zs"
	+ using "3.prems"(2) by auto
	+ thus ?thesis
	+ using Nil by force
	+ next
	+ case (Cons y ys')
	+ then obtain ys' where "x # slice 0 b xs = x # ys' @ zs" "x # ys' = ys"
	+ using "3.prems"(2) by auto
	+ thus ?thesis
	+ using "3.IH"[of ys' zs] by force
	+ qed
	+next
	+ case (4 a b x xs)
	+ thus ?case
	+ by (auto, metis slice.simps(4) Suc_le_mono)
	+qed auto
	+
	+lemma slice_nth:
	+ "a < length xs \<Longrightarrow> slice a (a+1) xs = [xs!a]"
	+ unfolding slice_drop_take
	+ by (metis Cons_nth_drop_Suc One_nat_def diff_add_inverse drop_take take_Suc_Cons take_eq_Nil)
	+
	+lemma slice_append_nth:
	+ "a \<le> b \<Longrightarrow> b < length xs \<Longrightarrow> slice a b xs @ [xs!b] = slice a (b+1) xs"
	+ by (metis le_add1 slice_concat slice_nth)
	+
	+lemma slice_empty:
	+ "b \<le> a \<Longrightarrow> slice a b xs = []"
	+ by (simp add: slice_drop_take)
	+
	+lemma slice_id[simp]:
	+ "slice 0 (length xs) xs = xs"
	+ by (simp add: slice_drop_take)
	+
	+lemma slice_singleton:
	+ "b \<le> length xs \<Longrightarrow> [x] = slice a b xs \<Longrightarrow> b = a + 1"
	+ by (induction a b xs rule: slice.induct) (auto simp: slice_drop_take)
	+
	+
	+section \<open>Earley recognizer\<close>
	+
	+subsection \<open>Earley items\<close>
	+
	+definition rule_head :: "'a rule \<Rightarrow> 'a" where
	+ "rule_head \<equiv> fst"
	+
	+definition rule_body :: "'a rule \<Rightarrow> 'a list" where
	+ "rule_body \<equiv> snd"
	+
	+datatype 'a item =
	+ Item (item_rule: "'a rule") (item_dot : nat) (item_origin : nat) (item_end : nat)
	+
	+definition item_rule_head :: "'a item \<Rightarrow> 'a" where
	+ "item_rule_head x \<equiv> rule_head (item_rule x)"
	+
	+definition item_rule_body :: "'a item \<Rightarrow> 'a sentence" where
	+ "item_rule_body x \<equiv> rule_body (item_rule x)"
	+
	+definition item_\<alpha> :: "'a item \<Rightarrow> 'a sentence" where
	+ "item_\<alpha> x \<equiv> take (item_dot x) (item_rule_body x)"
	+
	+definition item_\<beta> :: "'a item \<Rightarrow> 'a sentence" where
	+ "item_\<beta> x \<equiv> drop (item_dot x) (item_rule_body x)"
	+
	+definition is_complete :: "'a item \<Rightarrow> bool" where
	+ "is_complete x \<equiv> item_dot x \<ge> length (item_rule_body x)"
	+
	+definition next_symbol :: "'a item \<Rightarrow> 'a option" where
	+ "next_symbol x \<equiv> if is_complete x then None else Some (item_rule_body x ! item_dot x)"
	+
	+lemmas item_defs = item_rule_head_def item_rule_body_def item_\<alpha>_def item_\<beta>_def rule_head_def rule_body_def
	+
	+definition is_finished :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item \<Rightarrow> bool" where
	+ "is_finished \<G> \<omega> x \<equiv>
	+ item_rule_head x = \<SS> \<G> \<and>
	+ item_origin x = 0 \<and>
	+ item_end x = length \<omega> \<and>
	+ is_complete x"
	+
	+definition recognizing :: "'a item set \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> bool" where
	+ "recognizing I \<G> \<omega> \<equiv> \<exists>x \<in> I. is_finished \<G> \<omega> x"
	+
	+inductive_set Earley :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set"
	+ for \<G> :: "'a cfg" and \<omega> :: "'a sentence" where
	+ Init: "r \<in> set (\<RR> \<G>) \<Longrightarrow> fst r = \<SS> \<G> \<Longrightarrow>
	+ Item r 0 0 0 \<in> Earley \<G> \<omega>"
	+ \| Scan: "x = Item r b i j \<Longrightarrow> x \<in> Earley \<G> \<omega> \<Longrightarrow>
	+ \<omega>!j = a \<Longrightarrow> j < length \<omega> \<Longrightarrow> next_symbol x = Some a \<Longrightarrow>
	+ Item r (b + 1) i (j + 1) \<in> Earley \<G> \<omega>"
	+ \| Predict: "x = Item r b i j \<Longrightarrow> x \<in> Earley \<G> \<omega> \<Longrightarrow>
	+ r' \<in> set (\<RR> \<G>) \<Longrightarrow> next_symbol x = Some (rule_head r') \<Longrightarrow>
	+ Item r' 0 j j \<in> Earley \<G> \<omega>"
	+ \| Complete: "x = Item r\<^sub>x b\<^sub>x i j \<Longrightarrow> x \<in> Earley \<G> \<omega> \<Longrightarrow> y = Item r\<^sub>y b\<^sub>y j k \<Longrightarrow> y \<in> Earley \<G> \<omega> \<Longrightarrow>
	+ is_complete y \<Longrightarrow> next_symbol x = Some (item_rule_head y) \<Longrightarrow>
	+ Item r\<^sub>x (b\<^sub>x + 1) i k \<in> Earley \<G> \<omega>"
	+
	+
	+subsection \<open>Well-formedness\<close>
	+
	+definition wf_item :: "'a cfg \<Rightarrow> 'a sentence => 'a item \<Rightarrow> bool" where
	+ "wf_item \<G> \<omega> x \<equiv>
	+ item_rule x \<in> set (\<RR> \<G>) \<and>
	+ item_dot x \<le> length (item_rule_body x) \<and>
	+ item_origin x \<le> item_end x \<and>
	+ item_end x \<le> length \<omega>"
	+
	+lemma wf_Init:
	+ assumes "r \<in> set (\<RR> \<G>)" "fst r = \<SS> \<G>"
	+ shows "wf_item \<G> \<omega> (Item r 0 0 0)"
	+ using assms unfolding wf_item_def by simp
	+
	+lemma wf_Scan:
	+ assumes "x = Item r b i j" "wf_item \<G> \<omega> x" "\<omega>!j = a" "j < length \<omega>" "next_symbol x = Some a"
	+ shows "wf_item \<G> \<omega> (Item r (b + 1) i (j+1))"
	+ using assms unfolding wf_item_def by (auto simp: item_defs is_complete_def next_symbol_def split: if_splits)
	+
	+lemma wf_Predict:
	+ assumes "x = Item r b i j" "wf_item \<G> \<omega> x" "r' \<in> set (\<RR> \<G>)" "next_symbol x = Some (rule_head r')"
	+ shows "wf_item \<G> \<omega> (Item r' 0 j j)"
	+ using assms unfolding wf_item_def by simp
	+
	+lemma wf_Complete:
	+ assumes "x = Item r\<^sub>x b\<^sub>x i j" "wf_item \<G> \<omega> x" "y = Item r\<^sub>y b\<^sub>y j k" "wf_item \<G> \<omega> y"
	+ assumes "is_complete y" "next_symbol x = Some (item_rule_head y)"
	+ shows "wf_item \<G> \<omega> (Item r\<^sub>x (b\<^sub>x + 1) i k)"
	+ using assms unfolding wf_item_def is_complete_def next_symbol_def item_rule_body_def
	+ by (auto split: if_splits)
	+
	+lemma wf_Earley:
	+ assumes "x \<in> Earley \<G> \<omega>"
	+ shows "wf_item \<G> \<omega> x"
	+ using assms wf_Init wf_Scan wf_Predict wf_Complete
	+ by (induction rule: Earley.induct) fast+
	+
	+
	+subsection \<open>Soundness\<close>
	+
	+definition sound_item :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item \<Rightarrow> bool" where
	+ "sound_item \<G> \<omega> x \<equiv> derives \<G> [item_rule_head x] (slice (item_origin x) (item_end x) \<omega> @ item_\<beta> x)"
	+
	+lemma sound_Init:
	+ assumes "r \<in> set (\<RR> \<G>)" "fst r = \<SS> \<G>"
	+ shows "sound_item \<G> \<omega> (Item r 0 0 0)"
	+proof -
	+ let ?x = "Item r 0 0 0"
	+ have "(item_rule_head ?x, item_\<beta> ?x) \<in> set (\<RR> \<G>)"
	+ using assms(1) by (simp add: item_defs)
	+ hence "derives \<G> [item_rule_head ?x] (item_\<beta> ?x)"
	+ using derives_if_valid_rule by metis
	+ thus "sound_item \<G> \<omega> ?x"
	+ unfolding sound_item_def by (simp add: slice_empty)
	+qed
	+
	+lemma sound_Scan:
	+ assumes "x = Item r b i j" "wf_item \<G> \<omega> x" "sound_item \<G> \<omega> x"
	+ assumes "\<omega>!j = a" "j < length \<omega>" "next_symbol x = Some a"
	+ shows "sound_item \<G> \<omega> (Item r (b+1) i (j+1))"
	+proof -
	+ define x' where [simp]: "x' = Item r (b+1) i (j+1)"
	+ obtain item_\<beta>' where *: "item_\<beta> x = a # item_\<beta>'" "item_\<beta> x' = item_\<beta>'"
	+ using assms(1,6) apply (auto simp: item_defs next_symbol_def is_complete_def split: if_splits)
	+ by (metis Cons_nth_drop_Suc leI)
	+ have "slice i j \<omega> @ item_\<beta> x = slice i (j+1) \<omega> @ item_\<beta>'"
	+ using * assms(1,2,4,5) by (auto simp: slice_append_nth wf_item_def)
	+ moreover have "derives \<G> [item_rule_head x] (slice i j \<omega> @ item_\<beta> x)"
	+ using assms(1,3) sound_item_def by force
	+ ultimately show ?thesis
	+ using assms(1) * by (auto simp: item_defs sound_item_def)
	+qed
	+
	+lemma sound_Predict:
	+ assumes "x = Item r b i j" "wf_item \<G> \<omega> x" "sound_item \<G> \<omega> x"
	+ assumes "r' \<in> set (\<RR> \<G>)" "next_symbol x = Some (rule_head r')"
	+ shows "sound_item \<G> \<omega> (Item r' 0 j j)"
	+ using assms by (auto simp: sound_item_def derives_if_valid_rule slice_empty item_defs)
	+
	+lemma sound_Complete:
	+ assumes "x = Item r\<^sub>x b\<^sub>x i j" "wf_item \<G> \<omega> x" "sound_item \<G> \<omega> x"
	+ assumes "y = Item r\<^sub>y b\<^sub>y j k" "wf_item \<G> \<omega> y" "sound_item \<G> \<omega> y"
	+ assumes "is_complete y" "next_symbol x = Some (item_rule_head y)"
	+ shows "sound_item \<G> \<omega> (Item r\<^sub>x (b\<^sub>x + 1) i k)"
	+proof -
	+ have "derives \<G> [item_rule_head y] (slice j k \<omega>)"
	+ using assms(4,6,7) by (auto simp: sound_item_def is_complete_def item_defs)
	+ then obtain E where E: "Derivation \<G> [item_rule_head y] E (slice j k \<omega>)"
	+ using derives_implies_Derivation by blast
	+ have "derives \<G> [item_rule_head x] (slice i j \<omega> @ item_\<beta> x)"
	+ using assms(1,3,4) by (auto simp: sound_item_def)
	+ moreover have 0: "item_\<beta> x = (item_rule_head y) # tl (item_\<beta> x)"
	+ using assms(8) apply (auto simp: next_symbol_def is_complete_def item_defs split: if_splits)
	+ by (metis drop_eq_Nil hd_drop_conv_nth leI list.collapse)
	+ ultimately obtain D where D:
	+ "Derivation \<G> [item_rule_head x] D (slice i j \<omega> @ [item_rule_head y] @ (tl (item_\<beta> x)))"
	+ using derives_implies_Derivation by (metis append_Cons append_Nil)
	+ obtain F where F:
	+ "Derivation \<G> [item_rule_head x] F (slice i j \<omega> @ slice j k \<omega> @ tl (item_\<beta> x))"
	+ using Derivation_append_rewrite D E by blast
	+ moreover have "i \<le> j"
	+ using assms(1,2) wf_item_def by force
	+ moreover have "j \<le> k"
	+ using assms(4,5) wf_item_def by force
	+ ultimately have "derives \<G> [item_rule_head x] (slice i k \<omega> @ tl (item_\<beta> x))"
	+ by (metis Derivation_implies_derives append.assoc slice_concat)
	+ thus "sound_item \<G> \<omega> (Item r\<^sub>x (b\<^sub>x + 1) i k)"
	+ using assms(1,4) by (auto simp: sound_item_def item_defs drop_Suc tl_drop)
	+qed
	+
	+lemma sound_Earley:
	+ assumes "x \<in> Earley \<G> \<omega>" "wf_item \<G> \<omega> x"
	+ shows "sound_item \<G> \<omega> x"
	+ using assms
	+proof (induction rule: Earley.induct)
	+ case (Init r)
	+ thus ?case
	+ using sound_Init by blast
	+next
	+ case (Scan x r b i j a)
	+ thus ?case
	+ using wf_Earley sound_Scan by fast
	+next
	+ case (Predict x r b i j r')
	+ thus ?case
	+ using wf_Earley sound_Predict by blast
	+next
	+ case (Complete x r\<^sub>x b\<^sub>x i j y r\<^sub>y b\<^sub>y k)
	+ thus ?case
	+ using wf_Earley sound_Complete by metis
	+qed
	+
	+theorem soundness_Earley:
	+ assumes "recognizing (Earley \<G> \<omega>) \<G> \<omega>"
	+ shows "derives \<G> [\<SS> \<G>] \<omega>"
	+proof -
	+ obtain x where x: "x \<in> Earley \<G> \<omega>" "is_finished \<G> \<omega> x"
	+ using assms recognizing_def by blast
	+ hence "sound_item \<G> \<omega> x"
	+ using wf_Earley sound_Earley by blast
	+ thus ?thesis
	+ unfolding sound_item_def using x by (auto simp: is_finished_def is_complete_def item_defs)
	+qed
	+
	+
	+subsection \<open>Completeness\<close>
	+
	+definition partially_completed :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set \<Rightarrow> ('a derivation \<Rightarrow> bool) \<Rightarrow> bool" where
	+ "partially_completed k \<G> \<omega> E P \<equiv> \<forall>r b i' i j x a D.
	+ i \<le> j \<and> j \<le> k \<and> k \<le> length \<omega> \<and>
	+ x = Item r b i' i \<and> x \<in> E \<and> next_symbol x = Some a \<and>
	+ Derivation \<G> [a] D (slice i j \<omega>) \<and> P D \<longrightarrow>
	+ Item r (b+1) i' j \<in> E"
	+
	+lemma partially_completed_upto:
	+ assumes "j \<le> k" "k \<le> length \<omega>"
	+ assumes "x = Item (N,\<alpha>) d i j" "x \<in> I" "\<forall>x \<in> I. wf_item \<G> \<omega> x"
	+ assumes "Derivation \<G> (item_\<beta> x) D (slice j k \<omega>)"
	+ assumes "partially_completed k \<G> \<omega> I (\<lambda>D'. length D' \<le> length D)"
	+ shows "Item (N,\<alpha>) (length \<alpha>) i k \<in> I"
	+ using assms
	+proof (induction "item_\<beta> x" arbitrary: d i j k N \<alpha> x D)
	+ case Nil
	+ have "item_\<alpha> x = \<alpha>"
	+ using Nil(1,4) unfolding item_\<alpha>_def item_\<beta>_def item_rule_body_def rule_body_def by simp
	+ hence "x = Item (N,\<alpha>) (length \<alpha>) i j"
	+ using Nil.hyps Nil.prems(3-5) unfolding wf_item_def item_defs by auto
	+ have "Derivation \<G> [] D (slice j k \<omega>)"
	+ using Nil.hyps Nil.prems(6) by auto
	+ hence "slice j k \<omega> = []"
	+ using Derivation_from_empty by blast
	+ hence "j = k"
	+ unfolding slice_drop_take using Nil.prems(1,2) by simp
	+ thus ?case
	+ using \<open>x = Item (N, \<alpha>) (length \<alpha>) i j\<close> Nil.prems(4) by blast
	+next
	+ case (Cons b bs)
	+ obtain j' E F where *:
	+ "Derivation \<G> [b] E (slice j j' \<omega>)"
	+ "Derivation \<G> bs F (slice j' k \<omega>)"
	+ "j \<le> j'" "j' \<le> k" "length E \<le> length D" "length F \<le> length D"
	+ using Derivation_concat_split[of \<G> "[b]" bs D "slice j k \<omega>"] slice_concat_Ex
	+ using Cons.hyps(2) Cons.prems(1,6)
	+ by (smt (verit, ccfv_threshold) Cons_eq_appendI append_self_conv2)
	+ have "next_symbol x = Some b"
	+ using Cons.hyps(2) unfolding item_defs(4) next_symbol_def is_complete_def by (auto, metis nth_via_drop)
	+ hence "Item (N, \<alpha>) (d+1) i j' \<in> I"
	+ using Cons.prems(7) unfolding partially_completed_def
	+ using Cons.prems(2,3,4) *(1,3-5) by blast
	+ moreover have "partially_completed k \<G> \<omega> I (\<lambda>D'. length D' \<le> length F)"
	+ using Cons.prems(7) *(6) unfolding partially_completed_def by fastforce
	+ moreover have "bs = item_\<beta> (Item (N,\<alpha>) (d+1) i j')"
	+ using Cons.hyps(2) Cons.prems(3) unfolding item_defs(4) item_rule_body_def
	+ by (auto, metis List.list.sel(3) drop_Suc drop_tl)
	+ ultimately show ?case
	+ using Cons.hyps(1) *(2,4) Cons.prems(2,3,5) wf_item_def by blast
	+qed
	+
	+lemma partially_completed_Earley_k:
	+ assumes "wf_\<G> \<G>"
	+ shows "partially_completed k \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>_. True)"
	+ unfolding partially_completed_def
	+proof (standard, standard, standard, standard, standard, standard, standard, standard, standard)
	+ fix r b i' i j x a D
	+ assume
	+ "i \<le> j \<and> j \<le> k \<and> k \<le> length \<omega> \<and>
	+ x = Item r b i' i \<and> x \<in> Earley \<G> \<omega> \<and>
	+ next_symbol x = Some a \<and>
	+ Derivation \<G> [a] D (slice i j \<omega>) \<and> True"
	+ thus "Item r (b + 1) i' j \<in> Earley \<G> \<omega>"
	+ proof (induction "length D" arbitrary: r b i' i j x a D rule: nat_less_induct)
	+ case 1
	+ show ?case
	+ proof cases
	+ assume "D = []"
	+ hence "[a] = slice i j \<omega>"
	+ using "1.prems" by force
	+ moreover have "j \<le> length \<omega>"
	+ using le_trans "1.prems" by blast
	+ ultimately have "j = i+1"
	+ using slice_singleton by metis
	+ hence "i < length \<omega>"
	+ using \<open>j \<le> length \<omega>\<close> discrete by blast
	+ hence "\<omega>!i = a"
	+ using slice_nth \<open>[a] = slice i j \<omega>\<close> \<open>j = i + 1\<close> by fastforce
	+ hence "Item r (b + 1) i' j \<in> Earley \<G> \<omega>"
	+ using Earley.Scan "1.prems" \<open>i < length \<omega>\<close> \<open>j = i + 1\<close> by metis
	+ thus ?thesis
	+ by (simp add: \<open>j = i + 1\<close>)
	+ next
	+ assume "\<not> D = []"
	+ then obtain d D' where "D = d # D'"
	+ by (meson List.list.exhaust)
	+ then obtain \<alpha> where *: "Derives1 \<G> [a] (fst d) (snd d) \<alpha>" "Derivation \<G> \<alpha> D' (slice i j \<omega>)"
	+ using "1.prems" by auto
	+ hence rule: "(a, \<alpha>) \<in> set (\<RR> \<G>)" "fst d = 0" "snd d = (a ,\<alpha>)"
	+ using *(1) unfolding Derives1_def by (simp add: Cons_eq_append_conv)+
	+ show ?thesis
	+ proof cases
	+ assume "is_terminal \<G> a"
	+ have "is_nonterminal \<G> a"
	+ using rule by (simp add: assms)
	+ thus ?thesis
	+ using \<open>is_terminal \<G> a\<close> is_terminal_nonterminal by (metis assms)
	+ next
	+ assume "\<not> is_terminal \<G> a"
	+ define y where y_def: "y = Item (a ,\<alpha>) 0 i i"
	+ have "length D' < length D"
	+ using \<open>D = d # D'\<close> by fastforce
	+ hence "partially_completed k \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>E. length E \<le> length D')"
	+ unfolding partially_completed_def using "1.hyps" order_le_less_trans by (smt (verit, best))
	+ hence "partially_completed j \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>E. length E \<le> length D')"
	+ unfolding partially_completed_def using "1.prems" by force
	+ moreover have "Derivation \<G> (item_\<beta> y) D' (slice i j \<omega>)"
	+ using *(2) by (auto simp: item_defs y_def)
	+ moreover have "y \<in> Earley \<G> \<omega>"
	+ using y_def "1.prems" rule by (auto simp: item_defs Earley.Predict)
	+ moreover have "j \<le> length \<omega>"
	+ using "1.prems" by simp
	+ ultimately have "Item (a,\<alpha>) (length \<alpha>) i j \<in> Earley \<G> \<omega>"
	+ using partially_completed_upto "1.prems" wf_Earley y_def by metis
	+ moreover have x: "x = Item r b i' i" "x \<in> Earley \<G> \<omega>"
	+ using "1.prems" by blast+
	+ moreover have "next_symbol x = Some a"
	+ using "1.prems" by linarith
	+ ultimately show ?thesis
	+ using Earley.Complete[OF x] by (auto simp: is_complete_def item_defs)
	+ qed
	+ qed
	+ qed
	+qed
	+
	+lemma partially_completed_Earley:
	+ "wf_\<G> \<G> \<Longrightarrow> partially_completed (length \<omega>) \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>_. True)"
	+ by (simp add: partially_completed_Earley_k)
	+
	+theorem completeness_Earley:
	+ assumes "derives \<G> [\<SS> \<G>] \<omega>" "is_word \<G> \<omega>" "wf_\<G> \<G>"
	+ shows "recognizing (Earley \<G> \<omega>) \<G> \<omega>"
	+proof -
	+ obtain \<alpha> D where *: "(\<SS> \<G> ,\<alpha>) \<in> set (\<RR> \<G>)" "Derivation \<G> \<alpha> D \<omega>"
	+ using Derivation_\<SS>1 assms derives_implies_Derivation by metis
	+ define x where x_def: "x = Item (\<SS> \<G>, \<alpha>) 0 0 0"
	+ have "partially_completed (length \<omega>) \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>_. True)"
	+ using assms(3) partially_completed_Earley by blast
	+ hence 0: "partially_completed (length \<omega>) \<G> \<omega> (Earley \<G> \<omega>) (\<lambda>D'. length D' \<le> length D)"
	+ unfolding partially_completed_def by blast
	+ have 1: "x \<in> Earley \<G> \<omega>"
	+ using x_def Earley.Init *(1) by fastforce
	+ have 2: "Derivation \<G> (item_\<beta> x) D (slice 0 (length \<omega>) \<omega>)"
	+ using *(2) x_def by (simp add: item_defs)
	+ have "Item (\<SS> \<G>,\<alpha>) (length \<alpha>) 0 (length \<omega>) \<in> Earley \<G> \<omega>"
	+ using partially_completed_upto[OF _ _ _ _ _ 2 0] wf_Earley 1 x_def by auto
	+ then show ?thesis
	+ unfolding recognizing_def is_finished_def by (auto simp: is_complete_def item_defs, force)
	+qed
	+
	+
	+subsection \<open>Correctness\<close>
	+
	+theorem correctness_Earley:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>"
	+ shows "recognizing (Earley \<G> \<omega>) \<G> \<omega> \<longleftrightarrow> derives \<G> [\<SS> \<G>] \<omega>"
	+ using assms soundness_Earley completeness_Earley by blast
	+
	+
	+subsection \<open>Finiteness\<close>
	+
	+lemma finiteness_empty:
	+ "set (\<RR> \<G>) = {} \<Longrightarrow> finite { x \| x. wf_item \<G> \<omega> x }"
	+ unfolding wf_item_def by simp
	+
	+fun item_intro :: "'a rule \<times> nat \<times> nat \<times> nat \<Rightarrow> 'a item" where
	+ "item_intro (rule, dot, origin, ends) = Item rule dot origin ends"
	+
	+lemma finiteness_nonempty:
	+ assumes "set (\<RR> \<G>) \<noteq> {}"
	+ shows "finite { x \| x. wf_item \<G> \<omega> x }"
	+proof -
	+ define M where "M = Max { length (rule_body r) \| r. r \<in> set (\<RR> \<G>) }"
	+ define Top where "Top = (set (\<RR> \<G>) \<times> {0..M} \<times> {0..length \<omega>} \<times> {0..length \<omega>})"
	+ hence "finite Top"
	+ using finite_cartesian_product finite by blast
	+ have "inj_on item_intro Top"
	+ unfolding Top_def inj_on_def by simp
	+ hence "finite (item_intro ` Top)"
	+ using finite_image_iff \<open>finite Top\<close> by auto
	+ have "{ x \| x. wf_item \<G> \<omega> x } \<subseteq> item_intro ` Top"
	+ proof standard
	+ fix x
	+ assume "x \<in> { x \| x. wf_item \<G> \<omega> x }"
	+ then obtain rule dot origin endp where *: "x = Item rule dot origin endp"
	+ "rule \<in> set (\<RR> \<G>)" "dot \<le> length (item_rule_body x)" "origin \<le> length \<omega>" "endp \<le> length \<omega>"
	+ unfolding wf_item_def using item.exhaust_sel le_trans by blast
	+ hence "length (rule_body rule) \<in> { length (rule_body r) \| r. r \<in> set (\<RR> \<G>) }"
	+ using *(1,2) item_rule_body_def by blast
	+ moreover have "finite { length (rule_body r) \| r. r \<in> set (\<RR> \<G>) }"
	+ using finite finite_image_set[of "\<lambda>x. x \<in> set (\<RR> \<G>)"] by fastforce
	+ ultimately have "M \<ge> length (rule_body rule)"
	+ unfolding M_def by simp
	+ hence "dot \<le> M"
	+ using *(1,3) item_rule_body_def by (metis item.sel(1) le_trans)
	+ hence "(rule, dot, origin, endp) \<in> Top"
	+ using *(2,4,5) unfolding Top_def by simp
	+ thus "x \<in> item_intro ` Top"
	+ using *(1) by force
	+ qed
	+ thus ?thesis
	+ using \<open>finite (item_intro ` Top)\<close> rev_finite_subset by auto
	+qed
	+
	+lemma finiteness_UNIV_wf_item:
	+ "finite { x \| x. wf_item \<G> \<omega> x }"
	+ using finiteness_empty finiteness_nonempty by fastforce
	+
	+theorem finiteness_Earley:
	+ "finite (Earley \<G> \<omega>)"
	+ using finiteness_UNIV_wf_item wf_Earley rev_finite_subset by (metis mem_Collect_eq subsetI)
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/Earley_Fixpoint.thy b/thys/Earley_Parser/Earley_Fixpoint.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Earley_Fixpoint.thy
	@@ -0,0 +1,508 @@
	+theory Earley_Fixpoint
	+ imports
	+ Earley
	+ Limit
	+begin
	+
	+section \<open>Earley recognizer\<close>
	+
	+subsection \<open>Earley fixpoint\<close>
	+
	+definition init_item :: "'a rule \<Rightarrow> nat \<Rightarrow> 'a item" where
	+ "init_item r k \<equiv> Item r 0 k k"
	+
	+definition inc_item :: "'a item \<Rightarrow> nat \<Rightarrow> 'a item" where
	+ "inc_item x k \<equiv> Item (item_rule x) (item_dot x + 1) (item_origin x) k"
	+
	+definition bin :: "'a item set \<Rightarrow> nat \<Rightarrow> 'a item set" where
	+ "bin I k \<equiv> { x . x \<in> I \<and> item_end x = k }"
	+
	+definition Init\<^sub>F :: "'a cfg \<Rightarrow> 'a item set" where
	+ "Init\<^sub>F \<G> \<equiv> { init_item r 0 \| r. r \<in> set (\<RR> \<G>) \<and> fst r = (\<SS> \<G>) }"
	+
	+definition Scan\<^sub>F :: "nat \<Rightarrow> 'a sentence \<Rightarrow> 'a item set \<Rightarrow> 'a item set" where
	+ "Scan\<^sub>F k \<omega> I \<equiv> { inc_item x (k+1) \| x a.
	+ x \<in> bin I k \<and>
	+ \<omega>!k = a \<and>
	+ k < length \<omega> \<and>
	+ next_symbol x = Some a }"
	+
	+definition Predict\<^sub>F :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a item set \<Rightarrow> 'a item set" where
	+ "Predict\<^sub>F k \<G> I \<equiv> { init_item r k \| r x.
	+ r \<in> set (\<RR> \<G>) \<and>
	+ x \<in> bin I k \<and>
	+ next_symbol x = Some (rule_head r) }"
	+
	+definition Complete\<^sub>F :: "nat \<Rightarrow> 'a item set \<Rightarrow> 'a item set" where
	+ "Complete\<^sub>F k I \<equiv> { inc_item x k \| x y.
	+ x \<in> bin I (item_origin y) \<and>
	+ y \<in> bin I k \<and>
	+ is_complete y \<and>
	+ next_symbol x = Some (item_rule_head y) }"
	+
	+definition Earley\<^sub>F_bin_step :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set \<Rightarrow> 'a item set" where
	+ "Earley\<^sub>F_bin_step k \<G> \<omega> I \<equiv> I \<union> Scan\<^sub>F k \<omega> I \<union> Complete\<^sub>F k I \<union> Predict\<^sub>F k \<G> I"
	+
	+definition Earley\<^sub>F_bin :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set \<Rightarrow> 'a item set" where
	+ "Earley\<^sub>F_bin k \<G> \<omega> I \<equiv> limit (Earley\<^sub>F_bin_step k \<G> \<omega>) I"
	+
	+fun Earley\<^sub>F_bins :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set" where
	+ "Earley\<^sub>F_bins 0 \<G> \<omega> = Earley\<^sub>F_bin 0 \<G> \<omega> (Init\<^sub>F \<G>)"
	+\| "Earley\<^sub>F_bins (Suc n) \<G> \<omega> = Earley\<^sub>F_bin (Suc n) \<G> \<omega> (Earley\<^sub>F_bins n \<G> \<omega>)"
	+
	+definition Earley\<^sub>F :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a item set" where
	+ "Earley\<^sub>F \<G> \<omega> \<equiv> Earley\<^sub>F_bins (length \<omega>) \<G> \<omega>"
	+
	+
	+subsection \<open>Monotonicity and Absorption\<close>
	+
	+lemma Earley\<^sub>F_bin_step_empty:
	+ "Earley\<^sub>F_bin_step k \<G> \<omega> {} = {}"
	+ unfolding Earley\<^sub>F_bin_step_def Scan\<^sub>F_def Complete\<^sub>F_def Predict\<^sub>F_def bin_def by blast
	+
	+lemma Earley\<^sub>F_bin_step_setmonotone:
	+ "setmonotone (Earley\<^sub>F_bin_step k \<G> \<omega>)"
	+ by (simp add: Un_assoc Earley\<^sub>F_bin_step_def setmonotone_def)
	+
	+lemma Earley\<^sub>F_bin_step_continuous:
	+ "continuous (Earley\<^sub>F_bin_step k \<G> \<omega>)"
	+ unfolding continuous_def
	+proof (standard, standard, standard)
	+ fix C :: "nat \<Rightarrow> 'a item set"
	+ assume "chain C"
	+ thus "chain (Earley\<^sub>F_bin_step k \<G> \<omega> \<circ> C)"
	+ unfolding chain_def Earley\<^sub>F_bin_step_def by (auto simp: Scan\<^sub>F_def Predict\<^sub>F_def Complete\<^sub>F_def bin_def subset_eq)
	+next
	+ fix C :: "nat \<Rightarrow> 'a item set"
	+ assume *: "chain C"
	+ show "Earley\<^sub>F_bin_step k \<G> \<omega> (natUnion C) = natUnion (Earley\<^sub>F_bin_step k \<G> \<omega> \<circ> C)"
	+ unfolding natUnion_def
	+ proof standard
	+ show "Earley\<^sub>F_bin_step k \<G> \<omega> (\<Union> {C n \|n. True}) \<subseteq> \<Union> {(Earley\<^sub>F_bin_step k \<G> \<omega> \<circ> C) n \|n. True}"
	+ proof standard
	+ fix x
	+ assume #: "x \<in> Earley\<^sub>F_bin_step k \<G> \<omega> (\<Union> {C n \|n. True})"
	+ show "x \<in> \<Union> {(Earley\<^sub>F_bin_step k \<G> \<omega> \<circ> C) n \|n. True}"
	+ proof (cases "x \<in> Complete\<^sub>F k (\<Union> {C n \|n. True})")
	+ case True
	+ then show ?thesis
	+ using * unfolding chain_def Earley\<^sub>F_bin_step_def Complete\<^sub>F_def bin_def
	+ proof clarsimp
	+ fix y :: "'a item" and z :: "'a item" and n :: nat and m :: nat
	+ assume a1: "is_complete z"
	+ assume a2: "item_end y = item_origin z"
	+ assume a3: "y \<in> C n"
	+ assume a4: "z \<in> C m"
	+ assume a5: "next_symbol y = Some (item_rule_head z)"
	+ assume "\<forall>i. C i \<subseteq> C (Suc i)"
	+ hence f6: "\<And>n m. \<not> n \<le> m \<or> C n \<subseteq> C m"
	+ by (meson lift_Suc_mono_le)
	+ hence f7: "\<And>n. \<not> m \<le> n \<or> z \<in> C n"
	+ using a4 by blast
	+ have "\<exists>n \<ge> m. y \<in> C n"
	+ using f6 a3 by (meson le_sup_iff subset_eq sup_ge1)
	+ thus "\<exists>I.
	+ (\<exists>n. I = C n \<union>
	+ Scan\<^sub>F (item_end z) \<omega> (C n) \<union>
	+ {inc_item i (item_end z) \|i.
	+ i \<in> C n \<and>
	+ (\<exists>j.
	+ item_end i = item_origin j \<and>
	+ j \<in> C n \<and>
	+ item_end j = item_end z \<and>
	+ is_complete j \<and>
	+ next_symbol i = Some (item_rule_head j))} \<union>
	+ Predict\<^sub>F (item_end z) \<G> (C n))
	+ \<and> inc_item y (item_end z) \<in> I"
	+ using f7 a5 a2 a1 by blast
	+ qed
	+ next
	+ case False
	+ thus ?thesis
	+ using # Un_iff by (auto simp: Earley\<^sub>F_bin_step_def Scan\<^sub>F_def Predict\<^sub>F_def bin_def; blast)
	+ qed
	+ qed
	+ next
	+ show "\<Union> {(Earley\<^sub>F_bin_step k \<G> \<omega> \<circ> C) n \|n. True} \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> (\<Union> {C n \|n. True})"
	+ unfolding Earley\<^sub>F_bin_step_def
	+ using * by (auto simp: Scan\<^sub>F_def Predict\<^sub>F_def Complete\<^sub>F_def chain_def bin_def, metis+)
	+ qed
	+qed
	+
	+lemma Earley\<^sub>F_bin_step_regular:
	+ "regular (Earley\<^sub>F_bin_step k \<G> \<omega>)"
	+ by (simp add: Earley\<^sub>F_bin_step_continuous Earley\<^sub>F_bin_step_setmonotone regular_def)
	+
	+lemma Earley\<^sub>F_bin_idem:
	+ "Earley\<^sub>F_bin k \<G> \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I) = Earley\<^sub>F_bin k \<G> \<omega> I"
	+ by (simp add: Earley\<^sub>F_bin_def Earley\<^sub>F_bin_step_regular limit_is_idempotent)
	+
	+lemma Scan\<^sub>F_bin_absorb:
	+ "Scan\<^sub>F k \<omega> (bin I k) = Scan\<^sub>F k \<omega> I"
	+ unfolding Scan\<^sub>F_def bin_def by simp
	+
	+lemma Predict\<^sub>F_bin_absorb:
	+ "Predict\<^sub>F k \<G> (bin I k) = Predict\<^sub>F k \<G> I"
	+ unfolding Predict\<^sub>F_def bin_def by simp
	+
	+lemma Scan\<^sub>F_Un:
	+ "Scan\<^sub>F k \<omega> (I \<union> J) = Scan\<^sub>F k \<omega> I \<union> Scan\<^sub>F k \<omega> J"
	+ unfolding Scan\<^sub>F_def bin_def by blast
	+
	+lemma Predict\<^sub>F_Un:
	+ "Predict\<^sub>F k \<G> (I \<union> J) = Predict\<^sub>F k \<G> I \<union> Predict\<^sub>F k \<G> J"
	+ unfolding Predict\<^sub>F_def bin_def by blast
	+
	+lemma Scan\<^sub>F_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> Scan\<^sub>F k \<omega> I \<subseteq> Scan\<^sub>F k \<omega> J"
	+ unfolding Scan\<^sub>F_def bin_def by blast
	+
	+lemma Predict\<^sub>F_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> Predict\<^sub>F k \<G> I \<subseteq> Predict\<^sub>F k \<G> J"
	+ unfolding Predict\<^sub>F_def bin_def by blast
	+
	+lemma Complete\<^sub>F_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> Complete\<^sub>F k I \<subseteq> Complete\<^sub>F k J"
	+ unfolding Complete\<^sub>F_def bin_def by blast
	+
	+lemma Earley\<^sub>F_bin_step_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> Earley\<^sub>F_bin_step k \<G> \<omega> I \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> J"
	+ unfolding Earley\<^sub>F_bin_step_def using Scan\<^sub>F_sub_mono Predict\<^sub>F_sub_mono Complete\<^sub>F_sub_mono by (metis sup.mono)
	+
	+lemma funpower_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) n I \<subseteq> funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) n J"
	+ by (induction n) (auto simp: Earley\<^sub>F_bin_step_sub_mono)
	+
	+lemma Earley\<^sub>F_bin_sub_mono:
	+ "I \<subseteq> J \<Longrightarrow> Earley\<^sub>F_bin k \<G> \<omega> I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> J"
	+proof standard
	+ fix x
	+ assume "I \<subseteq> J" "x \<in> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ then obtain n where "x \<in> funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) n I"
	+ unfolding Earley\<^sub>F_bin_def limit_def natUnion_def by blast
	+ hence "x \<in> funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) n J"
	+ using \<open>I \<subseteq> J\<close> funpower_sub_mono by blast
	+ thus "x \<in> Earley\<^sub>F_bin k \<G> \<omega> J"
	+ unfolding Earley\<^sub>F_bin_def limit_def natUnion_def by blast
	+qed
	+
	+lemma Scan\<^sub>F_Earley\<^sub>F_bin_step_mono:
	+ "Scan\<^sub>F k \<omega> I \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_step_def by blast
	+
	+lemma Predict\<^sub>F_Earley\<^sub>F_bin_step_mono:
	+ "Predict\<^sub>F k \<G> I \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_step_def by blast
	+
	+lemma Complete\<^sub>F_Earley\<^sub>F_bin_step_mono:
	+ "Complete\<^sub>F k I \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_step_def by blast
	+
	+lemma Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono:
	+ "Earley\<^sub>F_bin_step k \<G> \<omega> I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+proof -
	+ have "Earley\<^sub>F_bin_step k \<G> \<omega> I \<subseteq> funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) 1 I"
	+ by simp
	+ thus ?thesis
	+ by (metis Earley\<^sub>F_bin_def limit_elem subset_eq)
	+qed
	+
	+lemma Scan\<^sub>F_Earley\<^sub>F_bin_mono:
	+ "Scan\<^sub>F k \<omega> I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Scan\<^sub>F_Earley\<^sub>F_bin_step_mono Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by force
	+
	+lemma Predict\<^sub>F_Earley\<^sub>F_bin_mono:
	+ "Predict\<^sub>F k \<G> I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Predict\<^sub>F_Earley\<^sub>F_bin_step_mono Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by force
	+
	+lemma Complete\<^sub>F_Earley\<^sub>F_bin_mono:
	+ "Complete\<^sub>F k I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Complete\<^sub>F_Earley\<^sub>F_bin_step_mono Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by force
	+
	+lemma Earley\<^sub>F_bin_mono:
	+ "I \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_step_def by blast
	+
	+lemma Init\<^sub>F_sub_Earley\<^sub>F_bins:
	+ "Init\<^sub>F \<G> \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ by (induction n) (use Earley\<^sub>F_bin_mono in fastforce)+
	+
	+subsection \<open>Soundness\<close>
	+
	+lemma Init\<^sub>F_sub_Earley:
	+ "Init\<^sub>F \<G> \<subseteq> Earley \<G> \<omega>"
	+ unfolding Init\<^sub>F_def init_item_def using Init by blast
	+
	+lemma Scan\<^sub>F_sub_Earley:
	+ assumes "I \<subseteq> Earley \<G> \<omega>"
	+ shows "Scan\<^sub>F k \<omega> I \<subseteq> Earley \<G> \<omega>"
	+ unfolding Scan\<^sub>F_def inc_item_def bin_def using assms Scan
	+ by (smt (verit, ccfv_SIG) item.exhaust_sel mem_Collect_eq subsetD subsetI)
	+
	+lemma Predict\<^sub>F_sub_Earley:
	+ assumes "I \<subseteq> Earley \<G> \<omega>"
	+ shows "Predict\<^sub>F k \<G> I \<subseteq> Earley \<G> \<omega>"
	+ unfolding Predict\<^sub>F_def init_item_def bin_def using assms Predict
	+ using item.exhaust_sel by blast
	+
	+lemma Complete\<^sub>F_sub_Earley:
	+ assumes "I \<subseteq> Earley \<G> \<omega>"
	+ shows "Complete\<^sub>F k I \<subseteq> Earley \<G> \<omega>"
	+ unfolding Complete\<^sub>F_def inc_item_def bin_def using assms Complete
	+ by (smt (verit, del_insts) item.exhaust_sel mem_Collect_eq subset_eq)
	+
	+lemma Earley\<^sub>F_bin_step_sub_Earley:
	+ assumes "I \<subseteq> Earley \<G> \<omega>"
	+ shows "Earley\<^sub>F_bin_step k \<G> \<omega> I \<subseteq> Earley \<G> \<omega>"
	+ unfolding Earley\<^sub>F_bin_step_def using assms Complete\<^sub>F_sub_Earley Predict\<^sub>F_sub_Earley Scan\<^sub>F_sub_Earley by (metis le_supI)
	+
	+lemma Earley\<^sub>F_bin_sub_Earley:
	+ assumes "I \<subseteq> Earley \<G> \<omega>"
	+ shows "Earley\<^sub>F_bin k \<G> \<omega> I \<subseteq> Earley \<G> \<omega>"
	+ using assms Earley\<^sub>F_bin_step_sub_Earley by (metis Earley\<^sub>F_bin_def limit_upperbound)
	+
	+lemma Earley\<^sub>F_bins_sub_Earley:
	+ shows "Earley\<^sub>F_bins n \<G> \<omega> \<subseteq> Earley \<G> \<omega>"
	+ by (induction n) (auto simp: Earley\<^sub>F_bin_sub_Earley Init\<^sub>F_sub_Earley)
	+
	+lemma Earley\<^sub>F_sub_Earley:
	+ shows "Earley\<^sub>F \<G> \<omega> \<subseteq> Earley \<G> \<omega>"
	+ by (simp add: Earley\<^sub>F_bins_sub_Earley Earley\<^sub>F_def)
	+
	+theorem soundness_Earley\<^sub>F:
	+ assumes "recognizing (Earley\<^sub>F \<G> \<omega>) \<G> \<omega>"
	+ shows "derives \<G> [\<SS> \<G>] \<omega>"
	+ using soundness_Earley Earley\<^sub>F_sub_Earley assms recognizing_def by (metis subsetD)
	+
	+
	+subsection \<open>Completeness\<close>
	+
	+definition prev_symbol :: "'a item \<Rightarrow> 'a option" where
	+ "prev_symbol x \<equiv> if item_dot x = 0 then None else Some (item_rule_body x ! (item_dot x - 1))"
	+
	+definition base :: "'a sentence \<Rightarrow> 'a item set \<Rightarrow> nat \<Rightarrow> 'a item set" where
	+ "base \<omega> I k \<equiv> { x . x \<in> I \<and> item_end x = k \<and> k > 0 \<and> prev_symbol x = Some (\<omega>!(k-1)) }"
	+
	+lemma Earley\<^sub>F_bin_sub_Earley\<^sub>F_bin:
	+ assumes "Init\<^sub>F \<G> \<subseteq> I"
	+ assumes "\<forall>k' < k. bin (Earley \<G> \<omega>) k' \<subseteq> I"
	+ assumes "base \<omega> (Earley \<G> \<omega>) k \<subseteq> I"
	+ shows "bin (Earley \<G> \<omega>) k \<subseteq> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+proof standard
	+ fix x
	+ assume *: "x \<in> bin (Earley \<G> \<omega>) k"
	+ hence "x \<in> Earley \<G> \<omega>"
	+ using bin_def by blast
	+ thus "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+ using assms *
	+ proof (induction rule: Earley.induct)
	+ case (Init r)
	+ thus ?case
	+ unfolding Init\<^sub>F_def init_item_def bin_def using Earley\<^sub>F_bin_mono by fast
	+ next
	+ case (Scan x r b i j a)
	+ have "j+1 = k"
	+ using Scan.prems(4) bin_def by (metis (mono_tags, lifting) CollectD item.sel(4))
	+ have "prev_symbol (Item r (b+1) i (j+1)) = Some (\<omega>!(k-1))"
	+ using Scan.hyps(1,3,5) \<open>j+1 = k\<close> by (auto simp: next_symbol_def prev_symbol_def item_rule_body_def split: if_splits)
	+ hence "Item r (b+1) i (j+1) \<in> base \<omega> (Earley \<G> \<omega>) k"
	+ unfolding base_def using Scan.prems(4) bin_def by fastforce
	+ hence "Item r (b+1) i (j+1) \<in> I"
	+ using Scan.prems(3) by blast
	+ hence "Item r (b+1) i (j+1) \<in> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_mono by blast
	+ thus ?case
	+ using \<open>j+1 = k\<close> bin_def by fastforce
	+ next
	+ case (Predict x r b i j r')
	+ have "j = k"
	+ using Predict.prems(4) bin_def by (metis (mono_tags, lifting) CollectD item.sel(4))
	+ hence "x \<in> bin (Earley \<G> \<omega>) k"
	+ using Predict.hyps(1,2) bin_def by fastforce
	+ hence "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+ using Predict.IH Predict.prems(1-3) by blast
	+ hence "Item r' 0 j j \<in> Predict\<^sub>F k \<G> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ unfolding Predict\<^sub>F_def init_item_def using Predict.hyps(1,3,4) \<open>j = k\<close> by blast
	+ hence "Item r' 0 j j \<in> Earley\<^sub>F_bin_step k \<G> \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ using Predict\<^sub>F_Earley\<^sub>F_bin_step_mono by blast
	+ hence "Item r' 0 j j \<in> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_idem Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by blast
	+ thus ?case
	+ by (simp add: \<open>j = k\<close> bin_def)
	+ next
	+ case (Complete x r\<^sub>x b\<^sub>x i j y r\<^sub>y b\<^sub>y l)
	+ have "l = k"
	+ using Complete.prems(4) bin_def by (metis (mono_tags, lifting) CollectD item.sel(4))
	+ hence "y \<in> bin (Earley \<G> \<omega>) l"
	+ using Complete.hyps(3,4) bin_def by fastforce
	+ hence 0: "y \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+ using Complete.IH(2) Complete.prems(1-3) \<open>l = k\<close> by blast
	+ have 1: "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) (item_origin y)"
	+ proof (cases "j = k")
	+ case True
	+ hence "x \<in> bin (Earley \<G> \<omega>) k"
	+ using Complete.hyps(1,2) bin_def by fastforce
	+ hence "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+ using Complete.IH(1) Complete.prems(1-3) by blast
	+ thus ?thesis
	+ using Complete.hyps(3) True by simp
	+ next
	+ case False
	+ hence "j < k"
	+ using \<open>l = k\<close> wf_Earley wf_item_def Complete.hyps(3,4) by force
	+ moreover have "x \<in> bin (Earley \<G> \<omega>) j"
	+ using Complete.hyps(1,2) bin_def by force
	+ ultimately have "x \<in> I"
	+ using Complete.prems(2) by blast
	+ hence "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) j"
	+ using Complete.hyps(1) Earley\<^sub>F_bin_mono bin_def by fastforce
	+ thus ?thesis
	+ using Complete.hyps(3) by simp
	+ qed
	+ have "Item r\<^sub>x (b\<^sub>x + 1) i k \<in> Complete\<^sub>F k (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ unfolding Complete\<^sub>F_def inc_item_def using 0 1 Complete.hyps(1,5,6) by force
	+ hence "Item r\<^sub>x (b\<^sub>x + 1) i k \<in> Earley\<^sub>F_bin_step k \<G> \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ unfolding Earley\<^sub>F_bin_step_def by blast
	+ hence "Item r\<^sub>x (b\<^sub>x + 1) i k \<in> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_idem Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by blast
	+ thus ?case
	+ using bin_def \<open>l = k\<close> by fastforce
	+ qed
	+qed
	+
	+lemma Earley_base_sub_Earley\<^sub>F_bin:
	+ assumes "Init\<^sub>F \<G> \<subseteq> I"
	+ assumes "\<forall>k' < k. bin (Earley \<G> \<omega>) k' \<subseteq> I"
	+ assumes "base \<omega> (Earley \<G> \<omega>) k \<subseteq> I"
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>"
	+ shows "base \<omega> (Earley \<G> \<omega>) (k+1) \<subseteq> bin (Earley\<^sub>F_bin k \<G> \<omega> I) (k+1)"
	+proof standard
	+ fix x
	+ assume *: "x \<in> base \<omega> (Earley \<G> \<omega>) (k+1)"
	+ hence "x \<in> Earley \<G> \<omega>"
	+ using base_def by blast
	+ thus "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) (k+1)"
	+ using assms *
	+ proof (induction rule: Earley.induct)
	+ case (Init r)
	+ have "k = 0"
	+ using Init.prems(6) unfolding base_def by simp
	+ hence False
	+ using Init.prems(6) unfolding base_def by simp
	+ thus ?case
	+ by blast
	+ next
	+ case (Scan x r b i j a)
	+ have "j = k"
	+ using Scan.prems(6) base_def by (metis (mono_tags, lifting) CollectD add_right_cancel item.sel(4))
	+ hence "x \<in> bin (Earley\<^sub>F_bin k \<G> \<omega> I) k"
	+ using Earley\<^sub>F_bin_sub_Earley\<^sub>F_bin Scan.prems Scan.hyps(1,2) bin_def
	+ by (metis (mono_tags, lifting) CollectI item.sel(4) subsetD)
	+ hence "Item r (b+1) i (j+1) \<in> Scan\<^sub>F k \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ unfolding Scan\<^sub>F_def inc_item_def using Scan.hyps \<open>j = k\<close> by force
	+ hence "Item r (b+1) i (j+1) \<in> Earley\<^sub>F_bin_step k \<G> \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ using Scan\<^sub>F_Earley\<^sub>F_bin_step_mono by blast
	+ hence "Item r (b+1) i (j+1) \<in> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using Earley\<^sub>F_bin_idem Earley\<^sub>F_bin_step_Earley\<^sub>F_bin_mono by blast
	+ thus ?case
	+ using \<open>j = k\<close> bin_def by fastforce
	+ next
	+ case (Predict x r b i j r')
	+ have False
	+ using Predict.prems(6) unfolding base_def by (auto simp: prev_symbol_def)
	+ thus ?case
	+ by blast
	+ next
	+ case (Complete x r\<^sub>x b\<^sub>x i j y r\<^sub>y b\<^sub>y l)
	+ have "l-1 < length \<omega>"
	+ using Complete.prems(6) base_def wf_Earley wf_item_def
	+ by (metis (mono_tags, lifting) CollectD add.right_neutral add_Suc_right add_diff_cancel_right' item.sel(4) less_eq_Suc_le plus_1_eq_Suc)
	+ hence "is_terminal \<G> (\<omega>!(l-1))"
	+ using Complete.prems(5) is_word_is_terminal by blast
	+ moreover have "is_nonterminal \<G> (item_rule_head y)"
	+ using Complete.hyps(3,4) Complete.prems(4) wf_Earley wf_item_def
	+ by (metis item_rule_head_def prod.collapse rule_head_def rule_nonterminal_type)
	+ moreover have "prev_symbol (Item r\<^sub>x (b\<^sub>x+1) i l) = next_symbol x"
	+ using Complete.hyps(1,6)
	+ by (auto simp: next_symbol_def prev_symbol_def is_complete_def item_rule_body_def split: if_splits)
	+ moreover have "prev_symbol (Item r\<^sub>x (b\<^sub>x+1) i l) = Some (\<omega>!(l-1))"
	+ using Complete.prems(6) base_def by (metis (mono_tags, lifting) CollectD item.sel(4))
	+ ultimately have False
	+ using Complete.hyps(6) Complete.prems(4) is_terminal_nonterminal by fastforce
	+ thus ?case
	+ by blast
	+ qed
	+qed
	+
	+lemma Earley\<^sub>F_bin_k_sub_Earley\<^sub>F_bins:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "k \<le> n"
	+ shows "bin (Earley \<G> \<omega>) k \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ using assms
	+proof (induction n arbitrary: k)
	+ case 0
	+ have "bin (Earley \<G> \<omega>) 0 \<subseteq> bin (Earley\<^sub>F_bin 0 \<G> \<omega> (Init\<^sub>F \<G>)) 0"
	+ using Earley\<^sub>F_bin_sub_Earley\<^sub>F_bin base_def by fastforce
	+ thus ?case
	+ unfolding bin_def using "0.prems"(3) by auto
	+next
	+ case (Suc n)
	+ show ?case
	+ proof (cases "k \<le> n")
	+ case True
	+ thus ?thesis
	+ using Suc Earley\<^sub>F_bin_mono by force
	+ next
	+ case False
	+ hence "k = n+1"
	+ using Suc.prems(3) by force
	+ have 0: "\<forall>k' < k. bin (Earley \<G> \<omega>) k' \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ using Suc by simp
	+ moreover have "base \<omega> (Earley \<G> \<omega>) k \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ proof -
	+ have "\<forall>k' < k-1. bin (Earley \<G> \<omega>) k' \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ using Suc \<open>k = n + 1\<close> by auto
	+ moreover have "base \<omega> (Earley \<G> \<omega>) (k-1) \<subseteq> Earley\<^sub>F_bins n \<G> \<omega>"
	+ using 0 bin_def base_def False \<open>k = n+1\<close>
	+ by (smt (verit) Suc_eq_plus1 diff_Suc_1 linorder_not_less mem_Collect_eq subsetD subsetI)
	+ ultimately have "base \<omega> (Earley \<G> \<omega>) k \<subseteq> bin (Earley\<^sub>F_bin n \<G> \<omega> (Earley\<^sub>F_bins n \<G> \<omega>)) k"
	+ using Suc.prems(1,2) Earley_base_sub_Earley\<^sub>F_bin \<open>k = n + 1\<close> Init\<^sub>F_sub_Earley\<^sub>F_bins by (metis add_diff_cancel_right')
	+ hence "base \<omega> (Earley \<G> \<omega>) k \<subseteq> bin (Earley\<^sub>F_bins n \<G> \<omega>) k"
	+ by (metis Earley\<^sub>F_bins.elims Earley\<^sub>F_bin_idem)
	+ thus ?thesis
	+ using bin_def by blast
	+ qed
	+ ultimately have "bin (Earley \<G> \<omega>) k \<subseteq> bin (Earley\<^sub>F_bin k \<G> \<omega> (Earley\<^sub>F_bins n \<G> \<omega>)) k"
	+ using Earley\<^sub>F_bin_sub_Earley\<^sub>F_bin Init\<^sub>F_sub_Earley\<^sub>F_bins by metis
	+ thus ?thesis
	+ using Earley\<^sub>F_bins.simps(2) \<open>k = n + 1\<close> bin_def by auto
	+ qed
	+qed
	+
	+lemma Earley_sub_Earley\<^sub>F:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>"
	+ shows "Earley \<G> \<omega> \<subseteq> Earley\<^sub>F \<G> \<omega>"
	+proof -
	+ have "\<forall>k \<le> length \<omega>. bin (Earley \<G> \<omega>) k \<subseteq> Earley\<^sub>F \<G> \<omega>"
	+ by (simp add: Earley\<^sub>F_bin_k_sub_Earley\<^sub>F_bins Earley\<^sub>F_def assms)
	+ thus ?thesis
	+ using wf_Earley wf_item_def bin_def by blast
	+qed
	+
	+theorem completeness_Earley\<^sub>F:
	+ assumes "derives \<G> [\<SS> \<G>] \<omega>" "is_word \<G> \<omega>" "wf_\<G> \<G>"
	+ shows "recognizing (Earley\<^sub>F \<G> \<omega>) \<G> \<omega>"
	+ using assms Earley_sub_Earley\<^sub>F Earley\<^sub>F_sub_Earley completeness_Earley by (metis subset_antisym)
	+
	+subsection \<open>Correctness\<close>
	+
	+theorem Earley_eq_Earley\<^sub>F:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>"
	+ shows "Earley \<G> \<omega> = Earley\<^sub>F \<G> \<omega>"
	+ using Earley_sub_Earley\<^sub>F Earley\<^sub>F_sub_Earley assms by blast
	+
	+theorem correctness_Earley\<^sub>F:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>"
	+ shows "recognizing (Earley\<^sub>F \<G> \<omega>) \<G> \<omega> \<longleftrightarrow> derives \<G> [\<SS> \<G>] \<omega>"
	+ using assms Earley_eq_Earley\<^sub>F correctness_Earley by fastforce
	+
	+end
	diff --git a/thys/Earley_Parser/Earley_Parser.thy b/thys/Earley_Parser/Earley_Parser.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Earley_Parser.thy
	@@ -0,0 +1,2528 @@
	+theory Earley_Parser
	+ imports
	+ Earley_Recognizer
	+ "HOL-Library.Monad_Syntax"
	+begin
	+
	+section \<open>Earley parser\<close>
	+
	+subsection \<open>Pointer lemmas\<close>
	+
	+definition predicts :: "'a item \<Rightarrow> bool" where
	+ "predicts x \<equiv> item_origin x = item_end x \<and> item_dot x = 0"
	+
	+definition scans :: "'a sentence \<Rightarrow> nat \<Rightarrow> 'a item \<Rightarrow> 'a item \<Rightarrow> bool" where
	+ "scans \<omega> k x y \<equiv> y = inc_item x k \<and> (\<exists>a. next_symbol x = Some a \<and> \<omega>!(k-1) = a)"
	+
	+definition completes :: "nat \<Rightarrow> 'a item \<Rightarrow> 'a item \<Rightarrow> 'a item \<Rightarrow> bool" where
	+ "completes k x y z \<equiv> y = inc_item x k \<and> is_complete z \<and> item_origin z = item_end x \<and>
	+ (\<exists>N. next_symbol x = Some N \<and> N = item_rule_head z)"
	+
	+definition sound_null_ptr :: "'a entry \<Rightarrow> bool" where
	+ "sound_null_ptr e \<equiv> (pointer e = Null \<longrightarrow> predicts (item e))"
	+
	+definition sound_pre_ptr :: "'a sentence \<Rightarrow> 'a bins \<Rightarrow> nat \<Rightarrow> 'a entry \<Rightarrow> bool" where
	+ "sound_pre_ptr \<omega> bs k e \<equiv> \<forall>pre. pointer e = Pre pre \<longrightarrow>
	+ k > 0 \<and> pre < length (bs!(k-1)) \<and> scans \<omega> k (item (bs!(k-1)!pre)) (item e)"
	+
	+definition sound_prered_ptr :: "'a bins \<Rightarrow> nat \<Rightarrow> 'a entry \<Rightarrow> bool" where
	+ "sound_prered_ptr bs k e \<equiv> \<forall>p ps k' pre red. pointer e = PreRed p ps \<and> (k', pre, red) \<in> set (p#ps) \<longrightarrow>
	+ k' < k \<and> pre < length (bs!k') \<and> red < length (bs!k) \<and> completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+
	+definition sound_ptrs :: "'a sentence \<Rightarrow> 'a bins \<Rightarrow> bool" where
	+ "sound_ptrs \<omega> bs \<equiv> \<forall>k < length bs. \<forall>e \<in> set (bs!k).
	+ sound_null_ptr e \<and> sound_pre_ptr \<omega> bs k e \<and> sound_prered_ptr bs k e"
	+
	+definition mono_red_ptr :: "'a bins \<Rightarrow> bool" where
	+ "mono_red_ptr bs \<equiv> \<forall>k < length bs. \<forall>i < length (bs!k).
	+ \<forall>k' pre red ps. pointer (bs!k!i) = PreRed (k', pre, red) ps \<longrightarrow> red < i"
	+
	+lemma nth_item_bin_upd:
	+ "n < length es \<Longrightarrow> item (bin_upd e es ! n) = item (es!n)"
	+ by (induction es arbitrary: e n) (auto simp: less_Suc_eq_0_disj split: entry.splits pointer.splits)
	+
	+lemma bin_upd_append:
	+ "item e \<notin> set (items es) \<Longrightarrow> bin_upd e es = es @ [e]"
	+ by (induction es arbitrary: e) (auto simp: items_def split: entry.splits pointer.splits)
	+
	+lemma bin_upd_null_pre:
	+ "item e \<in> set (items es) \<Longrightarrow> pointer e = Null \<or> pointer e = Pre pre \<Longrightarrow> bin_upd e es = es"
	+ by (induction es arbitrary: e) (auto simp: items_def split: entry.splits)
	+
	+lemma bin_upd_prered_nop:
	+ assumes "distinct (items es)" "i < length es"
	+ assumes "item e = item (es!i)" "pointer e = PreRed p ps" "\<nexists>p ps. pointer (es!i) = PreRed p ps"
	+ shows "bin_upd e es = es"
	+ using assms
	+ by (induction es arbitrary: e i) (auto simp: less_Suc_eq_0_disj items_def split: entry.splits pointer.splits)
	+
	+lemma bin_upd_prered_upd:
	+ assumes "distinct (items es)" "i < length es"
	+ assumes "item e = item (es!i)" "pointer e = PreRed p rs" "pointer (es!i) = PreRed p' rs'" "bin_upd e es = es'"
	+ shows "pointer (es'!i) = PreRed p' (p#rs@rs') \<and> (\<forall>j < length es'. i\<noteq>j \<longrightarrow> es'!j = es!j) \<and> length (bin_upd e es) = length es"
	+ using assms
	+proof (induction es arbitrary: e i es')
	+ case (Cons e' es)
	+ show ?case
	+ proof cases
	+ assume *: "item e = item e'"
	+ show ?thesis
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> e' = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where ee': "e = Entry x (PreRed xp xs)" "e' = Entry y (PreRed yp ys)" "x = y"
	+ using * by auto
	+ have simp: "bin_upd e (e' # es') = Entry x (PreRed yp (xp # xs @ ys)) # es'"
	+ using True ee' by simp
	+ show ?thesis
	+ using Cons simp ee' apply (auto simp: items_def)
	+ using less_Suc_eq_0_disj by fastforce+
	+ next
	+ case False
	+ hence "bin_upd e (e' # es') = e' # es'"
	+ using * by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using False * Cons.prems(1,2,3,4,5) by (auto simp: less_Suc_eq_0_disj items_def split: entry.splits)
	+ qed
	+ next
	+ assume *: "item e \<noteq> item e'"
	+ have simp: "bin_upd e (e' # es) = e' # bin_upd e es"
	+ using * by (auto split: pointer.splits entry.splits)
	+ have 0: "distinct (items es)"
	+ using Cons.prems(1) unfolding items_def by simp
	+ have 1: "i-1 < length es"
	+ using Cons.prems(2,3) * by (metis One_nat_def leI less_diff_conv2 less_one list.size(4) nth_Cons_0)
	+ have 2: "item e = item (es!(i-1))"
	+ using Cons.prems(3) * by (metis nth_Cons')
	+ have 3: "pointer e = PreRed p rs"
	+ using Cons.prems(4) by simp
	+ have 4: "pointer (es!(i-1)) = PreRed p' rs' "
	+ using Cons.prems(3,5) * by (metis nth_Cons')
	+ have "pointer (bin_upd e es!(i-1)) = PreRed p' (p # rs @ rs') \<and>
	+ (\<forall>j < length (bin_upd e es). i-1 \<noteq> j \<longrightarrow> (bin_upd e es) ! j = es ! j)"
	+ using Cons.IH[OF 0 1 2 3 4] by blast
	+ hence "pointer ((e' # bin_upd e es) ! i) = PreRed p' (p # rs @ rs') \<and>
	+ (\<forall>j < length (e' # bin_upd e es). i \<noteq> j \<longrightarrow> (e' # bin_upd e es) ! j = (e' # es) ! j)"
	+ using * Cons.prems(2,3) less_Suc_eq_0_disj by auto
	+ moreover have "e' # bin_upd e es = es'"
	+ using Cons.prems(6) simp by auto
	+ ultimately show ?thesis
	+ by (metis 0 1 2 3 4 Cons.IH Cons.prems(6) length_Cons)
	+ qed
	+qed simp
	+
	+lemma sound_ptrs_bin_upd:
	+ assumes "sound_ptrs \<omega> bs" "k < length bs" "es = bs!k" "distinct (items es)"
	+ assumes "sound_null_ptr e" "sound_pre_ptr \<omega> bs k e" "sound_prered_ptr bs k e"
	+ shows "sound_ptrs \<omega> (bs[k := bin_upd e es])"
	+ unfolding sound_ptrs_def
	+proof (standard, standard, standard)
	+ fix idx elem
	+ let ?bs = "bs[k := bin_upd e es]"
	+ assume a0: "idx < length ?bs"
	+ assume a1: "elem \<in> set (?bs ! idx)"
	+ show "sound_null_ptr elem \<and> sound_pre_ptr \<omega> ?bs idx elem \<and> sound_prered_ptr ?bs idx elem"
	+ proof cases
	+ assume a2: "idx = k"
	+ have "elem \<in> set es \<Longrightarrow> sound_pre_ptr \<omega> bs idx elem"
	+ using a0 a2 assms(1-3) sound_ptrs_def by blast
	+ hence pre_es: "elem \<in> set es \<Longrightarrow> sound_pre_ptr \<omega> ?bs idx elem"
	+ using a2 unfolding sound_pre_ptr_def by force
	+ have "elem = e \<Longrightarrow> sound_pre_ptr \<omega> bs idx elem"
	+ using a2 assms(6) by auto
	+ hence pre_e: "elem = e \<Longrightarrow> sound_pre_ptr \<omega> ?bs idx elem"
	+ using a2 unfolding sound_pre_ptr_def by force
	+ have "elem \<in> set es \<Longrightarrow> sound_prered_ptr bs idx elem"
	+ using a0 a2 assms(1-3) sound_ptrs_def by blast
	+ hence prered_es: "elem \<in> set es \<Longrightarrow> sound_prered_ptr (bs[k := bin_upd e es]) idx elem"
	+ using a2 assms(2,3) length_bin_upd nth_item_bin_upd unfolding sound_prered_ptr_def
	+ by (smt (verit, ccfv_SIG) dual_order.strict_trans1 nth_list_update)
	+ have "elem = e \<Longrightarrow> sound_prered_ptr bs idx elem"
	+ using a2 assms(7) by auto
	+ hence prered_e: "elem = e \<Longrightarrow> sound_prered_ptr ?bs idx elem"
	+ using a2 assms(2,3) length_bin_upd nth_item_bin_upd unfolding sound_prered_ptr_def
	+ by (smt (verit, best) dual_order.strict_trans1 nth_list_update)
	+ consider (A) "item e \<notin> set (items es)" \|
	+ (B) "item e \<in> set (items es) \<and> (\<exists>pre. pointer e = Null \<or> pointer e = Pre pre)" \|
	+ (C) "item e \<in> set (items es) \<and> \<not> (\<exists>pre. pointer e = Null \<or> pointer e = Pre pre)"
	+ by blast
	+ thus ?thesis
	+ proof cases
	+ case A
	+ hence "elem \<in> set (es @ [e])"
	+ using a1 a2 bin_upd_append assms(2) by force
	+ thus ?thesis
	+ using assms(1-3,5) pre_e pre_es prered_e prered_es sound_ptrs_def by auto
	+ next
	+ case B
	+ hence "elem \<in> set es"
	+ using a1 a2 bin_upd_null_pre assms(2) by force
	+ thus ?thesis
	+ using assms(1-3) pre_es prered_es sound_ptrs_def by blast
	+ next
	+ case C
	+ then obtain i p ps where C: "i < length es \<and> item e = item (es!i) \<and> pointer e = PreRed p ps"
	+ by (metis assms(4) distinct_Ex1 items_def length_map nth_map pointer.exhaust)
	+ show ?thesis
	+ proof cases
	+ assume "\<nexists>p' ps'. pointer (es!i) = PreRed p' ps'"
	+ hence C: "elem \<in> set es"
	+ using a1 a2 C bin_upd_prered_nop assms(2,4) by (metis nth_list_update_eq)
	+ thus ?thesis
	+ using assms(1-3) sound_ptrs_def pre_es prered_es by blast
	+ next
	+ assume "\<not> (\<nexists>p' ps'. pointer (es!i) = PreRed p' ps')"
	+ then obtain p' ps' where D: "pointer (es!i) = PreRed p' ps'"
	+ by blast
	+ hence 0: "pointer (bin_upd e es!i) = PreRed p' (p#ps@ps') \<and> (\<forall>j < length (bin_upd e es). i\<noteq>j \<longrightarrow> bin_upd e es!j = es!j)"
	+ using C assms(4) bin_upd_prered_upd by blast
	+ obtain j where 1: "j < length es \<and> elem = bin_upd e es!j"
	+ using a1 a2 assms(2) C items_def bin_eq_items_bin_upd by (metis in_set_conv_nth length_map nth_list_update_eq nth_map)
	+ show ?thesis
	+ proof cases
	+ assume a3: "i=j"
	+ hence a3: "pointer elem = PreRed p' (p#ps@ps')"
	+ using 0 1 by blast
	+ have "sound_null_ptr elem"
	+ using a3 unfolding sound_null_ptr_def by simp
	+ moreover have "sound_pre_ptr \<omega> ?bs idx elem"
	+ using a3 unfolding sound_pre_ptr_def by simp
	+ moreover have "sound_prered_ptr ?bs idx elem"
	+ unfolding sound_prered_ptr_def
	+ proof (standard, standard, standard, standard, standard, standard)
	+ fix P PS k' pre red
	+ assume a4: "pointer elem = PreRed P PS \<and> (k', pre, red) \<in> set (P#PS)"
	+ show "k' < idx \<and> pre < length (bs[k := bin_upd e es]!k') \<and> red < length (bs[k := bin_upd e es]!idx) \<and>
	+ completes idx (item (bs[k := bin_upd e es]!k'!pre)) (item elem) (item (bs[k := bin_upd e es]!idx!red))"
	+ proof cases
	+ assume a5: "(k', pre, red) \<in> set (p#ps)"
	+ show ?thesis
	+ using 0 1 C a2 a4 a5 prered_es assms(2,3,7) sound_prered_ptr_def length_bin_upd nth_item_bin_upd
	+ by (smt (verit) dual_order.strict_trans1 nth_list_update_eq nth_list_update_neq nth_mem)
	+ next
	+ assume a5: "(k', pre, red) \<notin> set (p#ps)"
	+ hence a5: "(k', pre, red) \<in> set (p'#ps')"
	+ using a3 a4 by auto
	+ have "k' < idx \<and> pre < length (bs!k') \<and> red < length (bs!idx) \<and>
	+ completes idx (item (bs!k'!pre)) (item e) (item (bs!idx!red))"
	+ using assms(1-3) C D a2 a5 unfolding sound_ptrs_def sound_prered_ptr_def by (metis nth_mem)
	+ thus ?thesis
	+ using 0 1 C a4 assms(2,3) length_bin_upd nth_item_bin_upd prered_es sound_prered_ptr_def
	+ by (smt (verit, best) dual_order.strict_trans1 nth_list_update_eq nth_list_update_neq nth_mem)
	+ qed
	+ qed
	+ ultimately show ?thesis
	+ by blast
	+ next
	+ assume a3: "i\<noteq>j"
	+ hence "elem \<in> set es"
	+ using 0 1 by (metis length_bin_upd nth_mem order_less_le_trans)
	+ thus ?thesis
	+ using assms(1-3) pre_es prered_es sound_ptrs_def by blast
	+ qed
	+ qed
	+ qed
	+ next
	+ assume a2: "idx \<noteq> k"
	+ have null: "sound_null_ptr elem"
	+ using a0 a1 a2 assms(1) sound_ptrs_def by auto
	+ have "sound_pre_ptr \<omega> bs idx elem"
	+ using a0 a1 a2 assms(1,2) unfolding sound_ptrs_def by simp
	+ hence pre: "sound_pre_ptr \<omega> ?bs idx elem"
	+ using assms(2,3) length_bin_upd nth_item_bin_upd unfolding sound_pre_ptr_def
	+ using dual_order.strict_trans1 nth_list_update by fastforce
	+ have "sound_prered_ptr bs idx elem"
	+ using a0 a1 a2 assms(1,2) unfolding sound_ptrs_def by simp
	+ hence prered: "sound_prered_ptr ?bs idx elem"
	+ using assms(2,3) length_bin_upd nth_item_bin_upd unfolding sound_prered_ptr_def
	+ by (smt (verit, best) dual_order.strict_trans1 nth_list_update)
	+ show ?thesis
	+ using null pre prered by blast
	+ qed
	+qed
	+
	+lemma mono_red_ptr_bin_upd:
	+ assumes "mono_red_ptr bs" "k < length bs" "es = bs!k" "distinct (items es)"
	+ assumes "\<forall>k' pre red ps. pointer e = PreRed (k', pre, red) ps \<longrightarrow> red < length es"
	+ shows "mono_red_ptr (bs[k := bin_upd e es])"
	+ unfolding mono_red_ptr_def
	+proof (standard, standard)
	+ fix idx
	+ let ?bs = "bs[k := bin_upd e es]"
	+ assume a0: "idx < length ?bs"
	+ show "\<forall>i < length (?bs!idx). \<forall>k' pre red ps. pointer (?bs!idx!i) = PreRed (k', pre, red) ps \<longrightarrow> red < i"
	+ proof cases
	+ assume a1: "idx=k"
	+ consider (A) "item e \<notin> set (items es)" \|
	+ (B) "item e \<in> set (items es) \<and> (\<exists>pre. pointer e = Null \<or> pointer e = Pre pre)" \|
	+ (C) "item e \<in> set (items es) \<and> \<not> (\<exists>pre. pointer e = Null \<or> pointer e = Pre pre)"
	+ by blast
	+ thus ?thesis
	+ proof cases
	+ case A
	+ hence "bin_upd e es = es @ [e]"
	+ using bin_upd_append by blast
	+ thus ?thesis
	+ using a1 assms(1-3,5) mono_red_ptr_def
	+ by (metis length_append_singleton less_antisym nth_append nth_append_length nth_list_update_eq)
	+ next
	+ case B
	+ hence "bin_upd e es = es"
	+ using bin_upd_null_pre by blast
	+ thus ?thesis
	+ using a1 assms(1-3) mono_red_ptr_def by force
	+ next
	+ case C
	+ then obtain i p ps where C: "i < length es" "item e = item (es!i)" "pointer e = PreRed p ps"
	+ by (metis in_set_conv_nth items_def length_map nth_map pointer.exhaust)
	+ show ?thesis
	+ proof cases
	+ assume "\<nexists>p' ps'. pointer (es!i) = PreRed p' ps'"
	+ hence "bin_upd e es = es"
	+ using bin_upd_prered_nop C assms(4) by blast
	+ thus ?thesis
	+ using a1 assms(1-3) mono_red_ptr_def by (metis nth_list_update_eq)
	+ next
	+ assume "\<not> (\<nexists>p' ps'. pointer (es!i) = PreRed p' ps')"
	+ then obtain p' ps' where D: "pointer (es!i) = PreRed p' ps'"
	+ by blast
	+ have 0: "pointer (bin_upd e es!i) = PreRed p' (p#ps@ps') \<and>
	+ (\<forall>j < length (bin_upd e es). i \<noteq> j \<longrightarrow> bin_upd e es!j = es!j) \<and>
	+ length (bin_upd e es) = length es"
	+ using C D assms(4) bin_upd_prered_upd by blast
	+ show ?thesis
	+ proof (standard, standard, standard, standard, standard, standard, standard)
	+ fix j k' pre red PS
	+ assume a2: "j < length (?bs!idx)"
	+ assume a3: "pointer (?bs!idx!j) = PreRed (k', pre, red) PS"
	+ have 1: "?bs!idx = bin_upd e es"
	+ by (simp add: a1 assms(2))
	+ show "red < j"
	+ proof cases
	+ assume a4: "i=j"
	+ show ?thesis
	+ using 0 1 C(1) D a3 a4 assms(1-3) unfolding mono_red_ptr_def by (metis pointer.inject(2))
	+ next
	+ assume a4: "i\<noteq>j"
	+ thus ?thesis
	+ using 0 1 a2 a3 assms(1) assms(2) assms(3) mono_red_ptr_def by force
	+ qed
	+ qed
	+ qed
	+ qed
	+ next
	+ assume a1: "idx\<noteq>k"
	+ show ?thesis
	+ using a0 a1 assms(1) mono_red_ptr_def by fastforce
	+ qed
	+qed
	+
	+lemma sound_mono_ptrs_bin_upds:
	+ assumes "sound_ptrs \<omega> bs" "mono_red_ptr bs" "k < length bs" "b = bs!k" "distinct (items b)" "distinct (items es)"
	+ assumes "\<forall>e \<in> set es. sound_null_ptr e \<and> sound_pre_ptr \<omega> bs k e \<and> sound_prered_ptr bs k e"
	+ assumes "\<forall>e \<in> set es. \<forall>k' pre red ps. pointer e = PreRed (k', pre, red) ps \<longrightarrow> red < length b"
	+ shows "sound_ptrs \<omega> (bs[k := bin_upds es b]) \<and> mono_red_ptr (bs[k := bin_upds es b])"
	+ using assms
	+proof (induction es arbitrary: b bs)
	+ case (Cons e es)
	+ let ?bs = "bs[k := bin_upd e b]"
	+ have 0: "sound_ptrs \<omega> ?bs"
	+ using sound_ptrs_bin_upd Cons.prems(1,3-5,7) by (metis list.set_intros(1))
	+ have 1: "mono_red_ptr ?bs"
	+ using mono_red_ptr_bin_upd Cons.prems(2-5,8) by auto
	+ have 2: "k < length ?bs"
	+ using Cons.prems(3) by simp
	+ have 3: "bin_upd e b = ?bs!k"
	+ using Cons.prems(3) by simp
	+ have 4: "\<forall>e' \<in> set es. sound_null_ptr e' \<and> sound_pre_ptr \<omega> ?bs k e' \<and> sound_prered_ptr ?bs k e'"
	+ using Cons.prems(3,4,7) length_bin_upd nth_item_bin_upd sound_pre_ptr_def sound_prered_ptr_def
	+ by (smt (verit, ccfv_threshold) list.set_intros(2) nth_list_update order_less_le_trans)
	+ have 5: "\<forall>e' \<in> set es. \<forall>k' pre red ps. pointer e' = PreRed (k', pre, red) ps \<longrightarrow> red < length (bin_upd e b)"
	+ by (meson Cons.prems(8) length_bin_upd order_less_le_trans set_subset_Cons subsetD)
	+ have "sound_ptrs \<omega> ((bs[k := bin_upd e b])[k := bin_upds es (bin_upd e b)]) \<and>
	+ mono_red_ptr (bs[k := bin_upd e b, k := bin_upds es (bin_upd e b)])"
	+ using Cons.IH[OF 0 1 2 3 _ _ 4 5] distinct_bin_upd Cons.prems(4,5,6) items_def by (metis distinct.simps(2) list.simps(9))
	+ thus ?case
	+ by simp
	+qed simp
	+
	+lemma sound_mono_ptrs_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "sound_ptrs \<omega> bs" "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ assumes "mono_red_ptr bs"
	+ assumes "nonempty_derives \<G>" "wf_\<G> \<G>"
	+ shows "sound_ptrs \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<and> mono_red_ptr (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Complete\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items (Complete\<^sub>L k x bs i)). sound_item \<G> \<omega> x"
	+ using sound_Complete\<^sub>L Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items x
	+ by (metis dual_order.refl)
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ by (metis Complete\<^sub>F.prems(1,3) UnE bins_bins_upd wf_earley_input_elim)
	+ have 0: "k < length bs"
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim by auto
	+ have 1: "\<forall>e \<in> set (Complete\<^sub>L k x bs i). sound_null_ptr e"
	+ unfolding Complete\<^sub>L_def sound_null_ptr_def by auto
	+ have 2: "\<forall>e \<in> set (Complete\<^sub>L k x bs i). sound_pre_ptr \<omega> bs k e"
	+ unfolding Complete\<^sub>L_def sound_pre_ptr_def by auto
	+ {
	+ fix e
	+ assume a0: "e \<in> set (Complete\<^sub>L k x bs i)"
	+ fix p ps k' pre red
	+ assume a1: "pointer e = PreRed p ps" "(k', pre, red) \<in> set (p#ps)"
	+ have "k' = item_origin x"
	+ using a0 a1 unfolding Complete\<^sub>L_def by auto
	+ moreover have "wf_item \<G> \<omega> x" "item_end x = k"
	+ using Complete\<^sub>F.prems(1) x wf_earley_input_elim wf_bins_kth_bin by blast+
	+ ultimately have 0: "k' \<le> k"
	+ using wf_item_def by blast
	+ have 1: "k' \<noteq> k"
	+ proof (rule ccontr)
	+ assume "\<not> k' \<noteq> k"
	+ have "sound_item \<G> \<omega> x"
	+ using Complete\<^sub>F.prems(1,3) x kth_bin_sub_bins wf_earley_input_elim by (metis subset_eq)
	+ moreover have "is_complete x"
	+ using Complete\<^sub>F.hyps(3) by (auto simp: next_symbol_def split: if_splits)
	+ moreover have "item_origin x = k"
	+ using \<open>\<not> k' \<noteq> k\<close> \<open>k' = item_origin x\<close> by auto
	+ ultimately show False
	+ using impossible_complete_item Complete\<^sub>F.prems(1,5) wf_earley_input_elim \<open>item_end x = k\<close> \<open>wf_item \<G> \<omega> x\<close> by blast
	+ qed
	+ have 2: "pre < length (bs!k')"
	+ using a0 a1 index_filter_with_index_lt_length unfolding Complete\<^sub>L_def by (auto simp: items_def; fastforce)
	+ have 3: "red < i+1"
	+ using a0 a1 unfolding Complete\<^sub>L_def by auto
	+
	+ have "item e = inc_item (item (bs!k'!pre)) k"
	+ using a0 a1 0 2 Complete\<^sub>F.hyps(1,2,3) Complete\<^sub>F.prems(1) \<open>k' = item_origin x\<close> unfolding Complete\<^sub>L_def
	+ by (auto simp: items_def, metis filter_with_index_nth nth_map)
	+ moreover have "is_complete (item (bs!k!red))"
	+ using a0 a1 0 2 Complete\<^sub>F.hyps(1,2,3) Complete\<^sub>F.prems(1) \<open>k' = item_origin x\<close> unfolding Complete\<^sub>L_def
	+ by (auto simp: next_symbol_def items_def split: if_splits)
	+ moreover have "item_origin (item (bs!k!red)) = item_end (item (bs!k'!pre))"
	+ using a0 a1 0 2 Complete\<^sub>F.hyps(1,2,3) Complete\<^sub>F.prems(1) \<open>k' = item_origin x\<close> unfolding Complete\<^sub>L_def
	+ apply (clarsimp simp: items_def)
	+ by (metis dual_order.strict_trans index_filter_with_index_lt_length items_def le_neq_implies_less nth_map nth_mem wf_bins_kth_bin wf_earley_input_elim)
	+ moreover have "(\<exists>N. next_symbol (item (bs ! k' ! pre)) = Some N \<and> N = item_rule_head (item (bs ! k ! red)))"
	+ using a0 a1 0 2 Complete\<^sub>F.hyps(1,2,3) Complete\<^sub>F.prems(1) \<open>k' = item_origin x\<close> unfolding Complete\<^sub>L_def
	+ by (auto simp: items_def, metis (mono_tags, lifting) filter_with_index_P filter_with_index_nth nth_map)
	+ ultimately have 4: "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ unfolding completes_def by blast
	+ have "k' < k" "pre < length (bs!k')" "red < i+1" "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ using 0 1 2 3 4 by simp_all
	+ }
	+ hence "\<forall>e \<in> set (Complete\<^sub>L k x bs i). \<forall>p ps k' pre red. pointer e = PreRed p ps \<and> (k', pre, red) \<in> set (p#ps) \<longrightarrow>
	+ k' < k \<and> pre < length (bs!k') \<and> red < i+1 \<and> completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ by force
	+ hence 3: "\<forall>e \<in> set (Complete\<^sub>L k x bs i). sound_prered_ptr bs k e"
	+ unfolding sound_prered_ptr_def using Complete\<^sub>F.hyps(1) items_def by (smt (verit) discrete dual_order.strict_trans1 leI length_map)
	+ have 4: "distinct (items (Complete\<^sub>L k x bs i))"
	+ using distinct_Complete\<^sub>L x Complete\<^sub>F.prems(1) wf_earley_input_elim wf_bin_def wf_bin_items_def wf_bins_def wf_item_def
	+ by (metis order_le_less_trans)
	+ have "sound_ptrs \<omega> ?bs' \<and> mono_red_ptr ?bs'"
	+ using sound_mono_ptrs_bin_upds[OF Complete\<^sub>F.prems(2) Complete\<^sub>F.prems(4) 0] 1 2 3 4 sound_prered_ptr_def
	+ Complete\<^sub>F.prems(1) bins_upd_def wf_earley_input_elim wf_bin_def wf_bins_def
	+ by (smt (verit, ccfv_SIG) list.set_intros(1))
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ ultimately have "sound_ptrs \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) \<and> mono_red_ptr (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Complete\<^sub>F.IH Complete\<^sub>F.prems(4-6) sound by blast
	+ thus ?case
	+ using Complete\<^sub>F.hyps by simp
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Scan\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items (Scan\<^sub>L k \<omega> a x i)). sound_item \<G> \<omega> x"
	+ using sound_Scan\<^sub>L Scan\<^sub>F.hyps(3,5) Scan\<^sub>F.prems(1,2,3) wf_earley_input_elim wf_bins_impl_wf_items wf_bins_impl_wf_items by fast
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1,3) bins_bins_upd wf_earley_input_elim
	+ by (metis UnE add_less_cancel_right)
	+ have 0: "k+1 < length bs"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1) wf_earley_input_elim by force
	+ have 1: "\<forall>e \<in> set (Scan\<^sub>L k \<omega> a x i). sound_null_ptr e"
	+ unfolding Scan\<^sub>L_def sound_null_ptr_def by auto
	+ have 2: "\<forall>e \<in> set (Scan\<^sub>L k \<omega> a x i). sound_pre_ptr \<omega> bs (k+1) e"
	+ using Scan\<^sub>F.hyps(1,2,3) unfolding sound_pre_ptr_def Scan\<^sub>L_def scans_def items_def by auto
	+ have 3: "\<forall>e \<in> set (Scan\<^sub>L k \<omega> a x i). sound_prered_ptr bs (k+1) e"
	+ unfolding Scan\<^sub>L_def sound_prered_ptr_def by simp
	+ have 4: "distinct (items (Scan\<^sub>L k \<omega> a x i))"
	+ using distinct_Scan\<^sub>L by fast
	+ have "sound_ptrs \<omega> ?bs' \<and> mono_red_ptr ?bs'"
	+ using sound_mono_ptrs_bin_upds[OF Scan\<^sub>F.prems(2) Scan\<^sub>F.prems(4) 0] 0 1 2 3 4 sound_prered_ptr_def
	+ Scan\<^sub>F.prems(1) bins_upd_def wf_earley_input_elim wf_bin_def wf_bins_def
	+ by (smt (verit, ccfv_threshold) list.set_intros(1))
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ ultimately have "sound_ptrs \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) \<and> mono_red_ptr (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Scan\<^sub>F.IH Scan\<^sub>F.prems(4-6) sound by blast
	+ thus ?case
	+ using Scan\<^sub>F.hyps by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Predict\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items(Predict\<^sub>L k \<G> a)). sound_item \<G> \<omega> x"
	+ using sound_Predict\<^sub>L Predict\<^sub>F.hyps(3) Predict\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items by fast
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ using Predict\<^sub>F.prems(1,3) UnE bins_bins_upd wf_earley_input_elim by metis
	+ have 0: "k < length bs"
	+ using Predict\<^sub>F.prems(1) wf_earley_input_elim by force
	+ have 1: "\<forall>e \<in> set (Predict\<^sub>L k \<G> a). sound_null_ptr e"
	+ unfolding sound_null_ptr_def Predict\<^sub>L_def predicts_def by (auto simp: init_item_def)
	+ have 2: "\<forall>e \<in> set (Predict\<^sub>L k \<G> a). sound_pre_ptr \<omega> bs k e"
	+ unfolding sound_pre_ptr_def Predict\<^sub>L_def by simp
	+ have 3: "\<forall>e \<in> set (Predict\<^sub>L k \<G> a). sound_prered_ptr bs k e"
	+ unfolding sound_prered_ptr_def Predict\<^sub>L_def by simp
	+ have 4: "distinct (items (Predict\<^sub>L k \<G> a))"
	+ using Predict\<^sub>F.prems(6) distinct_Predict\<^sub>L by fast
	+ have "sound_ptrs \<omega> ?bs' \<and> mono_red_ptr ?bs'"
	+ using sound_mono_ptrs_bin_upds[OF Predict\<^sub>F.prems(2) Predict\<^sub>F.prems(4) 0] 0 1 2 3 4 sound_prered_ptr_def
	+ Predict\<^sub>F.prems(1) bins_upd_def wf_earley_input_elim wf_bin_def wf_bins_def
	+ by (smt (verit, ccfv_threshold) list.set_intros(1))
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ ultimately have "sound_ptrs \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) \<and> mono_red_ptr (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Predict\<^sub>F.IH Predict\<^sub>F.prems(4-6) sound by blast
	+ thus ?case
	+ using Predict\<^sub>F.hyps by simp
	+qed simp_all
	+
	+lemma sound_mono_ptrs_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "sound_ptrs \<omega> bs" "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ assumes "mono_red_ptr bs"
	+ assumes "nonempty_derives \<G>" "wf_\<G> \<G>"
	+ shows "sound_ptrs \<omega> (Earley\<^sub>L_bin k \<G> \<omega> bs) \<and> mono_red_ptr (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms sound_mono_ptrs_Earley\<^sub>L_bin' Earley\<^sub>L_bin_def by metis
	+
	+lemma sound_ptrs_Init\<^sub>L:
	+ "sound_ptrs \<omega> (Init\<^sub>L \<G> \<omega>)"
	+ unfolding sound_ptrs_def sound_null_ptr_def sound_pre_ptr_def sound_prered_ptr_def
	+ predicts_def scans_def completes_def Init\<^sub>L_def
	+ by (auto simp: init_item_def less_Suc_eq_0_disj)
	+
	+lemma mono_red_ptr_Init\<^sub>L:
	+ "mono_red_ptr (Init\<^sub>L \<G> \<omega>)"
	+ unfolding mono_red_ptr_def Init\<^sub>L_def
	+ by (auto simp: init_item_def less_Suc_eq_0_disj)
	+
	+lemma sound_mono_ptrs_Earley\<^sub>L_bins:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>" "nonempty_derives \<G>" "wf_\<G> \<G>"
	+ shows "sound_ptrs \<omega> (Earley\<^sub>L_bins k \<G> \<omega>) \<and> mono_red_ptr (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ using assms
	+proof (induction k)
	+ case 0
	+ have "(0, \<G>, \<omega>, (Init\<^sub>L \<G> \<omega>)) \<in> wf_earley_input"
	+ using assms(2) wf_earley_input_Init\<^sub>L by blast
	+ moreover have "\<forall>x \<in> bins (Init\<^sub>L \<G> \<omega>). sound_item \<G> \<omega> x"
	+ by (metis Init\<^sub>L_eq_Init\<^sub>F Init\<^sub>F_sub_Earley sound_Earley subsetD wf_Earley)
	+ ultimately show ?case
	+ using sound_mono_ptrs_Earley\<^sub>L_bin sound_ptrs_Init\<^sub>L mono_red_ptr_Init\<^sub>L "0.prems"(2,3) by fastforce
	+next
	+ case (Suc k)
	+ have "(Suc k, \<G>, \<omega>, Earley\<^sub>L_bins k \<G> \<omega>) \<in> wf_earley_input"
	+ by (simp add: Suc.prems(1) Suc_leD assms(2) wf_earley_input_intro)
	+ moreover have "sound_ptrs \<omega> (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ using Suc by simp
	+ moreover have "\<forall>x \<in> bins (Earley\<^sub>L_bins k \<G> \<omega>). sound_item \<G> \<omega> x"
	+ by (meson Suc.prems(1) Suc_leD Earley\<^sub>L_bins_sub_Earley\<^sub>F_bins Earley\<^sub>F_bins_sub_Earley assms(2)
	+ sound_Earley subsetD wf_bins_Earley\<^sub>L_bins wf_bins_impl_wf_items)
	+ ultimately show ?case
	+ using Suc.prems(1,3,4) sound_mono_ptrs_Earley\<^sub>L_bin Suc.IH by fastforce
	+qed
	+
	+lemma sound_mono_ptrs_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "nonempty_derives \<G>"
	+ shows "sound_ptrs \<omega> (Earley\<^sub>L \<G> \<omega>) \<and> mono_red_ptr (Earley\<^sub>L \<G> \<omega>)"
	+ using assms sound_mono_ptrs_Earley\<^sub>L_bins Earley\<^sub>L_def by (metis dual_order.refl)
	+
	+
	+subsection \<open>Common Definitions\<close>
	+
	+datatype 'a tree =
	+ Leaf 'a
	+ \| Branch 'a "'a tree list"
	+
	+fun yield_tree :: "'a tree \<Rightarrow> 'a sentence" where
	+ "yield_tree (Leaf a) = [a]"
	+\| "yield_tree (Branch _ ts) = concat (map yield_tree ts)"
	+
	+fun root_tree :: "'a tree \<Rightarrow> 'a" where
	+ "root_tree (Leaf a) = a"
	+\| "root_tree (Branch N _) = N"
	+
	+fun wf_rule_tree :: "'a cfg \<Rightarrow> 'a tree \<Rightarrow> bool" where
	+ "wf_rule_tree _ (Leaf a) \<longleftrightarrow> True"
	+\| "wf_rule_tree \<G> (Branch N ts) \<longleftrightarrow> (
	+ (\<exists>r \<in> set (\<RR> \<G>). N = rule_head r \<and> map root_tree ts = rule_body r) \<and>
	+ (\<forall>t \<in> set ts. wf_rule_tree \<G> t))"
	+
	+fun wf_item_tree :: "'a cfg \<Rightarrow> 'a item \<Rightarrow> 'a tree \<Rightarrow> bool" where
	+ "wf_item_tree \<G> _ (Leaf a) \<longleftrightarrow> True"
	+\| "wf_item_tree \<G> x (Branch N ts) \<longleftrightarrow> (
	+ N = item_rule_head x \<and> map root_tree ts = take (item_dot x) (item_rule_body x) \<and>
	+ (\<forall>t \<in> set ts. wf_rule_tree \<G> t))"
	+
	+definition wf_yield_tree :: "'a sentence \<Rightarrow> 'a item \<Rightarrow> 'a tree \<Rightarrow> bool" where
	+ "wf_yield_tree \<omega> x t \<longleftrightarrow> yield_tree t = slice (item_origin x) (item_end x) \<omega>"
	+
	+datatype 'a forest =
	+ FLeaf 'a
	+ \| FBranch 'a "'a forest list list"
	+
	+fun combinations :: "'a list list \<Rightarrow> 'a list list" where
	+ "combinations [] = [[]]"
	+\| "combinations (xs#xss) = [ x#cs . x <- xs, cs <- combinations xss ]"
	+
	+fun trees :: "'a forest \<Rightarrow> 'a tree list" where
	+ "trees (FLeaf a) = [Leaf a]"
	+\| "trees (FBranch N fss) = (
	+ let tss = (map (\<lambda>fs. concat (map (\<lambda>f. trees f) fs)) fss) in
	+ map (\<lambda>ts. Branch N ts) (combinations tss)
	+ )"
	+
	+lemma list_comp_flatten:
	+ "[ f xs . xs <- [ g xs ys . xs <- as, ys <- bs ] ] = [ f (g xs ys) . xs <- as, ys <- bs ]"
	+ by (induction as) auto
	+
	+lemma list_comp_flatten_Cons:
	+ "[ x#xs . x <- as, xs <- [ xs @ ys. xs <- bs, ys <- cs ] ] = [ x#xs@ys. x <- as, xs <- bs, ys <- cs ]"
	+ by (induction as) (auto simp: list_comp_flatten)
	+
	+lemma list_comp_flatten_append:
	+ "[ xs@ys . xs <- [ x#xs . x <- as, xs <- bs ], ys <- cs ] = [ x#xs@ys . x <- as, xs <- bs, ys <- cs ]"
	+ by (induction as) (auto simp: o_def, meson append_Cons map_eq_conv)
	+
	+lemma combinations_append:
	+ "combinations (xss @ yss) = [ xs @ ys . xs <- combinations xss, ys <- combinations yss ]"
	+ by (induction xss) (auto simp: list_comp_flatten_Cons list_comp_flatten_append map_idI)
	+
	+lemma trees_append:
	+ "trees (FBranch N (xss @ yss)) = (
	+ let xtss = (map (\<lambda>xs. concat (map (\<lambda>f. trees f) xs)) xss) in
	+ let ytss = (map (\<lambda>ys. concat (map (\<lambda>f. trees f) ys)) yss) in
	+ map (\<lambda>ts. Branch N ts) [ xs @ ys . xs <- combinations xtss, ys <- combinations ytss ])"
	+ using combinations_append by (metis map_append trees.simps(2))
	+
	+lemma trees_append_singleton:
	+ "trees (FBranch N (xss @ [ys])) = (
	+ let xtss = (map (\<lambda>xs. concat (map (\<lambda>f. trees f) xs)) xss) in
	+ let ytss = [concat (map trees ys)] in
	+ map (\<lambda>ts. Branch N ts) [ xs @ ys . xs <- combinations xtss, ys <- combinations ytss ])"
	+ by (subst trees_append, simp)
	+
	+lemma trees_append_single_singleton:
	+ "trees (FBranch N (xss @ [[y]])) = (
	+ let xtss = (map (\<lambda>xs. concat (map (\<lambda>f. trees f) xs)) xss) in
	+ map (\<lambda>ts. Branch N ts) [ xs @ ys . xs <- combinations xtss, ys <- [ [t] . t <- trees y ] ])"
	+ by (subst trees_append_singleton, auto)
	+
	+
	+subsection \<open>foldl lemmas\<close>
	+
	+lemma foldl_add_nth:
	+ "k < length xs \<Longrightarrow> foldl (+) z (map length (take k xs)) + length (xs!k) = foldl (+) z (map length (take (k+1) xs))"
	+proof (induction xs arbitrary: k z)
	+ case (Cons x xs)
	+ then show ?case
	+ proof (cases "k = 0")
	+ case False
	+ thus ?thesis
	+ using Cons by (auto simp add: take_Cons')
	+ qed simp
	+qed simp
	+
	+lemma foldl_acc_mono:
	+ "a \<le> b \<Longrightarrow> foldl (+) a xs \<le> foldl (+) b xs" for a :: nat
	+ by (induction xs arbitrary: a b) auto
	+
	+lemma foldl_ge_z_nth:
	+ "j < length xs \<Longrightarrow> z + length (xs!j) \<le> foldl (+) z (map length (take (j+1) xs))"
	+proof (induction xs arbitrary: j z)
	+ case (Cons x xs)
	+ show ?case
	+ proof (cases "j = 0")
	+ case False
	+ have "z + length ((x # xs) ! j) = z + length (xs!(j-1))"
	+ using False by simp
	+ also have "... \<le> foldl (+) z (map length (take (j-1+1) xs))"
	+ using Cons False by (metis add_diff_inverse_nat length_Cons less_one nat_add_left_cancel_less plus_1_eq_Suc)
	+ also have "... = foldl (+) z (map length (take j xs))"
	+ using False by simp
	+ also have "... \<le> foldl (+) (z + length x) (map length (take j xs))"
	+ using foldl_acc_mono by force
	+ also have "... = foldl (+) z (map length (take (j+1) (x#xs)))"
	+ by simp
	+ finally show ?thesis
	+ by blast
	+ qed simp
	+qed simp
	+
	+lemma foldl_add_nth_ge:
	+ "i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> foldl (+) z (map length (take i xs)) + length (xs!j) \<le> foldl (+) z (map length (take (j+1) xs))"
	+proof (induction xs arbitrary: i j z)
	+ case (Cons x xs)
	+ show ?case
	+ proof (cases "i = 0")
	+ case True
	+ have "foldl (+) z (map length (take i (x # xs))) + length ((x # xs) ! j) = z + length ((x # xs) ! j)"
	+ using True by simp
	+ also have "... \<le> foldl (+) z (map length (take (j+1) (x#xs)))"
	+ using foldl_ge_z_nth Cons.prems(2) by blast
	+ finally show ?thesis
	+ by blast
	+ next
	+ case False
	+ have "i-1 \<le> j-1"
	+ by (simp add: Cons.prems(1) diff_le_mono)
	+ have "j-1 < length xs"
	+ using Cons.prems(1,2) False by fastforce
	+ have "foldl (+) z (map length (take i (x # xs))) + length ((x # xs) ! j) =
	+ foldl (+) (z + length x) (map length (take (i-1) xs)) + length ((x#xs)!j)"
	+ using False by (simp add: take_Cons')
	+ also have "... = foldl (+) (z + length x) (map length (take (i-1) xs)) + length (xs!(j-1))"
	+ using Cons.prems(1) False by auto
	+ also have "... \<le> foldl (+) (z + length x) (map length (take (j-1+1) xs))"
	+ using Cons.IH \<open>i - 1 \<le> j - 1\<close> \<open>j - 1 < length xs\<close> by blast
	+ also have "... = foldl (+) (z + length x) (map length (take j xs))"
	+ using Cons.prems(1) False by fastforce
	+ also have "... = foldl (+) z (map length (take (j+1) (x#xs)))"
	+ by fastforce
	+ finally show ?thesis
	+ by blast
	+ qed
	+qed simp
	+
	+lemma foldl_ge_acc:
	+ "foldl (+) z (map length xs) \<ge> z"
	+ by (induction xs arbitrary: z) (auto elim: add_leE)
	+
	+lemma foldl_take_mono:
	+ "i \<le> j \<Longrightarrow> foldl (+) z (map length (take i xs)) \<le> foldl (+) z (map length (take j xs))"
	+proof (induction xs arbitrary: z i j)
	+ case (Cons x xs)
	+ show ?case
	+ proof (cases "i = 0")
	+ case True
	+ have "foldl (+) z (map length (take i (x # xs))) = z"
	+ using True by simp
	+ also have "... \<le> foldl (+) z (map length (take j (x # xs)))"
	+ by (simp add: foldl_ge_acc)
	+ ultimately show ?thesis
	+ by simp
	+ next
	+ case False
	+ then show ?thesis
	+ using Cons by (simp add: take_Cons')
	+ qed
	+qed simp
	+
	+
	+subsection \<open>Parse tree\<close>
	+
	+partial_function (option) build_tree' :: "'a bins \<Rightarrow> 'a sentence \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a tree option" where
	+ "build_tree' bs \<omega> k i = (
	+ let e = bs!k!i in (
	+ case pointer e of
	+ Null \<Rightarrow> Some (Branch (item_rule_head (item e)) []) \<comment>\<open>start building sub-tree\<close>
	+ \| Pre pre \<Rightarrow> ( \<comment>\<open>add sub-tree starting from terminal\<close>
	+ do {
	+ t \<leftarrow> build_tree' bs \<omega> (k-1) pre;
	+ case t of
	+ Branch N ts \<Rightarrow> Some (Branch N (ts @ [Leaf (\<omega>!(k-1))]))
	+ \| _ \<Rightarrow> undefined \<comment>\<open>impossible case\<close>
	+ })
	+ \| PreRed (k', pre, red) _ \<Rightarrow> ( \<comment>\<open>add sub-tree starting from non-terminal\<close>
	+ do {
	+ t \<leftarrow> build_tree' bs \<omega> k' pre;
	+ case t of
	+ Branch N ts \<Rightarrow>
	+ do {
	+ t \<leftarrow> build_tree' bs \<omega> k red;
	+ Some (Branch N (ts @ [t]))
	+ }
	+ \| _ \<Rightarrow> undefined \<comment>\<open>impossible case\<close>
	+ })
	+ ))"
	+
	+declare build_tree'.simps [code]
	+
	+definition build_tree :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins \<Rightarrow> 'a tree option" where
	+ "build_tree \<G> \<omega> bs = (
	+ let k = length bs - 1 in (
	+ case filter_with_index (\<lambda>x. is_finished \<G> \<omega> x) (items (bs!k)) of
	+ [] \<Rightarrow> None
	+ \| (_, i)#_ \<Rightarrow> build_tree' bs \<omega> k i
	+ ))"
	+
	+lemma build_tree'_simps[simp]:
	+ "e = bs!k!i \<Longrightarrow> pointer e = Null \<Longrightarrow> build_tree' bs \<omega> k i = Some (Branch (item_rule_head (item e)) [])"
	+ "e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> build_tree' bs \<omega> (k-1) pre = None \<Longrightarrow>
	+ build_tree' bs \<omega> k i = None"
	+ "e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> build_tree' bs \<omega> (k-1) pre = Some (Branch N ts) \<Longrightarrow>
	+ build_tree' bs \<omega> k i = Some (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ "e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> build_tree' bs \<omega> (k-1) pre = Some (Leaf a) \<Longrightarrow>
	+ build_tree' bs \<omega> k i = undefined"
	+ "e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) reds \<Longrightarrow> build_tree' bs \<omega> k' pre = None \<Longrightarrow>
	+ build_tree' bs \<omega> k i = None"
	+ "e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) reds \<Longrightarrow> build_tree' bs \<omega> k' pre = Some (Branch N ts) \<Longrightarrow>
	+ build_tree' bs \<omega> k red = None \<Longrightarrow> build_tree' bs \<omega> k i = None"
	+ "e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) reds \<Longrightarrow> build_tree' bs \<omega> k' pre = Some (Leaf a) \<Longrightarrow>
	+ build_tree' bs \<omega> k i = undefined"
	+ "e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) reds \<Longrightarrow> build_tree' bs \<omega> k' pre = Some (Branch N ts) \<Longrightarrow>
	+ build_tree' bs \<omega> k red = Some t \<Longrightarrow>
	+ build_tree' bs \<omega> k i = Some (Branch N (ts @ [t]))"
	+ by (subst build_tree'.simps, simp)+
	+
	+definition wf_tree_input :: "('a bins \<times> 'a sentence \<times> nat \<times> nat) set" where
	+ "wf_tree_input = {
	+ (bs, \<omega>, k, i) \| bs \<omega> k i.
	+ sound_ptrs \<omega> bs \<and>
	+ mono_red_ptr bs \<and>
	+ k < length bs \<and>
	+ i < length (bs!k)
	+ }"
	+
	+fun build_tree'_measure :: "('a bins \<times> 'a sentence \<times> nat \<times> nat) \<Rightarrow> nat" where
	+ "build_tree'_measure (bs, \<omega>, k, i) = foldl (+) 0 (map length (take k bs)) + i"
	+
	+lemma wf_tree_input_pre:
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "e = bs!k!i" "pointer e = Pre pre"
	+ shows "(bs, \<omega>, (k-1), pre) \<in> wf_tree_input"
	+ using assms unfolding wf_tree_input_def
	+ using less_imp_diff_less nth_mem by (fastforce simp: sound_ptrs_def sound_pre_ptr_def)
	+
	+lemma wf_tree_input_prered_pre:
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "e = bs!k!i" "pointer e = PreRed (k', pre, red) ps"
	+ shows "(bs, \<omega>, k', pre) \<in> wf_tree_input"
	+ using assms unfolding wf_tree_input_def
	+ apply (auto simp: sound_ptrs_def sound_prered_ptr_def)
	+ apply metis+
	+ apply (metis dual_order.strict_trans nth_mem)
	+ by (metis nth_mem)
	+
	+lemma wf_tree_input_prered_red:
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "e = bs!k!i" "pointer e = PreRed (k', pre, red) ps"
	+ shows "(bs, \<omega>, k, red) \<in> wf_tree_input"
	+ using assms unfolding wf_tree_input_def
	+ apply (auto simp add: sound_ptrs_def sound_prered_ptr_def)
	+ apply (metis nth_mem)+
	+ done
	+
	+lemma build_tree'_induct:
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "\<And>bs \<omega> k i.
	+ (\<And>e pre. e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> P bs \<omega> (k-1) pre) \<Longrightarrow>
	+ (\<And>e k' pre red ps. e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) ps \<Longrightarrow> P bs \<omega> k' pre) \<Longrightarrow>
	+ (\<And>e k' pre red ps. e = bs!k!i \<Longrightarrow> pointer e = PreRed (k', pre, red) ps \<Longrightarrow> P bs \<omega> k red) \<Longrightarrow>
	+ P bs \<omega> k i"
	+ shows "P bs \<omega> k i"
	+ using assms(1)
	+proof (induction n\<equiv>"build_tree'_measure (bs, \<omega>, k, i)" arbitrary: k i rule: nat_less_induct)
	+ case 1
	+ obtain e where entry: "e = bs!k!i"
	+ by simp
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>k' pre red reds. pointer e = PreRed (k', pre, red) reds"
	+ by (metis pointer.exhaust surj_pair)
	+ thus ?case
	+ proof cases
	+ case Null
	+ thus ?thesis
	+ using assms(2) entry by fastforce
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ define n where n: "n = build_tree'_measure (bs, \<omega>, (k-1), pre)"
	+ have "0 < k" "pre < length (bs!(k-1))"
	+ using 1(2) entry pre unfolding wf_tree_input_def sound_ptrs_def sound_pre_ptr_def
	+ by (smt (verit) mem_Collect_eq nth_mem prod.inject)+
	+ have "k < length bs"
	+ using 1(2) unfolding wf_tree_input_def by blast+
	+ have "foldl (+) 0 (map length (take k bs)) + i - (foldl (+) 0 (map length (take (k-1) bs)) + pre) =
	+ foldl (+) 0 (map length (take (k-1) bs)) + length (bs!(k-1)) + i - (foldl (+) 0 (map length (take (k-1) bs)) + pre)"
	+ using foldl_add_nth[of \<open>k-1\<close> bs 0] by (simp add: \<open>0 < k\<close> \<open>k < length bs\<close> less_imp_diff_less)
	+ also have "... = length (bs!(k-1)) + i - pre"
	+ by simp
	+ also have "... > 0"
	+ using \<open>pre < length (bs!(k-1))\<close> by auto
	+ finally have "build_tree'_measure (bs, \<omega>, k, i) - build_tree'_measure (bs, \<omega>, (k-1), pre) > 0"
	+ by simp
	+ hence "P bs \<omega> (k-1) pre"
	+ using 1 n wf_tree_input_pre entry pre zero_less_diff by blast
	+ thus ?thesis
	+ using assms(2) entry pre pointer.distinct(5) pointer.inject(1) by presburger
	+ next
	+ case PreRed
	+ then obtain k' pre red ps where prered: "pointer e = PreRed (k', pre, red) ps"
	+ by blast
	+ have "k' < k" "pre < length (bs!k')"
	+ using 1(2) entry prered unfolding wf_tree_input_def sound_ptrs_def sound_prered_ptr_def
	+ apply simp_all
	+ apply (metis nth_mem)+
	+ done
	+ have "red < i"
	+ using 1(2) entry prered unfolding wf_tree_input_def mono_red_ptr_def by blast
	+ have "k < length bs" "i < length (bs!k)"
	+ using 1(2) unfolding wf_tree_input_def by blast+
	+ define n_pre where n_pre: "n_pre = build_tree'_measure (bs, \<omega>, k', pre)"
	+ have "0 < length (bs!k') + i - pre"
	+ by (simp add: \<open>pre < length (bs!k')\<close> add.commute trans_less_add2)
	+ also have "... = foldl (+) 0 (map length (take k' bs)) + length (bs!k') + i - (foldl (+) 0 (map length (take k' bs)) + pre)"
	+ by simp
	+ also have "... \<le> foldl (+) 0 (map length (take (k'+1) bs)) + i - (foldl (+) 0 (map length (take k' bs)) + pre)"
	+ using foldl_add_nth_ge[of k' k' bs 0] \<open>k < length bs\<close> \<open>k' < k\<close> by simp
	+ also have "... \<le> foldl (+) 0 (map length (take k bs)) + i - (foldl (+) 0 (map length (take k' bs)) + pre)"
	+ using foldl_take_mono by (metis Suc_eq_plus1 Suc_leI \<open>k' < k\<close> add.commute add_le_cancel_left diff_le_mono)
	+ finally have "build_tree'_measure (bs, \<omega>, k, i) - build_tree'_measure (bs, \<omega>, k', pre) > 0"
	+ by simp
	+ hence x: "P bs \<omega> k' pre"
	+ using 1(1) zero_less_diff by (metis "1.prems" entry prered wf_tree_input_prered_pre)
	+ define n_red where n_red: "n_red = build_tree'_measure (bs, \<omega>, k, red)"
	+ have "build_tree'_measure (bs, \<omega>, k, i) - build_tree'_measure (bs, \<omega>, k, red) > 0"
	+ using \<open>red < i\<close> by simp
	+ hence y: "P bs \<omega> k red"
	+ using "1.hyps" "1.prems" entry prered wf_tree_input_prered_red zero_less_diff by blast
	+ show ?thesis
	+ using assms(2) x y entry prered
	+ by (smt (verit, best) Pair_inject filter_cong pointer.distinct(5) pointer.inject(2))
	+ qed
	+qed
	+
	+lemma build_tree'_termination:
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ shows "\<exists>N ts. build_tree' bs \<omega> k i = Some (Branch N ts)"
	+proof -
	+ have "\<exists>N ts. build_tree' bs \<omega> k i = Some (Branch N ts)"
	+ apply (induction rule: build_tree'_induct[OF assms(1)])
	+ subgoal premises IH for bs \<omega> k i
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>k' pre red ps. pointer e = PreRed (k', pre, red) ps"
	+ by (metis pointer.exhaust surj_pair)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ thus ?thesis
	+ using build_tree'_simps(1) entry by simp
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ obtain N ts where Nts: "build_tree' bs \<omega> (k-1) pre = Some (Branch N ts)"
	+ using IH(1) entry pre by blast
	+ have "build_tree' bs \<omega> k i = Some (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ using build_tree'_simps(3) entry pre Nts by simp
	+ thus ?thesis
	+ by simp
	+ next
	+ case PreRed
	+ then obtain k' pre red ps where prered: "pointer e = PreRed (k', pre, red) ps"
	+ by blast
	+ then obtain N ts where Nts: "build_tree' bs \<omega> k' pre = Some (Branch N ts)"
	+ using IH(2) entry prered by blast
	+ obtain t_red where t_red: "build_tree' bs \<omega> k red = Some t_red"
	+ using IH(3) entry prered Nts by (metis option.exhaust)
	+ have "build_tree' bs \<omega> k i = Some (Branch N (ts @ [t_red]))"
	+ using build_tree'_simps(8) entry prered Nts t_red by auto
	+ thus ?thesis
	+ by blast
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ by blast
	+qed
	+
	+lemma wf_item_tree_build_tree':
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "wf_bins \<G> \<omega> bs"
	+ assumes "k < length bs" "i < length (bs!k)"
	+ assumes "build_tree' bs \<omega> k i = Some t"
	+ shows "wf_item_tree \<G> (item (bs!k!i)) t"
	+proof -
	+ have "wf_item_tree \<G> (item (bs!k!i)) t"
	+ using assms
	+ apply (induction arbitrary: t rule: build_tree'_induct[OF assms(1)])
	+ subgoal premises prems for bs \<omega> k i t
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>k' pre red ps. pointer e = PreRed (k', pre, red) ps"
	+ by (metis pointer.exhaust surj_pair)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ hence "build_tree' bs \<omega> k i = Some (Branch (item_rule_head (item e)) [])"
	+ using entry by simp
	+ have simp: "t = Branch (item_rule_head (item e)) []"
	+ using build_tree'_simps(1) Null prems(8) entry by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_tree_input_def by blast
	+ hence "predicts (item e)"
	+ using Null prems(6,7) nth_mem entry unfolding sound_ptrs_def sound_null_ptr_def by blast
	+ hence "item_dot (item e) = 0"
	+ unfolding predicts_def by blast
	+ thus ?thesis
	+ using simp entry by simp
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ obtain N ts where Nts: "build_tree' bs \<omega> (k-1) pre = Some (Branch N ts)"
	+ using build_tree'_termination entry pre prems(4) wf_tree_input_pre by blast
	+ have simp: "build_tree' bs \<omega> k i = Some (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ using build_tree'_simps(3) entry pre Nts by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_tree_input_def by blast
	+ hence "pre < length (bs!(k-1))"
	+ using entry pre prems(6,7) unfolding sound_ptrs_def sound_pre_ptr_def by (metis nth_mem)
	+ moreover have "k-1 < length bs"
	+ by (simp add: prems(6) less_imp_diff_less)
	+ ultimately have IH: "wf_item_tree \<G> (item (bs!(k-1)!pre)) (Branch N ts)"
	+ using prems(1,2,4,5) entry pre Nts by (meson wf_tree_input_pre)
	+ have scans: "scans \<omega> k (item (bs!(k-1)!pre)) (item e)"
	+ using entry pre prems(6-7) \<open>sound_ptrs \<omega> bs\<close> unfolding sound_ptrs_def sound_pre_ptr_def by simp
	+ hence *:
	+ "item_rule_head (item (bs!(k-1)!pre)) = item_rule_head (item e)"
	+ "item_rule_body (item (bs!(k-1)!pre)) = item_rule_body (item e)"
	+ "item_dot (item (bs!(k-1)!pre)) + 1 = item_dot (item e)"
	+ "next_symbol (item (bs!(k-1)!pre)) = Some (\<omega>!(k-1))"
	+ unfolding scans_def inc_item_def by (simp_all add: item_rule_head_def item_rule_body_def)
	+ have "map root_tree (ts @ [Leaf (\<omega>!(k-1))]) = map root_tree ts @ [\<omega>!(k-1)]"
	+ by simp
	+ also have "... = take (item_dot (item (bs!(k-1)!pre))) (item_rule_body (item (bs!(k-1)!pre))) @ [\<omega>!(k-1)]"
	+ using IH by simp
	+ also have "... = take (item_dot (item (bs!(k-1)!pre))) (item_rule_body (item e)) @ [\<omega>!(k-1)]"
	+ using *(2) by simp
	+ also have "... = take (item_dot (item e)) (item_rule_body (item e))"
	+ using *(2-4) by (auto simp: next_symbol_def is_complete_def split: if_splits; metis leI take_Suc_conv_app_nth)
	+ finally have "map root_tree (ts @ [Leaf (\<omega>!(k-1))]) = take (item_dot (item e)) (item_rule_body (item e))" .
	+ hence "wf_item_tree \<G> (item e) (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ using IH *(1) by simp
	+ thus ?thesis
	+ using entry prems(8) simp by auto
	+ next
	+ case PreRed
	+ then obtain k' pre red ps where prered: "pointer e = PreRed (k', pre, red) ps"
	+ by blast
	+ obtain N ts where Nts: "build_tree' bs \<omega> k' pre = Some (Branch N ts)"
	+ using build_tree'_termination entry prems(4) prered wf_tree_input_prered_pre by blast
	+ obtain N_red ts_red where Nts_red: "build_tree' bs \<omega> k red = Some (Branch N_red ts_red)"
	+ using build_tree'_termination entry prems(4) prered wf_tree_input_prered_red by blast
	+ have simp: "build_tree' bs \<omega> k i = Some (Branch N (ts @ [Branch N_red ts_red]))"
	+ using build_tree'_simps(8) entry prered Nts Nts_red by auto
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) wf_tree_input_def by fastforce
	+ have bounds: "k' < k" "pre < length (bs!k')" "red < length (bs!k)"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_prered_ptr_def sound_ptrs_def by fastforce+
	+ have completes: "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by fastforce
	+ have *:
	+ "item_rule_head (item (bs!k'!pre)) = item_rule_head (item e)"
	+ "item_rule_body (item (bs!k'!pre)) = item_rule_body (item e)"
	+ "item_dot (item (bs!k'!pre)) + 1 = item_dot (item e)"
	+ "next_symbol (item (bs!k'!pre)) = Some (item_rule_head (item (bs!k!red)))"
	+ "is_complete (item (bs!k!red))"
	+ using completes unfolding completes_def inc_item_def
	+ by (auto simp: item_rule_head_def item_rule_body_def is_complete_def)
	+ have "(bs, \<omega>, k', pre) \<in> wf_tree_input"
	+ using wf_tree_input_prered_pre[OF prems(4) entry prered] by blast
	+ hence IH_pre: "wf_item_tree \<G> (item (bs!k'!pre)) (Branch N ts)"
	+ using prems(2)[OF entry prered _ prems(5)] Nts bounds(1,2) order_less_trans prems(6) by blast
	+ have "(bs, \<omega>, k, red) \<in> wf_tree_input"
	+ using wf_tree_input_prered_red[OF prems(4) entry prered] by blast
	+ hence IH_r: "wf_item_tree \<G> (item (bs!k!red)) (Branch N_red ts_red)"
	+ using bounds(3) entry prems(3,5,6) prered Nts_red by blast
	+ have "map root_tree (ts @ [Branch N_red ts_red]) = map root_tree ts @ [root_tree (Branch N_red ts)]"
	+ by simp
	+ also have "... = take (item_dot (item (bs!k'!pre))) (item_rule_body (item (bs!k'!pre))) @ [root_tree (Branch N_red ts)]"
	+ using IH_pre by simp
	+ also have "... = take (item_dot (item (bs!k'!pre))) (item_rule_body (item (bs!k'!pre))) @ [item_rule_head (item (bs!k!red))]"
	+ using IH_r by simp
	+ also have "... = take (item_dot (item e)) (item_rule_body (item e))"
	+ using * by (auto simp: next_symbol_def is_complete_def split: if_splits; metis leI take_Suc_conv_app_nth)
	+ finally have roots: "map root_tree (ts @ [Branch N_red ts]) = take (item_dot (item e)) (item_rule_body (item e))" by simp
	+ have "wf_item \<G> \<omega> (item (bs!k!red))"
	+ using prems(5,6) bounds(3) unfolding wf_bins_def wf_bin_def wf_bin_items_def by (auto simp: items_def)
	+ moreover have "N_red = item_rule_head (item (bs!k!red))"
	+ using IH_r by fastforce
	+ moreover have "map root_tree ts_red = item_rule_body (item (bs!k!red))"
	+ using IH_r *(5) by (auto simp: is_complete_def)
	+ ultimately have "\<exists>r \<in> set (\<RR> \<G>). N_red = rule_head r \<and> map root_tree ts_red = rule_body r"
	+ unfolding wf_item_def item_rule_body_def item_rule_head_def by blast
	+ hence "wf_rule_tree \<G> (Branch N_red ts_red)"
	+ using IH_r by simp
	+ hence "wf_item_tree \<G> (item (bs!k!i)) (Branch N (ts @ [Branch N_red ts_red]))"
	+ using "*"(1) roots IH_pre entry by simp
	+ thus ?thesis
	+ using Nts_red prems(8) simp by auto
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ using assms(2) by blast
	+qed
	+
	+lemma wf_yield_tree_build_tree':
	+ assumes "(bs, \<omega>, k, i) \<in> wf_tree_input"
	+ assumes "wf_bins \<G> \<omega> bs"
	+ assumes "k < length bs" "i < length (bs!k)" "k \<le> length \<omega>"
	+ assumes "build_tree' bs \<omega> k i = Some t"
	+ shows "wf_yield_tree \<omega> (item (bs!k!i)) t"
	+proof -
	+ have "wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ using assms
	+ apply (induction arbitrary: t rule: build_tree'_induct[OF assms(1)])
	+ subgoal premises prems for bs \<omega> k i t
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>k' pre red reds. pointer e = PreRed (k', pre, red) reds"
	+ by (metis pointer.exhaust surj_pair)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ hence "build_tree' bs \<omega> k i = Some (Branch (item_rule_head (item e)) [])"
	+ using entry by simp
	+ have simp: "t = Branch (item_rule_head (item e)) []"
	+ using build_tree'_simps(1) Null prems(9) entry by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_tree_input_def by blast
	+ hence "predicts (item e)"
	+ using Null prems(6,7) nth_mem entry unfolding sound_ptrs_def sound_null_ptr_def by blast
	+ thus ?thesis
	+ unfolding wf_yield_tree_def predicts_def using simp entry by (auto simp: slice_empty)
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ obtain N ts where Nts: "build_tree' bs \<omega> (k-1) pre = Some (Branch N ts)"
	+ using build_tree'_termination entry pre prems(4) wf_tree_input_pre by blast
	+ have simp: "build_tree' bs \<omega> k i = Some (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ using build_tree'_simps(3) entry pre Nts by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_tree_input_def by blast
	+ hence bounds: "k > 0" "pre < length (bs!(k-1))"
	+ using entry pre prems(6,7) unfolding sound_ptrs_def sound_pre_ptr_def by (metis nth_mem)+
	+ moreover have "k-1 < length bs"
	+ by (simp add: prems(6) less_imp_diff_less)
	+ ultimately have IH: "wf_yield_tree \<omega> (item (bs!(k-1)!pre)) (Branch N ts)"
	+ using prems(1) entry pre Nts wf_tree_input_pre prems(4,5,8) by fastforce
	+ have scans: "scans \<omega> k (item (bs!(k-1)!pre)) (item e)"
	+ using entry pre prems(6-7) \<open>sound_ptrs \<omega> bs\<close> unfolding sound_ptrs_def sound_pre_ptr_def by simp
	+ have wf:
	+ "item_origin (item (bs!(k-1)!pre)) \<le> item_end (item (bs!(k-1)!pre))"
	+ "item_end (item (bs!(k-1)!pre)) = k-1"
	+ "item_end (item e) = k"
	+ using entry prems(5,6,7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def wf_item_def
	+ by (auto, meson less_imp_diff_less nth_mem)
	+ have "yield_tree (Branch N (ts @ [Leaf (\<omega>!(k-1))])) = concat (map yield_tree (ts @ [Leaf (\<omega>!(k-1))]))"
	+ by simp
	+ also have "... = concat (map yield_tree ts) @ [\<omega>!(k-1)]"
	+ by simp
	+ also have "... = slice (item_origin (item (bs!(k-1)!pre))) (item_end (item (bs!(k-1)!pre))) \<omega> @ [\<omega>!(k-1)]"
	+ using IH by (simp add: wf_yield_tree_def)
	+ also have "... = slice (item_origin (item (bs!(k-1)!pre))) (item_end (item (bs!(k-1)!pre)) + 1) \<omega>"
	+ using slice_append_nth wf \<open>k > 0\<close> prems(8)
	+ by (metis diff_less le_eq_less_or_eq less_imp_diff_less less_numeral_extra(1))
	+ also have "... = slice (item_origin (item e)) (item_end (item (bs!(k-1)!pre)) + 1) \<omega>"
	+ using scans unfolding scans_def inc_item_def by simp
	+ also have "... = slice (item_origin (item e)) k \<omega>"
	+ using scans wf unfolding scans_def by (metis Suc_diff_1 Suc_eq_plus1 bounds(1))
	+ also have "... = slice (item_origin (item e)) (item_end (item e)) \<omega>"
	+ using wf by auto
	+ finally show ?thesis
	+ using wf_yield_tree_def entry prems(9) simp by force
	+ next
	+ case PreRed
	+ then obtain k' pre red ps where prered: "pointer e = PreRed (k', pre, red) ps"
	+ by blast
	+ obtain N ts where Nts: "build_tree' bs \<omega> k' pre = Some (Branch N ts)"
	+ using build_tree'_termination entry prems(4) prered wf_tree_input_prered_pre by blast
	+ obtain N_red ts_red where Nts_red: "build_tree' bs \<omega> k red = Some (Branch N_red ts_red)"
	+ using build_tree'_termination entry prems(4) prered wf_tree_input_prered_red by blast
	+ have simp: "build_tree' bs \<omega> k i = Some (Branch N (ts @ [Branch N_red ts_red]))"
	+ using build_tree'_simps(8) entry prered Nts Nts_red by auto
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) wf_tree_input_def by fastforce
	+ have bounds: "k' < k" "pre < length (bs!k')" "red < length (bs!k)"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by fastforce+
	+ have completes: "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by fastforce
	+ have "(bs, \<omega>, k', pre) \<in> wf_tree_input"
	+ using wf_tree_input_prered_pre[OF prems(4) entry prered] by blast
	+ hence IH_pre: "wf_yield_tree \<omega> (item (bs!k'!pre)) (Branch N ts)"
	+ using prems(2)[OF entry prered _ prems(5)] Nts bounds(1,2) prems(6-8)
	+ by (meson dual_order.strict_trans1 nat_less_le)
	+ have "(bs, \<omega>, k, red) \<in> wf_tree_input"
	+ using wf_tree_input_prered_red[OF prems(4) entry prered] by blast
	+ hence IH_r: "wf_yield_tree \<omega> (item (bs!k!red)) (Branch N_red ts_red)"
	+ using bounds(3) entry prems(3,5,6,8) prered Nts_red by blast
	+ have wf1:
	+ "item_origin (item (bs!k'!pre)) \<le> item_end (item (bs!k'!pre))"
	+ "item_origin (item (bs!k!red)) \<le> item_end (item (bs!k!red))"
	+ using prems(5-7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def wf_item_def
	+ by (metis length_map nth_map nth_mem order_less_trans)+
	+ have wf2:
	+ "item_end (item (bs!k!red)) = k"
	+ "item_end (item (bs!k!i)) = k"
	+ using prems(5-7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def by simp_all
	+ have "yield_tree (Branch N (ts @ [Branch N_red ts_red])) = concat (map yield_tree (ts @ [Branch N_red ts_red]))"
	+ by (simp add: Nts_red)
	+ also have "... = concat (map yield_tree ts) @ yield_tree (Branch N_red ts_red)"
	+ by simp
	+ also have "... = slice (item_origin (item (bs!k'!pre))) (item_end (item (bs!k'!pre))) \<omega> @
	+ slice (item_origin (item (bs!k!red))) (item_end (item (bs!k!red))) \<omega>"
	+ using IH_pre IH_r by (simp add: wf_yield_tree_def)
	+ also have "... = slice (item_origin (item (bs!k'!pre))) (item_end (item (bs!k!red))) \<omega>"
	+ using slice_concat wf1 completes_def completes by (metis (no_types, lifting))
	+ also have "... = slice (item_origin (item e)) (item_end (item (bs!k!red))) \<omega>"
	+ using completes unfolding completes_def inc_item_def by simp
	+ also have "... = slice (item_origin (item e)) (item_end (item e)) \<omega>"
	+ using wf2 entry by presburger
	+ finally show ?thesis
	+ using wf_yield_tree_def entry prems(9) simp by force
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ using assms(2) by blast
	+qed
	+
	+theorem wf_rule_root_yield_tree_build_forest:
	+ assumes "wf_bins \<G> \<omega> bs" "sound_ptrs \<omega> bs" "mono_red_ptr bs" "length bs = length \<omega> + 1"
	+ assumes "build_tree \<G> \<omega> bs = Some t"
	+ shows "wf_rule_tree \<G> t \<and> root_tree t = \<SS> \<G> \<and> yield_tree t = \<omega>"
	+proof -
	+ let ?k = "length bs - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items (bs!?k))"
	+ then obtain x i where *: "(x,i) \<in> set finished" "Some t = build_tree' bs \<omega> ?k i"
	+ using assms(5) unfolding finished_def build_tree_def by (auto simp: Let_def split: list.splits, presburger)
	+ have k: "?k < length bs" "?k \<le> length \<omega>"
	+ using assms(4) by simp_all
	+ have i: "i < length (bs!?k)"
	+ using index_filter_with_index_lt_length * items_def finished_def by (metis length_map)
	+ have x: "x = item (bs!?k!i)"
	+ using * i filter_with_index_nth items_def nth_map finished_def by metis
	+ have finished: "is_finished \<G> \<omega> x"
	+ using * filter_with_index_P finished_def by metis
	+ have wf_trees_input: "(bs, \<omega>, ?k, i) \<in> wf_tree_input"
	+ unfolding wf_tree_input_def using assms(2,3) i k(1) by blast
	+ hence wf_item_tree: "wf_item_tree \<G> x t"
	+ using wf_item_tree_build_tree' assms(1,2) i k(1) x *(2) by metis
	+ have wf_item: "wf_item \<G> \<omega> (item (bs!?k!i))"
	+ using k(1) i assms(1) unfolding wf_bins_def wf_bin_def wf_bin_items_def by (simp add: items_def)
	+ obtain N ts where t: "t = Branch N ts"
	+ by (metis *(2) build_tree'_termination option.inject wf_trees_input)
	+ hence "N = item_rule_head x"
	+ "map root_tree ts = item_rule_body x"
	+ using finished wf_item_tree by (auto simp: is_finished_def is_complete_def)
	+ hence "\<exists>r \<in> set (\<RR> \<G>). N = rule_head r \<and> map root_tree ts = rule_body r"
	+ using wf_item x unfolding wf_item_def item_rule_body_def item_rule_head_def by blast
	+ hence wf_rule: "wf_rule_tree \<G> t"
	+ using wf_item_tree t by simp
	+ have root: "root_tree t = \<SS> \<G>"
	+ using finished t \<open>N = item_rule_head x\<close> by (auto simp: is_finished_def)
	+ have "yield_tree t = slice (item_origin (item (bs!?k!i))) (item_end (item (bs!?k!i))) \<omega>"
	+ using k i assms(1) wf_trees_input wf_yield_tree_build_tree' wf_yield_tree_def *(2) by (metis (no_types, lifting))
	+ hence yield: "yield_tree t = \<omega>"
	+ using finished x unfolding is_finished_def by simp
	+ show ?thesis
	+ using * wf_rule root yield assms(4) unfolding build_tree_def by simp
	+qed
	+
	+corollary wf_rule_root_yield_tree_build_tree_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "nonempty_derives \<G>"
	+ assumes "build_tree \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = Some t"
	+ shows "wf_rule_tree \<G> t \<and> root_tree t = \<SS> \<G> \<and> yield_tree t = \<omega>"
	+ using assms wf_rule_root_yield_tree_build_forest wf_bins_Earley\<^sub>L sound_mono_ptrs_Earley\<^sub>L Earley\<^sub>L_def
	+ length_Earley\<^sub>L_bins length_bins_Init\<^sub>L by (metis nle_le)
	+
	+theorem correctness_build_tree_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "(\<exists>t. build_tree \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = Some t) \<longleftrightarrow> derives \<G> [\<SS> \<G>] \<omega>" (is "?L \<longleftrightarrow> ?R")
	+proof standard
	+ assume *: ?L
	+ let ?k = "length (Earley\<^sub>L \<G> \<omega>) - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items ((Earley\<^sub>L \<G> \<omega>)!?k))"
	+ then obtain t x i where *: "(x,i) \<in> set finished" "Some t = build_tree' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i"
	+ using * unfolding finished_def build_tree_def by (auto simp: Let_def split: list.splits, presburger)
	+ have k: "?k < length (Earley\<^sub>L \<G> \<omega>)" "?k \<le> length \<omega>"
	+ by (simp_all add: Earley\<^sub>L_def assms(1))
	+ have i: "i < length ((Earley\<^sub>L \<G> \<omega>) ! ?k)"
	+ using index_filter_with_index_lt_length * items_def finished_def by (metis length_map)
	+ have x: "x = item ((Earley\<^sub>L \<G> \<omega>)!?k!i)"
	+ using * i filter_with_index_nth items_def nth_map finished_def by metis
	+ have finished: "is_finished \<G> \<omega> x"
	+ using * filter_with_index_P finished_def by metis
	+ moreover have "x \<in> set (items ((Earley\<^sub>L \<G> \<omega>) ! ?k))"
	+ using x by (auto simp: items_def; metis One_nat_def i imageI nth_mem)
	+ ultimately have "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega>"
	+ by (meson k(1) kth_bin_sub_bins recognizing_def subsetD)
	+ thus ?R
	+ using correctness_Earley\<^sub>L assms by blast
	+next
	+ assume *: ?R
	+ let ?k = "length (Earley\<^sub>L \<G> \<omega>) - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items ((Earley\<^sub>L \<G> \<omega>)!?k))"
	+ have "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega>"
	+ using assms * correctness_Earley\<^sub>L by blast
	+ moreover have "?k = length \<omega>"
	+ by (simp add: Earley\<^sub>L_def assms(1))
	+ ultimately have "\<exists>x \<in> set (items ((Earley\<^sub>L \<G> \<omega>)!?k)). is_finished \<G> \<omega> x"
	+ unfolding recognizing_def using assms(1) is_finished_def wf_bins_Earley\<^sub>L wf_item_in_kth_bin by metis
	+ then obtain x i xs where xis: "finished = (x,i)#xs"
	+ using filter_with_index_Ex_first by (metis finished_def)
	+ hence simp: "build_tree \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = build_tree' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i"
	+ unfolding build_tree_def finished_def by auto
	+ have "(x,i) \<in> set finished"
	+ using xis by simp
	+ hence "i < length ((Earley\<^sub>L \<G> \<omega>)!?k)"
	+ using index_filter_with_index_lt_length by (metis finished_def items_def length_map)
	+ moreover have "?k < length (Earley\<^sub>L \<G> \<omega>)"
	+ by (simp add: Earley\<^sub>L_def assms(1))
	+ ultimately have "(Earley\<^sub>L \<G> \<omega>, \<omega>, ?k, i) \<in> wf_tree_input"
	+ unfolding wf_tree_input_def using sound_mono_ptrs_Earley\<^sub>L assms(1,3) by blast
	+ then obtain N ts where "build_tree' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i = Some (Branch N ts)"
	+ using build_tree'_termination by blast
	+ thus ?L
	+ using simp by simp
	+qed
	+
	+
	+subsection \<open>those, map, map option lemmas\<close>
	+
	+lemma those_map_exists:
	+ "Some ys = those (map f xs) \<Longrightarrow> y \<in> set ys \<Longrightarrow> \<exists>x. x \<in> set xs \<and> Some y \<in> set (map f xs)"
	+proof (induction xs arbitrary: ys)
	+ case (Cons a xs)
	+ then show ?case
	+ apply (clarsimp split: option.splits)
	+ by (smt (verit, best) map_option_eq_Some set_ConsD)
	+qed auto
	+
	+lemma those_Some:
	+ "(\<forall>x \<in> set xs. \<exists>a. x = Some a) \<longleftrightarrow> (\<exists>ys. those xs = Some ys)"
	+ by (induction xs) (auto split: option.splits)
	+
	+lemma those_Some_P:
	+ assumes "\<forall>x \<in> set xs. \<exists>ys. x = Some ys \<and> (\<forall>y \<in> set ys. P y)"
	+ shows "\<exists>yss. those xs = Some yss \<and> (\<forall>ys \<in> set yss. \<forall>y \<in> set ys. P y)"
	+ using assms by (induction xs) auto
	+
	+lemma map_Some_P:
	+ assumes "z \<in> set (map f xs)"
	+ assumes "\<forall>x \<in> set xs. \<exists>ys. f x = Some ys \<and> (\<forall>y \<in> set ys. P y)"
	+ shows "\<exists>ys. z = Some ys \<and> (\<forall>y \<in> set ys. P y)"
	+ using assms by (induction xs) auto
	+
	+lemma those_map_FBranch_only:
	+ assumes "g = (\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) \| _ \<Rightarrow> None)"
	+ assumes "Some fs = those (map g pres)" "f \<in> set fs"
	+ assumes "\<forall>f \<in> set pres. \<exists>N fss. f = FBranch N fss"
	+ shows "\<exists>f_pre N fss. f = FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]]) \<and> f_pre = FBranch N fss \<and> f_pre \<in> set pres"
	+ using assms
	+ apply (induction pres arbitrary: fs f)
	+ apply (auto)
	+ by (smt (verit, best) list.inject list.set_cases map_option_eq_Some)
	+
	+lemma those_map_Some_concat_exists:
	+ assumes "y \<in> set (concat ys)"
	+ assumes "Some ys = those (map f xs)"
	+ shows "\<exists>ys x. Some ys = f x \<and> y \<in> set ys \<and> x \<in> set xs"
	+ using assms
	+ apply (induction xs arbitrary: ys y)
	+ apply (auto split: option.splits)
	+ by (smt (verit, ccfv_threshold) list.inject list.set_cases map_option_eq_Some)
	+
	+lemma map_option_concat_those_map_exists:
	+ assumes "Some fs = map_option concat (those (map F xs))"
	+ assumes "f \<in> set fs"
	+ shows "\<exists>fss fs'. Some fss = those (map F xs) \<and> fs' \<in> set fss \<and> f \<in> set fs'"
	+ using assms
	+ apply (induction xs arbitrary: fs f)
	+ apply (auto split: option.splits)
	+ by (smt (verit, best) UN_E map_option_eq_Some set_concat)
	+
	+lemma [partial_function_mono]:
	+ "monotone option.le_fun option_ord
	+ (\<lambda>f. map_option concat (those (map (\<lambda>((k', pre), reds).
	+ f ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. f ((((r, s), t), red), b \<union> {red})) reds) \<bind>
	+ (\<lambda>rss. those (map (\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss])) \| _ \<Rightarrow> None) pres))))
	+ xs)))"
	+proof -
	+ let ?f = "
	+ (\<lambda>f. map_option concat (those (map (\<lambda>((k', pre), reds).
	+ f ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. f ((((r, s), t), red), b \<union> {red})) reds) \<bind>
	+ (\<lambda>rss. those (map (\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss])) \| _ \<Rightarrow> None) pres))))
	+ xs)))"
	+ have 0: "\<And>x y. option.le_fun x y \<Longrightarrow> option_ord (?f x) (?f y)"
	+ apply (auto simp: flat_ord_def fun_ord_def option.leq_refl split: option.splits forest.splits)
	+ subgoal premises prems for x y
	+ proof -
	+ let ?t = "those (map (\<lambda>((k', pre), reds).
	+ x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))))
	+ xs) = None"
	+ show ?t
	+ proof (rule ccontr)
	+ assume a: "\<not>?t"
	+ obtain fss where fss: "those (map (\<lambda>((k', pre), reds).
	+ x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))))
	+ xs) = Some fss"
	+ using a by blast
	+ {
	+ fix k' pre reds
	+ assume *: "((k', pre), reds) \<in> set xs"
	+ obtain pres where pres: "x ((((r, s), k'), pre), {pre}) = Some pres"
	+ using fss * those_Some by force
	+ have "\<exists>fs. Some fs = those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))"
	+ proof (rule ccontr)
	+ assume "\<nexists>fs. Some fs =
	+ those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))"
	+ hence "None =
	+ those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))"
	+ by (smt (verit) not_None_eq)
	+ hence "None = x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))"
	+ by (simp add: pres)
	+ hence "\<exists>((k', pre), reds) \<in> set xs. None = x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))"
	+ using * by blast
	+ thus False
	+ using fss those_Some by force
	+ qed
	+ then obtain fs where fs: "Some fs = those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))"
	+ by blast
	+ obtain rss where rss: "those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) = Some rss"
	+ using fs by force
	+ have "x ((((r, s), k'), pre), {pre}) = y ((((r, s), k'), pre), {pre})"
	+ using pres prems(1) by (metis option.distinct(1))
	+ moreover have "those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))
	+ = those (map (\<lambda>red. y ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))"
	+ proof -
	+ have "\<forall>red \<in> set reds. x ((((r, s), t), red), insert red b) = y ((((r, s), t), red), insert red b)"
	+ proof standard
	+ fix red
	+ assume "red \<in> set reds"
	+ have "\<forall>x\<in>set (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) . \<exists>a. x = Some a"
	+ using rss those_Some by blast
	+ then obtain f where "x ((((r, s), t), red), insert red b) = Some f"
	+ using \<open>red \<in> set reds\<close> by auto
	+ thus "x ((((r, s), t), red), insert red b) = y ((((r, s), t), red), insert red b)"
	+ using prems(1) by (metis option.distinct(1))
	+ qed
	+ thus ?thesis
	+ by (smt (verit, best) map_eq_conv)
	+ qed
	+ ultimately have " x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))
	+ = y ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. y ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))"
	+ by (metis bind.bind_lunit pres)
	+ }
	+ hence "\<forall>((k', pre), reds) \<in> set xs. x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))
	+ = y ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. y ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres)))"
	+ by blast
	+ hence "those (map (\<lambda>((k', pre), reds).
	+ x ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. x ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))))
	+ xs) = those (map (\<lambda>((k', pre), reds).
	+ y ((((r, s), k'), pre), {pre}) \<bind>
	+ (\<lambda>pres. those (map (\<lambda>red. y ((((r, s), t), red), insert red b)) reds) \<bind>
	+ (\<lambda>rss. those (map (case_forest Map.empty (\<lambda>N fss. Some (FBranch N (fss @ [concat rss])))) pres))))
	+ xs)"
	+ using prems(1) by (smt (verit, best) case_prod_conv map_eq_conv split_cong)
	+ thus False
	+ using prems(2) by simp
	+ qed
	+ qed
	+ done
	+ show ?thesis
	+ using monotoneI[of option.le_fun option_ord ?f, OF 0] by blast
	+qed
	+
	+subsection \<open>Parse trees\<close>
	+
	+fun insert_group :: "('a \<Rightarrow> 'k) \<Rightarrow> ('a \<Rightarrow> 'v) \<Rightarrow> 'a \<Rightarrow> ('k \<times> 'v list) list \<Rightarrow> ('k \<times> 'v list) list" where
	+ "insert_group K V a [] = [(K a, [V a])]"
	+\| "insert_group K V a ((k, vs)#xs) = (
	+ if K a = k then (k, V a # vs) # xs
	+ else (k, vs) # insert_group K V a xs
	+ )"
	+
	+fun group_by :: "('a \<Rightarrow> 'k) \<Rightarrow> ('a \<Rightarrow> 'v) \<Rightarrow> 'a list \<Rightarrow> ('k \<times> 'v list) list" where
	+ "group_by K V [] = []"
	+\| "group_by K V (x#xs) = insert_group K V x (group_by K V xs)"
	+
	+lemma insert_group_cases:
	+ assumes "(k, vs) \<in> set (insert_group K V a xs)"
	+ shows "(k = K a \<and> vs = [V a]) \<or> (k, vs) \<in> set xs \<or> (\<exists>(k', vs') \<in> set xs. k' = k \<and> k = K a \<and> vs = V a # vs')"
	+ using assms by (induction xs) (auto split: if_splits)
	+
	+lemma group_by_exists_kv:
	+ "(k, vs) \<in> set (group_by K V xs) \<Longrightarrow> \<exists>x \<in> set xs. k = K x \<and> (\<exists>v \<in> set vs. v = V x)"
	+ using insert_group_cases by (induction xs) (simp, force)
	+
	+lemma group_by_forall_v_exists_k:
	+ "(k, vs) \<in> set (group_by K V xs) \<Longrightarrow> v \<in> set vs \<Longrightarrow> \<exists>x \<in> set xs. k = K x \<and> v = V x"
	+proof (induction xs arbitrary: vs)
	+ case (Cons x xs)
	+ show ?case
	+ proof (cases "(k, vs) \<in> set (group_by K V xs)")
	+ case True
	+ thus ?thesis
	+ using Cons by simp
	+ next
	+ case False
	+ hence "(k, vs) \<in> set (insert_group K V x (group_by K V xs))"
	+ using Cons.prems(1) by force
	+ then consider (A) "(k = K x \<and> vs = [V x])"
	+ \| (B) "(k, vs) \<in> set (group_by K V xs)"
	+ \| (C) "(\<exists>(k', vs') \<in> set (group_by K V xs). k' = k \<and> k = K x \<and> vs = V x # vs')"
	+ using insert_group_cases by fastforce
	+ thus ?thesis
	+ proof cases
	+ case A
	+ thus ?thesis
	+ using Cons.prems(2) by auto
	+ next
	+ case B
	+ then show ?thesis
	+ using False by linarith
	+ next
	+ case C
	+ then show ?thesis
	+ using Cons.IH Cons.prems(2) by fastforce
	+ qed
	+ qed
	+qed simp
	+
	+partial_function (option) build_trees' :: "'a bins \<Rightarrow> 'a sentence \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> 'a forest list option" where
	+ "build_trees' bs \<omega> k i I = (
	+ let e = bs!k!i in (
	+ case pointer e of
	+ Null \<Rightarrow> Some ([FBranch (item_rule_head (item e)) []]) \<comment>\<open>start building sub-trees\<close>
	+ \| Pre pre \<Rightarrow> ( \<comment>\<open>add sub-trees starting from terminal\<close>
	+ do {
	+ pres \<leftarrow> build_trees' bs \<omega> (k-1) pre {pre};
	+ those (map (\<lambda>f.
	+ case f of
	+ FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]]))
	+ \| _ \<Rightarrow> None \<comment>\<open>impossible case\<close>
	+ ) pres)
	+ })
	+ \| PreRed p ps \<Rightarrow> ( \<comment>\<open>add sub-trees starting from non-terminal\<close>
	+ let ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps) in
	+ let gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps' in
	+ map_option concat (those (map (\<lambda>((k', pre), reds).
	+ do {
	+ pres \<leftarrow> build_trees' bs \<omega> k' pre {pre};
	+ rss \<leftarrow> those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds);
	+ those (map (\<lambda>f.
	+ case f of
	+ FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss]))
	+ \| _ \<Rightarrow> None \<comment>\<open>impossible case\<close>
	+ ) pres)
	+ }
	+ ) gs))
	+ )
	+ ))"
	+
	+declare build_trees'.simps [code]
	+
	+definition build_trees :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins \<Rightarrow> 'a forest list option" where
	+ "build_trees \<G> \<omega> bs = (
	+ let k = length bs - 1 in
	+ let finished = filter_with_index (\<lambda>x. is_finished \<G> \<omega> x) (items (bs!k)) in
	+ map_option concat (those (map (\<lambda>(_, i). build_trees' bs \<omega> k i {i}) finished))
	+ )"
	+
	+lemma build_forest'_simps[simp]:
	+ "e = bs!k!i \<Longrightarrow> pointer e = Null \<Longrightarrow> build_trees' bs \<omega> k i I = Some ([FBranch (item_rule_head (item e)) []])"
	+ "e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> build_trees' bs \<omega> (k-1) pre {pre} = None \<Longrightarrow> build_trees' bs \<omega> k i I = None"
	+ "e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> build_trees' bs \<omega> (k-1) pre {pre} = Some pres \<Longrightarrow>
	+ build_trees' bs \<omega> k i I = those (map (\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) \| _ \<Rightarrow> None) pres)"
	+ by (subst build_trees'.simps, simp)+
	+
	+definition wf_trees_input :: "('a bins \<times> 'a sentence \<times> nat \<times> nat \<times> nat set) set" where
	+ "wf_trees_input = {
	+ (bs, \<omega>, k, i, I) \| bs \<omega> k i I.
	+ sound_ptrs \<omega> bs \<and>
	+ k < length bs \<and>
	+ i < length (bs!k) \<and>
	+ I \<subseteq> {0..<length (bs!k)} \<and>
	+ i \<in> I
	+ }"
	+
	+fun build_forest'_measure :: "('a bins \<times> 'a sentence \<times> nat \<times> nat \<times> nat set) \<Rightarrow> nat" where
	+ "build_forest'_measure (bs, \<omega>, k, i, I) = foldl (+) 0 (map length (take (k+1) bs)) - card I"
	+
	+lemma wf_trees_input_pre:
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "e = bs!k!i" "pointer e = Pre pre"
	+ shows "(bs, \<omega>, (k-1), pre, {pre}) \<in> wf_trees_input"
	+ using assms unfolding wf_trees_input_def
	+ apply (auto simp: sound_ptrs_def sound_pre_ptr_def)
	+ apply (metis nth_mem)
	+ done
	+
	+lemma wf_trees_input_prered_pre:
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "e = bs!k!i" "pointer e = PreRed p ps"
	+ assumes "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ assumes "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ assumes "((k', pre), reds) \<in> set gs"
	+ shows "(bs, \<omega>, k', pre, {pre}) \<in> wf_trees_input"
	+proof -
	+ obtain red where "(k', pre, red) \<in> set ps'"
	+ using assms(5,6) group_by_exists_kv by fast
	+ hence *: "(k', pre, red) \<in> set (p#ps)"
	+ using assms(4) by (meson filter_is_subset in_mono)
	+ have "k < length bs" "e \<in> set (bs!k)"
	+ using assms(1,2) unfolding wf_trees_input_def by auto
	+ hence "k' < k" "pre < length (bs!k')"
	+ using assms(1,3) * unfolding wf_trees_input_def sound_ptrs_def sound_prered_ptr_def by blast+
	+ thus ?thesis
	+ using assms by (simp add: wf_trees_input_def)
	+qed
	+
	+lemma wf_trees_input_prered_red:
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "e = bs!k!i" "pointer e = PreRed p ps"
	+ assumes "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ assumes "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ assumes "((k', pre), reds) \<in> set gs" "red \<in> set reds"
	+ shows "(bs, \<omega>, k, red, I \<union> {red}) \<in> wf_trees_input"
	+proof -
	+ have "(k', pre, red) \<in> set ps'"
	+ using assms(5,6,7) group_by_forall_v_exists_k by fastforce
	+ hence *: "(k', pre, red) \<in> set (p#ps)"
	+ using assms(4) by (meson filter_is_subset in_mono)
	+ have "k < length bs" "e \<in> set (bs!k)"
	+ using assms(1,2) unfolding wf_trees_input_def by auto
	+ hence 0: "red < length (bs!k)"
	+ using assms(1,3) * unfolding wf_trees_input_def sound_ptrs_def sound_prered_ptr_def by blast
	+ moreover have "I \<subseteq> {0..<length (bs!k)}"
	+ using assms(1) unfolding wf_trees_input_def by blast
	+ ultimately have 1: "I \<union> {red} \<subseteq> {0..<length (bs!k)}"
	+ by simp
	+ show ?thesis
	+ using 0 1 assms(1) unfolding wf_trees_input_def by blast
	+qed
	+
	+lemma build_trees'_induct:
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "\<And>bs \<omega> k i I.
	+ (\<And>e pre. e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> P bs \<omega> (k-1) pre {pre}) \<Longrightarrow>
	+ (\<And>e p ps ps' gs k' pre reds. e = bs!k!i \<Longrightarrow> pointer e = PreRed p ps \<Longrightarrow>
	+ ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps) \<Longrightarrow>
	+ gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps' \<Longrightarrow>
	+ ((k', pre), reds) \<in> set gs \<Longrightarrow> P bs \<omega> k' pre {pre}) \<Longrightarrow>
	+ (\<And>e p ps ps' gs k' pre red reds reds'. e = bs!k!i \<Longrightarrow> pointer e = PreRed p ps \<Longrightarrow>
	+ ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps) \<Longrightarrow>
	+ gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps' \<Longrightarrow>
	+ ((k', pre), reds) \<in> set gs \<Longrightarrow> red \<in> set reds \<Longrightarrow> P bs \<omega> k red (I \<union> {red})) \<Longrightarrow>
	+ P bs \<omega> k i I"
	+ shows "P bs \<omega> k i I"
	+ using assms(1)
	+proof (induction n\<equiv>"build_forest'_measure (bs, \<omega>, k, i, I)" arbitrary: k i I rule: nat_less_induct)
	+ case 1
	+ obtain e where entry: "e = bs!k!i"
	+ by simp
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>p ps. pointer e = PreRed p ps"
	+ by (metis pointer.exhaust)
	+ thus ?case
	+ proof cases
	+ case Null
	+ thus ?thesis
	+ using assms(2) entry by fastforce
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ define n where n: "n = build_forest'_measure (bs, \<omega>, (k-1), pre, {pre})"
	+ have "0 < k" "pre < length (bs!(k-1))"
	+ using 1(2) entry pre unfolding wf_trees_input_def sound_ptrs_def sound_pre_ptr_def
	+ by (smt (verit) mem_Collect_eq nth_mem prod.inject)+
	+ have "k < length bs" "i < length (bs!k)" "I \<subseteq> {0..<length (bs!k)}" "i \<in> I"
	+ using 1(2) unfolding wf_trees_input_def by blast+
	+ have "length (bs!(k-1)) > 0"
	+ using \<open>pre < length (bs!(k-1))\<close> by force
	+ hence "foldl (+) 0 (map length (take k bs)) > 0"
	+ by (smt (verit, del_insts) foldl_add_nth \<open>0 < k\<close> \<open>k < length bs\<close>
	+ add.commute add_diff_inverse_nat less_imp_diff_less less_one zero_eq_add_iff_both_eq_0)
	+ have "card I \<le> length (bs!k)"
	+ by (simp add: \<open>I \<subseteq> {0..<length (bs ! k)}\<close> subset_eq_atLeast0_lessThan_card)
	+ have "card I + (foldl (+) 0 (map length (take (Suc (k - Suc 0)) bs)) - Suc 0) =
	+ card I + (foldl (+) 0 (map length (take k bs)) - 1)"
	+ using \<open>0 < k\<close> by simp
	+ also have "... = card I + foldl (+) 0 (map length (take k bs)) - 1"
	+ using \<open>0 < foldl (+) 0 (map length (take k bs))\<close> by auto
	+ also have "... < card I + foldl (+) 0 (map length (take k bs))"
	+ by (simp add: \<open>0 < foldl (+) 0 (map length (take k bs))\<close>)
	+ also have "... \<le> foldl (+) 0 (map length (take k bs)) + length (bs!k)"
	+ by (simp add: \<open>card I \<le> length (bs ! k)\<close>)
	+ also have "... = foldl (+) 0 (map length (take (k+1) bs))"
	+ using foldl_add_nth \<open>k < length bs\<close> by blast
	+ finally have "build_forest'_measure (bs, \<omega>, k, i, I) - build_forest'_measure (bs, \<omega>, (k-1), pre, {pre}) > 0"
	+ by simp
	+ hence "P bs \<omega> (k-1) pre {pre}"
	+ using 1 n wf_trees_input_pre entry pre zero_less_diff by blast
	+ thus ?thesis
	+ using assms(2) entry pre pointer.distinct(5) pointer.inject(1) by presburger
	+ next
	+ case PreRed
	+ then obtain p ps where pps: "pointer e = PreRed p ps"
	+ by blast
	+ define ps' where ps': "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ define gs where gs: "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ have 0: "\<forall>(k', pre, red) \<in> set ps'. k' < k \<and> pre < length (bs!k') \<and> red < length (bs!k)"
	+ using entry pps ps' 1(2) unfolding wf_trees_input_def sound_ptrs_def sound_prered_ptr_def
	+ apply (auto simp del: filter.simps)
	+ apply (metis nth_mem prod_cases3)+
	+ done
	+ hence sound_gs: "\<forall>((k', pre), reds) \<in> set gs. k' < k \<and> pre < length (bs!k')"
	+ using gs group_by_exists_kv by fast
	+ have sound_gs2: "\<forall>((k', pre), reds) \<in> set gs. \<forall>red \<in> set reds. red < length (bs!k)"
	+ proof (standard, standard, standard, standard)
	+ fix x a b k' pre red
	+ assume "x \<in> set gs" "x = (a, b)" "(k', pre) = a" "red \<in> set b"
	+ hence "\<exists>x \<in> set ps'. red = (\<lambda>(k', pre, red). red) x"
	+ using group_by_forall_v_exists_k gs ps' by meson
	+ thus "red < length (bs!k)"
	+ using 0 by fast
	+ qed
	+ {
	+ fix k' pre reds red
	+ assume a0: "((k', pre), reds) \<in> set gs"
	+ define n_pre where n_pre: "n_pre = build_forest'_measure (bs, \<omega>, k', pre, {pre})"
	+ have "k < length bs" "i < length (bs!k)" "I \<subseteq> {0..<length (bs!k)}" "i \<in> I"
	+ using 1(2) unfolding wf_trees_input_def by blast+
	+ have "k' < k" "pre < length (bs!k')"
	+ using sound_gs a0 by fastforce+
	+ have "length (bs!k') > 0"
	+ using \<open>pre < length (bs!k')\<close> by force
	+ hence gt0: "foldl (+) 0 (map length (take (k'+1) bs)) > 0"
	+ by (smt (verit, del_insts) foldl_add_nth \<open>k < length bs\<close> \<open>k' < k\<close> add_gr_0 order.strict_trans)
	+ have card_bound: "card I \<le> length (bs!k)"
	+ by (simp add: \<open>I \<subseteq> {0..<length (bs ! k)}\<close> subset_eq_atLeast0_lessThan_card)
	+ have "card I + (foldl (+) 0 (map length (take (Suc k') bs)) - Suc 0) =
	+ card I + foldl (+) 0 (map length (take (Suc k') bs)) - 1"
	+ by (metis Nat.add_diff_assoc One_nat_def Suc_eq_plus1 Suc_leI \<open>0 < foldl (+) 0 (map length (take (k' + 1) bs))\<close>)
	+ also have "... < card I + foldl (+) 0 (map length (take (Suc k') bs))"
	+ using gt0 by auto
	+ also have "... \<le> foldl (+) 0 (map length (take (Suc k') bs)) + length (bs!k)"
	+ using card_bound by simp
	+ also have "... \<le> foldl (+) 0 (map length (take (k+1) bs))"
	+ using foldl_add_nth_ge Suc_leI \<open>k < length bs\<close> \<open>k' < k\<close> by blast
	+ finally have "build_forest'_measure (bs, \<omega>, k, i, I) - build_forest'_measure (bs, \<omega>, k', pre, {pre}) > 0"
	+ by simp
	+ hence "P bs \<omega> k' pre {pre}"
	+ using 1(1) wf_trees_input_prered_pre[OF "1.prems"(1) entry pps ps' gs a0] zero_less_diff by blast
	+ }
	+ moreover {
	+ fix k' pre reds red
	+ assume a0: "((k', pre), reds) \<in> set gs"
	+ assume a1: "red \<in> set reds"
	+ define n_red where n_red: "n_red = build_forest'_measure (bs, \<omega>, k, red, I \<union> {red})"
	+ have "k < length bs" "i < length (bs!k)" "I \<subseteq> {0..<length (bs!k)}" "i \<in> I"
	+ using 1(2) unfolding wf_trees_input_def by blast+
	+ have "k' < k" "pre < length (bs!k')" "red < length (bs!k)"
	+ using a0 a1 sound_gs sound_gs2 by fastforce+
	+ have "red \<notin> I"
	+ using a0 a1 unfolding gs ps'
	+ by (smt (verit, best) group_by_forall_v_exists_k case_prodE case_prod_conv mem_Collect_eq set_filter)
	+ have card_bound: "card I \<le> length (bs!k)"
	+ by (simp add: \<open>I \<subseteq> {0..<length (bs ! k)}\<close> subset_eq_atLeast0_lessThan_card)
	+ have "length (bs!k') > 0"
	+ using \<open>pre < length (bs!k')\<close> by force
	+ hence gt0: "foldl (+) 0 (map length (take (k'+1) bs)) > 0"
	+ by (smt (verit, del_insts) foldl_add_nth \<open>k < length bs\<close> \<open>k' < k\<close> add_gr_0 order.strict_trans)
	+ have lt: "foldl (+) 0 (map length (take (Suc k') bs)) + length (bs!k) \<le> foldl (+) 0 (map length (take (k+1) bs))"
	+ using foldl_add_nth_ge Suc_leI \<open>k < length bs\<close> \<open>k' < k\<close> by blast
	+ have "card I + (foldl (+) 0 (map length (take (Suc k) bs)) - card (insert red I)) =
	+ card I + (foldl (+) 0 (map length (take (Suc k) bs)) - card I - 1)"
	+ using \<open>I \<subseteq> {0..<length (bs ! k)}\<close> \<open>red \<notin> I\<close> finite_subset by fastforce
	+ also have "... = foldl (+) 0 (map length (take (Suc k) bs)) - 1"
	+ using gt0 card_bound lt by force
	+ also have "... < foldl (+) 0 (map length (take (Suc k) bs))"
	+ using gt0 lt by auto
	+ finally have "build_forest'_measure (bs, \<omega>, k, i, I) - build_forest'_measure (bs, \<omega>, k, red, I \<union> {red}) > 0"
	+ by simp
	+ moreover have "(bs, \<omega>, k, red, I \<union> {red}) \<in> wf_trees_input"
	+ using wf_trees_input_prered_red[OF "1"(2) entry pps ps' gs] a0 a1 by blast
	+ ultimately have "P bs \<omega> k red (I \<union> {red})"
	+ using 1(1) zero_less_diff by blast
	+ }
	+ moreover have "(\<And>e pre. e = bs!k!i \<Longrightarrow> pointer e = Pre pre \<Longrightarrow> P bs \<omega> (k-1) pre {pre})"
	+ using entry pps by fastforce
	+ ultimately show ?thesis
	+ using assms(2) entry gs pointer.inject(2) pps ps' by presburger
	+ qed
	+qed
	+
	+lemma build_trees'_termination:
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ shows "\<exists>fs. build_trees' bs \<omega> k i I = Some fs \<and> (\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss)"
	+proof -
	+ have "\<exists>fs. build_trees' bs \<omega> k i I = Some fs \<and> (\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss)"
	+ apply (induction rule: build_trees'_induct[OF assms(1)])
	+ subgoal premises IH for bs \<omega> k i I
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>k' pre red reds. pointer e = PreRed (k', pre, red) reds"
	+ by (metis pointer.exhaust surj_pair)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ have "build_trees' bs \<omega> k i I = Some ([FBranch (item_rule_head (item e)) []])"
	+ using build_forest'_simps(1) Null entry by simp
	+ thus ?thesis
	+ by simp
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ obtain fs where fs: "build_trees' bs \<omega> (k-1) pre {pre} = Some fs"
	+ "\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss"
	+ using IH(1) entry pre by blast
	+ let ?g = "\<lambda>f. case f of FLeaf a \<Rightarrow> None
	+ \| FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]]))"
	+ have simp: "build_trees' bs \<omega> k i I = those (map ?g fs)"
	+ using build_forest'_simps(3) entry pre fs by blast
	+ moreover have "\<forall>f \<in> set (map ?g fs). \<exists>a. f = Some a"
	+ using fs(2) by auto
	+ ultimately obtain fs' where fs': "build_trees' bs \<omega> k i I = Some fs'"
	+ using those_Some by (smt (verit, best))
	+ moreover have "\<forall>f \<in> set fs'. \<exists>N fss. f = FBranch N fss"
	+ proof standard
	+ fix f
	+ assume "f \<in> set fs'"
	+ then obtain x where "x \<in> set fs" "Some f \<in> set (map ?g fs)"
	+ using those_map_exists by (metis (no_types, lifting) fs' simp)
	+ thus "\<exists>N fss. f = FBranch N fss"
	+ using fs(2) by auto
	+ qed
	+ ultimately show ?thesis
	+ by blast
	+ next
	+ case PreRed
	+ then obtain p ps where pps: "pointer e = PreRed p ps"
	+ by blast
	+ define ps' where ps': "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ define gs where gs: "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ let ?g = "\<lambda>((k', pre), reds).
	+ do {
	+ pres \<leftarrow> build_trees' bs \<omega> k' pre {pre};
	+ rss \<leftarrow> those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds);
	+ those (map (\<lambda>f.
	+ case f of
	+ FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss]))
	+ \| _ \<Rightarrow> None \<comment>\<open>impossible case\<close>
	+ ) pres)
	+ }"
	+ have simp: "build_trees' bs \<omega> k i I = map_option concat (those (map ?g gs))"
	+ using entry pps ps' gs by (subst build_trees'.simps) (auto simp del: filter.simps)
	+ have "\<forall>fso \<in> set (map ?g gs). \<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss)"
	+ proof standard
	+ fix fso
	+ assume "fso \<in> set (map ?g gs)"
	+ moreover have "\<forall>ps \<in> set gs. \<exists>fs. ?g ps = Some fs \<and> (\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss)"
	+ proof standard
	+ fix ps
	+ assume "ps \<in> set gs"
	+ then obtain k' pre reds where *: "((k', pre), reds) \<in> set gs" "((k', pre), reds) = ps"
	+ by (metis surj_pair)
	+ then obtain pres where pres: "build_trees' bs \<omega> k' pre {pre} = Some pres"
	+ "\<forall>f \<in> set pres. \<exists>N fss. f = FBranch N fss"
	+ using IH(2) entry pps ps' gs by blast
	+ have "\<forall>f \<in> set (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds). \<exists>a. f = Some a"
	+ using IH(3)[OF entry pps ps' gs *(1)] by auto
	+ then obtain rss where rss: "Some rss = those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds)"
	+ using those_Some by (metis (full_types))
	+ let ?h = "\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss])) \| _ \<Rightarrow> None"
	+ have "\<forall>x \<in> set (map ?h pres). \<exists>a. x = Some a"
	+ using pres(2) by auto
	+ then obtain fs where fs: "Some fs = those (map ?h pres)"
	+ using those_Some by (smt (verit, best))
	+ have "\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss"
	+ proof standard
	+ fix f
	+ assume *: "f \<in> set fs"
	+ hence "\<exists>x. x \<in> set pres \<and> Some f \<in> set (map ?h pres)"
	+ using those_map_exists[OF fs *] by blast
	+ then obtain x where x: "x \<in> set pres \<and> Some f \<in> set (map ?h pres)"
	+ by blast
	+ thus "\<exists>N fss. f = FBranch N fss"
	+ using pres(2) by auto
	+ qed
	+ moreover have "?g ps = Some fs"
	+ using fs pres rss * by (auto, metis bind.bind_lunit)
	+ ultimately show "\<exists>fs. ?g ps = Some fs \<and> (\<forall>f\<in>set fs. \<exists>N fss. f = FBranch N fss)"
	+ by blast
	+ qed
	+ ultimately show "\<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss)"
	+ using map_Some_P by auto
	+ qed
	+ then obtain fss where "those (map ?g gs) = Some fss" "\<forall>fs \<in> set fss. \<forall>f \<in> set fs. \<exists>N fss. f = FBranch N fss"
	+ using those_Some_P by blast
	+ hence "build_trees' bs \<omega> k i I = Some (concat fss)" "\<forall>f \<in> set (concat fss). \<exists>N fss. f = FBranch N fss"
	+ using simp by auto
	+ thus ?thesis
	+ by blast
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ by blast
	+qed
	+
	+lemma wf_item_tree_build_trees':
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "wf_bins \<G> \<omega> bs"
	+ assumes "k < length bs" "i < length (bs!k)"
	+ assumes "build_trees' bs \<omega> k i I = Some fs"
	+ assumes "f \<in> set fs"
	+ assumes "t \<in> set (trees f)"
	+ shows "wf_item_tree \<G> (item (bs!k!i)) t"
	+proof -
	+ have "wf_item_tree \<G> (item (bs!k!i)) t"
	+ using assms
	+ apply (induction arbitrary: fs f t rule: build_trees'_induct[OF assms(1)])
	+ subgoal premises prems for bs \<omega> k i I fs f t
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>p ps. pointer e = PreRed p ps"
	+ by (metis pointer.exhaust)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ hence simp: "build_trees' bs \<omega> k i I = Some ([FBranch (item_rule_head (item e)) []])"
	+ using entry by simp
	+ moreover have "f = FBranch (item_rule_head (item e)) []"
	+ using build_forest'_simps(1) Null prems(8,9) entry by auto
	+ ultimately have simp: "t = Branch (item_rule_head (item e)) []"
	+ using prems(10) by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ hence "predicts (item e)"
	+ using Null prems(6,7) nth_mem entry unfolding sound_ptrs_def sound_null_ptr_def by blast
	+ hence "item_dot (item e) = 0"
	+ unfolding predicts_def by blast
	+ thus ?thesis
	+ using simp entry by simp
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ have sound: "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ have scans: "scans \<omega> k (item (bs!(k-1)!pre)) (item e)"
	+ using entry pre prems(6-7) \<open>sound_ptrs \<omega> bs\<close> unfolding sound_ptrs_def sound_pre_ptr_def by simp
	+ hence *:
	+ "item_rule_head (item (bs!(k-1)!pre)) = item_rule_head (item e)"
	+ "item_rule_body (item (bs!(k-1)!pre)) = item_rule_body (item e)"
	+ "item_dot (item (bs!(k-1)!pre)) + 1 = item_dot (item e)"
	+ "next_symbol (item (bs!(k-1)!pre)) = Some (\<omega>!(k-1))"
	+ unfolding scans_def inc_item_def by (simp_all add: item_rule_head_def item_rule_body_def)
	+ have wf: "(bs, \<omega>, k-1, pre, {pre}) \<in> wf_trees_input"
	+ using entry pre prems(4) wf_trees_input_pre by blast
	+ then obtain pres where pres: "build_trees' bs \<omega> (k-1) pre {pre} = Some pres"
	+ "\<forall>f \<in> set pres. \<exists>N fss. f = FBranch N fss"
	+ using build_trees'_termination wf by blast
	+ let ?g = "\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) \| _ \<Rightarrow> None"
	+ have "build_trees' bs \<omega> k i I = those (map ?g pres)"
	+ using entry pre pres by simp
	+ hence fs: "Some fs = those (map ?g pres)"
	+ using prems(8) by simp
	+ then obtain f_pre N fss where Nfss: "f = FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])"
	+ "f_pre = FBranch N fss" "f_pre \<in> set pres"
	+ using those_map_FBranch_only fs pres(2) prems(9) by blast
	+ define tss where tss: "tss = map (\<lambda>fs. concat (map (\<lambda>f. trees f) fs)) fss"
	+ have "trees (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) =
	+ map (\<lambda>ts. Branch N ts) [ ts @ [Leaf (\<omega>!(k-1))] . ts <- combinations tss ]"
	+ by (subst tss, subst trees_append_single_singleton, simp)
	+ moreover have "t \<in> set (trees (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])))"
	+ using Nfss(1) prems(10) by blast
	+ ultimately obtain ts where ts: "t = Branch N (ts @ [Leaf (\<omega>!(k-1))]) \<and> ts \<in> set (combinations tss)"
	+ by auto
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ hence "pre < length (bs!(k-1))"
	+ using entry pre prems(6,7) unfolding sound_ptrs_def sound_pre_ptr_def by (metis nth_mem)
	+ moreover have "k - 1 < length bs"
	+ by (simp add: prems(6) less_imp_diff_less)
	+ moreover have "Branch N ts \<in> set (trees (FBranch N fss))"
	+ using ts tss by simp
	+ ultimately have IH: "wf_item_tree \<G> (item (bs!(k-1)!pre)) (Branch N ts)"
	+ using prems(1,2,4,5) entry pre Nfss(2,3) wf pres(1) by blast
	+ have "map root_tree (ts @ [Leaf (\<omega>!(k-1))]) = map root_tree ts @ [\<omega>!(k-1)]"
	+ by simp
	+ also have "... = take (item_dot (item (bs!(k-1)!pre))) (item_rule_body (item (bs!(k-1)!pre))) @ [\<omega>!(k-1)]"
	+ using IH by simp
	+ also have "... = take (item_dot (item (bs!(k-1)!pre))) (item_rule_body (item e)) @ [\<omega>!(k-1)]"
	+ using *(2) by simp
	+ also have "... = take (item_dot (item e)) (item_rule_body (item e))"
	+ using *(2-4) by (auto simp: next_symbol_def is_complete_def split: if_splits; metis leI take_Suc_conv_app_nth)
	+ finally have "map root_tree (ts @ [Leaf (\<omega>!(k-1))]) = take (item_dot (item e)) (item_rule_body (item e))" .
	+ hence "wf_item_tree \<G> (item e) (Branch N (ts @ [Leaf (\<omega>!(k-1))]))"
	+ using IH *(1) by simp
	+ thus ?thesis
	+ using ts entry by fastforce
	+ next
	+ case PreRed
	+ then obtain p ps where prered: "pointer e = PreRed p ps"
	+ by blast
	+ define ps' where ps': "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ define gs where gs: "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ let ?g = "\<lambda>((k', pre), reds).
	+ do {
	+ pres \<leftarrow> build_trees' bs \<omega> k' pre {pre};
	+ rss \<leftarrow> those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds);
	+ those (map (\<lambda>f.
	+ case f of
	+ FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss]))
	+ \| _ \<Rightarrow> None \<comment>\<open>impossible case\<close>
	+ ) pres)
	+ }"
	+ have simp: "build_trees' bs \<omega> k i I = map_option concat (those (map ?g gs))"
	+ using entry prered ps' gs by (subst build_trees'.simps) (auto simp del: filter.simps)
	+ have "\<forall>fso \<in> set (map ?g gs). \<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t)"
	+ proof standard
	+ fix fso
	+ assume "fso \<in> set (map ?g gs)"
	+ moreover have "\<forall>ps \<in> set gs. \<exists>fs. ?g ps = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t)"
	+ proof standard
	+ fix g
	+ assume "g \<in> set gs"
	+ then obtain k' pre reds where g: "((k', pre), reds) \<in> set gs" "((k', pre), reds) = g"
	+ by (metis surj_pair)
	+ moreover have wf_pre: "(bs, \<omega>, k', pre, {pre}) \<in> wf_trees_input"
	+ using wf_trees_input_prered_pre[OF prems(4) entry prered ps' gs g(1)] by blast
	+ ultimately obtain pres where pres: "build_trees' bs \<omega> k' pre {pre} = Some pres"
	+ "\<forall>f_pre \<in> set pres. \<exists>N fss. f_pre = FBranch N fss"
	+ using build_trees'_termination by blast
	+ have wf_reds: "\<forall>red \<in> set reds. (bs, \<omega>, k, red, I \<union> {red}) \<in> wf_trees_input"
	+ using wf_trees_input_prered_red[OF prems(4) entry prered ps' gs g(1)] by blast
	+ hence "\<forall>f \<in> set (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds). \<exists>a. f = Some a"
	+ using build_trees'_termination by fastforce
	+ then obtain rss where rss: "Some rss = those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds)"
	+ using those_Some by (metis (full_types))
	+ let ?h = "\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss])) \| _ \<Rightarrow> None"
	+ have "\<forall>x \<in> set (map ?h pres). \<exists>a. x = Some a"
	+ using pres(2) by auto
	+ then obtain fs where fs: "Some fs = those (map ?h pres)"
	+ using those_Some by (smt (verit, best))
	+ have "\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t"
	+ proof (standard, standard)
	+ fix f t
	+ assume ft: "f \<in> set fs" "t \<in> set (trees f)"
	+ hence "\<exists>x. x \<in> set pres \<and> Some f \<in> set (map ?h pres)"
	+ using those_map_exists[OF fs ft(1)] by blast
	+ then obtain f_pre N fss where f_pre: "f_pre \<in> set pres" "f_pre = FBranch N fss"
	+ "f = FBranch N (fss @ [concat rss])"
	+ using pres(2) by force
	+ define tss where tss: "tss = map (\<lambda>fs. concat (map (\<lambda>f. trees f) fs)) fss"
	+ have "trees (FBranch N (fss @ [concat rss])) =
	+ map (\<lambda>ts. Branch N ts) [ ts0 @ ts1 . ts0 <- combinations tss,
	+ ts1 <- combinations [concat (map (\<lambda>f. trees f) (concat rss)) ] ]"
	+ by (subst tss, subst trees_append_singleton, simp)
	+ moreover have "t \<in> set (trees (FBranch N (fss @ [concat rss])))"
	+ using ft(2) f_pre(3) by blast
	+ ultimately obtain ts0 ts1 where tsx: "t = Branch N (ts0 @ [ts1])" "ts0 \<in> set (combinations tss)"
	+ "ts1 \<in> set (concat (map (\<lambda>f. trees f) (concat rss)))"
	+ by fastforce
	+ then obtain f_red where f_red: "f_red \<in> set (concat rss)" "ts1 \<in> set (trees f_red)"
	+ by auto
	+ obtain fs_red red where red: "Some fs_red = build_trees' bs \<omega> k red (I \<union> {red})"
	+ "f_red \<in> set fs_red" "red \<in> set reds"
	+ using f_red(1) rss those_map_Some_concat_exists by fast
	+ then obtain N_red fss_red where "f_red = FBranch N_red fss_red"
	+ using build_trees'_termination wf_reds by (metis option.inject)
	+ then obtain ts where ts: "Branch N_red ts = ts1"
	+ using tsx(3) f_red by auto
	+ have "(k', pre, red) \<in> set ps'"
	+ using group_by_forall_v_exists_k \<open>((k', pre), reds) \<in> set gs\<close> gs \<open>red \<in> set reds\<close> by fast
	+ hence mem: "(k', pre, red) \<in> set (p#ps)"
	+ using ps' by (metis filter_set member_filter)
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) wf_trees_input_def by fastforce
	+ have bounds: "k' < k" "pre < length (bs!k')" "red < length (bs!k)"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by (meson mem nth_mem)+
	+ have completes: "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by (metis mem nth_mem)
	+ have transform:
	+ "item_rule_head (item (bs!k'!pre)) = item_rule_head (item e)"
	+ "item_rule_body (item (bs!k'!pre)) = item_rule_body (item e)"
	+ "item_dot (item (bs!k'!pre)) + 1 = item_dot (item e)"
	+ "next_symbol (item (bs!k'!pre)) = Some (item_rule_head (item (bs!k!red)))"
	+ "is_complete (item (bs!k!red))"
	+ using completes unfolding completes_def inc_item_def
	+ by (auto simp: item_rule_head_def item_rule_body_def is_complete_def)
	+ have "Branch N ts0 \<in> set (trees (FBranch N fss))"
	+ using tss tsx(2) by simp
	+ hence IH_pre: "wf_item_tree \<G> (item (bs!k'!pre)) (Branch N ts0)"
	+ using prems(2)[OF entry prered ps' gs \<open>((k', pre), reds) \<in> set gs\<close> wf_pre prems(5)]
	+ pres(1) f_pre f_pre(3) bounds(1,2) prems(6) by fastforce
	+ have IH_r: "wf_item_tree \<G> (item (bs!k!red)) (Branch N_red ts)"
	+ using prems(3)[OF entry prered ps' gs \<open>((k', pre), reds) \<in> set gs\<close> \<open>red \<in> set reds\<close> _ prems(5)]
	+ bounds(3) f_red(2) red ts wf_reds prems(6) by metis
	+ have "map root_tree (ts0 @ [Branch N_red ts]) = map root_tree ts0 @ [root_tree (Branch N_red ts)]"
	+ by simp
	+ also have "... = take (item_dot (item (bs!k'!pre))) (item_rule_body (item (bs!k'!pre))) @ [root_tree (Branch N_red ts)]"
	+ using IH_pre by simp
	+ also have "... = take (item_dot (item (bs!k'!pre))) (item_rule_body (item (bs!k'!pre))) @ [item_rule_head (item (bs!k!red))]"
	+ using IH_r by simp
	+ also have "... = take (item_dot (item e)) (item_rule_body (item e))"
	+ using transform by (auto simp: next_symbol_def is_complete_def split: if_splits; metis leI take_Suc_conv_app_nth)
	+ finally have roots: "map root_tree (ts0 @ [Branch N_red ts]) = take (item_dot (item e)) (item_rule_body (item e))" .
	+ have "wf_item \<G> \<omega> (item (bs!k!red))"
	+ using prems(5,6) bounds(3) unfolding wf_bins_def wf_bin_def wf_bin_items_def by (auto simp: items_def)
	+ moreover have "N_red = item_rule_head (item (bs!k!red))"
	+ using IH_r by fastforce
	+ moreover have "map root_tree ts = item_rule_body (item (bs!k!red))"
	+ using IH_r transform(5) by (auto simp: is_complete_def)
	+ ultimately have "\<exists>r \<in> set (\<RR> \<G>). N_red = rule_head r \<and> map root_tree ts = rule_body r"
	+ unfolding wf_item_def item_rule_body_def item_rule_head_def by blast
	+ hence "wf_rule_tree \<G> (Branch N_red ts)"
	+ using IH_r by simp
	+ hence "wf_item_tree \<G> (item (bs!k!i)) (Branch N (ts0 @ [Branch N_red ts]))"
	+ using transform(1) roots IH_pre entry by simp
	+ thus "wf_item_tree \<G> (item (bs!k!i)) t"
	+ using tsx(1) red ts by blast
	+ qed
	+ moreover have "?g g = Some fs"
	+ using fs pres rss g by (auto, metis bind.bind_lunit)
	+ ultimately show "\<exists>fs. ?g g = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t)"
	+ by blast
	+ qed
	+ ultimately show "\<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t)"
	+ using map_Some_P by auto
	+ qed
	+ then obtain fss where "those (map ?g gs) = Some fss" "\<forall>fs \<in> set fss. \<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t"
	+ using those_Some_P by blast
	+ hence "build_trees' bs \<omega> k i I = Some (concat fss)" "\<forall>f \<in> set (concat fss). \<forall>t \<in> set (trees f). wf_item_tree \<G> (item (bs!k!i)) t"
	+ using simp by auto
	+ thus ?thesis
	+ using prems(8-10) by auto
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ by blast
	+qed
	+
	+lemma wf_yield_tree_build_trees':
	+ assumes "(bs, \<omega>, k, i, I) \<in> wf_trees_input"
	+ assumes "wf_bins \<G> \<omega> bs"
	+ assumes "k < length bs" "i < length (bs!k)" "k \<le> length \<omega>"
	+ assumes "build_trees' bs \<omega> k i I = Some fs"
	+ assumes "f \<in> set fs"
	+ assumes "t \<in> set (trees f)"
	+ shows "wf_yield_tree \<omega> (item (bs!k!i)) t"
	+proof -
	+ have "wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ using assms
	+ apply (induction arbitrary: fs f t rule: build_trees'_induct[OF assms(1)])
	+ subgoal premises prems for bs \<omega> k i I fs f t
	+ proof -
	+ define e where entry: "e = bs!k!i"
	+ consider (Null) "pointer e = Null"
	+ \| (Pre) "\<exists>pre. pointer e = Pre pre"
	+ \| (PreRed) "\<exists>p ps. pointer e = PreRed p ps"
	+ by (metis pointer.exhaust)
	+ thus ?thesis
	+ proof cases
	+ case Null
	+ hence simp: "build_trees' bs \<omega> k i I = Some ([FBranch (item_rule_head (item e)) []])"
	+ using entry by simp
	+ moreover have "f = FBranch (item_rule_head (item e)) []"
	+ using build_forest'_simps(1) Null prems(9,10) entry by auto
	+ ultimately have simp: "t = Branch (item_rule_head (item e)) []"
	+ using prems(11) by simp
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ hence "predicts (item e)"
	+ using Null prems(6,7) nth_mem entry unfolding sound_ptrs_def sound_null_ptr_def by blast
	+ thus ?thesis
	+ unfolding wf_yield_tree_def predicts_def using simp entry by (auto simp: slice_empty)
	+ next
	+ case Pre
	+ then obtain pre where pre: "pointer e = Pre pre"
	+ by blast
	+ have sound: "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ hence bounds: "k > 0" "pre < length (bs!(k-1))"
	+ using entry pre prems(6,7) unfolding sound_ptrs_def sound_pre_ptr_def by (metis nth_mem)+
	+ have scans: "scans \<omega> k (item (bs!(k-1)!pre)) (item e)"
	+ using entry pre prems(6-7) \<open>sound_ptrs \<omega> bs\<close> unfolding sound_ptrs_def sound_pre_ptr_def by simp
	+ have wf: "(bs, \<omega>, k-1, pre, {pre}) \<in> wf_trees_input"
	+ using entry pre prems(4) wf_trees_input_pre by blast
	+ then obtain pres where pres: "build_trees' bs \<omega> (k-1) pre {pre} = Some pres"
	+ "\<forall>f \<in> set pres. \<exists>N fss. f = FBranch N fss"
	+ using build_trees'_termination wf by blast
	+ let ?g = "\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) \| _ \<Rightarrow> None"
	+ have "build_trees' bs \<omega> k i I = those (map ?g pres)"
	+ using entry pre pres by simp
	+ hence fs: "Some fs = those (map ?g pres)"
	+ using prems(9) by simp
	+ then obtain f_pre N fss where Nfss: "f = FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])"
	+ "f_pre = FBranch N fss" "f_pre \<in> set pres"
	+ using those_map_FBranch_only fs pres(2) prems(10) by blast
	+ define tss where tss: "tss = map (\<lambda>fs. concat (map (\<lambda>f. trees f) fs)) fss"
	+ have "trees (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])) =
	+ map (\<lambda>ts. Branch N ts) [ ts @ [Leaf (\<omega>!(k-1))] . ts <- combinations tss ]"
	+ by (subst tss, subst trees_append_single_singleton, simp)
	+ moreover have "t \<in> set (trees (FBranch N (fss @ [[FLeaf (\<omega>!(k-1))]])))"
	+ using Nfss(1) prems(11) by blast
	+ ultimately obtain ts where ts: "t = Branch N (ts @ [Leaf (\<omega>!(k-1))]) \<and> ts \<in> set (combinations tss)"
	+ by auto
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) unfolding wf_trees_input_def by blast
	+ hence "pre < length (bs!(k-1))"
	+ using entry pre prems(6,7) unfolding sound_ptrs_def sound_pre_ptr_def by (metis nth_mem)
	+ moreover have "k-1 < length bs"
	+ by (simp add: prems(6) less_imp_diff_less)
	+ moreover have "Branch N ts \<in> set (trees (FBranch N fss))"
	+ using ts tss by simp
	+ ultimately have IH: "wf_yield_tree \<omega> (item (bs!(k-1)!pre)) (Branch N ts)"
	+ using prems(1,2,4,5,8) entry pre Nfss(2,3) wf pres(1) by simp
	+ have transform:
	+ "item_origin (item (bs!(k-1)!pre)) \<le> item_end (item (bs!(k-1)!pre))"
	+ "item_end (item (bs!(k-1)!pre)) = k-1"
	+ "item_end (item e) = k"
	+ using entry prems(5,6,7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def wf_item_def
	+ by (auto, meson less_imp_diff_less nth_mem)
	+ have "yield_tree t = concat (map yield_tree (ts @ [Leaf (\<omega>!(k-1))]))"
	+ by (simp add: ts)
	+ also have "... = concat (map yield_tree ts) @ [\<omega>!(k-1)]"
	+ by simp
	+ also have "... = slice (item_origin (item (bs!(k-1)!pre))) (item_end (item (bs!(k-1)!pre))) \<omega> @ [\<omega>!(k-1)]"
	+ using IH by (simp add: wf_yield_tree_def)
	+ also have "... = slice (item_origin (item (bs!(k-1)!pre))) (item_end (item (bs!(k-1)!pre)) + 1) \<omega>"
	+ using slice_append_nth transform \<open>k > 0\<close> prems(8)
	+ by (metis diff_less le_eq_less_or_eq less_imp_diff_less less_numeral_extra(1))
	+ also have "... = slice (item_origin (item e)) (item_end (item (bs!(k-1)!pre)) + 1) \<omega>"
	+ using scans unfolding scans_def inc_item_def by simp
	+ also have "... = slice (item_origin (item e)) k \<omega>"
	+ using scans transform unfolding scans_def by (metis Suc_diff_1 Suc_eq_plus1 bounds(1))
	+ also have "... = slice (item_origin (item e)) (item_end (item e)) \<omega>"
	+ using transform by auto
	+ finally show ?thesis
	+ using wf_yield_tree_def entry by blast
	+ next
	+ case PreRed
	+ then obtain p ps where prered: "pointer e = PreRed p ps"
	+ by blast
	+ define ps' where ps': "ps' = filter (\<lambda>(k', pre, red). red \<notin> I) (p#ps)"
	+ define gs where gs: "gs = group_by (\<lambda>(k', pre, red). (k', pre)) (\<lambda>(k', pre, red). red) ps'"
	+ let ?g = "\<lambda>((k', pre), reds).
	+ do {
	+ pres \<leftarrow> build_trees' bs \<omega> k' pre {pre};
	+ rss \<leftarrow> those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds);
	+ those (map (\<lambda>f.
	+ case f of
	+ FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss]))
	+ \| _ \<Rightarrow> None \<comment>\<open>impossible case\<close>
	+ ) pres)
	+ }"
	+ have simp: "build_trees' bs \<omega> k i I = map_option concat (those (map ?g gs))"
	+ using entry prered ps' gs by (subst build_trees'.simps) (auto simp del: filter.simps)
	+ have "\<forall>fso \<in> set (map ?g gs). \<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t)"
	+ proof standard
	+ fix fso
	+ assume "fso \<in> set (map ?g gs)"
	+ moreover have "\<forall>ps \<in> set gs. \<exists>fs. ?g ps = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t)"
	+ proof standard
	+ fix g
	+ assume "g \<in> set gs"
	+ then obtain k' pre reds where g: "((k', pre), reds) \<in> set gs" "((k', pre), reds) = g"
	+ by (metis surj_pair)
	+ moreover have wf_pre: "(bs, \<omega>, k', pre, {pre}) \<in> wf_trees_input"
	+ using wf_trees_input_prered_pre[OF prems(4) entry prered ps' gs g(1)] by blast
	+ ultimately obtain pres where pres: "build_trees' bs \<omega> k' pre {pre} = Some pres"
	+ "\<forall>f_pre \<in> set pres. \<exists>N fss. f_pre = FBranch N fss"
	+ using build_trees'_termination by blast
	+ have wf_reds: "\<forall>red \<in> set reds. (bs, \<omega>, k, red, I \<union> {red}) \<in> wf_trees_input"
	+ using wf_trees_input_prered_red[OF prems(4) entry prered ps' gs g(1)] by blast
	+ hence "\<forall>f \<in> set (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds). \<exists>a. f = Some a"
	+ using build_trees'_termination by fastforce
	+ then obtain rss where rss: "Some rss = those (map (\<lambda>red. build_trees' bs \<omega> k red (I \<union> {red})) reds)"
	+ using those_Some by (metis (full_types))
	+ let ?h = "\<lambda>f. case f of FBranch N fss \<Rightarrow> Some (FBranch N (fss @ [concat rss])) \| _ \<Rightarrow> None"
	+ have "\<forall>x \<in> set (map ?h pres). \<exists>a. x = Some a"
	+ using pres(2) by auto
	+ then obtain fs where fs: "Some fs = those (map ?h pres)"
	+ using those_Some by (smt (verit, best))
	+ have "\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ proof (standard, standard)
	+ fix f t
	+ assume ft: "f \<in> set fs" "t \<in> set (trees f)"
	+ hence "\<exists>x. x \<in> set pres \<and> Some f \<in> set (map ?h pres)"
	+ using those_map_exists[OF fs ft(1)] by blast
	+ then obtain f_pre N fss where f_pre: "f_pre \<in> set pres" "f_pre = FBranch N fss"
	+ "f = FBranch N (fss @ [concat rss])"
	+ using pres(2) by force
	+ define tss where tss: "tss = map (\<lambda>fs. concat (map (\<lambda>f. trees f) fs)) fss"
	+ have "trees (FBranch N (fss @ [concat rss])) =
	+ map (\<lambda>ts. Branch N ts) [ ts0 @ ts1 . ts0 <- combinations tss,
	+ ts1 <- combinations [concat (map (\<lambda>f. trees f) (concat rss)) ] ]"
	+ by (subst tss, subst trees_append_singleton, simp)
	+ moreover have "t \<in> set (trees (FBranch N (fss @ [concat rss])))"
	+ using ft(2) f_pre(3) by blast
	+ ultimately obtain ts0 ts1 where tsx: "t = Branch N (ts0 @ [ts1])" "ts0 \<in> set (combinations tss)"
	+ "ts1 \<in> set (concat (map (\<lambda>f. trees f) (concat rss)))"
	+ by fastforce
	+ then obtain f_red where f_red: "f_red \<in> set (concat rss)" "ts1 \<in> set (trees f_red)"
	+ by auto
	+ obtain fs_red red where red: "Some fs_red = build_trees' bs \<omega> k red (I \<union> {red})"
	+ "f_red \<in> set fs_red" "red \<in> set reds"
	+ using f_red(1) rss those_map_Some_concat_exists by fast
	+ then obtain N_red fss_red where "f_red = FBranch N_red fss_red"
	+ using build_trees'_termination wf_reds by (metis option.inject)
	+ then obtain ts where ts: "Branch N_red ts = ts1"
	+ using tsx(3) f_red by auto
	+ have "(k', pre, red) \<in> set ps'"
	+ using group_by_forall_v_exists_k \<open>((k', pre), reds) \<in> set gs\<close> gs \<open>red \<in> set reds\<close> by fast
	+ hence mem: "(k', pre, red) \<in> set (p#ps)"
	+ using ps' by (metis filter_set member_filter)
	+ have "sound_ptrs \<omega> bs"
	+ using prems(4) wf_trees_input_def by fastforce
	+ have bounds: "k' < k" "pre < length (bs!k')" "red < length (bs!k)"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by (meson mem nth_mem)+
	+ have completes: "completes k (item (bs!k'!pre)) (item e) (item (bs!k!red))"
	+ using prered entry prems(6,7) \<open>sound_ptrs \<omega> bs\<close>
	+ unfolding sound_ptrs_def sound_prered_ptr_def by (metis mem nth_mem)
	+ have transform:
	+ "item_rule_head (item (bs!k'!pre)) = item_rule_head (item e)"
	+ "item_rule_body (item (bs!k'!pre)) = item_rule_body (item e)"
	+ "item_dot (item (bs!k'!pre)) + 1 = item_dot (item e)"
	+ "next_symbol (item (bs!k'!pre)) = Some (item_rule_head (item (bs!k!red)))"
	+ "is_complete (item (bs!k!red))"
	+ using completes unfolding completes_def inc_item_def
	+ by (auto simp: item_rule_head_def item_rule_body_def is_complete_def)
	+ have "Branch N ts0 \<in> set (trees (FBranch N fss))"
	+ using tss tsx(2) by simp
	+ hence IH_pre: "wf_yield_tree \<omega> (item (bs!k'!pre)) (Branch N ts0)"
	+ using prems(2)[OF entry prered ps' gs \<open>((k', pre), reds) \<in> set gs\<close> wf_pre prems(5)]
	+ pres(1) f_pre f_pre(3) bounds(1,2) prems(6,8) by simp
	+ have IH_r: "wf_yield_tree \<omega> (item (bs!k!red)) (Branch N_red ts)"
	+ using prems(3)[OF entry prered ps' gs \<open>((k', pre), reds) \<in> set gs\<close> \<open>red \<in> set reds\<close> _ prems(5)]
	+ bounds(3) f_red(2) red ts wf_reds prems(6,8) by metis
	+ have wf1:
	+ "item_origin (item (bs!k'!pre)) \<le> item_end (item (bs!k'!pre))"
	+ "item_origin (item (bs!k!red)) \<le> item_end (item (bs!k!red))"
	+ using prems(5-7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def wf_item_def
	+ by (metis length_map nth_map nth_mem order_less_trans)+
	+ have wf2:
	+ "item_end (item (bs!k!red)) = k"
	+ "item_end (item (bs!k!i)) = k"
	+ using prems(5-7) bounds unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def by simp_all
	+ have "yield_tree t = concat (map yield_tree (ts0 @ [Branch N_red ts]))"
	+ by (simp add: ts tsx(1))
	+ also have "... = concat (map yield_tree ts0) @ yield_tree (Branch N_red ts)"
	+ by simp
	+ also have "... = slice (item_origin (item (bs!k'!pre))) (item_end (item (bs!k'!pre))) \<omega> @
	+ slice (item_origin (item (bs!k!red))) (item_end (item (bs!k!red))) \<omega>"
	+ using IH_pre IH_r by (simp add: wf_yield_tree_def)
	+ also have "... = slice (item_origin (item (bs!k'!pre))) (item_end (item (bs!k!red))) \<omega>"
	+ using slice_concat wf1 completes_def completes by (metis (no_types, lifting))
	+ also have "... = slice (item_origin (item e)) (item_end (item (bs!k!red))) \<omega>"
	+ using completes unfolding completes_def inc_item_def by simp
	+ also have "... = slice (item_origin (item e)) (item_end (item e)) \<omega>"
	+ using wf2 entry by presburger
	+ finally show "wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ using wf_yield_tree_def entry by blast
	+ qed
	+ moreover have "?g g = Some fs"
	+ using fs pres rss g by (auto, metis bind.bind_lunit)
	+ ultimately show "\<exists>fs. ?g g = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t)"
	+ by blast
	+ qed
	+ ultimately show "\<exists>fs. fso = Some fs \<and> (\<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t)"
	+ using map_Some_P by auto
	+ qed
	+ then obtain fss where "those (map ?g gs) = Some fss" "\<forall>fs \<in> set fss. \<forall>f \<in> set fs. \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ using those_Some_P by blast
	+ hence "build_trees' bs \<omega> k i I = Some (concat fss)" "\<forall>f \<in> set (concat fss). \<forall>t \<in> set (trees f). wf_yield_tree \<omega> (item (bs!k!i)) t"
	+ using simp by auto
	+ thus ?thesis
	+ using prems(9-11) by auto
	+ qed
	+ qed
	+ done
	+ thus ?thesis
	+ using assms(2) by blast
	+qed
	+
	+theorem wf_rule_root_yield_tree_build_trees:
	+ assumes "wf_bins \<G> \<omega> bs" "sound_ptrs \<omega> bs" "length bs = length \<omega> + 1"
	+ assumes "build_trees \<G> \<omega> bs = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "wf_rule_tree \<G> t \<and> root_tree t = \<SS> \<G> \<and> yield_tree t = \<omega>"
	+proof -
	+ let ?k = "length bs - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items (bs!?k))"
	+ have #: "Some fs = map_option concat (those (map (\<lambda>(_, i). build_trees' bs \<omega> ?k i {i}) finished))"
	+ using assms(4) build_trees_def finished_def by (metis (full_types))
	+ then obtain fss fs' where fss: "Some fss = those (map (\<lambda>(_, i). build_trees' bs \<omega> ?k i {i}) finished)"
	+ "fs' \<in> set fss" "f \<in> set fs'"
	+ using map_option_concat_those_map_exists assms(5) by fastforce
	+ then obtain x i where *: "(x,i) \<in> set finished" "Some fs' = build_trees' bs \<omega> (length bs - 1) i {i}"
	+ using those_map_exists[OF fss(1,2)] by auto
	+ have k: "?k < length bs" "?k \<le> length \<omega>"
	+ using assms(3) by simp_all
	+ have i: "i < length (bs!?k)"
	+ using index_filter_with_index_lt_length * items_def finished_def by (metis (no_types, opaque_lifting) length_map)
	+ have x: "x = item (bs!?k!i)"
	+ using * i filter_with_index_nth items_def nth_map finished_def assms(3) by metis
	+ have finished: "is_finished \<G> \<omega> x"
	+ using * filter_with_index_P finished_def by metis
	+ have "{i} \<subseteq> {0..<length (bs!?k)}"
	+ using atLeastLessThan_iff i by blast
	+ hence wf: "(bs, \<omega>, ?k, i, {i}) \<in> wf_trees_input"
	+ unfolding wf_trees_input_def using assms(2) i k(1) by simp
	+ hence wf_item_tree: "wf_item_tree \<G> (item (bs!?k!i)) t"
	+ using wf_item_tree_build_trees' assms(1,2,5,6) i k(1) x *(2) fss(3) by metis
	+ have wf_item: "wf_item \<G> \<omega> (item (bs!?k!i))"
	+ using k(1) i assms(1) unfolding wf_bins_def wf_bin_def wf_bin_items_def by (simp add: items_def)
	+ obtain N fss where Nfss: "f = FBranch N fss"
	+ using build_trees'_termination[OF wf] by (metis "*"(2) fss(3) option.inject)
	+ then obtain ts where ts: "t = Branch N ts"
	+ using assms(6) by auto
	+ hence "N = item_rule_head x"
	+ "map root_tree ts = item_rule_body x"
	+ using finished wf_item_tree x by (auto simp: is_finished_def is_complete_def)
	+ hence "\<exists>r \<in> set (\<RR> \<G>). N = rule_head r \<and> map root_tree ts = rule_body r"
	+ using wf_item x unfolding wf_item_def item_rule_body_def item_rule_head_def by blast
	+ hence wf_rule: "wf_rule_tree \<G> t"
	+ using wf_item_tree ts by simp
	+ have root: "root_tree t = \<SS> \<G>"
	+ using finished ts \<open>N = item_rule_head x\<close> by (auto simp: is_finished_def)
	+ have "yield_tree t = slice (item_origin (item (bs!?k!i))) (item_end (item (bs!?k!i))) \<omega>"
	+ using k i assms(1,6) wf wf_yield_tree_build_trees' wf_yield_tree_def *(2) fss(3) by (smt (verit, best))
	+ hence yield: "yield_tree t = \<omega>"
	+ using finished x unfolding is_finished_def by simp
	+ show ?thesis
	+ using * wf_rule root yield assms(4) unfolding build_trees_def by simp
	+qed
	+
	+corollary wf_rule_root_yield_tree_build_trees_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "nonempty_derives \<G>"
	+ assumes "build_trees \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "wf_rule_tree \<G> t \<and> root_tree t = \<SS> \<G> \<and> yield_tree t = \<omega>"
	+ using assms wf_rule_root_yield_tree_build_trees wf_bins_Earley\<^sub>L Earley\<^sub>L_def
	+ length_Earley\<^sub>L_bins length_bins_Init\<^sub>L sound_mono_ptrs_Earley\<^sub>L
	+ by (metis dual_order.eq_iff )
	+
	+theorem soundness_build_trees_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ assumes "build_trees \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "derives \<G> [\<SS> \<G>] \<omega>"
	+proof -
	+ let ?k = "length (Earley\<^sub>L \<G> \<omega>) - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items ((Earley\<^sub>L \<G> \<omega>)!?k))"
	+ have #: "Some fs = map_option concat (those (map (\<lambda>(_, i). build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}) finished))"
	+ using assms(4) build_trees_def finished_def by (metis (full_types))
	+ then obtain fss fs' where fss: "Some fss = those (map (\<lambda>(_, i). build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}) finished)"
	+ "fs' \<in> set fss" "f \<in> set fs'"
	+ using map_option_concat_those_map_exists assms(5) by fastforce
	+ then obtain x i where *: "(x,i) \<in> set finished" "Some fs' = build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}"
	+ using those_map_exists[OF fss(1,2)] by auto
	+ have k: "?k < length (Earley\<^sub>L \<G> \<omega>)" "?k \<le> length \<omega>"
	+ by (simp_all add: Earley\<^sub>L_def assms(1))
	+ have i: "i < length ((Earley\<^sub>L \<G> \<omega>) ! ?k)"
	+ using index_filter_with_index_lt_length * items_def finished_def by (metis length_map)
	+ have x: "x = item ((Earley\<^sub>L \<G> \<omega>)!?k!i)"
	+ using * i filter_with_index_nth items_def nth_map finished_def by metis
	+ have finished: "is_finished \<G> \<omega> x"
	+ using * filter_with_index_P finished_def by metis
	+ moreover have "x \<in> set (items ((Earley\<^sub>L \<G> \<omega>) ! ?k))"
	+ using x by (auto simp: items_def; metis One_nat_def i imageI nth_mem)
	+ ultimately have "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega>"
	+ by (meson k(1) kth_bin_sub_bins recognizing_def subsetD)
	+ thus ?thesis
	+ using correctness_Earley\<^sub>L assms by blast
	+qed
	+
	+theorem termination_build_tree_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "nonempty_derives \<G>" "derives \<G> [\<SS> \<G>] \<omega>"
	+ shows "\<exists>fs. build_trees \<G> \<omega> (Earley\<^sub>L \<G> \<omega>) = Some fs"
	+proof -
	+ let ?k = "length (Earley\<^sub>L \<G> \<omega>) - 1"
	+ define finished where finished_def: "finished = filter_with_index (is_finished \<G> \<omega>) (items ((Earley\<^sub>L \<G> \<omega>)!?k))"
	+ have "\<forall>f \<in> set finished. (Earley\<^sub>L \<G> \<omega>, \<omega>, ?k, snd f, {snd f}) \<in> wf_trees_input"
	+ proof standard
	+ fix f
	+ assume a: "f \<in> set finished"
	+ then obtain x i where *: "(x,i) = f"
	+ by (metis surj_pair)
	+ have "sound_ptrs \<omega> (Earley\<^sub>L \<G> \<omega>)"
	+ using sound_mono_ptrs_Earley\<^sub>L assms by blast
	+ moreover have "?k < length (Earley\<^sub>L \<G> \<omega>)"
	+ by (simp add: Earley\<^sub>L_def assms(1))
	+ moreover have "i < length ((Earley\<^sub>L \<G> \<omega>)!?k)"
	+ using index_filter_with_index_lt_length a * items_def finished_def by (metis length_map)
	+ ultimately show "(Earley\<^sub>L \<G> \<omega>, \<omega>, ?k, snd f, {snd f}) \<in> wf_trees_input"
	+ using * unfolding wf_trees_input_def by auto
	+ qed
	+ hence "\<forall>fso \<in> set (map (\<lambda>(_, i). build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}) finished). \<exists>fs. fso = Some fs"
	+ using build_trees'_termination by fastforce
	+ then obtain fss where fss: "Some fss = those (map (\<lambda>(_, i). build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}) finished)"
	+ by (smt (verit, best) those_Some)
	+ then obtain fs where fs: "Some fs = map_option concat (those (map (\<lambda>(_, i). build_trees' (Earley\<^sub>L \<G> \<omega>) \<omega> ?k i {i}) finished))"
	+ by (metis map_option_eq_Some)
	+ show ?thesis
	+ using finished_def fss fs build_trees_def by (metis (full_types))
	+qed
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/Earley_Recognizer.thy b/thys/Earley_Parser/Earley_Recognizer.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Earley_Recognizer.thy
	@@ -0,0 +1,2331 @@
	+theory Earley_Recognizer
	+ imports
	+ Earley_Fixpoint
	+begin
	+
	+section \<open>Earley recognizer\<close>
	+
	+subsection \<open>List auxilaries\<close>
	+
	+fun filter_with_index' :: "nat \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> ('a \<times> nat) list" where
	+ "filter_with_index' _ _ [] = []"
	+\| "filter_with_index' i P (x#xs) = (
	+ if P x then (x,i) # filter_with_index' (i+1) P xs
	+ else filter_with_index' (i+1) P xs)"
	+
	+definition filter_with_index :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> ('a \<times> nat) list" where
	+ "filter_with_index P xs = filter_with_index' 0 P xs"
	+
	+lemma filter_with_index'_P:
	+ "(x, n) \<in> set (filter_with_index' i P xs) \<Longrightarrow> P x"
	+ by (induction xs arbitrary: i) (auto split: if_splits)
	+
	+lemma filter_with_index_P:
	+ "(x, n) \<in> set (filter_with_index P xs) \<Longrightarrow> P x"
	+ by (metis filter_with_index'_P filter_with_index_def)
	+
	+lemma filter_with_index'_cong_filter:
	+ "map fst (filter_with_index' i P xs) = filter P xs"
	+ by (induction xs arbitrary: i) auto
	+
	+lemma filter_with_index_cong_filter:
	+ "map fst (filter_with_index P xs) = filter P xs"
	+ by (simp add: filter_with_index'_cong_filter filter_with_index_def)
	+
	+lemma size_index_filter_with_index':
	+ "(x, n) \<in> set (filter_with_index' i P xs) \<Longrightarrow> n \<ge> i"
	+ by (induction xs arbitrary: i) (auto simp: Suc_leD split: if_splits)
	+
	+lemma index_filter_with_index'_lt_length:
	+ "(x, n) \<in> set (filter_with_index' i P xs) \<Longrightarrow> n-i < length xs"
	+ by (induction xs arbitrary: i)(auto simp: less_Suc_eq_0_disj split: if_splits; metis Suc_diff_Suc leI)+
	+
	+lemma index_filter_with_index_lt_length:
	+ "(x, n) \<in> set (filter_with_index P xs) \<Longrightarrow> n < length xs"
	+ by (metis filter_with_index_def index_filter_with_index'_lt_length minus_nat.diff_0)
	+
	+lemma filter_with_index'_nth:
	+ "(x, n) \<in> set (filter_with_index' i P xs) \<Longrightarrow> xs ! (n-i) = x"
	+proof (induction xs arbitrary: i)
	+ case (Cons y xs)
	+ show ?case
	+ proof (cases "x = y")
	+ case True
	+ thus ?thesis
	+ using Cons by (auto simp: nth_Cons' split: if_splits)
	+ next
	+ case False
	+ hence "(x, n) \<in> set (filter_with_index' (i+1) P xs)"
	+ using Cons.prems by (cases xs) (auto split: if_splits)
	+ hence "n \<ge> i + 1" "xs ! (n - i - 1) = x"
	+ by (auto simp: size_index_filter_with_index' Cons.IH)
	+ thus ?thesis
	+ by simp
	+ qed
	+qed simp
	+
	+lemma filter_with_index_nth:
	+ "(x, n) \<in> set (filter_with_index P xs) \<Longrightarrow> xs ! n = x"
	+ by (metis diff_zero filter_with_index'_nth filter_with_index_def)
	+
	+lemma filter_with_index_nonempty:
	+ "x \<in> set xs \<Longrightarrow> P x \<Longrightarrow> filter_with_index P xs \<noteq> []"
	+ by (metis filter_empty_conv filter_with_index_cong_filter list.map(1))
	+
	+lemma filter_with_index'_Ex_first:
	+ "(\<exists>x i xs'. filter_with_index' n P xs = (x, i)#xs') \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
	+ by (induction xs arbitrary: n) auto
	+
	+lemma filter_with_index_Ex_first:
	+ "(\<exists>x i xs'. filter_with_index P xs = (x, i)#xs') \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
	+ using filter_with_index'_Ex_first filter_with_index_def by metis
	+
	+
	+subsection \<open>Definitions\<close>
	+
	+datatype pointer =
	+ Null
	+ \| Pre nat \<comment>\<open>pre\<close>
	+ \| PreRed "nat \<times> nat \<times> nat" "(nat \<times> nat \<times> nat) list" \<comment>\<open>k', pre, red\<close>
	+
	+datatype 'a entry =
	+ Entry (item : "'a item") (pointer : pointer)
	+
	+type_synonym 'a bin = "'a entry list"
	+type_synonym 'a bins = "'a bin list"
	+
	+definition items :: "'a bin \<Rightarrow> 'a item list" where
	+ "items b \<equiv> map item b"
	+
	+definition pointers :: "'a bin \<Rightarrow> pointer list" where
	+ "pointers b \<equiv> map pointer b"
	+
	+definition bins_eq_items :: "'a bins \<Rightarrow> 'a bins \<Rightarrow> bool" where
	+ "bins_eq_items bs0 bs1 \<equiv> map items bs0 = map items bs1"
	+
	+definition bins :: "'a bins \<Rightarrow> 'a item set" where
	+ "bins bs \<equiv> \<Union> { set (items (bs!k)) \| k. k < length bs }"
	+
	+definition bin_upto :: "'a bin \<Rightarrow> nat \<Rightarrow> 'a item set" where
	+ "bin_upto b i \<equiv> { items b ! j \| j. j < i \<and> j < length (items b) }"
	+
	+definition bins_upto :: "'a bins \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a item set" where
	+ "bins_upto bs k i \<equiv> \<Union> { set (items (bs ! l)) \| l. l < k } \<union> bin_upto (bs ! k) i"
	+
	+definition wf_bin_items :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> nat \<Rightarrow> 'a item list \<Rightarrow> bool" where
	+ "wf_bin_items \<G> \<omega> k xs \<equiv> \<forall>x \<in> set xs. wf_item \<G> \<omega> x \<and> item_end x = k"
	+
	+definition wf_bin :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> nat \<Rightarrow> 'a bin \<Rightarrow> bool" where
	+ "wf_bin \<G> \<omega> k b \<equiv> distinct (items b) \<and> wf_bin_items \<G> \<omega> k (items b)"
	+
	+definition wf_bins :: "'a cfg \<Rightarrow> 'a list \<Rightarrow> 'a bins \<Rightarrow> bool" where
	+ "wf_bins \<G> \<omega> bs \<equiv> \<forall>k < length bs. wf_bin \<G> \<omega> k (bs!k)"
	+
	+definition nonempty_derives :: "'a cfg \<Rightarrow> bool" where
	+ "nonempty_derives \<G> \<equiv> \<forall>N. N \<in> set (\<NN> \<G>) \<longrightarrow> \<not> derives \<G> [N] []"
	+
	+definition Init\<^sub>L :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins" where
	+ "Init\<^sub>L \<G> \<omega> \<equiv>
	+ let rs = filter (\<lambda>r. rule_head r = \<SS> \<G>) (\<RR> \<G>) in
	+ let b0 = map (\<lambda>r. (Entry (init_item r 0) Null)) rs in
	+ let bs = replicate (length \<omega> + 1) ([]) in
	+ bs[0 := b0]"
	+
	+definition Scan\<^sub>L :: "nat \<Rightarrow> 'a sentence \<Rightarrow> 'a \<Rightarrow> 'a item \<Rightarrow> nat \<Rightarrow> 'a entry list" where
	+ "Scan\<^sub>L k \<omega> a x pre \<equiv>
	+ if \<omega>!k = a then
	+ let x' = inc_item x (k+1) in
	+ [Entry x' (Pre pre)]
	+ else []"
	+
	+definition Predict\<^sub>L :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a \<Rightarrow> 'a entry list" where
	+ "Predict\<^sub>L k \<G> X \<equiv>
	+ let rs = filter (\<lambda>r. rule_head r = X) (\<RR> \<G>) in
	+ map (\<lambda>r. (Entry (init_item r k) Null)) rs"
	+
	+definition Complete\<^sub>L :: "nat \<Rightarrow> 'a item \<Rightarrow> 'a bins \<Rightarrow> nat \<Rightarrow> 'a entry list" where
	+ "Complete\<^sub>L k y bs red \<equiv>
	+ let orig = bs ! (item_origin y) in
	+ let is = filter_with_index (\<lambda>x. next_symbol x = Some (item_rule_head y)) (items orig) in
	+ map (\<lambda>(x, pre). (Entry (inc_item x k) (PreRed (item_origin y, pre, red) []))) is"
	+
	+fun bin_upd :: "'a entry \<Rightarrow> 'a bin \<Rightarrow> 'a bin" where
	+ "bin_upd e' [] = [e']"
	+\| "bin_upd e' (e#es) = (
	+ case (e', e) of
	+ (Entry x (PreRed px xs), Entry y (PreRed py ys)) \<Rightarrow>
	+ if x = y then Entry x (PreRed py (px#xs@ys)) # es
	+ else e # bin_upd e' es
	+ \| _ \<Rightarrow>
	+ if item e' = item e then e # es
	+ else e # bin_upd e' es)"
	+
	+fun bin_upds :: "'a entry list \<Rightarrow> 'a bin \<Rightarrow> 'a bin" where
	+ "bin_upds [] b = b"
	+\| "bin_upds (e#es) b = bin_upds es (bin_upd e b)"
	+
	+definition bins_upd :: "'a bins \<Rightarrow> nat \<Rightarrow> 'a entry list \<Rightarrow> 'a bins" where
	+ "bins_upd bs k es \<equiv> bs[k := bin_upds es (bs!k)]"
	+
	+partial_function (tailrec) Earley\<^sub>L_bin' :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins \<Rightarrow> nat \<Rightarrow> 'a bins" where
	+ "Earley\<^sub>L_bin' k \<G> \<omega> bs i = (
	+ if i \<ge> length (items (bs ! k)) then bs
	+ else
	+ let x = items (bs!k) ! i in
	+ let bs' =
	+ case next_symbol x of
	+ Some a \<Rightarrow>
	+ if is_terminal \<G> a then
	+ if k < length \<omega> then bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)
	+ else bs
	+ else bins_upd bs k (Predict\<^sub>L k \<G> a)
	+ \| None \<Rightarrow> bins_upd bs k (Complete\<^sub>L k x bs i)
	+ in Earley\<^sub>L_bin' k \<G> \<omega> bs' (i+1))"
	+
	+declare Earley\<^sub>L_bin'.simps[code]
	+
	+definition Earley\<^sub>L_bin :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins \<Rightarrow> 'a bins" where
	+ "Earley\<^sub>L_bin k \<G> \<omega> bs \<equiv> Earley\<^sub>L_bin' k \<G> \<omega> bs 0"
	+
	+fun Earley\<^sub>L_bins :: "nat \<Rightarrow> 'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins" where
	+ "Earley\<^sub>L_bins 0 \<G> \<omega> = Earley\<^sub>L_bin 0 \<G> \<omega> (Init\<^sub>L \<G> \<omega>)"
	+\| "Earley\<^sub>L_bins (Suc n) \<G> \<omega> = Earley\<^sub>L_bin (Suc n) \<G> \<omega> (Earley\<^sub>L_bins n \<G> \<omega>)"
	+
	+definition Earley\<^sub>L :: "'a cfg \<Rightarrow> 'a sentence \<Rightarrow> 'a bins" where
	+ "Earley\<^sub>L \<G> \<omega> \<equiv> Earley\<^sub>L_bins (length \<omega>) \<G> \<omega>"
	+
	+
	+subsection \<open>Bin lemmas\<close>
	+
	+lemma length_bins_upd[simp]:
	+ "length (bins_upd bs k es) = length bs"
	+ unfolding bins_upd_def by simp
	+
	+lemma length_bin_upd:
	+ "length (bin_upd e b) \<ge> length b"
	+ by (induction e b rule: bin_upd.induct) (auto split: pointer.splits entry.splits)
	+
	+lemma length_bin_upds:
	+ "length (bin_upds es b) \<ge> length b"
	+ by (induction es arbitrary: b) (auto, meson le_trans length_bin_upd)
	+
	+lemma length_nth_bin_bins_upd:
	+ "length (bins_upd bs k es ! n) \<ge> length (bs ! n)"
	+ unfolding bins_upd_def using length_bin_upds
	+ by (metis linorder_not_le list_update_beyond nth_list_update_eq nth_list_update_neq order_refl)
	+
	+lemma nth_idem_bins_upd:
	+ "k \<noteq> n \<Longrightarrow> bins_upd bs k es ! n = bs ! n"
	+ unfolding bins_upd_def by simp
	+
	+lemma items_nth_idem_bin_upd:
	+ "n < length b \<Longrightarrow> items (bin_upd e b) ! n = items b ! n"
	+ by (induction b arbitrary: e n) (auto simp: items_def less_Suc_eq_0_disj split!: entry.split pointer.split)
	+
	+lemma items_nth_idem_bin_upds:
	+ "n < length b \<Longrightarrow> items (bin_upds es b) ! n = items b ! n"
	+ by (induction es arbitrary: b)
	+ (auto, metis items_def items_nth_idem_bin_upd length_bin_upd nth_map order.strict_trans2)
	+
	+lemma items_nth_idem_bins_upd:
	+ "n < length (bs ! k) \<Longrightarrow> items (bins_upd bs k es ! k) ! n = items (bs ! k) ! n"
	+ unfolding bins_upd_def using items_nth_idem_bin_upds
	+ by (metis linorder_not_less list_update_beyond nth_list_update_eq)
	+
	+lemma bin_upto_eq_set_items:
	+ "i \<ge> length b \<Longrightarrow> bin_upto b i = set (items b)"
	+ by (auto simp: bin_upto_def items_def, metis in_set_conv_nth nth_map order_le_less order_less_trans)
	+
	+lemma bins_upto_empty:
	+ "bins_upto bs 0 0 = {}"
	+ unfolding bins_upto_def bin_upto_def by simp
	+
	+lemma set_items_bin_upd:
	+ "set (items (bin_upd e b)) = set (items b) \<union> {item e}"
	+proof (induction b arbitrary: e)
	+ case (Cons b bs)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> b = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where "e = Entry x (PreRed xp xs)" "b = Entry y (PreRed yp ys)"
	+ by blast
	+ thus ?thesis
	+ using Cons.IH by (auto simp: items_def)
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item b"
	+ hence "bin_upd e (b # bs) = b # bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * by (auto simp: items_def)
	+ next
	+ assume *: "\<not> item e = item b"
	+ hence "bin_upd e (b # bs) = b # bin_upd e bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons.IH by (auto simp: items_def)
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma set_items_bin_upds:
	+ "set (items (bin_upds es b)) = set (items b) \<union> set (items es)"
	+ using set_items_bin_upd by (induction es arbitrary: b) (auto simp: items_def, blast, force+)
	+
	+lemma bins_bins_upd:
	+ assumes "k < length bs"
	+ shows "bins (bins_upd bs k es) = bins bs \<union> set (items es)"
	+proof -
	+ let ?bs = "bins_upd bs k es"
	+ have "bins (bins_upd bs k es) = \<Union> {set (items (?bs ! k)) \|k. k < length ?bs}"
	+ unfolding bins_def by blast
	+ also have "... = \<Union> {set (items (bs ! l)) \|l. l < length bs \<and> l \<noteq> k} \<union> set (items (?bs ! k))"
	+ unfolding bins_upd_def using assms by (auto, metis nth_list_update)
	+ also have "... = \<Union> {set (items (bs ! l)) \|l. l < length bs \<and> l \<noteq> k} \<union> set (items (bs ! k)) \<union> set (items es)"
	+ using set_items_bin_upds[of es "bs!k"] by (simp add: assms bins_upd_def sup_assoc)
	+ also have "... = \<Union> {set (items (bs ! k)) \|k. k < length bs} \<union> set (items es)"
	+ using assms by blast
	+ also have "... = bins bs \<union> set (items es)"
	+ unfolding bins_def by blast
	+ finally show ?thesis .
	+qed
	+
	+lemma kth_bin_sub_bins:
	+ "k < length bs \<Longrightarrow> set (items (bs ! k)) \<subseteq> bins bs"
	+ unfolding bins_def bins_upto_def bin_upto_def by blast+
	+
	+lemma bin_upto_Cons_0:
	+ "bin_upto (e#es) 0 = {}"
	+ by (auto simp: bin_upto_def)
	+
	+lemma bin_upto_Cons:
	+ assumes "0 < n"
	+ shows "bin_upto (e#es) n = { item e } \<union> bin_upto es (n-1)"
	+proof -
	+ have "bin_upto (e#es) n = { items (e#es) ! j \| j. j < n \<and> j < length (items (e#es)) }"
	+ unfolding bin_upto_def by blast
	+ also have "... = { item e } \<union> { items es ! j \| j. j < (n-1) \<and> j < length (items es) }"
	+ using assms by (cases n) (auto simp: items_def nth_Cons', metis One_nat_def Zero_not_Suc diff_Suc_1 not_less_eq nth_map)
	+ also have "... = { item e } \<union> bin_upto es (n-1)"
	+ unfolding bin_upto_def by blast
	+ finally show ?thesis .
	+qed
	+
	+lemma bin_upto_nth_idem_bin_upd:
	+ "n < length b \<Longrightarrow> bin_upto (bin_upd e b) n = bin_upto b n"
	+proof (induction b arbitrary: e n)
	+ case (Cons b bs)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> b = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where "e = Entry x (PreRed xp xs)" "b = Entry y (PreRed yp ys)"
	+ by blast
	+ thus ?thesis
	+ using Cons bin_upto_Cons_0
	+ by (cases n) (auto simp: items_def bin_upto_Cons, blast+)
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item b"
	+ hence "bin_upd e (b # bs) = b # bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * by (auto simp: items_def)
	+ next
	+ assume *: "\<not> item e = item b"
	+ hence "bin_upd e (b # bs) = b # bin_upd e bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons bin_upto_Cons_0
	+ by (cases n) (auto simp: items_def bin_upto_Cons, blast+)
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma bin_upto_nth_idem_bin_upds:
	+ "n < length b \<Longrightarrow> bin_upto (bin_upds es b) n = bin_upto b n"
	+ using bin_upto_nth_idem_bin_upd length_bin_upd
	+ apply (induction es arbitrary: b)
	+ apply auto
	+ using order.strict_trans2 order.strict_trans1 by blast+
	+
	+lemma bins_upto_kth_nth_idem:
	+ assumes "l < length bs" "k \<le> l" "n < length (bs ! k)"
	+ shows "bins_upto (bins_upd bs l es) k n = bins_upto bs k n"
	+proof -
	+ let ?bs = "bins_upd bs l es"
	+ have "bins_upto ?bs k n = \<Union> {set (items (?bs ! l)) \|l. l < k} \<union> bin_upto (?bs ! k) n"
	+ unfolding bins_upto_def by blast
	+ also have "... = \<Union> {set (items (bs ! l)) \|l. l < k} \<union> bin_upto (?bs ! k) n"
	+ unfolding bins_upd_def using assms(1,2) by auto
	+ also have "... = \<Union> {set (items (bs ! l)) \|l. l < k} \<union> bin_upto (bs ! k) n"
	+ unfolding bins_upd_def using assms(1,3) bin_upto_nth_idem_bin_upds
	+ by (metis (no_types, lifting) nth_list_update)
	+ also have "... = bins_upto bs k n"
	+ unfolding bins_upto_def by blast
	+ finally show ?thesis .
	+qed
	+
	+lemma bins_upto_sub_bins:
	+ "k < length bs \<Longrightarrow> bins_upto bs k n \<subseteq> bins bs"
	+ unfolding bins_def bins_upto_def bin_upto_def using less_trans by (auto, blast)
	+
	+lemma bins_upto_Suc_Un:
	+ "n < length (bs ! k) \<Longrightarrow> bins_upto bs k (n+1) = bins_upto bs k n \<union> { items (bs ! k) ! n }"
	+ unfolding bins_upto_def bin_upto_def using less_Suc_eq by (auto simp: items_def, metis nth_map)
	+
	+lemma bins_bin_exists:
	+ "x \<in> bins bs \<Longrightarrow> \<exists>k < length bs. x \<in> set (items (bs ! k))"
	+ unfolding bins_def by blast
	+
	+lemma distinct_bin_upd:
	+ "distinct (items b) \<Longrightarrow> distinct (items (bin_upd e b))"
	+proof (induction b arbitrary: e)
	+ case (Cons b bs)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> b = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where "e = Entry x (PreRed xp xs)" "b = Entry y (PreRed yp ys)"
	+ by blast
	+ thus ?thesis
	+ using Cons
	+ apply (auto simp: items_def)
	+ by (metis Un_insert_right entry.sel(1) imageI items_def list.set_map list.simps(15) set_ConsD set_items_bin_upd sup_bot_right)
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item b"
	+ hence "bin_upd e (b # bs) = b # bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons.prems by (auto simp: items_def)
	+ next
	+ assume *: "\<not> item e = item b"
	+ hence "bin_upd e (b # bs) = b # bin_upd e bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ moreover have "distinct (items (bin_upd e bs))"
	+ using Cons by (auto simp: items_def)
	+ ultimately show ?thesis
	+ using * Cons.prems set_items_bin_upd
	+ by (metis Un_insert_right distinct.simps(2) insertE items_def list.simps(9) sup_bot_right)
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma wf_bins_kth_bin:
	+ "wf_bins \<G> \<omega> bs \<Longrightarrow> k < length bs \<Longrightarrow> x \<in> set (items (bs ! k)) \<Longrightarrow> wf_item \<G> \<omega> x \<and> item_end x = k"
	+ using wf_bin_def wf_bins_def wf_bin_items_def by blast
	+
	+lemma wf_bin_bin_upd:
	+ assumes "wf_bin \<G> \<omega> k b" "wf_item \<G> \<omega> (item e) \<and> item_end (item e) = k"
	+ shows "wf_bin \<G> \<omega> k (bin_upd e b)"
	+ using assms
	+proof (induction b arbitrary: e)
	+ case (Cons b bs)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> b = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where "e = Entry x (PreRed xp xs)" "b = Entry y (PreRed yp ys)"
	+ by blast
	+ thus ?thesis
	+ using Cons distinct_bin_upd wf_bin_def wf_bin_items_def set_items_bin_upd
	+ by (smt (verit, best) Un_insert_right insertE sup_bot.right_neutral)
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item b"
	+ hence "bin_upd e (b # bs) = b # bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons.prems by (auto simp: items_def)
	+ next
	+ assume *: "\<not> item e = item b"
	+ hence "bin_upd e (b # bs) = b # bin_upd e bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons.prems set_items_bin_upd distinct_bin_upd wf_bin_def wf_bin_items_def
	+ by (smt (verit, best) Un_insert_right insertE sup_bot_right)
	+ qed
	+ qed
	+qed (auto simp: items_def wf_bin_def wf_bin_items_def)
	+
	+lemma wf_bin_bin_upds:
	+ assumes "wf_bin \<G> \<omega> k b" "distinct (items es)"
	+ assumes "\<forall>x \<in> set (items es). wf_item \<G> \<omega> x \<and> item_end x = k"
	+ shows "wf_bin \<G> \<omega> k (bin_upds es b)"
	+ using assms by (induction es arbitrary: b) (auto simp: wf_bin_bin_upd items_def)
	+
	+lemma wf_bins_bins_upd:
	+ assumes "wf_bins \<G> \<omega> bs" "distinct (items es)"
	+ assumes "\<forall>x \<in> set (items es). wf_item \<G> \<omega> x \<and> item_end x = k"
	+ shows "wf_bins \<G> \<omega> (bins_upd bs k es)"
	+ unfolding bins_upd_def using assms wf_bin_bin_upds wf_bins_def
	+ by (metis length_list_update nth_list_update_eq nth_list_update_neq)
	+
	+lemma wf_bins_impl_wf_items:
	+ "wf_bins \<G> \<omega> bs \<Longrightarrow> \<forall>x \<in> (bins bs). wf_item \<G> \<omega> x"
	+ unfolding wf_bins_def wf_bin_def wf_bin_items_def bins_def by auto
	+
	+lemma bin_upds_eq_items:
	+ "set (items es) \<subseteq> set (items b) \<Longrightarrow> set (items (bin_upds es b)) = set (items b)"
	+ apply (induction es arbitrary: b)
	+ apply (auto simp: set_items_bin_upd set_items_bin_upds)
	+ apply (simp add: items_def)
	+ by (metis Un_iff Un_subset_iff items_def list.simps(9) set_subset_Cons)
	+
	+lemma bin_eq_items_bin_upd:
	+ "item e \<in> set (items b) \<Longrightarrow> items (bin_upd e b) = items b"
	+proof (induction b arbitrary: e)
	+ case (Cons b bs)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> b = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where "e = Entry x (PreRed xp xs)" "b = Entry y (PreRed yp ys)"
	+ by blast
	+ thus ?thesis
	+ using Cons by (auto simp: items_def)
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item b"
	+ hence "bin_upd e (b # bs) = b # bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons.prems by (auto simp: items_def)
	+ next
	+ assume *: "\<not> item e = item b"
	+ hence "bin_upd e (b # bs) = b # bin_upd e bs"
	+ using False by (auto split: pointer.splits entry.splits)
	+ thus ?thesis
	+ using * Cons by (auto simp: items_def)
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma bin_eq_items_bin_upds:
	+ assumes "set (items es) \<subseteq> set (items b)"
	+ shows "items (bin_upds es b) = items b"
	+ using assms
	+proof (induction es arbitrary: b)
	+ case (Cons e es)
	+ have "items (bin_upds es (bin_upd e b)) = items (bin_upd e b)"
	+ using Cons bin_upds_eq_items set_items_bin_upd set_items_bin_upds
	+ by (metis Un_upper2 bin_upds.simps(2) sup.coboundedI1)
	+ moreover have "items (bin_upd e b) = items b"
	+ using bin_eq_items_bin_upd Cons.prems by (auto simp: items_def)
	+ ultimately show ?case
	+ by simp
	+qed (auto simp: items_def)
	+
	+lemma bins_eq_items_bins_upd:
	+ assumes "set (items es) \<subseteq> set (items (bs!k))"
	+ shows "bins_eq_items (bins_upd bs k es) bs"
	+ unfolding bins_upd_def using assms bin_eq_items_bin_upds bins_eq_items_def
	+ by (metis list_update_id map_update)
	+
	+lemma bins_eq_items_imp_eq_bins:
	+ "bins_eq_items bs bs' \<Longrightarrow> bins bs = bins bs'"
	+ unfolding bins_eq_items_def bins_def items_def
	+ by (metis (no_types, lifting) length_map nth_map)
	+
	+lemma bin_eq_items_dist_bin_upd_bin:
	+ assumes "items a = items b"
	+ shows "items (bin_upd e a) = items (bin_upd e b)"
	+ using assms
	+proof (induction a arbitrary: e b)
	+ case (Cons a as)
	+ obtain b' bs where bs: "b = b' # bs" "item a = item b'" "items as = items bs"
	+ using Cons.prems by (auto simp: items_def)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> a = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where #: "e = Entry x (PreRed xp xs)" "a = Entry y (PreRed yp ys)"
	+ by blast
	+ show ?thesis
	+ proof cases
	+ assume *: "x = y"
	+ hence "items (bin_upd e (a # as)) = x # items as"
	+ using # by (auto simp: items_def)
	+ moreover have "items (bin_upd e (b' # bs)) = x # items bs"
	+ using bs # * by (auto simp: items_def split: pointer.splits entry.splits)
	+ ultimately show ?thesis
	+ using bs by simp
	+ next
	+ assume *: "\<not> x = y"
	+ hence "items (bin_upd e (a # as)) = y # items (bin_upd e as)"
	+ using # by (auto simp: items_def)
	+ moreover have "items (bin_upd e (b' # bs)) = y # items (bin_upd e bs)"
	+ using bs # * by (auto simp: items_def split: pointer.splits entry.splits)
	+ ultimately show ?thesis
	+ using bs Cons.IH by simp
	+ qed
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item a"
	+ hence "items (bin_upd e (a # as)) = item a # items as"
	+ using False by (auto simp: items_def split: pointer.splits entry.splits)
	+ moreover have "items (bin_upd e (b' # bs)) = item b' # items bs"
	+ using bs False * by (auto simp: items_def split: pointer.splits entry.splits)
	+ ultimately show ?thesis
	+ using bs by simp
	+ next
	+ assume *: "\<not> item e = item a"
	+ hence "items (bin_upd e (a # as)) = item a # items (bin_upd e as)"
	+ using False by (auto simp: items_def split: pointer.splits entry.splits)
	+ moreover have "items (bin_upd e (b' # bs)) = item b' # items (bin_upd e bs)"
	+ using bs False * by (auto simp: items_def split: pointer.splits entry.splits)
	+ ultimately show ?thesis
	+ using bs Cons by simp
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma bin_eq_items_dist_bin_upds_bin:
	+ assumes "items a = items b"
	+ shows "items (bin_upds es a) = items (bin_upds es b)"
	+ using assms
	+proof (induction es arbitrary: a b)
	+ case (Cons e es)
	+ hence "items (bin_upds es (bin_upd e a)) = items (bin_upds es (bin_upd e b))"
	+ using bin_eq_items_dist_bin_upd_bin by blast
	+ thus ?case
	+ by simp
	+qed simp
	+
	+lemma bin_eq_items_dist_bin_upd_entry:
	+ assumes "item e = item e'"
	+ shows "items (bin_upd e b) = items (bin_upd e' b)"
	+ using assms
	+proof (induction b arbitrary: e e')
	+ case (Cons a as)
	+ show ?case
	+ proof (cases "\<exists>x xp xs y yp ys. e = Entry x (PreRed xp xs) \<and> a = Entry y (PreRed yp ys)")
	+ case True
	+ then obtain x xp xs y yp ys where #: "e = Entry x (PreRed xp xs)" "a = Entry y (PreRed yp ys)"
	+ by blast
	+ show ?thesis
	+ proof cases
	+ assume *: "x = y"
	+ thus ?thesis
	+ using # Cons.prems by (auto simp: items_def split: pointer.splits entry.splits)
	+ next
	+ assume *: "\<not> x = y"
	+ thus ?thesis
	+ using # Cons.prems
	+ by (auto simp: items_def split!: pointer.splits entry.splits, metis Cons.IH Cons.prems items_def)+
	+ qed
	+ next
	+ case False
	+ then show ?thesis
	+ proof cases
	+ assume *: "item e = item a"
	+ thus ?thesis
	+ using Cons.prems by (auto simp: items_def split: pointer.splits entry.splits)
	+ next
	+ assume *: "\<not> item e = item a"
	+ thus ?thesis
	+ using Cons.prems
	+ by (auto simp: items_def split!: pointer.splits entry.splits, metis Cons.IH Cons.prems items_def)+
	+ qed
	+ qed
	+qed (auto simp: items_def)
	+
	+lemma bin_eq_items_dist_bin_upds_entries:
	+ assumes "items es = items es'"
	+ shows "items (bin_upds es b) = items (bin_upds es' b)"
	+ using assms
	+proof (induction es arbitrary: es' b)
	+ case (Cons e es)
	+ then obtain e' es'' where "item e = item e'" "items es = items es''" "es' = e' # es''"
	+ by (auto simp: items_def)
	+ hence "items (bin_upds es (bin_upd e b)) = items (bin_upds es'' (bin_upd e' b))"
	+ using Cons.IH
	+ by (metis bin_eq_items_dist_bin_upd_entry bin_eq_items_dist_bin_upds_bin)
	+ thus ?case
	+ by (simp add: \<open>es' = e' # es''\<close>)
	+qed (auto simp: items_def)
	+
	+lemma bins_eq_items_dist_bins_upd:
	+ assumes "bins_eq_items as bs" "items aes = items bes" "k < length as"
	+ shows "bins_eq_items (bins_upd as k aes) (bins_upd bs k bes)"
	+proof -
	+ have "k < length bs"
	+ using assms(1,3) bins_eq_items_def map_eq_imp_length_eq by metis
	+ hence "items (bin_upds (as!k) aes) = items (bin_upds (bs!k) bes)"
	+ using bin_eq_items_dist_bin_upds_entries bin_eq_items_dist_bin_upds_bin bins_eq_items_def assms
	+ by (metis (no_types, lifting) nth_map)
	+ thus ?thesis
	+ using \<open>k < length bs\<close> assms bin_eq_items_dist_bin_upds_bin bin_eq_items_dist_bin_upds_entries
	+ bins_eq_items_def bins_upd_def by (smt (verit) map_update nth_map)
	+qed
	+
	+subsection \<open>Well-formed bins\<close>
	+
	+lemma distinct_Scan\<^sub>L:
	+ "distinct (items (Scan\<^sub>L k \<omega> a x pre))"
	+ unfolding Scan\<^sub>L_def by (auto simp: items_def)
	+
	+lemma distinct_Predict\<^sub>L:
	+ "wf_\<G> \<G> \<Longrightarrow> distinct (items (Predict\<^sub>L k \<G> X))"
	+ unfolding Predict\<^sub>L_def wf_\<G>_defs by (auto simp: init_item_def rule_head_def distinct_map inj_on_def items_def)
	+
	+lemma inj_on_inc_item:
	+ "\<forall>x \<in> A. item_end x = l \<Longrightarrow> inj_on (\<lambda>x. inc_item x k) A"
	+ unfolding inj_on_def inc_item_def by (simp add: item.expand)
	+
	+lemma distinct_Complete\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "item_origin y < length bs"
	+ shows "distinct (items (Complete\<^sub>L k y bs red))"
	+proof -
	+ let ?orig = "bs ! (item_origin y)"
	+ let ?is = "filter_with_index (\<lambda>x. next_symbol x = Some (item_rule_head y)) (items ?orig)"
	+ let ?is' = "map (\<lambda>(x, pre). (Entry (inc_item x k) (PreRed (item_origin y, pre, red) []))) ?is"
	+ have wf: "wf_bin \<G> \<omega> (item_origin y) ?orig"
	+ using assms wf_bins_def by blast
	+ have 0: "\<forall>x \<in> set (map fst ?is). item_end x = (item_origin y)"
	+ using wf wf_bin_def wf_bin_items_def filter_is_subset filter_with_index_cong_filter by (metis in_mono)
	+ hence "distinct (items ?orig)"
	+ using wf unfolding wf_bin_def by blast
	+ hence "distinct (map fst ?is)"
	+ using filter_with_index_cong_filter distinct_filter by metis
	+ moreover have "items ?is' = map (\<lambda>x. inc_item x k) (map fst ?is)"
	+ by (induction ?is) (auto simp: items_def)
	+ moreover have "inj_on (\<lambda>x. inc_item x k) (set (map fst ?is))"
	+ using inj_on_inc_item 0 by blast
	+ ultimately have "distinct (items ?is')"
	+ using distinct_map by metis
	+ thus ?thesis
	+ unfolding Complete\<^sub>L_def by simp
	+qed
	+
	+lemma wf_bins_Scan\<^sub>L':
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "x \<in> set (items (bs ! k))"
	+ assumes "k < length \<omega>" "next_symbol x \<noteq> None" "y = inc_item x (k+1)"
	+ shows "wf_item \<G> \<omega> y \<and> item_end y = k+1"
	+ using assms wf_bins_kth_bin[OF assms(1-3)]
	+ unfolding wf_item_def inc_item_def next_symbol_def is_complete_def item_rule_body_def
	+ by (auto split: if_splits)
	+
	+lemma wf_bins_Scan\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "x \<in> set (items (bs ! k))" "k < length \<omega>" "next_symbol x \<noteq> None"
	+ shows "\<forall>y \<in> set (items (Scan\<^sub>L k \<omega> a x pre)). wf_item \<G> \<omega> y \<and> item_end y = (k+1)"
	+ using wf_bins_Scan\<^sub>L'[OF assms] by (simp add: Scan\<^sub>L_def items_def)
	+
	+lemma wf_bins_Predict\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "\<forall>y \<in> set (items (Predict\<^sub>L k \<G> X)). wf_item \<G> \<omega> y \<and> item_end y = k"
	+ using assms by (auto simp: Predict\<^sub>L_def wf_item_def wf_bins_def wf_bin_def init_item_def wf_\<G>_defs items_def)
	+
	+lemma wf_item_inc_item:
	+ assumes "wf_item \<G> \<omega> x" "next_symbol x = Some a" "item_origin x \<le> k" "k \<le> length \<omega>"
	+ shows "wf_item \<G> \<omega> (inc_item x k) \<and> item_end (inc_item x k) = k"
	+ using assms by (auto simp: wf_item_def inc_item_def item_rule_body_def next_symbol_def is_complete_def split: if_splits)
	+
	+lemma wf_bins_Complete\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "y \<in> set (items (bs ! k))"
	+ shows "\<forall>x \<in> set (items (Complete\<^sub>L k y bs red)). wf_item \<G> \<omega> x \<and> item_end x = k"
	+proof -
	+ let ?orig = "bs ! (item_origin y)"
	+ let ?is = "filter_with_index (\<lambda>x. next_symbol x = Some (item_rule_head y)) (items ?orig)"
	+ let ?is' = "map (\<lambda>(x, pre). (Entry (inc_item x k) (PreRed (item_origin y, pre, red) []))) ?is"
	+ {
	+ fix x
	+ assume *: "x \<in> set (map fst ?is)"
	+ have "item_end x = item_origin y"
	+ using * assms wf_bins_kth_bin wf_item_def filter_with_index_cong_filter
	+ by (metis dual_order.strict_trans2 filter_is_subset subsetD)
	+ have "wf_item \<G> \<omega> x"
	+ using * assms wf_bins_kth_bin wf_item_def filter_with_index_cong_filter
	+ by (metis dual_order.strict_trans2 filter_is_subset subsetD)
	+ moreover have "next_symbol x = Some (item_rule_head y)"
	+ using * filter_set filter_with_index_cong_filter member_filter by metis
	+ moreover have "item_origin x \<le> k"
	+ using \<open>item_end x = item_origin y\<close> \<open>wf_item \<G> \<omega> x\<close> assms wf_bins_kth_bin wf_item_def
	+ by (metis dual_order.order_iff_strict dual_order.strict_trans1)
	+ moreover have "k \<le> length \<omega>"
	+ using assms wf_bins_kth_bin wf_item_def by blast
	+ ultimately have "wf_item \<G> \<omega> (inc_item x k)" "item_end (inc_item x k) = k"
	+ by (simp_all add: wf_item_inc_item)
	+ }
	+ hence "\<forall>x \<in> set (items ?is'). wf_item \<G> \<omega> x \<and> item_end x = k"
	+ by (auto simp: items_def rev_image_eqI)
	+ thus ?thesis
	+ unfolding Complete\<^sub>L_def by presburger
	+qed
	+
	+lemma Ex_wf_bins:
	+ "\<exists>n bs \<omega> \<G>. n \<le> length \<omega> \<and> length bs = Suc (length \<omega>) \<and> wf_\<G> \<G> \<and> wf_bins \<G> \<omega> bs"
	+ apply (rule exI[where x="0"])
	+ apply (rule exI[where x="[[]]"])
	+ apply (rule exI[where x="[]"])
	+ apply (auto simp: wf_bins_def wf_bin_def wf_\<G>_defs wf_bin_items_def items_def split: prod.splits)
	+ by (metis cfg.sel distinct.simps(1) empty_iff empty_set inf_bot_right list.set_intros(1))
	+
	+definition wf_earley_input :: "(nat \<times> 'a cfg \<times> 'a sentence \<times> 'a bins) set" where
	+ "wf_earley_input = {
	+ (k, \<G>, \<omega>, bs) \| k \<G> \<omega> bs.
	+ k \<le> length \<omega> \<and>
	+ length bs = length \<omega> + 1 \<and>
	+ wf_\<G> \<G> \<and>
	+ wf_bins \<G> \<omega> bs
	+ }"
	+
	+typedef 'a wf_bins = "wf_earley_input::(nat \<times> 'a cfg \<times> 'a sentence \<times> 'a bins) set"
	+ morphisms from_wf_bins to_wf_bins
	+ using Ex_wf_bins by (auto simp: wf_earley_input_def)
	+
	+lemma wf_earley_input_elim:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "k \<le> length \<omega> \<and> k < length bs \<and> length bs = length \<omega> + 1 \<and> wf_\<G> \<G> \<and> wf_bins \<G> \<omega> bs"
	+ using assms(1) from_wf_bins wf_earley_input_def by (smt (verit) Suc_eq_plus1 less_Suc_eq_le mem_Collect_eq prod.sel(1) snd_conv)
	+
	+lemma wf_earley_input_intro:
	+ assumes "k \<le> length \<omega>" "length bs = length \<omega> + 1" "wf_\<G> \<G>" "wf_bins \<G> \<omega> bs"
	+ shows "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ by (simp add: assms wf_earley_input_def)
	+
	+lemma wf_earley_input_Complete\<^sub>L:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input" "\<not> length (items (bs ! k)) \<le> i"
	+ assumes "x = items (bs ! k) ! i" "next_symbol x = None"
	+ shows "(k, \<G>, \<omega>, bins_upd bs k (Complete\<^sub>L k x bs red)) \<in> wf_earley_input"
	+proof -
	+ have *: "k \<le> length \<omega>" "length bs = length \<omega> + 1" "wf_\<G> \<G>" "wf_bins \<G> \<omega> bs"
	+ using wf_earley_input_elim assms(1) by metis+
	+ have x: "x \<in> set (items (bs ! k))"
	+ using assms(2,3) by simp
	+ have "item_origin x < length bs"
	+ using x wf_bins_kth_bin *(1,2,4) wf_item_def
	+ by (metis One_nat_def add.right_neutral add_Suc_right dual_order.trans le_imp_less_Suc)
	+ hence "wf_bins \<G> \<omega> (bins_upd bs k (Complete\<^sub>L k x bs red))"
	+ using *(1,2,4) Suc_eq_plus1 distinct_Complete\<^sub>L le_imp_less_Suc wf_bins_Complete\<^sub>L wf_bins_bins_upd x by metis
	+ thus ?thesis
	+ by (simp add: *(1-3) wf_earley_input_def)
	+qed
	+
	+lemma wf_earley_input_Scan\<^sub>L:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input" "\<not> length (items (bs ! k)) \<le> i"
	+ assumes "x = items (bs ! k) ! i" "next_symbol x = Some a"
	+ assumes "is_terminal \<G> a" "k < length \<omega>"
	+ shows "(k, \<G>, \<omega>, bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x pre)) \<in> wf_earley_input"
	+proof -
	+ have *: "k \<le> length \<omega>" "length bs = length \<omega> + 1" "wf_\<G> \<G>" "wf_bins \<G> \<omega> bs"
	+ using wf_earley_input_elim assms(1) by metis+
	+ have x: "x \<in> set (items(bs ! k))"
	+ using assms(2,3) by simp
	+ have "wf_bins \<G> \<omega> (bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x pre))"
	+ using * x assms(1,4,6) distinct_Scan\<^sub>L wf_bins_Scan\<^sub>L wf_bins_bins_upd wf_earley_input_elim
	+ by (metis option.discI)
	+ thus ?thesis
	+ by (simp add: *(1-3) wf_earley_input_def)
	+qed
	+
	+lemma wf_earley_input_Predict\<^sub>L:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input" "\<not> length (items (bs ! k)) \<le> i"
	+ assumes "x = items (bs ! k) ! i" "next_symbol x = Some a" "\<not> is_terminal \<G> a"
	+ shows "(k, \<G>, \<omega>, bins_upd bs k (Predict\<^sub>L k \<G> a)) \<in> wf_earley_input"
	+proof -
	+ have *: "k \<le> length \<omega>" "length bs = length \<omega> + 1" "wf_\<G> \<G>" "wf_bins \<G> \<omega> bs"
	+ using wf_earley_input_elim assms(1) by metis+
	+ have x: "x \<in> set (items (bs ! k))"
	+ using assms(2,3) by simp
	+ hence "wf_bins \<G> \<omega> (bins_upd bs k (Predict\<^sub>L k \<G> a))"
	+ using * x assms(1,4) distinct_Predict\<^sub>L wf_bins_Predict\<^sub>L wf_bins_bins_upd wf_earley_input_elim by metis
	+ thus ?thesis
	+ by (simp add: *(1-3) wf_earley_input_def)
	+qed
	+
	+fun earley_measure :: "nat \<times> 'a cfg \<times> 'a sentence \<times> 'a bins \<Rightarrow> nat \<Rightarrow> nat" where
	+ "earley_measure (k, \<G>, \<omega>, bs) i = card { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k } - i"
	+
	+lemma Earley\<^sub>L_bin'_simps[simp]:
	+ "i \<ge> length (items (bs ! k)) \<Longrightarrow> Earley\<^sub>L_bin' k \<G> \<omega> bs i = bs"
	+ "\<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs!k) ! i \<Longrightarrow> next_symbol x = None \<Longrightarrow>
	+ Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd bs k (Complete\<^sub>L k x bs i)) (i+1)"
	+ "\<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs!k) ! i \<Longrightarrow> next_symbol x = Some a \<Longrightarrow>
	+ is_terminal \<G> a \<Longrightarrow> k < length \<omega> \<Longrightarrow> Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)) (i+1)"
	+ "\<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs!k) ! i \<Longrightarrow> next_symbol x = Some a \<Longrightarrow>
	+ is_terminal \<G> a \<Longrightarrow> \<not> k < length \<omega> \<Longrightarrow> Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> bs (i+1)"
	+ "\<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs!k) ! i \<Longrightarrow> next_symbol x = Some a \<Longrightarrow>
	+ \<not> is_terminal \<G> a \<Longrightarrow> Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd bs k (Predict\<^sub>L k \<G> a)) (i+1)"
	+ by (subst Earley\<^sub>L_bin'.simps, simp)+
	+
	+lemma Earley\<^sub>L_bin'_induct[case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F]:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes base: "\<And>k \<G> \<omega> bs i. i \<ge> length (items (bs ! k)) \<Longrightarrow> P k \<G> \<omega> bs i"
	+ assumes complete: "\<And>k \<G> \<omega> bs i x. \<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs ! k) ! i \<Longrightarrow>
	+ next_symbol x = None \<Longrightarrow> P k \<G> \<omega> (bins_upd bs k (Complete\<^sub>L k x bs i)) (i+1) \<Longrightarrow> P k \<G> \<omega> bs i"
	+ assumes scan: "\<And>k \<G> \<omega> bs i x a. \<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs ! k) ! i \<Longrightarrow>
	+ next_symbol x = Some a \<Longrightarrow> is_terminal \<G> a \<Longrightarrow> k < length \<omega> \<Longrightarrow>
	+ P k \<G> \<omega> (bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)) (i+1) \<Longrightarrow> P k \<G> \<omega> bs i"
	+ assumes pass: "\<And>k \<G> \<omega> bs i x a. \<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs ! k) ! i \<Longrightarrow>
	+ next_symbol x = Some a \<Longrightarrow> is_terminal \<G> a \<Longrightarrow> \<not> k < length \<omega> \<Longrightarrow>
	+ P k \<G> \<omega> bs (i+1) \<Longrightarrow> P k \<G> \<omega> bs i"
	+ assumes predict: "\<And>k \<G> \<omega> bs i x a. \<not> i \<ge> length (items (bs ! k)) \<Longrightarrow> x = items (bs ! k) ! i \<Longrightarrow>
	+ next_symbol x = Some a \<Longrightarrow> \<not> is_terminal \<G> a \<Longrightarrow>
	+ P k \<G> \<omega> (bins_upd bs k (Predict\<^sub>L k \<G> a)) (i+1) \<Longrightarrow> P k \<G> \<omega> bs i"
	+ shows "P k \<G> \<omega> bs i"
	+ using assms(1)
	+proof (induction n\<equiv>"earley_measure (k, \<G>, \<omega>, bs) i" arbitrary: bs i rule: nat_less_induct)
	+ case 1
	+ have wf: "k \<le> length \<omega>" "length bs = length \<omega> + 1" "wf_\<G> \<G>" "wf_bins \<G> \<omega> bs"
	+ using "1.prems" wf_earley_input_elim by metis+
	+ hence k: "k < length bs"
	+ by simp
	+ have fin: "finite { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k }"
	+ using finiteness_UNIV_wf_item by fastforce
	+ show ?case
	+ proof cases
	+ assume "i \<ge> length (items (bs ! k))"
	+ then show ?thesis
	+ by (simp add: base)
	+ next
	+ assume a1: "\<not> i \<ge> length (items (bs ! k))"
	+ let ?x = "items (bs ! k) ! i"
	+ have x: "?x \<in> set (items (bs ! k))"
	+ using a1 by fastforce
	+ show ?thesis
	+ proof cases
	+ assume a2: "next_symbol ?x = None"
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k ?x bs i)"
	+ have "item_origin ?x < length bs"
	+ using wf(4) k wf_bins_kth_bin wf_item_def x by (metis order_le_less_trans)
	+ hence wf_bins': "wf_bins \<G> \<omega> ?bs'"
	+ using wf_bins_Complete\<^sub>L distinct_Complete\<^sub>L wf(4) wf_bins_bins_upd k x by metis
	+ hence wf': "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using wf(1,2,3) wf_earley_input_intro by fastforce
	+ have sub: "set (items (?bs' ! k)) \<subseteq> { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k }"
	+ using wf(1,2) wf_bins' unfolding wf_bin_def wf_bins_def wf_bin_items_def using order_le_less_trans by auto
	+ have "i < length (items (?bs' ! k))"
	+ using a1 by (metis dual_order.strict_trans1 items_def leI length_map length_nth_bin_bins_upd)
	+ also have "... = card (set (items (?bs' ! k)))"
	+ using wf(1,2) wf_bins' distinct_card wf_bins_def wf_bin_def by (metis k length_bins_upd)
	+ also have "... \<le> card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k}"
	+ using card_mono fin sub by blast
	+ finally have "card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k} > i"
	+ by blast
	+ hence "earley_measure (k, \<G>, \<omega>, ?bs') (Suc i) < earley_measure (k, \<G>, \<omega>, bs) i"
	+ by simp
	+ thus ?thesis
	+ using 1 a1 a2 complete wf' by simp
	+ next
	+ assume a2: "\<not> next_symbol ?x = None"
	+ then obtain a where a_def: "next_symbol ?x = Some a"
	+ by blast
	+ show ?thesis
	+ proof cases
	+ assume a3: "is_terminal \<G> a"
	+ show ?thesis
	+ proof cases
	+ assume a4: "k < length \<omega>"
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a ?x i)"
	+ have wf_bins': "wf_bins \<G> \<omega> ?bs'"
	+ using wf_bins_Scan\<^sub>L distinct_Scan\<^sub>L wf(1,4) wf_bins_bins_upd a2 a4 k x by metis
	+ hence wf': "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using wf(1,2,3) wf_earley_input_intro by fastforce
	+ have sub: "set (items (?bs' ! k)) \<subseteq> { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k }"
	+ using wf(1,2) wf_bins' unfolding wf_bin_def wf_bins_def wf_bin_items_def using order_le_less_trans by auto
	+ have "i < length (items (?bs' ! k))"
	+ using a1 by (metis dual_order.strict_trans1 items_def leI length_map length_nth_bin_bins_upd)
	+ also have "... = card (set (items (?bs' ! k)))"
	+ using wf(1,2) wf_bins' distinct_card wf_bins_def wf_bin_def
	+ by (metis Suc_eq_plus1 le_imp_less_Suc length_bins_upd)
	+ also have "... \<le> card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k}"
	+ using card_mono fin sub by blast
	+ finally have "card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k} > i"
	+ by blast
	+ hence "earley_measure (k, \<G>, \<omega>, ?bs') (Suc i) < earley_measure (k, \<G>, \<omega>, bs) i"
	+ by simp
	+ thus ?thesis
	+ using 1 a1 a_def a3 a4 scan wf' by simp
	+ next
	+ assume a4: "\<not> k < length \<omega>"
	+ have sub: "set (items (bs ! k)) \<subseteq> { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k }"
	+ using wf(1,2,4) unfolding wf_bin_def wf_bins_def wf_bin_items_def using order_le_less_trans by auto
	+ have "i < length (items (bs ! k))"
	+ using a1 by simp
	+ also have "... = card (set (items (bs ! k)))"
	+ using wf(1,2,4) distinct_card wf_bins_def wf_bin_def by (metis Suc_eq_plus1 le_imp_less_Suc)
	+ also have "... \<le> card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k}"
	+ using card_mono fin sub by blast
	+ finally have "card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k} > i"
	+ by blast
	+ hence "earley_measure (k, \<G>, \<omega>, bs) (Suc i) < earley_measure (k, \<G>, \<omega>, bs) i"
	+ by simp
	+ thus ?thesis
	+ using 1 a1 a3 a4 a_def pass by simp
	+ qed
	+ next
	+ assume a3: "\<not> is_terminal \<G> a"
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have wf_bins': "wf_bins \<G> \<omega> ?bs'"
	+ using wf_bins_Predict\<^sub>L distinct_Predict\<^sub>L wf(1,3,4) wf_bins_bins_upd k x by metis
	+ hence wf': "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using wf(1,2,3) wf_earley_input_intro by fastforce
	+ have sub: "set (items (?bs' ! k)) \<subseteq> { x \| x. wf_item \<G> \<omega> x \<and> item_end x = k }"
	+ using wf(1,2) wf_bins' unfolding wf_bin_def wf_bins_def wf_bin_items_def using order_le_less_trans by auto
	+ have "i < length (items (?bs' ! k))"
	+ using a1 by (metis dual_order.strict_trans1 items_def leI length_map length_nth_bin_bins_upd)
	+ also have "... = card (set (items (?bs' ! k)))"
	+ using wf(1,2) wf_bins' distinct_card wf_bins_def wf_bin_def
	+ by (metis Suc_eq_plus1 le_imp_less_Suc length_bins_upd)
	+ also have "... \<le> card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k}"
	+ using card_mono fin sub by blast
	+ finally have "card {x \|x. wf_item \<G> \<omega> x \<and> item_end x = k} > i"
	+ by blast
	+ hence "earley_measure (k, \<G>, \<omega>, ?bs') (Suc i) < earley_measure (k, \<G>, \<omega>, bs) i"
	+ by simp
	+ thus ?thesis
	+ using 1 a1 a_def a3 a_def predict wf' by simp
	+ qed
	+ qed
	+ qed
	+qed
	+
	+lemma wf_earley_input_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "(k, \<G>, \<omega>, Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<in> wf_earley_input"
	+ using assms
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems wf_earley_input_Complete\<^sub>L by blast
	+ thus ?case
	+ using Complete\<^sub>F.IH Complete\<^sub>F.hyps by simp
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems wf_earley_input_Scan\<^sub>L by metis
	+ thus ?case
	+ using Scan\<^sub>F.IH Scan\<^sub>F.hyps by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems wf_earley_input_Predict\<^sub>L by metis
	+ thus ?case
	+ using Predict\<^sub>F.IH Predict\<^sub>F.hyps by simp
	+qed simp_all
	+
	+lemma wf_earley_input_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "(k, \<G>, \<omega>, Earley\<^sub>L_bin k \<G> \<omega> bs) \<in> wf_earley_input"
	+ using assms by (simp add: Earley\<^sub>L_bin_def wf_earley_input_Earley\<^sub>L_bin')
	+
	+lemma length_bins_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "length (Earley\<^sub>L_bin' k \<G> \<omega> bs i) = length bs"
	+ by (metis assms wf_earley_input_Earley\<^sub>L_bin' wf_earley_input_elim)
	+
	+lemma length_nth_bin_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "length (items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l)) \<ge> length (items (bs ! l))"
	+ using length_nth_bin_bins_upd order_trans
	+ by (induction i rule: Earley\<^sub>L_bin'_induct[OF assms]) (auto simp: items_def, blast+)
	+
	+lemma wf_bins_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "wf_bins \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms wf_earley_input_Earley\<^sub>L_bin' wf_earley_input_elim by blast
	+
	+lemma wf_bins_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "wf_bins \<G> \<omega> (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms Earley\<^sub>L_bin_def wf_bins_Earley\<^sub>L_bin' by metis
	+
	+lemma kth_Earley\<^sub>L_bin'_bins:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "j < length (items (bs ! l))"
	+ shows "items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l) ! j = items (bs ! l) ! j"
	+ using assms(2)
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have "items (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1) ! l) ! j = items (?bs' ! l) ! j"
	+ using Complete\<^sub>F.IH Complete\<^sub>F.prems length_nth_bin_bins_upd items_def order.strict_trans2 by (metis length_map)
	+ also have "... = items (bs ! l) ! j"
	+ using Complete\<^sub>F.prems items_nth_idem_bins_upd nth_idem_bins_upd length_map items_def by metis
	+ finally show ?case
	+ using Complete\<^sub>F.hyps by simp
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "items (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1) ! l) ! j = items (?bs' ! l) ! j"
	+ using Scan\<^sub>F.IH Scan\<^sub>F.prems length_nth_bin_bins_upd order.strict_trans2 items_def by (metis length_map)
	+ also have "... = items (bs ! l) ! j"
	+ using Scan\<^sub>F.prems items_nth_idem_bins_upd nth_idem_bins_upd length_map items_def by metis
	+ finally show ?case
	+ using Scan\<^sub>F.hyps by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "items (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1) ! l) ! j = items (?bs' ! l) ! j"
	+ using Predict\<^sub>F.IH Predict\<^sub>F.prems length_nth_bin_bins_upd order.strict_trans2 items_def by (metis length_map)
	+ also have "... = items (bs ! l) ! j"
	+ using Predict\<^sub>F.prems items_nth_idem_bins_upd nth_idem_bins_upd length_map items_def by metis
	+ finally show ?case
	+ using Predict\<^sub>F.hyps by simp
	+qed simp_all
	+
	+lemma nth_bin_sub_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "set (items (bs ! l)) \<subseteq> set (items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l))"
	+proof standard
	+ fix x
	+ assume "x \<in> set (items (bs ! l))"
	+ then obtain j where *: "j < length (items (bs ! l))" "items (bs ! l) ! j = x"
	+ using in_set_conv_nth by metis
	+ have "x = items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l) ! j"
	+ using kth_Earley\<^sub>L_bin'_bins assms * by metis
	+ moreover have "j < length (items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l))"
	+ using assms *(1) length_nth_bin_Earley\<^sub>L_bin' less_le_trans by blast
	+ ultimately show "x \<in> set (items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l))"
	+ by simp
	+qed
	+
	+lemma nth_Earley\<^sub>L_bin'_eq:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "l < k \<Longrightarrow> Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l = bs ! l"
	+ by (induction i rule: Earley\<^sub>L_bin'_induct[OF assms]) (auto simp: bins_upd_def)
	+
	+lemma set_items_Earley\<^sub>L_bin'_eq:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "l < k \<Longrightarrow> set (items (Earley\<^sub>L_bin' k \<G> \<omega> bs i ! l)) = set (items (bs ! l))"
	+ by (simp add: assms nth_Earley\<^sub>L_bin'_eq)
	+
	+lemma bins_upto_k0_Earley\<^sub>L_bin'_eq:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "bins_upto (Earley\<^sub>L_bin k \<G> \<omega> bs) k 0 = bins_upto bs k 0"
	+ unfolding bins_upto_def bin_upto_def Earley\<^sub>L_bin_def using set_items_Earley\<^sub>L_bin'_eq assms nth_Earley\<^sub>L_bin'_eq by fastforce
	+
	+lemma wf_earley_input_Init\<^sub>L:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "(k, \<G>, \<omega>, Init\<^sub>L \<G> \<omega>) \<in> wf_earley_input"
	+proof -
	+ let ?rs = "filter (\<lambda>r. rule_head r = \<SS> \<G>) (\<RR> \<G>)"
	+ let ?b0 = "map (\<lambda>r. (Entry (init_item r 0) Null)) ?rs"
	+ let ?bs = "replicate (length \<omega> + 1) ([])"
	+ have "distinct (items ?b0)"
	+ using assms unfolding wf_bin_def wf_item_def wf_\<G>_def distinct_rules_def items_def
	+ by (auto simp: init_item_def distinct_map inj_on_def)
	+ moreover have "\<forall>x \<in> set (items ?b0). wf_item \<G> \<omega> x \<and> item_end x = 0"
	+ using assms unfolding wf_bin_def wf_item_def by (auto simp: init_item_def items_def)
	+ moreover have "wf_bins \<G> \<omega> ?bs"
	+ unfolding wf_bins_def wf_bin_def wf_bin_items_def items_def using less_Suc_eq_0_disj by force
	+ ultimately show ?thesis
	+ using assms length_replicate wf_earley_input_intro
	+ unfolding wf_bin_def Init\<^sub>L_def wf_bin_def wf_bin_items_def wf_bins_def
	+ by (metis (no_types, lifting) length_list_update nth_list_update_eq nth_list_update_neq)
	+qed
	+
	+lemma length_bins_Init\<^sub>L[simp]:
	+ "length (Init\<^sub>L \<G> \<omega>) = length \<omega> + 1"
	+ by (simp add: Init\<^sub>L_def)
	+
	+lemma wf_earley_input_Earley\<^sub>L_bins[simp]:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "(k, \<G>, \<omega>, Earley\<^sub>L_bins k \<G> \<omega>) \<in> wf_earley_input"
	+ using assms
	+proof (induction k)
	+ case 0
	+ have "(k, \<G>, \<omega>, Init\<^sub>L \<G> \<omega>) \<in> wf_earley_input"
	+ using assms wf_earley_input_Init\<^sub>L by blast
	+ thus ?case
	+ by (simp add: assms(2) wf_earley_input_Init\<^sub>L wf_earley_input_Earley\<^sub>L_bin)
	+next
	+ case (Suc k)
	+ have "(Suc k, \<G>, \<omega>, Earley\<^sub>L_bins k \<G> \<omega>) \<in> wf_earley_input"
	+ using Suc.IH Suc.prems(1) Suc_leD assms(2) wf_earley_input_elim wf_earley_input_intro by metis
	+ thus ?case
	+ by (simp add: wf_earley_input_Earley\<^sub>L_bin)
	+qed
	+
	+lemma length_Earley\<^sub>L_bins[simp]:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "length (Earley\<^sub>L_bins k \<G> \<omega>) = length (Init\<^sub>L \<G> \<omega>)"
	+ using assms wf_earley_input_Earley\<^sub>L_bins wf_earley_input_elim by fastforce
	+
	+lemma wf_bins_Earley\<^sub>L_bins[simp]:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "wf_bins \<G> \<omega> (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ using assms wf_earley_input_Earley\<^sub>L_bins wf_earley_input_elim by fastforce
	+
	+lemma wf_bins_Earley\<^sub>L:
	+ "wf_\<G> \<G> \<Longrightarrow> wf_bins \<G> \<omega> (Earley\<^sub>L \<G> \<omega>)"
	+ by (simp add: Earley\<^sub>L_def)
	+
	+
	+subsection \<open>Soundness\<close>
	+
	+lemma Init\<^sub>L_eq_Init\<^sub>F:
	+ "bins (Init\<^sub>L \<G> \<omega>) = Init\<^sub>F \<G>"
	+proof -
	+ let ?rs = "filter (\<lambda>r. rule_head r = \<SS> \<G>) (\<RR> \<G>)"
	+ let ?b0 = "map (\<lambda>r. (Entry (init_item r 0) Null)) ?rs"
	+ let ?bs = "replicate (length \<omega> + 1) ([])"
	+ have "bins (?bs[0 := ?b0]) = set (items ?b0)"
	+ proof -
	+ have "bins (?bs[0 := ?b0]) = \<Union> {set (items ((?bs[0 := ?b0]) ! k)) \|k. k < length (?bs[0 := ?b0])}"
	+ unfolding bins_def by blast
	+ also have "... = set (items ((?bs[0 := ?b0]) ! 0)) \<union> \<Union> {set (items ((?bs[0 := ?b0]) ! k)) \|k. k < length (?bs[0 := ?b0]) \<and> k \<noteq> 0}"
	+ by fastforce
	+ also have "... = set (items (?b0))"
	+ by (auto simp: items_def)
	+ finally show ?thesis .
	+ qed
	+ also have "... = Init\<^sub>F \<G>"
	+ by (auto simp: Init\<^sub>F_def items_def rule_head_def)
	+ finally show ?thesis
	+ by (auto simp: Init\<^sub>L_def)
	+qed
	+
	+lemma Scan\<^sub>L_sub_Scan\<^sub>F:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "x \<in> set (items (bs ! k))" "k < length bs" "k < length \<omega>"
	+ assumes "next_symbol x = Some a"
	+ shows "set (items (Scan\<^sub>L k \<omega> a x pre)) \<subseteq> Scan\<^sub>F k \<omega> I"
	+proof standard
	+ fix y
	+ assume *: "y \<in> set (items (Scan\<^sub>L k \<omega> a x pre))"
	+ have "x \<in> bin I k"
	+ using kth_bin_sub_bins assms(1-4) items_def wf_bin_def wf_bins_def wf_bin_items_def bin_def by fastforce
	+ {
	+ assume #: "k < length \<omega>" "\<omega>!k = a"
	+ hence "y = inc_item x (k+1)"
	+ using * unfolding Scan\<^sub>L_def by (simp add: items_def)
	+ hence "y \<in> Scan\<^sub>F k \<omega> I"
	+ using \<open>x \<in> bin I k\<close> # assms(6) unfolding Scan\<^sub>F_def by blast
	+ }
	+ thus "y \<in> Scan\<^sub>F k \<omega> I"
	+ using * assms(5) unfolding Scan\<^sub>L_def by (auto simp: items_def)
	+qed
	+
	+lemma Predict\<^sub>L_sub_Predict\<^sub>F:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "x \<in> set (items (bs ! k))" "k < length bs"
	+ assumes "next_symbol x = Some X"
	+ shows "set (items (Predict\<^sub>L k \<G> X)) \<subseteq> Predict\<^sub>F k \<G> I"
	+proof standard
	+ fix y
	+ assume *: "y \<in> set (items (Predict\<^sub>L k \<G> X))"
	+ have "x \<in> bin I k"
	+ using kth_bin_sub_bins assms(1-4) items_def wf_bin_def wf_bins_def bin_def wf_bin_items_def by fast
	+ let ?rs = "filter (\<lambda>r. rule_head r = X) (\<RR> \<G>)"
	+ let ?xs = "map (\<lambda>r. init_item r k) ?rs"
	+ have "y \<in> set ?xs"
	+ using * unfolding Predict\<^sub>L_def items_def by simp
	+ then obtain r where "y = init_item r k" "rule_head r = X" "r \<in> set (\<RR> \<G>)" "next_symbol x = Some (rule_head r)"
	+ using assms(5) by auto
	+ thus "y \<in> Predict\<^sub>F k \<G> I"
	+ unfolding Predict\<^sub>F_def using \<open>x \<in> bin I k\<close> by blast
	+qed
	+
	+lemma Complete\<^sub>L_sub_Complete\<^sub>F:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "y \<in> set (items (bs ! k))" "k < length bs"
	+ assumes "next_symbol y = None"
	+ shows "set (items (Complete\<^sub>L k y bs red)) \<subseteq> Complete\<^sub>F k I"
	+proof standard
	+ fix x
	+ assume *: "x \<in> set (items (Complete\<^sub>L k y bs red))"
	+ have "y \<in> bin I k"
	+ using kth_bin_sub_bins assms items_def wf_bin_def wf_bins_def bin_def wf_bin_items_def by fast
	+ let ?orig = "bs ! item_origin y"
	+ let ?xs = "filter_with_index (\<lambda>x. next_symbol x = Some (item_rule_head y)) (items ?orig)"
	+ let ?xs' = "map (\<lambda>(x, pre). (Entry (inc_item x k) (PreRed (item_origin y, pre, red) []))) ?xs"
	+ have 0: "item_origin y < length bs"
	+ using wf_bins_def wf_bin_def wf_item_def wf_bin_items_def assms(1,3,4)
	+ by (metis Orderings.preorder_class.dual_order.strict_trans1 leD not_le_imp_less)
	+ {
	+ fix z
	+ assume *: "z \<in> set (map fst ?xs)"
	+ have "next_symbol z = Some (item_rule_head y)"
	+ using * by (simp add: filter_with_index_cong_filter)
	+ moreover have "z \<in> bin I (item_origin y)"
	+ using 0 * assms(1,2) bin_def kth_bin_sub_bins wf_bins_kth_bin filter_with_index_cong_filter
	+ by (metis (mono_tags, lifting) filter_is_subset in_mono mem_Collect_eq)
	+ ultimately have "next_symbol z = Some (item_rule_head y)" "z \<in> bin I (item_origin y)"
	+ by simp_all
	+ }
	+ hence 1: "\<forall>z \<in> set (map fst ?xs). next_symbol z = Some (item_rule_head y) \<and> z \<in> bin I (item_origin y)"
	+ by blast
	+ obtain z where z: "x = inc_item z k" "z \<in> set (map fst ?xs)"
	+ using * unfolding Complete\<^sub>L_def by (auto simp: rev_image_eqI items_def)
	+ moreover have "next_symbol z = Some (item_rule_head y)" "z \<in> bin I (item_origin y)"
	+ using 1 z by blast+
	+ ultimately show "x \<in> Complete\<^sub>F k I"
	+ using \<open>y \<in> bin I k\<close> assms(5) unfolding Complete\<^sub>F_def next_symbol_def by (auto split: if_splits)
	+qed
	+
	+lemma sound_Scan\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "x \<in> set (items (bs!k))" "k < length bs" "k < length \<omega>"
	+ assumes "next_symbol x = Some a" "\<forall>x \<in> I. wf_item \<G> \<omega> x" "\<forall>x \<in> I. sound_item \<G> \<omega> x"
	+ shows "\<forall>x \<in> set (items (Scan\<^sub>L k \<omega> a x i)). sound_item \<G> \<omega> x"
	+proof standard
	+ fix y
	+ assume "y \<in> set (items (Scan\<^sub>L k \<omega> a x i))"
	+ hence "y \<in> Scan\<^sub>F k \<omega> I"
	+ by (meson Scan\<^sub>L_sub_Scan\<^sub>F assms(1-6) in_mono)
	+ thus "sound_item \<G> \<omega> y"
	+ using sound_Scan assms(7,8) unfolding Scan\<^sub>F_def inc_item_def bin_def
	+ by (smt (verit, best) item.exhaust_sel mem_Collect_eq)
	+qed
	+
	+lemma sound_Predict\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "x \<in> set (items (bs!k))" "k < length bs"
	+ assumes "next_symbol x = Some X" "\<forall>x \<in> I. wf_item \<G> \<omega> x" "\<forall>x \<in> I. sound_item \<G> \<omega> x"
	+ shows "\<forall>x \<in> set (items (Predict\<^sub>L k \<G> X)). sound_item \<G> \<omega> x"
	+proof standard
	+ fix y
	+ assume "y \<in> set (items (Predict\<^sub>L k \<G> X))"
	+ hence "y \<in> Predict\<^sub>F k \<G> I"
	+ by (meson Predict\<^sub>L_sub_Predict\<^sub>F assms(1-5) subsetD)
	+ thus "sound_item \<G> \<omega> y"
	+ using sound_Predict assms(6,7) unfolding Predict\<^sub>F_def init_item_def bin_def
	+ by (smt (verit, del_insts) item.exhaust_sel mem_Collect_eq)
	+qed
	+
	+lemma sound_Complete\<^sub>L:
	+ assumes "wf_bins \<G> \<omega> bs" "bins bs \<subseteq> I" "y \<in> set (items (bs!k))" "k < length bs"
	+ assumes "next_symbol y = None" "\<forall>x \<in> I. wf_item \<G> \<omega> x" "\<forall>x \<in> I. sound_item \<G> \<omega> x"
	+ shows "\<forall>x \<in> set (items (Complete\<^sub>L k y bs i)). sound_item \<G> \<omega> x"
	+proof standard
	+ fix x
	+ assume "x \<in> set (items (Complete\<^sub>L k y bs i))"
	+ hence "x \<in> Complete\<^sub>F k I"
	+ using Complete\<^sub>L_sub_Complete\<^sub>F assms(1-5) by blast
	+ thus "sound_item \<G> \<omega> x"
	+ using sound_Complete assms(6,7) unfolding Complete\<^sub>F_def inc_item_def bin_def
	+ by (smt (verit, del_insts) item.exhaust_sel mem_Collect_eq)
	+qed
	+
	+lemma sound_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ shows "\<forall>x \<in> bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i). sound_item \<G> \<omega> x"
	+ using assms
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Complete\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items (Complete\<^sub>L k x bs i)). sound_item \<G> \<omega> x"
	+ using sound_Complete\<^sub>L Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items by fastforce
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ ultimately have "\<forall>x \<in> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1)). sound_item \<G> \<omega> x"
	+ using Complete\<^sub>F.IH Complete\<^sub>F.prems(2) length_bins_upd bins_bins_upd wf_earley_input_elim
	+ Suc_eq_plus1 Un_iff le_imp_less_Suc by metis
	+ thus ?case
	+ using Complete\<^sub>F.hyps by simp
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Scan\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items (Scan\<^sub>L k \<omega> a x i)). sound_item \<G> \<omega> x"
	+ using sound_Scan\<^sub>L Scan\<^sub>F.hyps(3,5) Scan\<^sub>F.prems(1,2) wf_earley_input_elim wf_bins_impl_wf_items by fast
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ ultimately have "\<forall>x \<in> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1)). sound_item \<G> \<omega> x"
	+ using Scan\<^sub>F.IH Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(2) length_bins_upd bins_bins_upd wf_earley_input_elim
	+ by (metis UnE add_less_cancel_right)
	+ thus ?case
	+ using Scan\<^sub>F.hyps by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Predict\<^sub>F.hyps(1,2) by force
	+ hence "\<forall>x \<in> set (items(Predict\<^sub>L k \<G> a)). sound_item \<G> \<omega> x"
	+ using sound_Predict\<^sub>L Predict\<^sub>F.hyps(3) Predict\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items by fast
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ ultimately have "\<forall>x \<in> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1)). sound_item \<G> \<omega> x"
	+ using Predict\<^sub>F.IH Predict\<^sub>F.prems(2) length_bins_upd bins_bins_upd wf_earley_input_elim
	+ by (metis Suc_eq_plus1 UnE)
	+ thus ?case
	+ using Predict\<^sub>F.hyps by simp
	+qed simp_all
	+
	+lemma sound_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ shows "\<forall>x \<in> bins (Earley\<^sub>L_bin k \<G> \<omega> bs). sound_item \<G> \<omega> x"
	+ using sound_Earley\<^sub>L_bin' assms Earley\<^sub>L_bin_def by metis
	+
	+lemma Earley\<^sub>L_bin'_sub_Earley\<^sub>F_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "bins bs \<subseteq> I"
	+ shows "bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using assms
	+proof (induction i arbitrary: I rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Base k \<G> \<omega> bs i)
	+ thus ?case
	+ using Earley\<^sub>F_bin_mono by fastforce
	+next
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Complete\<^sub>F.hyps(1,2) by force
	+ hence "bins ?bs' \<subseteq> I \<union> Complete\<^sub>F k I"
	+ using Complete\<^sub>L_sub_Complete\<^sub>F Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems(1,2) bins_bins_upd wf_earley_input_elim by blast
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ ultimately have "bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> (I \<union> Complete\<^sub>F k I)"
	+ using Complete\<^sub>F.IH Complete\<^sub>F.hyps by simp
	+ also have "... \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> (Earley\<^sub>F_bin k \<G> \<omega> I)"
	+ using Complete\<^sub>F_Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_sub_mono by (metis Un_subset_iff)
	+ finally show ?case
	+ using Earley\<^sub>F_bin_idem by blast
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Scan\<^sub>F.hyps(1,2) by force
	+ hence "bins ?bs' \<subseteq> I \<union> Scan\<^sub>F k \<omega> I"
	+ using Scan\<^sub>L_sub_Scan\<^sub>F Scan\<^sub>F.hyps(3,5) Scan\<^sub>F.prems bins_bins_upd wf_earley_input_elim
	+ by (metis add_mono1 sup_mono)
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ ultimately have "bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> (I \<union> Scan\<^sub>F k \<omega> I)"
	+ using Scan\<^sub>F.IH Scan\<^sub>F.hyps by simp
	+ thus ?case
	+ using Scan\<^sub>F_Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_sub_mono Earley\<^sub>F_bin_idem by (metis le_supI order_trans)
	+next
	+ case (Pass k \<G> \<omega> bs i x a)
	+ thus ?case
	+ by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "x \<in> set (items (bs ! k))"
	+ using Predict\<^sub>F.hyps(1,2) by force
	+ hence "bins ?bs' \<subseteq> I \<union> Predict\<^sub>F k \<G> I"
	+ using Predict\<^sub>L_sub_Predict\<^sub>F Predict\<^sub>F.hyps(3) Predict\<^sub>F.prems bins_bins_upd wf_earley_input_elim
	+ by (metis sup_mono)
	+ moreover have "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ ultimately have "bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i) \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> (I \<union> Predict\<^sub>F k \<G> I)"
	+ using Predict\<^sub>F.IH Predict\<^sub>F.hyps by simp
	+ thus ?case
	+ using Predict\<^sub>F_Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_mono Earley\<^sub>F_bin_sub_mono Earley\<^sub>F_bin_idem by (metis le_supI order_trans)
	+qed
	+
	+lemma Earley\<^sub>L_bin_sub_Earley\<^sub>F_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "bins bs \<subseteq> I"
	+ shows "bins (Earley\<^sub>L_bin k \<G> \<omega> bs) \<subseteq> Earley\<^sub>F_bin k \<G> \<omega> I"
	+ using assms Earley\<^sub>L_bin'_sub_Earley\<^sub>F_bin Earley\<^sub>L_bin_def by metis
	+
	+lemma Earley\<^sub>L_bins_sub_Earley\<^sub>F_bins:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ shows "bins (Earley\<^sub>L_bins k \<G> \<omega>) \<subseteq> Earley\<^sub>F_bins k \<G> \<omega>"
	+ using assms
	+proof (induction k)
	+ case 0
	+ have "(k, \<G>, \<omega>, Init\<^sub>L \<G> \<omega>) \<in> wf_earley_input"
	+ using assms(1) assms(2) wf_earley_input_Init\<^sub>L by blast
	+ thus ?case
	+ by (simp add: Init\<^sub>L_eq_Init\<^sub>F Earley\<^sub>L_bin_sub_Earley\<^sub>F_bin assms(2) wf_earley_input_Init\<^sub>L)
	+next
	+ case (Suc k)
	+ have "(Suc k, \<G>, \<omega>, Earley\<^sub>L_bins k \<G> \<omega>) \<in> wf_earley_input"
	+ by (simp add: Suc.prems(1) Suc_leD assms(2) wf_earley_input_intro)
	+ thus ?case
	+ by (simp add: Suc.IH Suc.prems(1) Suc_leD Earley\<^sub>L_bin_sub_Earley\<^sub>F_bin assms(2))
	+qed
	+
	+lemma Earley\<^sub>L_sub_Earley\<^sub>F:
	+ "wf_\<G> \<G> \<Longrightarrow> bins (Earley\<^sub>L \<G> \<omega>) \<subseteq> Earley\<^sub>F \<G> \<omega>"
	+ using Earley\<^sub>L_bins_sub_Earley\<^sub>F_bins Earley\<^sub>F_def Earley\<^sub>L_def by (metis dual_order.refl)
	+
	+theorem soundness_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega>"
	+ shows "derives \<G> [\<SS> \<G>] \<omega>"
	+ using assms Earley\<^sub>L_sub_Earley\<^sub>F recognizing_def soundness_Earley\<^sub>F by (meson subsetD)
	+
	+
	+subsection \<open>Completeness\<close>
	+
	+lemma bin_bins_upto_bins_eq:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "i \<ge> length (items (bs ! k))" "l \<le> k"
	+ shows "bin (bins_upto bs k i) l = bin (bins bs) l"
	+ unfolding bins_upto_def bins_def bin_def using assms nat_less_le
	+ apply (auto simp: nth_list_update bin_upto_eq_set_items wf_bins_kth_bin items_def)
	+ apply (metis imageI nle_le order_trans, fast)
	+ done
	+
	+lemma impossible_complete_item:
	+ assumes "wf_\<G> \<G>" "wf_item \<G> \<omega> x" "sound_item \<G> \<omega> x"
	+ assumes "is_complete x" "item_origin x = k" "item_end x = k" "nonempty_derives \<G>"
	+ shows False
	+proof -
	+ have "derives \<G> [item_rule_head x] []"
	+ using assms(3-6) by (simp add: slice_empty is_complete_def sound_item_def item_\<beta>_def )
	+ moreover have "is_nonterminal \<G> (item_rule_head x)"
	+ using assms(1,2) unfolding wf_item_def item_rule_head_def rule_head_def
	+ by (metis prod.collapse rule_nonterminal_type)
	+ ultimately show ?thesis
	+ using assms(7) nonempty_derives_def is_nonterminal_def by metis
	+qed
	+
	+lemma Complete\<^sub>F_Un_eq_terminal:
	+ assumes "next_symbol z = Some a" "is_terminal \<G> a" "\<forall>x \<in> I. wf_item \<G> \<omega> x" "wf_item \<G> \<omega> z" "wf_\<G> \<G>"
	+ shows "Complete\<^sub>F k (I \<union> {z}) = Complete\<^sub>F k I"
	+proof (rule ccontr)
	+ assume "Complete\<^sub>F k (I \<union> {z}) \<noteq> Complete\<^sub>F k I"
	+ hence "Complete\<^sub>F k I \<subset> Complete\<^sub>F k (I \<union> {z})"
	+ using Complete\<^sub>F_sub_mono by blast
	+ then obtain w x y where *:
	+ "w \<in> Complete\<^sub>F k (I \<union> {z})" "w \<notin> Complete\<^sub>F k I" "w = inc_item x k"
	+ "x \<in> bin (I \<union> {z}) (item_origin y)" "y \<in> bin (I \<union> {z}) k"
	+ "is_complete y" "next_symbol x = Some (item_rule_head y)"
	+ unfolding Complete\<^sub>F_def by fast
	+ show False
	+ proof (cases "x = z")
	+ case True
	+ have "is_nonterminal \<G> (item_rule_head y)"
	+ using *(5,6) assms(1,3-5)
	+ apply (clarsimp simp: wf_item_def bin_def item_rule_head_def rule_head_def next_symbol_def)
	+ by (metis prod.exhaust_sel rule_nonterminal_type)
	+ thus ?thesis
	+ using True *(7) assms(1,2,5) is_terminal_nonterminal by fastforce
	+ next
	+ case False
	+ thus ?thesis
	+ using * assms(1) by (auto simp: next_symbol_def Complete\<^sub>F_def bin_def)
	+ qed
	+qed
	+
	+lemma Complete\<^sub>F_Un_eq_nonterminal:
	+ assumes "wf_\<G> \<G>" "\<forall>x \<in> I. wf_item \<G> \<omega> x" "\<forall>x \<in> I. sound_item \<G> \<omega> x"
	+ assumes "nonempty_derives \<G>" "wf_item \<G> \<omega> z"
	+ assumes "item_end z = k" "next_symbol z \<noteq> None"
	+ shows "Complete\<^sub>F k (I \<union> {z}) = Complete\<^sub>F k I"
	+proof (rule ccontr)
	+ assume "Complete\<^sub>F k (I \<union> {z}) \<noteq> Complete\<^sub>F k I"
	+ hence "Complete\<^sub>F k I \<subset> Complete\<^sub>F k (I \<union> {z})"
	+ using Complete\<^sub>F_sub_mono by blast
	+ then obtain x x' y where *:
	+ "x \<in> Complete\<^sub>F k (I \<union> {z})" "x \<notin> Complete\<^sub>F k I" "x = inc_item x' k"
	+ "x' \<in> bin (I \<union> {z}) (item_origin y)" "y \<in> bin (I \<union> {z}) k"
	+ "is_complete y" "next_symbol x' = Some (item_rule_head y)"
	+ unfolding Complete\<^sub>F_def by fast
	+ consider (A) "x' = z" \| (B) "y = z"
	+ using *(2-7) Complete\<^sub>F_def by (auto simp: bin_def; blast)
	+ thus False
	+ proof cases
	+ case A
	+ have "item_origin y = k"
	+ using *(4) A bin_def assms(6) by (metis (mono_tags, lifting) mem_Collect_eq)
	+ moreover have "item_end y = k"
	+ using *(5) bin_def by blast
	+ moreover have "sound_item \<G> \<omega> y"
	+ using *(5,6) assms(3,7) by (auto simp: bin_def next_symbol_def sound_item_def)
	+ moreover have "wf_item \<G> \<omega> y"
	+ using *(5) assms(2,5) wf_item_def by (auto simp: bin_def)
	+ ultimately show ?thesis
	+ using impossible_complete_item *(6) assms(1,4) by blast
	+ next
	+ case B
	+ thus ?thesis
	+ using *(6) assms(7) by (auto simp: next_symbol_def)
	+ qed
	+qed
	+
	+lemma wf_item_in_kth_bin:
	+ "wf_bins \<G> \<omega> bs \<Longrightarrow> x \<in> bins bs \<Longrightarrow> item_end x = k \<Longrightarrow> x \<in> set (items (bs ! k))"
	+ using bins_bin_exists wf_bins_kth_bin wf_bins_def by blast
	+
	+lemma Complete\<^sub>F_bins_upto_eq_bins:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs" "i \<ge> length (items (bs ! k))"
	+ shows "Complete\<^sub>F k (bins_upto bs k i) = Complete\<^sub>F k (bins bs)"
	+proof -
	+ have "\<And>l. l \<le> k \<Longrightarrow> bin (bins_upto bs k i) l = bin (bins bs) l"
	+ using bin_bins_upto_bins_eq[OF assms] by blast
	+ moreover have "\<forall>x \<in> bins bs. wf_item \<G> \<omega> x"
	+ using assms(1) wf_bins_impl_wf_items by metis
	+ ultimately show ?thesis
	+ unfolding Complete\<^sub>F_def bin_def wf_item_def wf_item_def by auto
	+qed
	+
	+lemma Complete\<^sub>F_sub_bins_Un_Complete\<^sub>L:
	+ assumes "Complete\<^sub>F k I \<subseteq> bins bs" "I \<subseteq> bins bs" "is_complete z" "wf_bins \<G> \<omega> bs" "wf_item \<G> \<omega> z"
	+ shows "Complete\<^sub>F k (I \<union> {z}) \<subseteq> bins bs \<union> set (items (Complete\<^sub>L k z bs red))"
	+proof standard
	+ fix w
	+ assume "w \<in> Complete\<^sub>F k (I \<union> {z})"
	+ then obtain x y where *:
	+ "w = inc_item x k" "x \<in> bin (I \<union> {z}) (item_origin y)" "y \<in> bin (I \<union> {z}) k"
	+ "is_complete y" "next_symbol x = Some (item_rule_head y)"
	+ unfolding Complete\<^sub>F_def by blast
	+ consider (A) "x = z" \| (B) "y = z" \| "\<not> (x = z \<or> y = z)"
	+ by blast
	+ thus "w \<in> bins bs \<union> set (items (Complete\<^sub>L k z bs red))"
	+ proof cases
	+ case A
	+ thus ?thesis
	+ using *(5) assms(3) by (auto simp: next_symbol_def)
	+ next
	+ case B
	+ let ?orig = "bs ! item_origin z"
	+ let ?is = "filter_with_index (\<lambda>x. next_symbol x = Some (item_rule_head z)) (items ?orig)"
	+ have "x \<in> bin I (item_origin y)"
	+ using B (2) (5) assms(3) by (auto simp: next_symbol_def bin_def)
	+ moreover have "bin I (item_origin z) \<subseteq> set (items (bs ! item_origin z))"
	+ using wf_item_in_kth_bin assms(2,4) bin_def by blast
	+ ultimately have "x \<in> set (map fst ?is)"
	+ using *(5) B by (simp add: filter_with_index_cong_filter in_mono)
	+ thus ?thesis
	+ unfolding Complete\<^sub>L_def *(1) by (auto simp: rev_image_eqI items_def)
	+ next
	+ case 3
	+ thus ?thesis
	+ using * assms(1) Complete\<^sub>F_def by (auto simp: bin_def; blast)
	+ qed
	+qed
	+
	+lemma Complete\<^sub>L_eq_item_origin:
	+ "bs ! item_origin y = bs' ! item_origin y \<Longrightarrow> Complete\<^sub>L k y bs red = Complete\<^sub>L k y bs' red"
	+ by (auto simp: Complete\<^sub>L_def)
	+
	+lemma kth_bin_bins_upto_empty:
	+ assumes "wf_bins \<G> \<omega> bs" "k < length bs"
	+ shows "bin (bins_upto bs k 0) k = {}"
	+proof -
	+ {
	+ fix x
	+ assume "x \<in> bins_upto bs k 0"
	+ then obtain l where "x \<in> set (items (bs ! l))" "l < k"
	+ unfolding bins_upto_def bin_upto_def by blast
	+ hence "item_end x = l"
	+ using wf_bins_kth_bin assms by fastforce
	+ hence "item_end x < k"
	+ using \<open>l < k\<close> by blast
	+ }
	+ thus ?thesis
	+ by (auto simp: bin_def)
	+qed
	+
	+lemma Earley\<^sub>L_bin'_mono:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ shows "bins bs \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ hence "bins bs \<subseteq> bins ?bs'"
	+ using length_bins_upd bins_bins_upd wf_earley_input_elim by (metis Un_upper1)
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using wf Complete\<^sub>F.IH by blast
	+ finally show ?case
	+ using Complete\<^sub>F.hyps by simp
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ hence "bins bs \<subseteq> bins ?bs'"
	+ using Scan\<^sub>F.hyps(5) length_bins_upd bins_bins_upd wf_earley_input_elim
	+ by (metis add_mono1 sup_ge1)
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using wf Scan\<^sub>F.IH by blast
	+ finally show ?case
	+ using Scan\<^sub>F.hyps by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ hence "bins bs \<subseteq> bins ?bs'"
	+ using length_bins_upd bins_bins_upd wf_earley_input_elim by (metis sup_ge1)
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using wf Predict\<^sub>F.IH by blast
	+ finally show ?case
	+ using Predict\<^sub>F.hyps by simp
	+qed simp_all
	+
	+lemma Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin':
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "Earley\<^sub>F_bin_step k \<G> \<omega> (bins_upto bs k i) \<subseteq> bins bs"
	+ assumes "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley\<^sub>F_bin_step k \<G> \<omega> (bins bs) \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms
	+proof (induction i rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Base k \<G> \<omega> bs i)
	+ have "bin (bins bs) k = bin (bins_upto bs k i) k"
	+ using Base.hyps Base.prems(1) bin_bins_upto_bins_eq wf_earley_input_elim by blast
	+ thus ?case
	+ using Scan\<^sub>F_bin_absorb Predict\<^sub>F_bin_absorb Complete\<^sub>F_bins_upto_eq_bins wf_earley_input_elim
	+ Base.hyps Base.prems(1,2,3,5) Earley\<^sub>F_bin_step_def Complete\<^sub>F_Earley\<^sub>F_bin_step_mono
	+ Predict\<^sub>F_Earley\<^sub>F_bin_step_mono Scan\<^sub>F_Earley\<^sub>F_bin_step_mono Earley\<^sub>L_bin'_mono
	+ by (metis (no_types, lifting) Un_assoc sup.orderE)
	+next
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Complete\<^sub>F.hyps(1,2) by auto
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ hence sound: "\<forall>x \<in> set (items (Complete\<^sub>L k x bs i)). sound_item \<G> \<omega> x"
	+ using sound_Complete\<^sub>L Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items x
	+ by (metis dual_order.refl)
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) = Scan\<^sub>F k \<omega> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using Complete\<^sub>F.hyps(1) bins_upto_Suc_Un length_nth_bin_bins_upd items_def
	+ by (metis length_map linorder_not_less sup.boundedE sup.order_iff)
	+ also have "... = Scan\<^sub>F k \<omega> (bins_upto bs k i \<union> {x})"
	+ using Complete\<^sub>F.hyps(1,2) Complete\<^sub>F.prems(1) items_nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... \<subseteq> bins bs \<union> Scan\<^sub>F k \<omega> {x}"
	+ using Complete\<^sub>F.prems(2,3) Scan\<^sub>F_Un Scan\<^sub>F_Earley\<^sub>F_bin_step_mono by fastforce
	+ also have "... = bins bs"
	+ using Complete\<^sub>F.hyps(3) by (auto simp: Scan\<^sub>F_def bin_def)
	+ finally show ?thesis
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by blast
	+ qed
	+ moreover have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) = Predict\<^sub>F k \<G> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using Complete\<^sub>F.hyps(1) bins_upto_Suc_Un length_nth_bin_bins_upd
	+ by (metis dual_order.strict_trans1 items_def length_map not_le_imp_less)
	+ also have "... = Predict\<^sub>F k \<G> (bins_upto bs k i \<union> {x})"
	+ using Complete\<^sub>F.hyps(1,2) Complete\<^sub>F.prems(1) items_nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... \<subseteq> bins bs \<union> Predict\<^sub>F k \<G> {x}"
	+ using Complete\<^sub>F.prems(2,3) Predict\<^sub>F_Un Predict\<^sub>F_Earley\<^sub>F_bin_step_mono by blast
	+ also have "... = bins bs"
	+ using Complete\<^sub>F.hyps(3) by (auto simp: Predict\<^sub>F_def bin_def)
	+ finally show ?thesis
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by blast
	+ qed
	+ moreover have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) = Complete\<^sub>F k (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using bins_upto_Suc_Un length_nth_bin_bins_upd Complete\<^sub>F.hyps(1)
	+ by (metis (no_types, opaque_lifting) dual_order.trans items_def length_map not_le_imp_less)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i \<union> {x})"
	+ using items_nth_idem_bins_upd Complete\<^sub>F.hyps(1,2) bins_upto_kth_nth_idem Complete\<^sub>F.prems(1) wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... \<subseteq> bins bs \<union> set (items (Complete\<^sub>L k x bs i))"
	+ using Complete\<^sub>F_sub_bins_Un_Complete\<^sub>L Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems(1,2,3) next_symbol_def
	+ bins_upto_sub_bins wf_bins_kth_bin x Complete\<^sub>F_Earley\<^sub>F_bin_step_mono wf_earley_input_elim
	+ by (smt (verit, best) option.distinct(1) subset_trans)
	+ finally show ?thesis
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by blast
	+ qed
	+ ultimately have "Earley\<^sub>F_bin_step k \<G> \<omega> (bins ?bs') \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Complete\<^sub>F.IH Complete\<^sub>F.prems sound wf Earley\<^sub>F_bin_step_def bins_upto_sub_bins
	+ wf_earley_input_elim bins_bins_upd
	+ by (metis UnE sup.boundedI)
	+ thus ?case
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) Earley\<^sub>L_bin'_simps(2) Earley\<^sub>F_bin_step_sub_mono bins_bins_upd wf_earley_input_elim
	+ by (smt (verit, best) sup.coboundedI2 sup.orderE sup_ge1)
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Scan\<^sub>F.hyps(1,2) by auto
	+ hence sound: "\<forall>x \<in> set (items (Scan\<^sub>L k \<omega> a x i)). sound_item \<G> \<omega> x"
	+ using sound_Scan\<^sub>L Scan\<^sub>F.hyps(3,5) Scan\<^sub>F.prems(1,2,3) wf_earley_input_elim wf_bins_impl_wf_items x
	+ by (metis dual_order.refl)
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) = Scan\<^sub>F k \<omega> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using bins_upto_Suc_Un Scan\<^sub>F.hyps(1) nth_idem_bins_upd
	+ by (metis Suc_eq_plus1 items_def length_map lessI less_not_refl not_le_imp_less)
	+ also have "... = Scan\<^sub>F k \<omega> (bins_upto bs k i \<union> {x})"
	+ using Scan\<^sub>F.hyps(1,2,5) Scan\<^sub>F.prems(1,2) nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis add_mono_thms_linordered_field(1) items_def length_map less_add_one linorder_le_less_linear not_add_less1)
	+ also have "... \<subseteq> bins bs \<union> Scan\<^sub>F k \<omega> {x}"
	+ using Scan\<^sub>F.prems(2,3) Scan\<^sub>F_Un Scan\<^sub>F_Earley\<^sub>F_bin_step_mono by fastforce
	+ finally have *: "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins bs \<union> Scan\<^sub>F k \<omega> {x}" .
	+ show ?thesis
	+ proof cases
	+ assume a1: "\<omega>!k = a"
	+ hence "Scan\<^sub>F k \<omega> {x} = {inc_item x (k+1)}"
	+ using Scan\<^sub>F.hyps(1-3,5) Scan\<^sub>F.prems(1,2) wf_earley_input_elim apply (auto simp: Scan\<^sub>F_def bin_def)
	+ using wf_bins_kth_bin x by blast
	+ hence "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins bs \<union> {inc_item x (k+1)}"
	+ using * by blast
	+ also have "... = bins bs \<union> set (items (Scan\<^sub>L k \<omega> a x i))"
	+ using a1 Scan\<^sub>F.hyps(5) by (auto simp: Scan\<^sub>L_def items_def)
	+ also have "... = bins ?bs'"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by (metis add_mono1)
	+ finally show ?thesis .
	+ next
	+ assume a1: "\<not> \<omega>!k = a"
	+ hence "Scan\<^sub>F k \<omega> {x} = {}"
	+ using Scan\<^sub>F.hyps(3) by (auto simp: Scan\<^sub>F_def bin_def)
	+ hence "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins bs"
	+ using * by blast
	+ also have "... \<subseteq> bins ?bs'"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd
	+ by (metis Un_left_absorb add_strict_right_mono subset_Un_eq)
	+ finally show ?thesis .
	+ qed
	+ qed
	+ moreover have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) = Predict\<^sub>F k \<G> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using bins_upto_Suc_Un Scan\<^sub>F.hyps(1) nth_idem_bins_upd
	+ by (metis Suc_eq_plus1 dual_order.refl items_def length_map lessI linorder_not_less)
	+ also have "... = Predict\<^sub>F k \<G> (bins_upto bs k i \<union> {x})"
	+ using Scan\<^sub>F.hyps(1,2,5) Scan\<^sub>F.prems(1,2) nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis add_strict_right_mono items_def le_add1 length_map less_add_one linorder_not_le)
	+ also have "... \<subseteq> bins bs \<union> Predict\<^sub>F k \<G> {x}"
	+ using Scan\<^sub>F.prems(2,3) Predict\<^sub>F_Un Predict\<^sub>F_Earley\<^sub>F_bin_step_mono by fastforce
	+ also have "... = bins bs"
	+ using Scan\<^sub>F.hyps(3,4) Scan\<^sub>F.prems(1) is_terminal_nonterminal wf_earley_input_elim
	+ by (auto simp: Predict\<^sub>F_def bin_def rule_head_def, fastforce)
	+ finally show ?thesis
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1) by (simp add: bins_bins_upd sup.coboundedI1 wf_earley_input_elim)
	+ qed
	+ moreover have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) = Complete\<^sub>F k (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using bins_upto_Suc_Un Scan\<^sub>F.hyps(1) nth_idem_bins_upd
	+ by (metis Suc_eq_plus1 items_def length_map lessI less_not_refl not_le_imp_less)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i \<union> {x})"
	+ using Scan\<^sub>F.hyps(1,2,5) Scan\<^sub>F.prems(1,2) nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis add_mono1 items_def length_map less_add_one linorder_not_le not_add_less1)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i)"
	+ using Complete\<^sub>F_Un_eq_terminal Scan\<^sub>F.hyps(3,4) Scan\<^sub>F.prems bins_upto_sub_bins subset_iff
	+ wf_bins_impl_wf_items wf_bins_kth_bin wf_item_def x wf_earley_input_elim
	+ by (smt (verit, ccfv_threshold))
	+ finally show ?thesis
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1,2,3) Complete\<^sub>F_Earley\<^sub>F_bin_step_mono by (auto simp: bins_bins_upd wf_earley_input_elim, blast)
	+ qed
	+ ultimately have "Earley\<^sub>F_bin_step k \<G> \<omega> (bins ?bs') \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Scan\<^sub>F.IH Scan\<^sub>F.prems Scan\<^sub>F.hyps(5) sound wf Earley\<^sub>F_bin_step_def bins_upto_sub_bins wf_earley_input_elim
	+ bins_bins_upd by (metis UnE add_mono1 le_supI)
	+ thus ?case
	+ using Earley\<^sub>F_bin_step_sub_mono Earley\<^sub>L_bin'_simps(3) Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd
	+ by (smt (verit, ccfv_SIG) add_mono1 sup.cobounded1 sup.coboundedI2 sup.orderE)
	+next
	+ case (Pass k \<G> \<omega> bs i x a)
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Pass.hyps(1,2) by auto
	+ have "Scan\<^sub>F k \<omega> (bins_upto bs k (i + 1)) \<subseteq> bins bs"
	+ using Scan\<^sub>F_def Pass.hyps(5) by auto
	+ moreover have "Predict\<^sub>F k \<G> (bins_upto bs k (i + 1)) \<subseteq> bins bs"
	+ proof -
	+ have "Predict\<^sub>F k \<G> (bins_upto bs k (i + 1)) = Predict\<^sub>F k \<G> (bins_upto bs k i \<union> {items (bs ! k) ! i})"
	+ using bins_upto_Suc_Un Pass.hyps(1) by (metis items_def length_map not_le_imp_less)
	+ also have "... = Predict\<^sub>F k \<G> (bins_upto bs k i \<union> {x})"
	+ using Pass.hyps(1,2,5) nth_idem_bins_upd bins_upto_kth_nth_idem by simp
	+ also have "... \<subseteq> bins bs \<union> Predict\<^sub>F k \<G> {x}"
	+ using Pass.prems(2) Predict\<^sub>F_Un Predict\<^sub>F_Earley\<^sub>F_bin_step_mono by blast
	+ also have "... = bins bs"
	+ using Pass.hyps(3,4) Pass.prems(1) is_terminal_nonterminal wf_earley_input_elim
	+ by (auto simp: Predict\<^sub>F_def bin_def rule_head_def, fastforce)
	+ finally show ?thesis
	+ using bins_bins_upd Pass.hyps(5) Pass.prems(3) by auto
	+ qed
	+ moreover have "Complete\<^sub>F k (bins_upto bs k (i + 1)) \<subseteq> bins bs"
	+ proof -
	+ have "Complete\<^sub>F k (bins_upto bs k (i + 1)) = Complete\<^sub>F k (bins_upto bs k i \<union> {x})"
	+ using bins_upto_Suc_Un Pass.hyps(1,2)
	+ by (metis items_def length_map not_le_imp_less)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i)"
	+ using Complete\<^sub>F_Un_eq_terminal Pass.hyps Pass.prems bins_upto_sub_bins subset_iff
	+ wf_bins_impl_wf_items wf_item_def wf_bins_kth_bin x wf_earley_input_elim by (smt (verit, best))
	+ finally show ?thesis
	+ using Pass.prems(1,2) Complete\<^sub>F_Earley\<^sub>F_bin_step_mono wf_earley_input_elim by blast
	+ qed
	+ ultimately have "Earley\<^sub>F_bin_step k \<G> \<omega> (bins bs) \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> bs (i+1))"
	+ using Pass.IH Pass.prems Earley\<^sub>F_bin_step_def bins_upto_sub_bins wf_earley_input_elim
	+ by (metis le_sup_iff)
	+ thus ?case
	+ using bins_bins_upd Pass.hyps Pass.prems by simp
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "k \<ge> length \<omega> \<or> \<not> \<omega>!k = a"
	+ using Predict\<^sub>F.hyps(4) Predict\<^sub>F.prems(4) is_word_is_terminal leI by blast
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Predict\<^sub>F.hyps(1,2) by auto
	+ hence sound: "\<forall>x \<in> set (items(Predict\<^sub>L k \<G> a)). sound_item \<G> \<omega> x"
	+ using sound_Predict\<^sub>L Predict\<^sub>F.hyps(3) Predict\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items by fast
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ have len: "i < length (items (?bs' ! k))"
	+ using length_nth_bin_bins_upd Predict\<^sub>F.hyps(1)
	+ by (metis dual_order.strict_trans1 items_def length_map linorder_not_less)
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Scan\<^sub>F k \<omega> (bins_upto ?bs' k (i + 1)) = Scan\<^sub>F k \<omega> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using Predict\<^sub>F.hyps(1) bins_upto_Suc_Un by (metis items_def len length_map)
	+ also have "... = Scan\<^sub>F k \<omega> (bins_upto bs k i \<union> {x})"
	+ using Predict\<^sub>F.hyps(1,2) Predict\<^sub>F.prems(1) items_nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... \<subseteq> bins bs \<union> Scan\<^sub>F k \<omega> {x}"
	+ using Predict\<^sub>F.prems(2,3) Scan\<^sub>F_Un Scan\<^sub>F_Earley\<^sub>F_bin_step_mono by fastforce
	+ also have "... = bins bs"
	+ using Predict\<^sub>F.hyps(3) \<open>length \<omega> \<le> k \<or> \<omega> ! k \<noteq> a\<close> by (auto simp: Scan\<^sub>F_def bin_def)
	+ finally show ?thesis
	+ using Predict\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by blast
	+ qed
	+ moreover have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Predict\<^sub>F k \<G> (bins_upto ?bs' k (i + 1)) = Predict\<^sub>F k \<G> (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using Predict\<^sub>F.hyps(1) bins_upto_Suc_Un by (metis items_def len length_map)
	+ also have "... = Predict\<^sub>F k \<G> (bins_upto bs k i \<union> {x})"
	+ using Predict\<^sub>F.hyps(1,2) Predict\<^sub>F.prems(1) items_nth_idem_bins_upd bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... \<subseteq> bins bs \<union> Predict\<^sub>F k \<G> {x}"
	+ using Predict\<^sub>F.prems(2,3) Predict\<^sub>F_Un Predict\<^sub>F_Earley\<^sub>F_bin_step_mono by fastforce
	+ also have "... = bins bs \<union> set (items (Predict\<^sub>L k \<G> a))"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1-3) wf_earley_input_elim
	+ apply (auto simp: Predict\<^sub>F_def Predict\<^sub>L_def bin_def items_def)
	+ using wf_bins_kth_bin x by blast
	+ finally show ?thesis
	+ using Predict\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by blast
	+ qed
	+ moreover have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) \<subseteq> bins ?bs'"
	+ proof -
	+ have "Complete\<^sub>F k (bins_upto ?bs' k (i + 1)) = Complete\<^sub>F k (bins_upto ?bs' k i \<union> {items (?bs' ! k) ! i})"
	+ using bins_upto_Suc_Un len by (metis items_def length_map)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i \<union> {x})"
	+ using items_nth_idem_bins_upd Predict\<^sub>F.hyps(1,2) Predict\<^sub>F.prems(1) bins_upto_kth_nth_idem wf_earley_input_elim
	+ by (metis dual_order.refl items_def length_map not_le_imp_less)
	+ also have "... = Complete\<^sub>F k (bins_upto bs k i)"
	+ using Complete\<^sub>F_Un_eq_nonterminal Predict\<^sub>F.prems bins_upto_sub_bins Predict\<^sub>F.hyps(3)
	+ subset_eq wf_bins_kth_bin x wf_bins_impl_wf_items wf_item_def wf_earley_input_elim
	+ by (smt (verit, ccfv_SIG) option.simps(3))
	+ also have "... \<subseteq> bins bs"
	+ using Complete\<^sub>F_Earley\<^sub>F_bin_step_mono Predict\<^sub>F.prems(2) by blast
	+ finally show ?thesis
	+ using bins_bins_upd Predict\<^sub>F.prems(1,2,3) wf_earley_input_elim by (metis Un_upper1 dual_order.trans)
	+ qed
	+ ultimately have "Earley\<^sub>F_bin_step k \<G> \<omega> (bins ?bs') \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1))"
	+ using Predict\<^sub>F.IH Predict\<^sub>F.prems sound wf Earley\<^sub>F_bin_step_def bins_upto_sub_bins
	+ bins_bins_upd wf_earley_input_elim by (metis UnE le_supI)
	+ hence "Earley\<^sub>F_bin_step k \<G> \<omega> (bins ?bs') \<subseteq> bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using Predict\<^sub>F.hyps Earley\<^sub>L_bin'_simps(5) by simp
	+ moreover have "Earley\<^sub>F_bin_step k \<G> \<omega> (bins bs) \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> (bins ?bs')"
	+ using Earley\<^sub>F_bin_step_sub_mono Predict\<^sub>F.prems(1) wf_earley_input_elim bins_bins_upd by (metis Un_upper1)
	+ ultimately show ?case
	+ by blast
	+qed
	+
	+lemma Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "Earley\<^sub>F_bin_step k \<G> \<omega> (bins_upto bs k 0) \<subseteq> bins bs"
	+ assumes "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley\<^sub>F_bin_step k \<G> \<omega> (bins bs) \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin' Earley\<^sub>L_bin_def by metis
	+
	+lemma bins_eq_items_Complete\<^sub>L:
	+ assumes "bins_eq_items as bs" "item_origin x < length as"
	+ shows "items (Complete\<^sub>L k x as i) = items (Complete\<^sub>L k x bs i)"
	+proof -
	+ let ?orig_a = "as ! item_origin x"
	+ let ?orig_b = "bs ! item_origin x"
	+ have "items ?orig_a = items ?orig_b"
	+ using assms by (metis (no_types, opaque_lifting) bins_eq_items_def length_map nth_map)
	+ thus ?thesis
	+ unfolding Complete\<^sub>L_def by simp
	+qed
	+
	+lemma Earley\<^sub>L_bin'_bins_eq:
	+ assumes "(k, \<G>, \<omega>, as) \<in> wf_earley_input"
	+ assumes "bins_eq_items as bs" "wf_bins \<G> \<omega> as"
	+ shows "bins_eq_items (Earley\<^sub>L_bin' k \<G> \<omega> as i) (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms
	+proof (induction i arbitrary: bs rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Base k \<G> \<omega> as i)
	+ have "Earley\<^sub>L_bin' k \<G> \<omega> as i = as"
	+ by (simp add: Base.hyps)
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> bs i = bs"
	+ using Base.hyps Base.prems(1,2) unfolding bins_eq_items_def
	+ by (metis Earley\<^sub>L_bin'_simps(1) length_map nth_map wf_earley_input_elim)
	+ ultimately show ?case
	+ using Base.prems(2) by presburger
	+next
	+ case (Complete\<^sub>F k \<G> \<omega> as i x)
	+ let ?as' = "bins_upd as k (Complete\<^sub>L k x as i)"
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have k: "k < length as"
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim by blast
	+ hence wf_x: "wf_item \<G> \<omega> x"
	+ using Complete\<^sub>F.hyps(1,2) Complete\<^sub>F.prems(3) wf_bins_kth_bin by fastforce
	+ have "(k, \<G>, \<omega>, ?as') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ moreover have "bins_eq_items ?as' ?bs'"
	+ using Complete\<^sub>F.hyps(1,2) Complete\<^sub>F.prems(2,3) bins_eq_items_dist_bins_upd bins_eq_items_Complete\<^sub>L
	+ k wf_x wf_bins_kth_bin wf_item_def by (metis dual_order.strict_trans2 leI nth_mem)
	+ ultimately have "bins_eq_items (Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i + 1)) (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using Complete\<^sub>F.IH wf_earley_input_elim by blast
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> as i = Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i+1)"
	+ using Complete\<^sub>F.hyps by simp
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems unfolding bins_eq_items_def
	+ by (metis Earley\<^sub>L_bin'_simps(2) map_eq_imp_length_eq nth_map wf_earley_input_elim)
	+ ultimately show ?case
	+ by argo
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> as i x a)
	+ let ?as' = "bins_upd as (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have "(k, \<G>, \<omega>, ?as') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by fast
	+ moreover have "bins_eq_items ?as' ?bs'"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1,2) bins_eq_items_dist_bins_upd add_mono1 wf_earley_input_elim by metis
	+ ultimately have "bins_eq_items (Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i + 1)) (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using Scan\<^sub>F.IH wf_earley_input_elim by blast
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> as i = Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i+1)"
	+ using Scan\<^sub>F.hyps by simp
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems unfolding bins_eq_items_def
	+ by (smt (verit, ccfv_threshold) Earley\<^sub>L_bin'_simps(3) length_map nth_map wf_earley_input_elim)
	+ ultimately show ?case
	+ by argo
	+next
	+ case (Pass k \<G> \<omega> as i x a)
	+ have "bins_eq_items (Earley\<^sub>L_bin' k \<G> \<omega> as (i + 1)) (Earley\<^sub>L_bin' k \<G> \<omega> bs (i + 1))"
	+ using Pass.prems Pass.IH by blast
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> as i = Earley\<^sub>L_bin' k \<G> \<omega> as (i+1)"
	+ using Pass.hyps by simp
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> bs (i+1)"
	+ using Pass.hyps Pass.prems unfolding bins_eq_items_def
	+ by (metis Earley\<^sub>L_bin'_simps(4) map_eq_imp_length_eq nth_map wf_earley_input_elim)
	+ ultimately show ?case
	+ by argo
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> as i x a)
	+ let ?as' = "bins_upd as k (Predict\<^sub>L k \<G> a)"
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have "(k, \<G>, \<omega>, ?as') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by fast
	+ moreover have "bins_eq_items ?as' ?bs'"
	+ using Predict\<^sub>F.prems(1,2) bins_eq_items_dist_bins_upd wf_earley_input_elim by blast
	+ ultimately have "bins_eq_items (Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i + 1)) (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using Predict\<^sub>F.IH wf_earley_input_elim by blast
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> as i = Earley\<^sub>L_bin' k \<G> \<omega> ?as' (i+1)"
	+ using Predict\<^sub>F.hyps by simp
	+ moreover have "Earley\<^sub>L_bin' k \<G> \<omega> bs i = Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems unfolding bins_eq_items_def
	+ by (metis Earley\<^sub>L_bin'_simps(5) length_map nth_map wf_earley_input_elim)
	+ ultimately show ?case
	+ by argo
	+qed
	+
	+lemma Earley\<^sub>L_bin'_idem:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "i \<le> j" "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x" "nonempty_derives \<G>"
	+ shows "bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i) j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> bs i)"
	+ using assms
	+proof (induction i arbitrary: j rule: Earley\<^sub>L_bin'_induct[OF assms(1), case_names Base Complete\<^sub>F Scan\<^sub>F Pass Predict\<^sub>F])
	+ case (Complete\<^sub>F k \<G> \<omega> bs i x)
	+ let ?bs' = "bins_upd bs k (Complete\<^sub>L k x bs i)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Complete\<^sub>F.hyps(1,2) by auto
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) wf_earley_input_Complete\<^sub>L by blast
	+ hence "\<forall>x \<in> set (items (Complete\<^sub>L k x bs i)). sound_item \<G> \<omega> x"
	+ using sound_Complete\<^sub>L Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items x
	+ by (metis dual_order.refl)
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ by (metis Complete\<^sub>F.prems(1,3) UnE bins_bins_upd wf_earley_input_elim)
	+ show ?case
	+ proof cases
	+ assume "i+1 \<le> j"
	+ thus ?thesis
	+ using wf sound Complete\<^sub>F Earley\<^sub>L_bin'_simps(2) by metis
	+ next
	+ assume "\<not> i+1 \<le> j"
	+ hence "i = j"
	+ using Complete\<^sub>F.prems(2) by simp
	+ have "bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i) j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) j)"
	+ using Earley\<^sub>L_bin'_simps(2) Complete\<^sub>F.hyps(1-3) by simp
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) (j+1))"
	+ proof -
	+ let ?bs'' = "Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ have "length (items (?bs'' ! k)) \<ge> length (items (bs ! k))"
	+ using length_nth_bin_Earley\<^sub>L_bin' length_nth_bin_bins_upd order_trans wf Complete\<^sub>F.hyps Complete\<^sub>F.prems(1)
	+ by (smt (verit, ccfv_threshold) Earley\<^sub>L_bin'_simps(2))
	+ hence 0: "\<not> length (items (?bs'' ! k)) \<le> j"
	+ using \<open>i = j\<close> Complete\<^sub>F.hyps(1) by linarith
	+ have "x = items (?bs' ! k) ! j"
	+ using \<open>i = j\<close> items_nth_idem_bins_upd Complete\<^sub>F.hyps(1,2)
	+ by (metis items_def length_map not_le_imp_less)
	+ hence 1: "x = items (?bs'' ! k) ! j"
	+ using \<open>i = j\<close> kth_Earley\<^sub>L_bin'_bins Complete\<^sub>F.hyps Complete\<^sub>F.prems(1) Earley\<^sub>L_bin'_simps(2) leI by metis
	+ have "bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs'' j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd ?bs'' k (Complete\<^sub>L k x ?bs'' i)) (j+1))"
	+ using Earley\<^sub>L_bin'_simps(2) 0 1 Complete\<^sub>F.hyps(1,3) Complete\<^sub>F.prems(2) \<open>i = j\<close> by auto
	+ moreover have "bins_eq_items (bins_upd ?bs'' k (Complete\<^sub>L k x ?bs'' i)) ?bs''"
	+ proof -
	+ have "k < length bs"
	+ using Complete\<^sub>F.prems(1) wf_earley_input_elim by blast
	+ have 0: "set (Complete\<^sub>L k x bs i) = set (Complete\<^sub>L k x ?bs'' i)"
	+ proof (cases "item_origin x = k")
	+ case True
	+ thus ?thesis
	+ using impossible_complete_item kth_bin_sub_bins Complete\<^sub>F.hyps(3) Complete\<^sub>F.prems wf_earley_input_elim
	+ wf_bins_kth_bin x next_symbol_def by (metis option.distinct(1) subsetD)
	+ next
	+ case False
	+ hence "item_origin x < k"
	+ using x Complete\<^sub>F.prems(1) wf_bins_kth_bin wf_item_def nat_less_le by (metis wf_earley_input_elim)
	+ hence "bs ! item_origin x = ?bs'' ! item_origin x"
	+ using False nth_idem_bins_upd nth_Earley\<^sub>L_bin'_eq wf by metis
	+ thus ?thesis
	+ using Complete\<^sub>L_eq_item_origin by metis
	+ qed
	+ have "set (items (Complete\<^sub>L k x bs i)) \<subseteq> set (items (?bs' ! k))"
	+ by (simp add: \<open>k < length bs\<close> bins_upd_def set_items_bin_upds)
	+ hence "set (items (Complete\<^sub>L k x ?bs'' i)) \<subseteq> set (items (?bs' ! k))"
	+ using 0 by (simp add: items_def)
	+ also have "... \<subseteq> set (items (?bs'' ! k))"
	+ by (simp add: wf nth_bin_sub_Earley\<^sub>L_bin')
	+ finally show ?thesis
	+ using bins_eq_items_bins_upd by blast
	+ qed
	+ moreover have "(k, \<G>, \<omega>, bins_upd ?bs'' k (Complete\<^sub>L k x ?bs'' i)) \<in> wf_earley_input"
	+ using wf_earley_input_Earley\<^sub>L_bin' wf_earley_input_Complete\<^sub>L Complete\<^sub>F.hyps Complete\<^sub>F.prems(1)
	+ \<open>length (items (bs ! k)) \<le> length (items (?bs'' ! k))\<close> kth_Earley\<^sub>L_bin'_bins 0 1 by blast
	+ ultimately show ?thesis
	+ using Earley\<^sub>L_bin'_bins_eq bins_eq_items_imp_eq_bins wf_earley_input_elim by blast
	+ qed
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using Complete\<^sub>F.IH[OF wf _ sound Complete\<^sub>F.prems(4)] \<open>i = j\<close> by blast
	+ finally show ?thesis
	+ using Complete\<^sub>F.hyps by simp
	+ qed
	+next
	+ case (Scan\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs (k+1) (Scan\<^sub>L k \<omega> a x i)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Scan\<^sub>F.hyps(1,2) by auto
	+ hence "\<forall>x \<in> set (items (Scan\<^sub>L k \<omega> a x i)). sound_item \<G> \<omega> x"
	+ using sound_Scan\<^sub>L Scan\<^sub>F.hyps(3,5) Scan\<^sub>F.prems(1,2,3) wf_earley_input_elim wf_bins_impl_wf_items x
	+ by (metis dual_order.refl)
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems(1,3) bins_bins_upd wf_earley_input_elim
	+ by (metis UnE add_less_cancel_right)
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) wf_earley_input_Scan\<^sub>L by metis
	+ show ?case
	+ proof cases
	+ assume "i+1 \<le> j"
	+ thus ?thesis
	+ using sound Scan\<^sub>F by (metis Earley\<^sub>L_bin'_simps(3) wf_earley_input_Scan\<^sub>L)
	+ next
	+ assume "\<not> i+1 \<le> j"
	+ hence "i = j"
	+ using Scan\<^sub>F.prems(2) by auto
	+ have "bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i) j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) j)"
	+ using Scan\<^sub>F.hyps by simp
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) (j+1))"
	+ proof -
	+ let ?bs'' = "Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ have "length (items (?bs'' ! k)) \<ge> length (items (bs ! k))"
	+ using length_nth_bin_Earley\<^sub>L_bin' length_nth_bin_bins_upd order_trans Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) Earley\<^sub>L_bin'_simps(3)
	+ by (smt (verit, ccfv_SIG))
	+ hence "bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs'' j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd ?bs'' (k+1) (Scan\<^sub>L k \<omega> a x i)) (j+1))"
	+ using \<open>i = j\<close> kth_Earley\<^sub>L_bin'_bins nth_idem_bins_upd Earley\<^sub>L_bin'_simps(3) Scan\<^sub>F.hyps Scan\<^sub>F.prems(1) by (smt (verit, best) leI le_trans)
	+ moreover have "bins_eq_items (bins_upd ?bs'' (k+1) (Scan\<^sub>L k \<omega> a x i)) ?bs''"
	+ proof -
	+ have "k+1 < length bs"
	+ using Scan\<^sub>F.hyps(5) Scan\<^sub>F.prems wf_earley_input_elim by fastforce+
	+ hence "set (items (Scan\<^sub>L k \<omega> a x i)) \<subseteq> set (items (?bs' ! (k+1)))"
	+ by (simp add: bins_upd_def set_items_bin_upds)
	+ also have "... \<subseteq> set (items (?bs'' ! (k+1)))"
	+ using wf nth_bin_sub_Earley\<^sub>L_bin' by blast
	+ finally show ?thesis
	+ using bins_eq_items_bins_upd by blast
	+ qed
	+ moreover have "(k, \<G>, \<omega>, bins_upd ?bs'' (k+1) (Scan\<^sub>L k \<omega> a x i)) \<in> wf_earley_input"
	+ using wf_earley_input_Earley\<^sub>L_bin' wf_earley_input_Scan\<^sub>L Scan\<^sub>F.hyps Scan\<^sub>F.prems(1)
	+ \<open>length (items (bs ! k)) \<le> length (items (?bs'' ! k))\<close> kth_Earley\<^sub>L_bin'_bins
	+ by (smt (verit, ccfv_SIG) Earley\<^sub>L_bin'_simps(3) linorder_not_le order.trans)
	+ ultimately show ?thesis
	+ using Earley\<^sub>L_bin'_bins_eq bins_eq_items_imp_eq_bins wf_earley_input_elim by blast
	+ qed
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using \<open>i = j\<close> Scan\<^sub>F.IH Scan\<^sub>F.prems Scan\<^sub>F.hyps sound wf_earley_input_Scan\<^sub>L by fast
	+ finally show ?thesis
	+ using Scan\<^sub>F.hyps by simp
	+ qed
	+next
	+ case (Pass k \<G> \<omega> bs i x a)
	+ show ?case
	+ proof cases
	+ assume "i+1 \<le> j"
	+ thus ?thesis
	+ using Pass by (metis Earley\<^sub>L_bin'_simps(4))
	+ next
	+ assume "\<not> i+1 \<le> j"
	+ show ?thesis
	+ using Pass Earley\<^sub>L_bin'_simps(1,4) kth_Earley\<^sub>L_bin'_bins by (metis Suc_eq_plus1 Suc_leI antisym_conv2 not_le_imp_less)
	+ qed
	+next
	+ case (Predict\<^sub>F k \<G> \<omega> bs i x a)
	+ let ?bs' = "bins_upd bs k (Predict\<^sub>L k \<G> a)"
	+ have x: "x \<in> set (items (bs ! k))"
	+ using Predict\<^sub>F.hyps(1,2) by auto
	+ hence "\<forall>x \<in> set (items(Predict\<^sub>L k \<G> a)). sound_item \<G> \<omega> x"
	+ using sound_Predict\<^sub>L Predict\<^sub>F.hyps(3) Predict\<^sub>F.prems wf_earley_input_elim wf_bins_impl_wf_items by fast
	+ hence sound: "\<forall>x \<in> bins ?bs'. sound_item \<G> \<omega> x"
	+ using Predict\<^sub>F.prems(1,3) UnE bins_bins_upd wf_earley_input_elim by metis
	+ have wf: "(k, \<G>, \<omega>, ?bs') \<in> wf_earley_input"
	+ using Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) wf_earley_input_Predict\<^sub>L by metis
	+ have len: "i < length (items (?bs' ! k))"
	+ using length_nth_bin_bins_upd Predict\<^sub>F.hyps(1) Orderings.preorder_class.dual_order.strict_trans1 linorder_not_less
	+ by (metis items_def length_map)
	+ show ?case
	+ proof cases
	+ assume "i+1 \<le> j"
	+ thus ?thesis
	+ using sound wf Predict\<^sub>F by (metis Earley\<^sub>L_bin'_simps(5))
	+ next
	+ assume "\<not> i+1 \<le> j"
	+ hence "i = j"
	+ using Predict\<^sub>F.prems(2) by auto
	+ have "bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> bs i) j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) j)"
	+ using Predict\<^sub>F.hyps by simp
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)) (j+1))"
	+ proof -
	+ let ?bs'' = "Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i+1)"
	+ have "length (items (?bs'' ! k)) \<ge> length (items (bs ! k))"
	+ using length_nth_bin_Earley\<^sub>L_bin' length_nth_bin_bins_upd order_trans wf
	+ by (metis (no_types, lifting) items_def length_map)
	+ hence "bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs'' j) = bins (Earley\<^sub>L_bin' k \<G> \<omega> (bins_upd ?bs'' k (Predict\<^sub>L k \<G> a)) (j+1))"
	+ using \<open>i = j\<close> kth_Earley\<^sub>L_bin'_bins nth_idem_bins_upd Earley\<^sub>L_bin'_simps(5) Predict\<^sub>F.hyps Predict\<^sub>F.prems(1) length_bins_Earley\<^sub>L_bin'
	+ wf_bins_Earley\<^sub>L_bin' wf_bins_kth_bin wf_item_def x by (smt (verit, ccfv_SIG) linorder_not_le order.trans)
	+ moreover have "bins_eq_items (bins_upd ?bs'' k (Predict\<^sub>L k \<G> a)) ?bs''"
	+ proof -
	+ have "k < length bs"
	+ using wf_earley_input_elim[OF Predict\<^sub>F.prems(1)] by blast
	+ hence "set (items (Predict\<^sub>L k \<G> a)) \<subseteq> set (items (?bs' ! k))"
	+ by (simp add: bins_upd_def set_items_bin_upds)
	+ also have "... \<subseteq> set (items (?bs'' ! k))"
	+ using wf nth_bin_sub_Earley\<^sub>L_bin' by blast
	+ finally show ?thesis
	+ using bins_eq_items_bins_upd by blast
	+ qed
	+ moreover have "(k, \<G>, \<omega>, bins_upd ?bs'' k (Predict\<^sub>L k \<G> a)) \<in> wf_earley_input"
	+ using wf_earley_input_Earley\<^sub>L_bin' wf_earley_input_Predict\<^sub>L Predict\<^sub>F.hyps Predict\<^sub>F.prems(1)
	+ \<open>length (items (bs ! k)) \<le> length (items (?bs'' ! k))\<close> kth_Earley\<^sub>L_bin'_bins
	+ by (smt (verit, best) Earley\<^sub>L_bin'_simps(5) dual_order.trans not_le_imp_less)
	+ ultimately show ?thesis
	+ using Earley\<^sub>L_bin'_bins_eq bins_eq_items_imp_eq_bins wf_earley_input_elim by blast
	+ qed
	+ also have "... = bins (Earley\<^sub>L_bin' k \<G> \<omega> ?bs' (i + 1))"
	+ using \<open>i = j\<close> Predict\<^sub>F.IH Predict\<^sub>F.prems sound wf by (metis order_refl)
	+ finally show ?thesis
	+ using Predict\<^sub>F.hyps by simp
	+ qed
	+qed simp
	+
	+lemma Earley\<^sub>L_bin_idem:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x" "nonempty_derives \<G>"
	+ shows "bins (Earley\<^sub>L_bin k \<G> \<omega> (Earley\<^sub>L_bin k \<G> \<omega> bs)) = bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms Earley\<^sub>L_bin'_idem Earley\<^sub>L_bin_def le0 by metis
	+
	+lemma funpower_Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "Earley\<^sub>F_bin_step k \<G> \<omega> (bins_upto bs k 0) \<subseteq> bins bs" "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ assumes "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) n (bins bs) \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms
	+proof (induction n)
	+ case 0
	+ thus ?case
	+ using Earley\<^sub>L_bin'_mono Earley\<^sub>L_bin_def by (simp add: Earley\<^sub>L_bin'_mono Earley\<^sub>L_bin_def)
	+next
	+ case (Suc n)
	+ have 0: "Earley\<^sub>F_bin_step k \<G> \<omega> (bins_upto (Earley\<^sub>L_bin k \<G> \<omega> bs) k 0) \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using Earley\<^sub>L_bin'_mono bins_upto_k0_Earley\<^sub>L_bin'_eq assms(1,2) Earley\<^sub>L_bin_def order_trans
	+ by (metis (no_types, lifting))
	+ have "funpower (Earley\<^sub>F_bin_step k \<G> \<omega>) (Suc n) (bins bs) \<subseteq> Earley\<^sub>F_bin_step k \<G> \<omega> (bins (Earley\<^sub>L_bin k \<G> \<omega> bs))"
	+ using Earley\<^sub>F_bin_step_sub_mono Suc by (metis funpower.simps(2))
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> (Earley\<^sub>L_bin k \<G> \<omega> bs))"
	+ using Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin Suc.prems wf_bins_Earley\<^sub>L_bin sound_Earley\<^sub>L_bin 0 wf_earley_input_Earley\<^sub>L_bin by blast
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using Earley\<^sub>L_bin_idem Suc.prems by blast
	+ finally show ?case .
	+qed
	+
	+lemma Earley\<^sub>F_bin_sub_Earley\<^sub>L_bin:
	+ assumes "(k, \<G>, \<omega>, bs) \<in> wf_earley_input"
	+ assumes "Earley\<^sub>F_bin_step k \<G> \<omega> (bins_upto bs k 0) \<subseteq> bins bs" "\<forall>x \<in> bins bs. sound_item \<G> \<omega> x"
	+ assumes "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley\<^sub>F_bin k \<G> \<omega> (bins bs) \<subseteq> bins (Earley\<^sub>L_bin k \<G> \<omega> bs)"
	+ using assms funpower_Earley\<^sub>F_bin_step_sub_Earley\<^sub>L_bin Earley\<^sub>F_bin_def elem_limit_simp by fastforce
	+
	+lemma Earley\<^sub>F_bins_sub_Earley\<^sub>L_bins:
	+ assumes "k \<le> length \<omega>" "wf_\<G> \<G>"
	+ assumes "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley\<^sub>F_bins k \<G> \<omega> \<subseteq> bins (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ using assms
	+proof (induction k)
	+ case 0
	+ hence "Earley\<^sub>F_bin 0 \<G> \<omega> (Init\<^sub>F \<G>) \<subseteq> bins (Earley\<^sub>L_bin 0 \<G> \<omega> (Init\<^sub>L \<G> \<omega>))"
	+ using Earley\<^sub>F_bin_sub_Earley\<^sub>L_bin Init\<^sub>L_eq_Init\<^sub>F length_bins_Init\<^sub>L Init\<^sub>L_eq_Init\<^sub>F sound_Init bins_upto_empty
	+ Earley\<^sub>F_bin_step_empty bins_upto_sub_bins wf_earley_input_Init\<^sub>L wf_earley_input_elim
	+ by (smt (verit, ccfv_threshold) Init\<^sub>F_sub_Earley basic_trans_rules(31) sound_Earley wf_bins_impl_wf_items)
	+ thus ?case
	+ by simp
	+next
	+ case (Suc k)
	+ have wf: "(Suc k, \<G>, \<omega>, Earley\<^sub>L_bins k \<G> \<omega>) \<in> wf_earley_input"
	+ by (simp add: Suc.prems(1) Suc_leD assms(2) wf_earley_input_intro)
	+ have sub: "Earley\<^sub>F_bin_step (Suc k) \<G> \<omega> (bins_upto (Earley\<^sub>L_bins k \<G> \<omega>) (Suc k) 0) \<subseteq> bins (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ proof -
	+ have "bin (bins_upto (Earley\<^sub>L_bins k \<G> \<omega>) (Suc k) 0) (Suc k) = {}"
	+ using kth_bin_bins_upto_empty wf Suc.prems wf_earley_input_elim by blast
	+ hence "Earley\<^sub>F_bin_step (Suc k) \<G> \<omega> (bins_upto (Earley\<^sub>L_bins k \<G> \<omega>) (Suc k) 0) = bins_upto (Earley\<^sub>L_bins k \<G> \<omega>) (Suc k) 0"
	+ unfolding Earley\<^sub>F_bin_step_def Scan\<^sub>F_def Complete\<^sub>F_def Predict\<^sub>F_def bin_def by blast
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bins k \<G> \<omega>)"
	+ using wf Suc.prems bins_upto_sub_bins wf_earley_input_elim by blast
	+ finally show ?thesis .
	+ qed
	+ have sound: "\<forall>x \<in> bins (Earley\<^sub>L_bins k \<G> \<omega>). sound_item \<G> \<omega> x"
	+ using Suc Earley\<^sub>L_bins_sub_Earley\<^sub>F_bins by (metis Suc_leD Earley\<^sub>F_bins_sub_Earley in_mono sound_Earley wf_Earley)
	+ have "Earley\<^sub>F_bins (Suc k) \<G> \<omega> \<subseteq> Earley\<^sub>F_bin (Suc k) \<G> \<omega> (bins (Earley\<^sub>L_bins k \<G> \<omega>))"
	+ using Suc Earley\<^sub>F_bin_sub_mono by simp
	+ also have "... \<subseteq> bins (Earley\<^sub>L_bin (Suc k) \<G> \<omega> (Earley\<^sub>L_bins k \<G> \<omega>))"
	+ using Earley\<^sub>F_bin_sub_Earley\<^sub>L_bin wf sub sound Suc.prems by fastforce
	+ finally show ?case
	+ by simp
	+qed
	+
	+lemma Earley\<^sub>F_sub_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley\<^sub>F \<G> \<omega> \<subseteq> bins (Earley\<^sub>L \<G> \<omega>)"
	+ using assms Earley\<^sub>F_bins_sub_Earley\<^sub>L_bins Earley\<^sub>F_def Earley\<^sub>L_def by (metis le_refl)
	+
	+theorem completeness_Earley\<^sub>L:
	+ assumes "derives \<G> [\<SS> \<G>] \<omega>" "is_word \<G> \<omega>" "wf_\<G> \<G>" "nonempty_derives \<G>"
	+ shows "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega>"
	+ using assms Earley\<^sub>F_sub_Earley\<^sub>L Earley\<^sub>L_sub_Earley\<^sub>F completeness_Earley\<^sub>F by (metis subset_antisym)
	+
	+
	+subsection \<open>Correctness\<close>
	+
	+theorem Earley_eq_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "Earley \<G> \<omega> = bins (Earley\<^sub>L \<G> \<omega>)"
	+ using assms Earley\<^sub>F_sub_Earley\<^sub>L Earley\<^sub>L_sub_Earley\<^sub>F Earley_eq_Earley\<^sub>F by blast
	+
	+theorem correctness_Earley\<^sub>L:
	+ assumes "wf_\<G> \<G>" "is_word \<G> \<omega>" "nonempty_derives \<G>"
	+ shows "recognizing (bins (Earley\<^sub>L \<G> \<omega>)) \<G> \<omega> \<longleftrightarrow> derives \<G> [\<SS> \<G>] \<omega>"
	+ using assms Earley_eq_Earley\<^sub>L correctness_Earley by fastforce
	+
	+end
	diff --git a/thys/Earley_Parser/Examples.thy b/thys/Earley_Parser/Examples.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Examples.thy
	@@ -0,0 +1,196 @@
	+theory Examples
	+ imports Earley_Parser
	+begin
	+
	+section \<open>Epsilon productions\<close>
	+
	+definition \<epsilon>_free :: "'a cfg \<Rightarrow> bool" where
	+ "\<epsilon>_free \<G> \<longleftrightarrow> (\<forall>r \<in> set (\<RR> \<G>). rule_body r \<noteq> [])"
	+
	+lemma \<epsilon>_free_impl_non_empty_sentence_deriv:
	+ "\<epsilon>_free \<G> \<Longrightarrow> a \<noteq> [] \<Longrightarrow> \<not> Derivation \<G> a D []"
	+proof (induction "length D" arbitrary: a D rule: nat_less_induct)
	+ case 1
	+ show ?case
	+ proof (rule ccontr)
	+ assume assm: "\<not> \<not> Derivation \<G> a D []"
	+ show False
	+ proof (cases "D = []")
	+ case True
	+ then show ?thesis
	+ using "1.prems"(2) assm by auto
	+ next
	+ case False
	+ then obtain d D' \<alpha> where *:
	+ "D = d # D'" "Derives1 \<G> a (fst d) (snd d) \<alpha>" "Derivation \<G> \<alpha> D' []" "snd d \<in> set (\<RR> \<G>)"
	+ using list.exhaust assm Derives1_def by (metis Derivation.simps(2))
	+ show ?thesis
	+ proof cases
	+ assume "\<alpha> = []"
	+ thus ?thesis
	+ using *(2,4) Derives1_split \<epsilon>_free_def rule_body_def "1.prems"(1) by (metis append_is_Nil_conv)
	+ next
	+ assume "\<not> \<alpha> = []"
	+ thus ?thesis
	+ using *(1,3) "1.hyps" "1.prems"(1) by auto
	+ qed
	+ qed
	+ qed
	+qed
	+
	+lemma \<epsilon>_free_impl_non_empty_deriv:
	+ "\<epsilon>_free \<G> \<Longrightarrow> \<forall>N \<in> set (\<NN> \<G>). \<not> derives \<G> [N] []"
	+ using \<epsilon>_free_impl_non_empty_sentence_deriv derives_implies_Derivation by (metis not_Cons_self2)
	+
	+lemma nonempty_deriv_impl_\<epsilon>_free:
	+ assumes "\<forall>N \<in> set (\<NN> \<G>). \<not> derives \<G> [N] []" "\<forall>(N, \<alpha>) \<in> set (\<RR> \<G>). N \<in> set (\<NN> \<G>)"
	+ shows "\<epsilon>_free \<G>"
	+proof (rule ccontr)
	+ assume "\<not> \<epsilon>_free \<G>"
	+ then obtain N \<alpha> where *: "(N, \<alpha>) \<in> set (\<RR> \<G>)" "rule_body (N, \<alpha>) = []"
	+ unfolding \<epsilon>_free_def by auto
	+ hence "derives1 \<G> [N] []"
	+ unfolding derives1_def rule_body_def by auto
	+ hence "derives \<G> [N] []"
	+ by auto
	+ moreover have "N \<in> set (\<NN> \<G>)"
	+ using *(1) assms(2) by blast
	+ ultimately show False
	+ using assms(1) by blast
	+qed
	+
	+lemma nonempty_deriv_iff_\<epsilon>_free:
	+ assumes "\<forall>(N, \<alpha>) \<in> set (\<RR> \<G>). N \<in> set (\<NN> \<G>)"
	+ shows "(\<forall>N \<in> set (\<NN> \<G>). \<not> derives \<G> [N] []) \<longleftrightarrow> \<epsilon>_free \<G>"
	+ using \<epsilon>_free_impl_non_empty_deriv nonempty_deriv_impl_\<epsilon>_free[OF _ assms] by blast
	+
	+section \<open>Example 1: Addition\<close>
	+
	+datatype t1 = x \| plus
	+datatype n1 = S
	+datatype s1 = Terminal t1 \| Nonterminal n1
	+
	+definition nonterminals1 :: "s1 list" where
	+ "nonterminals1 = [Nonterminal S]"
	+
	+definition terminals1 :: "s1 list" where
	+ "terminals1 = [Terminal x, Terminal plus]"
	+
	+definition rules1 :: "s1 rule list" where
	+ "rules1 = [
	+ (Nonterminal S, [Terminal x]),
	+ (Nonterminal S, [Nonterminal S, Terminal plus, Nonterminal S])
	+ ]"
	+
	+definition start_symbol1 :: s1 where
	+ "start_symbol1 = Nonterminal S"
	+
	+definition cfg1 :: "s1 cfg" where
	+ "cfg1 = CFG nonterminals1 terminals1 rules1 start_symbol1"
	+
	+definition inp1 :: "s1 list" where
	+ "inp1 = [Terminal x, Terminal plus, Terminal x, Terminal plus, Terminal x]"
	+
	+lemmas cfg1_defs = cfg1_def nonterminals1_def terminals1_def rules1_def start_symbol1_def
	+
	+lemma wf_\<G>1:
	+ "wf_\<G> cfg1"
	+ by (auto simp: wf_\<G>_defs cfg1_defs)
	+
	+lemma is_word_inp1:
	+ "is_word cfg1 inp1"
	+ by (auto simp: is_word_def is_terminal_def cfg1_defs inp1_def)
	+
	+lemma nonempty_derives1:
	+ "nonempty_derives cfg1"
	+ by (auto simp: \<epsilon>_free_def cfg1_defs rule_body_def nonempty_derives_def \<epsilon>_free_impl_non_empty_deriv)
	+
	+lemma correctness1:
	+ "recognizing (bins (Earley\<^sub>L cfg1 inp1)) cfg1 inp1 \<longleftrightarrow> derives cfg1 [\<SS> cfg1] inp1"
	+ using correctness_Earley\<^sub>L wf_\<G>1 is_word_inp1 nonempty_derives1 by blast
	+
	+lemma wf_tree1:
	+ assumes "build_tree cfg1 inp1 (Earley\<^sub>L cfg1 inp1) = Some t"
	+ shows "wf_rule_tree cfg1 t \<and> root_tree t = \<SS> cfg1 \<and> yield_tree t = inp1"
	+ using assms nonempty_derives1 wf_\<G>1 wf_rule_root_yield_tree_build_tree_Earley\<^sub>L by blast
	+
	+lemma correctness_tree1:
	+ "(\<exists>t. build_tree cfg1 inp1 (Earley\<^sub>L cfg1 inp1) = Some t) \<longleftrightarrow> derives cfg1 [\<SS> cfg1] inp1"
	+ using correctness_build_tree_Earley\<^sub>L is_word_inp1 nonempty_derives1 wf_\<G>1 by blast
	+
	+lemma wf_trees1:
	+ assumes "build_trees cfg1 inp1 (Earley\<^sub>L cfg1 inp1) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "wf_rule_tree cfg1 t \<and> root_tree t = \<SS> cfg1 \<and> yield_tree t = inp1"
	+ using assms nonempty_derives1 wf_\<G>1 wf_rule_root_yield_tree_build_trees_Earley\<^sub>L by blast
	+
	+lemma soundness_trees1:
	+ assumes "build_trees cfg1 inp1 (Earley\<^sub>L cfg1 inp1) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "derives cfg1 [\<SS> cfg1] inp1"
	+ using assms is_word_inp1 nonempty_derives1 soundness_build_trees_Earley\<^sub>L wf_\<G>1 by blast
	+
	+section \<open>Example 2: Cyclic reduction pointers\<close>
	+
	+datatype t2 = x
	+datatype n2 = A \| B
	+datatype s2 = Terminal t2 \| Nonterminal n2
	+
	+definition nonterminals2 :: "s2 list" where
	+ "nonterminals2 = [Nonterminal A, Nonterminal B]"
	+
	+definition terminals2 :: "s2 list" where
	+ "terminals2 = [Terminal x]"
	+
	+definition rules2 :: "s2 rule list" where
	+ "rules2 = [
	+ (Nonterminal B, [Nonterminal A]),
	+ (Nonterminal A, [Nonterminal B]),
	+ (Nonterminal A, [Terminal x])
	+ ]"
	+
	+definition start_symbol2 :: s2 where
	+ "start_symbol2 = Nonterminal A"
	+
	+definition cfg2 :: "s2 cfg" where
	+ "cfg2 = CFG nonterminals2 terminals2 rules2 start_symbol2"
	+
	+definition inp2 :: "s2 list" where
	+ "inp2 = [Terminal x]"
	+
	+lemmas cfg2_defs = cfg2_def nonterminals2_def terminals2_def rules2_def start_symbol2_def
	+
	+lemma wf_\<G>2:
	+ "wf_\<G> cfg2"
	+ by (auto simp: wf_\<G>_defs cfg2_defs)
	+
	+lemma is_word_inp2:
	+ "is_word cfg2 inp2"
	+ by (auto simp: is_word_def is_terminal_def cfg2_defs inp2_def)
	+
	+lemma nonempty_derives2:
	+ "nonempty_derives cfg2"
	+ by (auto simp: \<epsilon>_free_def cfg2_defs rule_body_def nonempty_derives_def \<epsilon>_free_impl_non_empty_deriv)
	+
	+lemma correctness2:
	+ "recognizing (bins (Earley\<^sub>L cfg2 inp2)) cfg2 inp2 \<longleftrightarrow> derives cfg2 [\<SS> cfg2] inp2"
	+ using correctness_Earley\<^sub>L wf_\<G>2 is_word_inp2 nonempty_derives2 by blast
	+
	+lemma wf_tree2:
	+ assumes "build_tree cfg2 inp2 (Earley\<^sub>L cfg2 inp2) = Some t"
	+ shows "wf_rule_tree cfg2 t \<and> root_tree t = \<SS> cfg2 \<and> yield_tree t = inp2"
	+ using assms nonempty_derives2 wf_\<G>2 wf_rule_root_yield_tree_build_tree_Earley\<^sub>L by blast
	+
	+lemma correctness_tree2:
	+ "(\<exists>t. build_tree cfg2 inp2 (Earley\<^sub>L cfg2 inp2) = Some t) \<longleftrightarrow> derives cfg2 [\<SS> cfg2] inp2"
	+ using correctness_build_tree_Earley\<^sub>L is_word_inp2 nonempty_derives2 wf_\<G>2 by blast
	+
	+lemma wf_trees2:
	+ assumes "build_trees cfg2 inp2 (Earley\<^sub>L cfg2 inp2) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "wf_rule_tree cfg2 t \<and> root_tree t = \<SS> cfg2 \<and> yield_tree t = inp2"
	+ using assms nonempty_derives2 wf_\<G>2 wf_rule_root_yield_tree_build_trees_Earley\<^sub>L by blast
	+
	+lemma soundness_trees2:
	+ assumes "build_trees cfg2 inp2 (Earley\<^sub>L cfg2 inp2) = Some fs" "f \<in> set fs" "t \<in> set (trees f)"
	+ shows "derives cfg2 [\<SS> cfg2] inp2"
	+ using assms is_word_inp2 nonempty_derives2 soundness_build_trees_Earley\<^sub>L wf_\<G>2 by blast
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/Limit.thy b/thys/Earley_Parser/Limit.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/Limit.thy
	@@ -0,0 +1,146 @@
	+theory Limit
	+ imports Main
	+begin
	+
	+section \<open>Slightly adjusted content from AFP/LocalLexing\<close>
	+
	+fun funpower :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a \<Rightarrow> 'a)" where
	+ "funpower f 0 x = x"
	+\| "funpower f (Suc n) x = f (funpower f n x)"
	+
	+definition natUnion :: "(nat \<Rightarrow> 'a set) \<Rightarrow> 'a set" where
	+ "natUnion f = \<Union> { f n \| n. True }"
	+
	+definition limit :: "('a set \<Rightarrow> 'a set) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
	+ "limit f x = natUnion (\<lambda> n. funpower f n x)"
	+
	+definition setmonotone :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool" where
	+ "setmonotone f = (\<forall> X. X \<subseteq> f X)"
	+
	+lemma subset_setmonotone: "setmonotone f \<Longrightarrow> X \<subseteq> f X"
	+ by (simp add: setmonotone_def)
	+
	+lemma funpower_id [simp]: "funpower id n = id"
	+ by (rule ext, induct n, simp_all)
	+
	+lemma limit_id [simp]: "limit id = id"
	+ by (rule ext, auto simp add: limit_def natUnion_def)
	+
	+definition chain :: "(nat \<Rightarrow> 'a set) \<Rightarrow> bool"
	+where
	+ "chain C = (\<forall> i. C i \<subseteq> C (i + 1))"
	+
	+definition continuous :: "('a set \<Rightarrow> 'b set) \<Rightarrow> bool"
	+where
	+ "continuous f = (\<forall> C. chain C \<longrightarrow> (chain (f o C) \<and> f (natUnion C) = natUnion (f o C)))"
	+
	+lemma natUnion_upperbound:
	+ "(\<And> n. f n \<subseteq> G) \<Longrightarrow> (natUnion f) \<subseteq> G"
	+by (auto simp add: natUnion_def)
	+
	+lemma funpower_upperbound:
	+ "(\<And> I. I \<subseteq> G \<Longrightarrow> f I \<subseteq> G) \<Longrightarrow> I \<subseteq> G \<Longrightarrow> funpower f n I \<subseteq> G"
	+proof (induct n)
	+ case 0 thus ?case by simp
	+next
	+ case (Suc n) thus ?case by simp
	+qed
	+
	+lemma limit_upperbound:
	+ "(\<And> I. I \<subseteq> G \<Longrightarrow> f I \<subseteq> G) \<Longrightarrow> I \<subseteq> G \<Longrightarrow> limit f I \<subseteq> G"
	+by (simp add: funpower_upperbound limit_def natUnion_upperbound)
	+
	+lemma elem_limit_simp: "x \<in> limit f X = (\<exists> n. x \<in> funpower f n X)"
	+by (auto simp add: limit_def natUnion_def)
	+
	+definition pointwise :: "('a set \<Rightarrow> 'b set) \<Rightarrow> bool" where
	+ "pointwise f = (\<forall> X. f X = \<Union> { f {x} \| x. x \<in> X})"
	+
	+lemma natUnion_elem: "x \<in> f n \<Longrightarrow> x \<in> natUnion f"
	+using natUnion_def by fastforce
	+
	+lemma limit_elem: "x \<in> funpower f n X \<Longrightarrow> x \<in> limit f X"
	+by (simp add: limit_def natUnion_elem)
	+
	+definition pointbase :: "('a set \<Rightarrow> 'b set) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
	+ "pointbase F I = \<Union> { F X \| X. finite X \<and> X \<subseteq> I }"
	+
	+definition pointbased :: "('a set \<Rightarrow> 'b set) \<Rightarrow> bool" where
	+ "pointbased f = (\<exists> F. f = pointbase F)"
	+
	+lemma chain_implies_mono: "chain C \<Longrightarrow> mono C"
	+by (simp add: chain_def mono_iff_le_Suc)
	+
	+lemma setmonotone_implies_chain_funpower:
	+ assumes setmonotone: "setmonotone f"
	+ shows "chain (\<lambda> n. funpower f n I)"
	+by (simp add: chain_def setmonotone subset_setmonotone)
	+
	+lemma natUnion_subset: "(\<And> n. \<exists> m. f n \<subseteq> g m) \<Longrightarrow> natUnion f \<subseteq> natUnion g"
	+ by (meson natUnion_elem natUnion_upperbound subset_iff)
	+
	+lemma natUnion_eq[case_names Subset Superset]:
	+ "(\<And> n. \<exists> m. f n \<subseteq> g m) \<Longrightarrow> (\<And> n. \<exists> m. g n \<subseteq> f m) \<Longrightarrow> natUnion f = natUnion g"
	+by (simp add: natUnion_subset subset_antisym)
	+
	+lemma natUnion_shift[symmetric]:
	+ assumes chain: "chain C"
	+ shows "natUnion C = natUnion (\<lambda> n. C (n + m))"
	+proof (induct rule: natUnion_eq)
	+ case (Subset n)
	+ show ?case using chain chain_implies_mono le_add1 mono_def by blast
	+next
	+ case (Superset n)
	+ show ?case by blast
	+qed
	+
	+definition regular :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool"
	+where
	+ "regular f = (setmonotone f \<and> continuous f)"
	+
	+lemma regular_fixpoint:
	+ assumes regular: "regular f"
	+ shows "f (limit f I) = limit f I"
	+proof -
	+ have setmonotone: "setmonotone f" using regular regular_def by blast
	+ have continuous: "continuous f" using regular regular_def by blast
	+
	+ let ?C = "\<lambda> n. funpower f n I"
	+ have chain: "chain ?C"
	+ by (simp add: setmonotone setmonotone_implies_chain_funpower)
	+ have "f (limit f I) = f (natUnion ?C)"
	+ using limit_def by metis
	+ also have "f (natUnion ?C) = natUnion (f o ?C)"
	+ by (metis continuous continuous_def chain)
	+ also have "natUnion (f o ?C) = natUnion (\<lambda> n. f(funpower f n I))"
	+ by (meson comp_apply)
	+ also have "natUnion (\<lambda> n. f(funpower f n I)) = natUnion (\<lambda> n. ?C (n + 1))"
	+ by simp
	+ also have "natUnion (\<lambda> n. ?C(n + 1)) = natUnion ?C"
	+ by (metis (no_types, lifting) Limit.chain_def chain natUnion_eq)
	+ finally show ?thesis by (simp add: limit_def)
	+qed
	+
	+lemma fix_is_fix_of_limit:
	+ assumes fixpoint: "f I = I"
	+ shows "limit f I = I"
	+proof -
	+ have funpower: "\<And> n. funpower f n I = I"
	+ proof -
	+ fix n :: nat
	+ from fixpoint show "funpower f n I = I"
	+ by (induct n, auto)
	+ qed
	+ show ?thesis by (simp add: limit_def funpower natUnion_def)
	+qed
	+
	+lemma limit_is_idempotent: "regular f \<Longrightarrow> limit f (limit f I) = limit f I"
	+by (simp add: fix_is_fix_of_limit regular_fixpoint)
	+
	+definition mk_regular1 :: "('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
	+ "mk_regular1 P F I = I \<union> { F q x \| q x. x \<in> I \<and> P q x }"
	+
	+definition mk_regular2 :: "('b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
	+ "mk_regular2 P F I = I \<union> { F q x y \| q x y. x \<in> I \<and> y \<in> I \<and> P q x y }"
	+
	+end
	\ No newline at end of file
	diff --git a/thys/Earley_Parser/ROOT b/thys/Earley_Parser/ROOT
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/ROOT
	@@ -0,0 +1,19 @@
	+chapter AFP
	+
	+session Earley_Parser (AFP) = HOL +
	+ options [timeout = 600]
	+ sessions
	+ "HOL-Library"
	+ theories
	+ Limit
	+ CFG
	+ Derivations
	+ Earley
	+ Earley_Fixpoint
	+ Earley_Recognizer
	+ Earley_Parser
	+ Examples
	+ document_files
	+ "root.tex"
	+ "root.bib"
	+
	diff --git a/thys/Earley_Parser/document/root.bib b/thys/Earley_Parser/document/root.bib
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/document/root.bib
	@@ -0,0 +1,65 @@
	+@inproceedings{Jones:1972,
	+author = {Jones, C B},
	+title = {Formal Development of Correct Algorithms: An Example Based on Earley's Recogniser},
	+year = {1972},
	+isbn = {9781450378918},
	+publisher = {Association for Computing Machinery},
	+address = {New York, NY, USA},
	+url = {https://doi.org/10.1145/800235.807083},
	+doi = {10.1145/800235.807083},
	+booktitle = {Proceedings of ACM Conference on Proving Assertions about Programs},
	+pages = {150–169},
	+numpages = {20},
	+location = {Las Cruces, New Mexico, USA}
	+}
	+
	+@article{Earley:1970,
	+author = {Earley, Jay},
	+title = {An Efficient Context-Free Parsing Algorithm},
	+year = {1970},
	+publisher = {Association for Computing Machinery},
	+address = {New York, NY, USA},
	+volume = {13},
	+number = {2},
	+issn = {0001-0782},
	+url = {https://doi.org/10.1145/362007.362035},
	+doi = {10.1145/362007.362035},
	+journal = {Commun. ACM},
	+pages = {94–102},
	+numpages = {9},
	+keywords = {computational complexity, parsing, syntax analysis, context-free grammar, compilers}
	+}
	+
	+@article{Scott:2008,
	+title = {SPPF-Style Parsing From Earley Recognisers},
	+journal = {Electronic Notes in Theoretical Computer Science},
	+volume = {203},
	+number = {2},
	+pages = {53-67},
	+year = {2008},
	+note = {Proceedings of the Seventh Workshop on Language Descriptions, Tools, and Applications (LDTA 2007)},
	+issn = {1571-0661},
	+doi = {https://doi.org/10.1016/j.entcs.2008.03.044},
	+url = {https://www.sciencedirect.com/science/article/pii/S1571066108001497},
	+author = {Elizabeth Scott},
	+keywords = {Earley parsing, cubic generalised parsing, context free languages},
	+}
	+
	+@misc{Obua:2017,
	+ title={Local Lexing},
	+ author={Steven Obua and Phil Scott and Jacques Fleuriot},
	+ year={2017},
	+ eprint={1702.03277},
	+ archivePrefix={arXiv},
	+ primaryClass={cs.LO}
	+}
	+
	+@article{LocalLexing-AFP,
	+ author = {Steven Obua},
	+ title = {Local Lexing},
	+ journal = {Archive of Formal Proofs},
	+ year = {2017},
	+ note = {\url{https://isa-afp.org/entries/LocalLexing.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}
	diff --git a/thys/Earley_Parser/document/root.tex b/thys/Earley_Parser/document/root.tex
	new file mode 100644
	--- /dev/null
	+++ b/thys/Earley_Parser/document/root.tex
	@@ -0,0 +1,59 @@
	+\documentclass[11pt,a4paper]{article}
	+\usepackage[T1]{fontenc}
	+\usepackage{isabelle,isabellesym}
	+
	+% further packages required for unusual symbols (see also
	+% isabellesym.sty), use only when needed
	+
	+\usepackage{amssymb}
	+ %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
	+ %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
	+ %\<triangleq>, \<yen>, \<lozenge>
	+
	+%\usepackage{eurosym}
	+ %for \<euro>
	+
	+%\usepackage[only,bigsqcap,bigparallel,fatsemi,interleave,sslash]{stmaryrd}
	+ %for \<Sqinter>, \<Parallel>, \<Zsemi>, \<Parallel>, \<sslash>
	+
	+%\usepackage{eufrak}
	+ %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
	+
	+%\usepackage{textcomp}
	+ %for \<onequarter>, \<onehalf>, \<threequarters>, \<degree>, \<cent>,
	+ %\<currency>
	+
	+% this should be the last package used
	+\usepackage{pdfsetup}
	+
	+% urls in roman style, theory text in math-similar italics
	+\urlstyle{rm}
	+\isabellestyle{it}
	+
	+
	+\begin{document}
	+
	+\title{Earley}
	+\author{Martin Rau}
	+\maketitle
	+
	+\begin{abstract}
	+In 1968 Earley \cite{Earley:1970} introduced his parsing algorithm capable of parsing all context-free grammars in cubic
	+space and time. This entry contains a formalization of an executable Earley parser. We base our development on Jones' \cite{Jones:1972}
	+extensive paper proof of Earley's recognizer and the formalization of context-free grammars
	+and derivations of Obua \cite{Obua:2017} \cite{LocalLexing-AFP}. We implement and prove correct a functional recognizer modeling Earley's
	+original imperative implementation and extend it with the necessary data structures to enable the construction
	+of parse trees following the work of Scott \cite{Scott:2008}. We then develop a functional algorithm that
	+builds a single parse tree and prove its correctness. Finally, we generalize this approach to an algorithm
	+for a complete parse forest and prove soundness.
	+\end{abstract}
	+
	+\tableofcontents
	+
	+% include generated text of all theories
	+\input{session}
	+
	+\bibliographystyle{abbrv}
	+\bibliography{root}
	+
	+\end{document}
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Cauchy.thy b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Cauchy.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Cauchy.thy
	@@ -0,0 +1,732 @@
	+subsection \<open> Cauchy's Polygonal Number Theorem \<close>
	+
	+text \<open>We will use the definition of the polygonal numbers from the Gauss Theorem theory file
	+which also imports the Three Squares Theorem AFP entry \cite{Three_Squares-AFP}.\<close>
	+
	+theory Polygonal_Number_Theorem_Cauchy
	+ imports Polygonal_Number_Theorem_Gauss
	+begin
	+
	+text\<open>The following lemma shows there are two consecutive odd integers in any four consecutive
	+integers.\<close>
	+
	+lemma two_consec_odd:
	+ fixes a1 a2 a3 a4 :: int
	+ assumes "a1-a2 = 1"
	+ assumes "a2-a3 = 1"
	+ assumes "a3-a4 = 1"
	+ shows "\<exists>k1 k2 :: int. {k1, k2} \<subseteq> {a1, a2, a3, a4} \<and> (k2 = k1+2) \<and> odd k1"
	+
	+proof -
	+ have c1:"\<exists>k1 k2 :: int. {k1, k2} \<subseteq> {a1, a2, a3, a4} \<and> (k2 = k1+2) \<and> odd k1"
	+ if odd_case:"odd a4"
	+ proof-
	+ define k1 where k1_def:"k1 = a4"
	+ define k2 where k2_def:"k2 = k1 + 2"
	+ have 0:"k2 = a2" using k2_def k1_def assms by simp
	+ have 1:"odd k1" using k1_def odd_case by simp
	+ show "\<exists>k1 k2 :: int. {k1, k2} \<subseteq> {a1, a2, a3, a4} \<and> (k2 = k1+2) \<and> odd k1"
	+ using 0 1 k1_def k2_def by auto
	+ qed
	+
	+ have c2:"\<exists>k1 k2 :: int. {k1, k2} \<subseteq> {a1, a2, a3, a4} \<and> (k2 = k1+2) \<and> odd k1"
	+ if even_case:"even a4"
	+ proof -
	+ define k1 where k1_def:"k1 = a3"
	+ define k2 where k2_def:"k2 = k1 + 2"
	+ have 2:"odd k1" using even_case assms k1_def by presburger
	+ have 3:"k2 = a1" using k1_def k2_def assms by simp
	+ show "\<exists>k1 k2 :: int. {k1, k2} \<subseteq> {a1, a2, a3, a4} \<and> (k2 = k1+2) \<and> odd k1"
	+ using 2 3 k1_def k2_def by auto
	+ qed
	+ show ?thesis using c1 c2 by auto
	+qed
	+
	+text \<open>This lemma proves that for two consecutive integers $b_1$ and $b_2$, and $r \in \{0,1,\dots,m-3\}$,
	+numbers of the form $b_1+r$ and $b_2+r$ can cover all the congruence classes modulo $m$.\<close>
	+
	+lemma cong_classes:
	+ fixes b1 b2 :: int
	+ fixes m :: nat
	+ assumes "m \<ge> 4"
	+ assumes "odd b1"
	+ assumes "b2 = b1 + 2"
	+ shows "\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+
	+proof -
	+ have first:"\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if first_assum:"b1 mod m \<ge> 3"
	+ proof -
	+ define k1 where k1_def:"k1 = b1 mod m"
	+ define l where l_def:"l = m - k1"
	+ have k1_size:"k1\<ge>3" using first_assum k1_def by simp
	+ have l_size:"l \<le> m-3" using first_assum k1_def l_def by auto
	+ have "(l+k1) mod m = 0" using l_def by auto
	+ hence "(l+b1) mod m = 0" using k1_def l_def by (metis mod_add_right_eq)
	+ define w where w_def:"w = m-3-l"
	+ have w_size:"w\<ge>0 \<and> w\<le>m-3" using w_def l_size l_def k1_def first_assum
	+ by (smt (verit, best) Euclidean_Division.pos_mod_bound assms(1) le_antisym numeral_neq_zero
	+ of_nat_0_less_iff order_trans_rules(22) verit_comp_simplify(3) zero_le_numeral)
	+ have "k1 = w+3" using w_def k1_def l_def w_size first_assum by linarith
	+ hence "w+2 = k1-1" by auto
	+ hence "w+2 = (b1-1) mod m" using first_assum k1_def
	+ by (smt (verit, del_insts) Euclidean_Division.pos_mod_bound assms(1)
	+ mod_diff_eq mod_pos_pos_trivial of_nat_le_0_iff verit_comp_simplify(8))
	+ hence w_cover:"w+2 = k1-1" using k1_def using \<open>w + 2 = k1 - 1\<close> by fastforce
	+
	+ have "\<exists>r::nat. (r\<le>m-3) \<and> [N=b1+r] (mod m)" if asm1:"N mod m \<ge> k1 \<and> N mod m \<le> m-1" for N
	+ proof -
	+ have "m - (N mod m) \<le> l" using asm1 l_def k1_def by linarith
	+ hence "\<exists>d::nat. d\<le>l \<and> [N = k1+d] (mod m)" using asm1 k1_def l_def
	+ by (metis add.commute add_le_cancel_left cong_mod_left cong_refl diff_add_cancel
	+ diff_le_self le_trans of_nat_mod of_nat_mono zle_iff_zadd)
	+ hence "\<exists>d::nat. d\<le>l \<and> [N=b1+d] (mod m)" using k1_def
	+ by (metis mod_add_left_eq unique_euclidean_semiring_class.cong_def)
	+ thus "\<exists>r::nat. (r\<le>m-3) \<and> [N=b1+r] (mod m)" using l_size
	+ by (smt (verit, best) nat_leq_as_int)
	+ qed
	+ hence c1:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)" if asm1:"N mod m \<ge> k1 \<and> N mod m \<le> m-1" for N using asm1 by blast
	+
	+ have c2:"\<exists>r::nat. (r\<le>m-3) \<and> [N=b1+r] (mod m)" if asm2:"N mod m =0" for N using l_def k1_def
	+ by (smt (verit, ccfv_threshold) \<open>(l + b1) mod int m = 0\<close> add_diff_cancel_left' cong_0_iff
	+ cong_sym cong_trans diff_add_cancel diff_ge_0_iff_ge dvd_eq_mod_eq_0 int_dvd_int_iff nat_0_le
	+of_nat_le_iff that w_def w_size)
	+ hence c2:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm2:"N mod m = 0" for N using asm2 by metis
	+
	+ have "\<exists>r::nat. (r\<le>m-3) \<and> [N=b1+r] (mod m)" if asm3:"N mod m > 0 \<and> N mod m \<le>w" for N
	+ proof -
	+ have "l+ (N mod m) \<le> m-3" using asm3 w_def by auto
	+ hence "\<exists>d::nat. (d\<le>w) \<and> [N = k1+l+d] (mod m)" using asm3 w_def k1_def l_def
	+ by (smt (verit, ccfv_threshold) minus_mod_self2 mod_mod_trivial of_nat_mod
	+ unique_euclidean_semiring_class.cong_def)
	+ hence "\<exists>d::nat. (d\<le>w) \<and> [N = b1+l+d] (mod m)" using k1_def by (metis (mono_tags,
	+ opaque_lifting) mod_add_left_eq unique_euclidean_semiring_class.cong_def)
	+ hence "\<exists>r::nat. (r\<le>w+l) \<and> [N = b1+r] (mod m)" by (smt (verit) add.commute
	+ add.left_commute le_add_same_cancel2 of_nat_0_le_iff w_def w_size zero_le_imp_eq_int)
	+ thus "\<exists>r::nat. (r\<le>m-3) \<and> [N = b1+r] (mod m)" using w_def by auto
	+ qed
	+ hence "\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm3:"N mod m > 0 \<and> N mod m \<le>w" for N using asm3 by blast
	+ hence c3:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm8:"N mod m > 0 \<and> N mod m \<le>k1-3" using asm8 w_cover by auto
	+
	+ have "\<exists>r::nat. (r\<le>m-3) \<and> [N=b2+r] (mod m)" if asm4:"N mod m = w+1 \<or> N mod m = w+2" for N
	+ proof -
	+ have c4_1:"[N = b2+(m-3)] (mod m)" if asm5:"N mod m=w+2" for N using asm5 w_def assms(3) l_def
	+ by (smt (verit) \<open>w + 2 = (b1 - 1) mod int m\<close> \<open>w + 2 = k1 - 1\<close> mod_add_self1 of_nat_mod
	+ unique_euclidean_semiring_class.cong_def)
	+ hence "[N-1 = b2+(m-4)] (mod m)" if asm5:"N mod m = w+2" for N
	+ by (smt (verit, ccfv_threshold) Num.of_nat_simps(2) \<open>w + 2 = k1 - 1\<close> asm5 assms(1)
	+ cong_iff_lin first_assum k1_def l_def mod_less_eq_dividend numeral_Bit0 of_nat_diff of_nat_le_iff
	+of_nat_numeral semiring_norm(172) w_def)
	+ hence "[N = b2+(m-4)] (mod m)" if asm6:"N mod m = w+1" for N using asm6
	+ by (metis \<open>w + 2 = (b1 - 1) mod int m\<close> add_diff_cancel_right' arith_special(3) int_ops(4)
	+ is_num_normalize(1) mod_add_left_eq mod_diff_left_eq of_nat_mod)
	+ thus ?thesis using c4_1 by (metis asm4 diff_le_mono2 nat_le_linear numeral_le_iff
	+ verit_comp_simplify(10) verit_comp_simplify(13))
	+ qed
	+ hence "\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm4:"N mod m = w+1 \<or> N mod m = w+2" for N using asm4 by blast
	+ hence c4:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm7:"N mod m = k1-2 \<or> N mod m = k1-1" for N using w_cover asm7 by auto
	+
	+ have "\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm10:"N mod m \<ge> 0 \<and> N mod m \<le>w" for N using c2 c3 asm10
	+ using \<open>\<And>N. 0 < N mod m \<and> int (N mod m) \<le> w \<Longrightarrow> \<exists>b r. r \<le> m - 3 \<and> [int N = b + int r]
	+ (mod int m) \<and> (b = b1 \<or> b = b2)\<close> by blast
	+ hence c5:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm11:"N mod m \<ge> 0 \<and> N mod m \<le>k1-3" for N using w_cover using asm11 by force
	+
	+ have c6:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm9:"(N mod m \<ge>0 \<and> N mod m \<le> k1-3) \<or> N mod m = k1-2 \<or> N mod m = k1-1" for N
	+ using c5 c4 asm9 by blast
	+
	+ hence c7:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if asm12:"(N mod m \<ge>0 \<and> N mod m \<le> k1-3) \<or> N mod m = k1-2 \<or> N mod m = k1-1 \<or>
	+ (N mod m \<ge> k1 \<and> N mod m \<le> m-1)" for N using asm12 c1 by blast
	+
	+ have "\<forall>N::nat. (N mod m \<ge>0 \<and> N mod m\<le> k1-3) \<or> N mod m = k1-2 \<or> N mod m = k1-1 \<or>
	+ (N mod m \<ge> k1 \<and> N mod m \<le> m-1)" using k1_def
	+ by (smt (verit, best) Suc_pred' assms(1) bot_nat_0.extremum le_simps(2) mod_less_divisor
	+not_numeral_le_zero of_nat_0_less_iff of_nat_le_0_iff)
	+
	+ thus ?thesis using c7 by auto
	+ qed
	+
	+ have second:"\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if second_assum:"b1 mod m \<ge>0 \<and> b1 mod m \<le>2"
	+ proof -
	+ have case1:"\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case1_assum:"b1 mod m = 0"
	+ proof -
	+ have "\<exists>r::nat. (r \<le> m-3) \<and> [N = b1+r] (mod m)" if case1_1_assum:"N mod m \<le>m-3" for N
	+ using case1_assum case1_1_assum
	+ by (metis cong_add_rcancel_0 cong_mod_left cong_refl cong_sym_eq zmod_int)
	+ hence case1_1:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case1_1_assum:"N mod m \<le>m-3" for N using case1_1_assum by blast
	+
	+ have "[N = b1+(m-2)] (mod m)" if case1_2_assum:"N mod m = m-2" for N using case1_2_assum
	+ case1_assum by (metis (no_types, opaque_lifting) add.commute cong_add_lcancel_0
	+ cong_mod_right of_nat_mod unique_euclidean_semiring_class.cong_def)
	+ hence "[N = b2+(m-4)] (mod m)" if case1_2_assum:"N mod m = m-2" for N using case1_2_assum assms(3)
	+ by (smt (verit, best) add_leD2 assms(1) int_ops(2) numeral_Bit0 of_nat_diff of_nat_numeral
	+ semiring_norm(172))
	+ hence "\<exists>r::nat. (r \<le> m-3) \<and> [N = b2+r] (mod m)" if case1_2_assum:"N mod m = m-2" for N
	+ using case1_2_assum
	+ by (meson diff_le_mono2 less_num_simps(2) numeral_le_iff verit_comp_simplify(15))
	+ hence case1_2:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case1_2_assum:"N mod m = m-2" for N using case1_2_assum by blast
	+
	+ have "[N = b1+(m-1)] (mod m)" if case1_3_assum:"N mod m = m-1" for N using case1_3_assum
	+ case1_assum by (metis (no_types, opaque_lifting) add.commute cong_add_lcancel_0
	+ cong_mod_right of_nat_mod unique_euclidean_semiring_class.cong_def)
	+ hence "[N = b2+(m-3)] (mod m)" if case1_3_assum:"N mod m = m-1" for N using case1_3_assum assms(3)
	+ by (smt (verit, best) assms(1) int_ops(2) int_ops(6) numeral_Bit0 numeral_Bit1 of_nat_mono
	+ of_nat_numeral semiring_norm(172))
	+ hence "\<exists>r::nat. (r \<le> m-3) \<and> [N = b2+r] (mod m)" if case1_3_assum:"N mod m = m-1" for N
	+ using case1_3_assum by blast
	+ hence case1_3:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case1_3_assum:"N mod m = m-1" for N using case1_3_assum by blast
	+
	+ have "\<forall>N::nat. (N mod m = m-1) \<or> (N mod m = m-2) \<or> (N mod m \<le> m-3)"
	+ by (smt (verit, ccfv_threshold) Suc_pred' assms(1) bot_nat_0.not_eq_extremum diff_diff_add
	+diff_is_0_eq' le_simps(2) mod_less_divisor nat_1_add_1 nat_less_le not_numeral_le_zero
	+numeral.simps(3) semiring_norm(172))
	+ thus ?thesis using case1_1 case1_2 case1_3 by blast
	+ qed
	+
	+ have case2:"\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case2_assum:"b1 mod m = 1"
	+ proof -
	+ have case2b2:"b2 mod m = 3" using case2_assum assms(3) by (smt (verit) assms(1) int_ops(2)
	+ mod_add_eq mod_pos_pos_trivial numeral_Bit0 of_nat_mono of_nat_numeral semiring_norm(172))
	+
	+ have "\<exists>r::nat. (r\<le>m-3) \<and> [N = b2+r] (mod m)" if case2_1_assum:"N mod m = m-1" for N
	+ proof -
	+ have "[N = 3+(m-4)] (mod m)" using case2_1_assum
	+ by (metis (mono_tags, lifting) Suc_eq_plus1 Suc_numeral add_diff_cancel_left
	+ arithmetic_simps(1) arithmetic_simps(7) assms(1)mod_mod_trivial
	+ ordered_cancel_comm_monoid_diff_class.diff_add_assoc unique_euclidean_semiring_class.cong_def)
	+ hence "[N = b2+(m-4)] (mod m)" using case2b2
	+ by (metis (mono_tags, lifting) Num.of_nat_simps(4) mod_add_left_eq
	+ of_nat_mod of_nat_numeral unique_euclidean_semiring_class.cong_def)
	+ thus ?thesis using le_diff_conv by fastforce
	+ qed
	+ hence case2_1:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case2_1_assum:"N mod m = m-1" for N using case2_1_assum by blast
	+
	+ have "\<exists>r::nat. (r\<le>m-3) \<and> [N = b2+r] (mod m)" if case2_2_assum:"N mod m =0" for N
	+ proof -
	+ have "(3+(m-3)) mod m = 0" using assms(1) by fastforce
	+ hence "(b2+(m-3)) mod m = 0" using case2b2 by (metis Num.of_nat_simps(1)
	+ Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
	+ thus ?thesis using case2_2_assum
	+ by (metis int_ops(1) nat_le_linear of_nat_mod unique_euclidean_semiring_class.cong_def)
	+ qed
	+ hence case2_2:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case2_2_assum:"N mod m =0" for N using case2_2_assum by metis
	+
	+ have "\<exists>r::nat. (r \<le> m-3) \<and> [N = b1+r] (mod m)" if
	+ case2_3_assum:"N mod m \<le>m-2 \<and> N mod m \<ge>1" for N
	+ proof -
	+ have "\<exists>r::nat. (r\<le>m-3)\<and>((b1+r) mod m = l)" if asml:"l\<ge>1 \<and> l\<le>m-2" for l
	+ proof -
	+ define r where r_def:"r = l-1"
	+ from asml have r_range:"r\<ge>0 \<and> r\<le>m-3" using r_def by linarith
	+ have "(1+r) mod m = l" using asml r_def r_range by fastforce
	+ hence "(b1+r) mod m = l" using case2_assum
	+ by (metis Num.of_nat_simps(3) int_ops(9) mod_add_left_eq plus_1_eq_Suc)
	+ thus ?thesis using asml r_range by blast
	+ qed
	+ thus ?thesis using case2_3_assum
	+ by (metis case2_3_assum of_nat_mod unique_euclidean_semiring_class.cong_def)
	+ qed
	+ hence case2_3:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case2_3_assum:"N mod m \<le>m-2 \<and> N mod m \<ge>1" for N using case2_3_assum by blast
	+
	+ have "\<forall>N::nat. N mod m = 0 \<or> (N mod m \<ge>1 \<and> N mod m \<le>m-1)" by (metis One_nat_def Suc_pred
	+assms(1) bot_nat_0.extremum_uniqueI leI less_Suc_eq_le mod_less_divisor not_numeral_le_zero)
	+ hence "\<forall>N::nat. N mod m = 0 \<or> (N mod m \<ge>1 \<and> N mod m \<le>m-2) \<or> N mod m = m-1"
	+ by (smt (verit) arithmetic_simps(68) diff_diff_eq le_add_diff_inverse le_neq_implies_less le_simps(2)
	+ le_trans plus_1_eq_Suc)
	+ thus ?thesis using case2_1 case2_2 case2_3 by (metis \<open>\<And>N. N mod m \<le> m - 2 \<and> 1 \<le> N mod m
	+ \<Longrightarrow> \<exists>r\<le>m - 3. [int N = b1 + int r] (mod int m)\<close>)
	+ qed
	+
	+ have case3:"\<forall>N::nat. \<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case3_assum:"b1 mod m = 2"
	+ proof -
	+ have case3b2:"b2 mod m = 4" using assms case3_assum by (smt (verit, ccfv_SIG)
	+ Euclidean_Division.pos_mod_sign dvd_mod_imp_dvd even_numeral int_ops(2) int_ops(4)
	+mod_diff_eq mod_pos_pos_trivial nat_1_add_1 numeral_Bit0 of_nat_le_iff of_nat_numeral plus_1_eq_Suc)
	+
	+ have "\<exists>r::nat. (r\<le>m-3)\<and> [N = b2+r] (mod m)" if case3_1_assum:"N mod m = 0 \<or> N mod m =1" for N
	+ proof -
	+ have "(4+(m-3)) mod m = (4+m-3) mod m" using assms(1) by auto
	+ have "(4+m-3) mod m = (1+m) mod m" by simp
	+ hence "(4+(m-3)) mod m = 1" using \<open>(4+(m-3)) mod m = (4+m-3) mod m\<close>
	+ by (smt (verit, best) Euclidean_Division.pos_mod_bound add_lessD1 arith_special(2) assms(1) case3b2
	+ landau_product_preprocess(4) mod_add_self2 mod_less numeral_Bit0 of_nat_numeral order_le_less)
	+ hence caseone:"(b2+(m-3)) mod m = 1" using case3b2
	+ by (metis Num.of_nat_simps(2) Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
	+
	+ have "(4+(m-4)) mod m = 0" using assms(1) by auto
	+ hence casezero:"(b2+(m-4)) mod m = 0" using case3b2
	+ by (metis (full_types) Num.of_nat_simps(1) Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
	+
	+ show ?thesis using caseone casezero case3_1_assum
	+ by (metis cong_int cong_mod_right cong_refl diff_le_mono2 nat_le_linear numeral_le_iff
	+ of_nat_0 of_nat_1 semiring_norm(69) semiring_norm(72))
	+ qed
	+ hence case3_1:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case3_1_assum:"N mod m = 0 \<or> N mod m =1" for N using case3_1_assum by metis
	+
	+ have "\<exists>r::nat.(r\<le>m-3)\<and> [N = b1+r] (mod m)" if case3_2_assum:"N mod m \<ge>2\<and>N mod m \<le> m-1" for N
	+ proof -
	+ have "\<exists>r::nat. (r\<le>m-3)\<and>((b1+r) mod m = l)" if asml1:"l\<ge>2 \<and> l\<le>m-1" for l
	+ proof -
	+ define r1 where r1_def:"r1 = l-2"
	+ from asml1 have r1_range:"r1\<ge>0 \<and> r1\<le>m-2" using r1_def by linarith
	+ have "(2+r1) mod m = l" using asml1 r1_def r1_range by fastforce
	+ hence "(b1+r1) mod m = l" using case3_assum
	+ by (metis Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
	+ thus ?thesis using asml1 r1_range by (metis One_nat_def diff_diff_add diff_le_mono
	+ nat_1_add_1 numeral_3_eq_3 plus_1_eq_Suc r1_def)
	+ qed
	+ thus ?thesis using case3_2_assum
	+ by (metis case3_2_assum of_nat_mod unique_euclidean_semiring_class.cong_def)
	+ qed
	+ hence case3_2:"\<exists>b::int. \<exists>r::nat. (r \<le> m-3) \<and> [N=b+r] (mod m) \<and> (b = b1 \<or> b = b2)"
	+ if case3_2_assum:"N mod m \<ge>2\<and>N mod m \<le> m-1" for N using case3_2_assum by blast
	+
	+ have "\<forall>N::nat. N mod m = 0 \<or> (N mod m \<ge>1 \<and> N mod m \<le>m-1)" by (metis Suc_pred' assms(1)
	+bot_nat_0.not_eq_extremum less_one mod_Suc_le_divisor rel_simps(76) verit_comp_simplify1(3))
	+ hence "\<forall>N::nat. N mod m = 0 \<or> N mod m = 1 \<or> (N mod m \<ge>2 \<and> N mod m \<le>m-1)"
	+ by (metis Suc_eq_plus1 le_neq_implies_less le_simps(3) nat_1_add_1)
	+
	+ thus ?thesis using case3_1 case3_2 by blast
	+ qed
	+
	+ show ?thesis using case1 case2 case3 using that by fastforce
	+ qed
	+
	+ show ?thesis using first second using assms(1) by force
	+qed
	+
	+text\<open>The strong form of Cauchy's polygonal number theorem shows for a natural number $N$ satisfying
	+certain conditions, it may be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four
	+of which are different from 0 or 1. This corresponds to Theorem 1.9 in \cite{nathanson1996}.\<close>
	+
	+theorem Strong_Form_of_Cauchy_Polygonal_Number_Theorem_1:
	+ fixes m N :: nat
	+ assumes "m\<ge>4"
	+ assumes "N\<ge>108*m"
	+ shows "\<exists> xs :: nat list. (length xs = m+1) \<and> (sum_list xs = N) \<and> (\<forall>k\<le>3. \<exists>a. xs! k = polygonal_number m a)
	+ \<and> (\<forall> k \<in> {4..m} . xs! k = 0 \<or> xs! k = 1)"
	+
	+proof -
	+ define L where L_def:"L = (2/3 + sqrt (8N/m - 8)) - (1/2 + sqrt (6N/m - 3))"
	+ from assms L_def have "L>4" using interval_length_greater_than_four
	+ apply(rule_tac N = "of_nat N" and m = "of_nat m" in interval_length_greater_than_four)
	+ by auto
	+ define c1 where c1_def:"c1 = \<lceil>1/2 + sqrt (6*N/m - 3)\<rceil>"
	+ define c2 where c2_def:"c2 = c1+1"
	+ define c3 where c3_def:"c3 = c1+2"
	+ define c4 where c4_def:"c4 = c1+3"
	+ from \<open>L>4\<close> c1_def c2_def c3_def c4_def L_def have "c4<(2/3 + sqrt (8*N/m - 8))" by linarith
	+
	+ have "N/m \<ge> 108" using assms using le_divide_eq by fastforce
	+ hence "sqrt(6*N/m - 3)\<ge>1" by simp
	+ hence "1/2 + sqrt(6*N/m - 3) \<ge>1" by linarith
	+ hence "c1 \<ge>1" using c1_def by simp
	+
	+ obtain b1 b2 where bproperties:"{b1, b2} \<subseteq> {c1, c2, c3, c4} \<and> (b2 = b1+2) \<and> odd b1"
	+ using two_consec_odd c1_def c2_def c3_def c4_def by (metis (no_types, opaque_lifting) Groups.add_ac(2)
	+empty_subsetI even_plus_one_iff insert_commute insert_mono nat_arith.add1 numeral.simps(2) numeral.simps(3))
	+ have b1andb2:"odd b1 \<and> b2 = b1+2" using bproperties by auto
	+ have b1pos:"b1 \<ge>1" using \<open>c1\<ge>1\<close> c2_def c3_def c4_def bproperties by auto
	+ hence b2pos:"b2 \<ge>3" using bproperties by simp
	+ have b2odd:"odd b2" using bproperties by simp
	+
	+ obtain b r where b_r:"r\<le>m-3 \<and> (b = b1 \<or> b = b2) \<and> [int N = b+r] (mod m)" using b1andb2 assms(1)
	+ cong_classes by meson
	+ have bpos:"b\<ge>1" using b1pos b2pos b_r by auto
	+ have bodd:"odd b" using b_r bproperties by auto
	+
	+ define a where a_def:"a = b+2*(N-b-r) div m"
	+ have m_div_num:"m dvd (N-b-r)" using b_r
	+ by (simp add: diff_diff_add mod_eq_dvd_iff unique_euclidean_semiring_class.cong_def)
	+ hence "(N-b-r)/m = (N-b-r) div m" by (simp add: real_of_int_div)
	+ hence a_def1:"a = b+2*(N-b-r)/m" using a_def by (metis \<open>int m dvd int N - b - int r\<close>
	+ dvd_add_right_iff mult_2 of_int_add of_int_of_nat_eq real_of_int_div)
	+ have "N-m>0" using assms by linarith
	+ hence "N-r>0" using b_r by force
	+ hence "(N-b-r) = (N-r)-b" by linarith
	+ hence "(N-b-r)/m = (N-r)/m - b/m" by (metis diff_divide_distrib int_of_reals(3) of_int_of_nat_eq)
	+ hence "a = b+2*((N-r)/m - b/m)" using a_def1 by (metis int_of_reals(6) of_int_mult times_divide_eq_right)
	+ hence a_def2:"a = b- b2/m+2(N-r)/m " by simp
	+ have "b(1-2/m) = b1-b*(2/m)" by (simp add: Rings.ring_distribs(4))
	+ hence a_def3:"a = b(1-2/m) + 2(N-r)/m" using a_def2 by simp
	+ have "1-2/m>0" using assms(1) by simp
	+ hence size1:"b*(1-2/m)>0" using bpos by simp
	+ have "N-r>0" using b_r assms by auto
	+ hence size2:"2*(N-r)/m>0" using assms(1) by simp
	+ have apos:"a\<ge>1" using size1 size2 a_def3 by simp
	+
	+ have "odd (b+2*(N-b-r) div m)" using m_div_num b_r b2odd bproperties
	+ by (metis div_mult_swap zdvd_reduce)
	+ hence aodd:"odd a" using a_def by simp
	+
	+ from a_def1 have "a-b = 2*(N-b-r)/m" by simp
	+ hence "m*(a-b)/2 = N-b-r" using assms(1) by simp
	+ hence N_expr:"N = r+b+m*(a-b)/2" by simp
	+
	+ have "b1 \<ge> c1" using bproperties c2_def c3_def c4_def by force
	+ hence "b1 \<ge> 1/2 + sqrt (6*N/m - 3)" using c1_def using ceiling_le_iff by blast
	+ have b_ineq1:"b \<ge> 1/2 + sqrt (6*N/m - 3)" using b_r bproperties
	+ using \<open>1 / 2 + sqrt (real (6 * N) / real m - 3) \<le> real_of_int b1\<close> by fastforce
	+
	+ have "b2 \<le> c4" using bproperties c1_def c2_def c3_def c4_def by fastforce
	+ hence "b2 \<le> (2/3 + sqrt (8*N/m - 8))"
	+ using \<open>real_of_int c4 < 2 / 3 + sqrt (real (8 * N) / real m - 8)\<close> by linarith
	+ hence b_ineq2:"b\<le>(2/3 + sqrt (8*N/m - 8))" using b_r bproperties by linarith
	+
	+ define Nr where "Nr = real_of_nat N"
	+ define mr where "mr = real m"
	+ define ar where "ar = real_of_int a"
	+ define br where "br = real_of_int b"
	+ define rr where "rr = real_of_nat r"
	+ from assms(1) have "mr \<ge>3" using mr_def by auto
	+ from assms(2) have "N\<ge>2*m" by simp
	+ hence "Nr \<ge> 2mr" using Nr_def mr_def \<open>N \<ge> 2 m\<close> by auto
	+ moreover have "br\<ge>0" using br_def bpos by auto
	+ moreover have "mr\<ge>3" using mr_def assms by auto
	+ moreover have "ar\<ge>0" using ar_def apos by auto
	+ moreover have "rr\<ge>0" using rr_def b_r by auto
	+ moreover have "mr > rr" using mr_def rr_def b_r assms(1) by linarith
	+ moreover have "Nr = mr*(ar-br)/2+br+rr" using Nr_def mr_def ar_def br_def N_expr rr_def by auto
	+ moreover have "1/2+sqrt(6Nr/mr-3)\<le>br \<and> br\<le>2/3+sqrt(8Nr/mr-8)" using Nr_def mr_def br_def b_ineq1 b_ineq2 by auto
	+ ultimately have "br^2<4ar \<and> 3ar<br^2+2*br+4" using Cauchy_lemma by auto
	+ hence real_ineq:"(real_of_int b)^2 < 4(real_of_int a) \<and> 3(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
	+ using br_def ar_def by auto
	+ hence int_ineq1: "b^2<4*a" using of_int_less_iff by fastforce
	+ from real_ineq have int_ineq2: "3a<b^2+2b+4" using of_int_less_iff by fastforce
	+
	+ have con1:"nat a \<ge>1" using apos by auto
	+ have con2:"nat b \<ge>1" using bpos by auto
	+ have con3:"odd (nat a)" using aodd apos even_nat_iff by auto
	+ have con4:"odd (nat b)" using bodd bpos even_nat_iff by auto
	+ have "(nat b)^2 = b^2" using \<open>nat b \<ge>1\<close> by auto
	+ hence con5:"(nat b)^2<4*(nat a)" using int_ineq1 by linarith
	+ have con6:"3(nat a)<(nat b)^2+2(nat b)+4" using \<open>(nat b)^2 = b^2\<close> int_ineq2 by linarith
	+ obtain s t u v where stuv:"s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> int(nat a) = s^2 + t^2 + u^2 + v^2 \<and>
	+ int(nat b) = s+t+u+v" using four_nonneg_int_sum con1 con2 con3 con4 con5 con6 by presburger
	+ have a_expr:"a = s^2 + t^2 + u^2 + v^2" using apos stuv by linarith
	+ have b_expr:"b = s+t+u+v" using bpos stuv by linarith
	+
	+ from N_expr have "N = m/2*(s^2-s+t^2-t+u^2-u+v^2-v)+r+(s+t+u+v)" using a_expr b_expr by simp
	+ hence N_expr2:"N = m/2(s^2-s)+ m/2(t^2-t)+ m/2(u^2-u)+ m/2(v^2-v)+ r+(s+t+u+v)"
	+ by (metis (no_types, opaque_lifting) add_diff_eq nat_distrib(2) of_int_add)
	+ have s_div2:"m/2(s^2-s) = m(s^2-s) div 2" using real_of_int_div by auto
	+ have t_div2:"m/2(t^2-t) = m(t^2-t) div 2" using real_of_int_div by auto
	+ have u_div2:"m/2(u^2-u) = m(u^2-u) div 2" using real_of_int_div by auto
	+ have v_div2:"m/2(v^2-v) = m(v^2-v) div 2" using real_of_int_div by auto
	+ have N_expr3:"N = (m(s^2-s) div 2+s)+(m(t^2-t) div 2+t)+(m(u^2-u) div 2+u)+(m(v^2-v) div 2+v)+r"
	+ using s_div2 t_div2 u_div2 v_div2 N_expr2 by simp
	+
	+ define sn where "sn = nat s"
	+ define tn where "tn = nat t"
	+ define un where "un = nat u"
	+ define vn where "vn = nat v"
	+ have seqsn:"s^2-s = sn^2 - sn" using stuv sn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have teqtn:"t^2-t = tn^2 - tn" using stuv tn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have uequn:"u^2-u = un^2 - un" using stuv un_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have veqvn:"v^2-v = vn^2 - vn" using stuv vn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+
	+ from N_expr3 have
	+ "N = (m(sn^2-sn) div 2+s)+(m(tn^2-tn) div 2+t)+(m(un^2-un) div 2+u)+(m(vn^2-vn) div 2+ v)+r"
	+ using seqsn teqtn uequn veqvn by (metis (mono_tags, lifting) int_ops(2) int_ops(4) int_ops(7)
	+ numeral_Bit0 numeral_code(1) plus_1_eq_Suc zdiv_int)
	+ hence "N = (m(sn^2-sn) div 2+sn)+(m(tn^2-tn) div 2+tn)+(m(un^2-un) div 2+un)+(m(vn^2-vn) div 2+ v)+r"
	+ using sn_def tn_def un_def stuv int_nat_eq int_ops(5) by presburger
	+ hence "N = (m(sn^2-sn) div 2+sn)+(m(tn^2-tn) div 2+tn)+(m(un^2-un) div 2+un)+(m(vn^2-vn) div 2+ vn)+r"
	+ using vn_def stuv by (smt (verit, del_insts) Num.of_nat_simps(4) int_nat_eq of_nat_eq_iff)
	+ hence "N = (m* sn(sn-1) div 2+sn)+(mtn(tn-1) div 2+tn)+(mun(un-1) div 2+un)+(m vn*(vn-1) div 2+ vn)+r"
	+ by (smt (verit, ccfv_threshold) more_arith_simps(11) mult.right_neutral power2_eq_square right_diff_distrib')
	+ hence N_expr4:"N = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn +r"
	+ using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger
	+
	+ define T where T_def:"T = [polygonal_number m sn,polygonal_number m tn,polygonal_number m un,polygonal_number m vn]"
	+ define ones where ones_def:"ones = replicate r (1::nat)"
	+ define zeros where zeros_def:"zeros = replicate (m+1-4-r) (0::nat)"
	+ define final where final_def:"final = T@ones@zeros"
	+
	+ have "m+1-4-r\<ge>0" using assms(1) b_r by force
	+ hence "4+r+(m+1-4-r) = m+1" using assms(1) b_r by force
	+ have "length final = 4+r+(m+1-4-r)" using final_def T_def ones_def zeros_def by auto
	+ hence final_length:"length final = m+1" using \<open>4+r+(m+1-4-r) = m+1\<close> by simp
	+ have T_sum:"sum_list T = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn" by (simp add: T_def)
	+ have ones_sum:"sum_list ones = r" using ones_def by (simp add: sum_list_replicate)
	+ have zeros_sum:"sum_list zeros = 0" using zeros_def by simp
	+ have "sum_list final = sum_list T + sum_list ones + sum_list zeros" using final_def by simp
	+ hence final_sum:"sum_list final = N" using N_expr4 by (simp add: T_sum ones_sum zeros_sum)
	+
	+ have final_0th:"final! 0 = polygonal_number m sn" using final_def T_def by simp
	+ have final_1st:"final! 1 = polygonal_number m tn" using final_def T_def by simp
	+ have final_2nd:"final! 2 = polygonal_number m un" using final_def T_def by simp
	+ have final_3rd:"final! 3 = polygonal_number m vn" using final_def T_def by simp
	+
	+ have first_four:"\<forall>k\<le>3. \<exists>a. final! k = polygonal_number m a" using final_0th final_1st final_2nd final_3rd
	+ by (metis Suc_eq_plus1 add_leD2 arith_simps(50) le_simps(2) numeral_Bit0 numeral_Bit1
	+ numeral_One verit_comp_simplify1(3) verit_la_disequality)
	+
	+ have "length T = 4" using T_def by simp
	+ have "\<forall>k<length (ones@zeros). (ones@zeros)! k =1 \<or> (ones@zeros)! k =0" using ones_def zeros_def
	+ by (simp add: nth_append)
	+ hence "final! k = 1 \<or> final! k = 0" if "k\<ge>4 \<and> k<(length final)" for k
	+ using \<open>length T = 4\<close> final_def that by (metis add_less_cancel_left le_add_diff_inverse
	+ length_append nth_append verit_comp_simplify1(3))
	+ hence other_terms:"\<forall> k \<in> {4..m} . final! k = 0 \<or> final! k = 1" using final_length
	+ by (metis Suc_eq_plus1 atLeastAtMost_iff le_simps(2))
	+
	+ show ?thesis using final_length final_sum first_four other_terms by auto
	+qed
	+
	+
	+theorem Strong_Form_of_Cauchy_Polygonal_Number_Theorem_2:
	+ fixes N :: nat
	+ assumes "N\<ge>324"
	+ shows "\<exists> p1 p2 p3 p4 r ::nat. N = p1+p2+p3+p4+r \<and> (\<exists>k1. p1 = polygonal_number 3 k1) \<and> (\<exists>k2. p2 = polygonal_number 3 k2)
	+\<and> (\<exists>k3. p3 = polygonal_number 3 k3) \<and> (\<exists>k4. p4 = polygonal_number 3 k4) \<and> (r = 0 \<or> r = 1)"
	+
	+proof -
	+ define L where L_def:"L = (2/3 + sqrt (8N/3 - 8)) - (1/2 + sqrt (6N/3 - 3))" (Now m = 3)
	+ from assms L_def have "L>4" using interval_length_greater_than_four
	+ apply -
	+ apply(rule interval_length_greater_than_four[where N = "of_nat N" and m = "of_nat 3"])
	+ by auto
	+ define c1 where c1_def:"c1 = \<lceil>1/2 + sqrt (6*N/3 - 3)\<rceil>"
	+ define c2 where c2_def:"c2 = c1+1"
	+ define c3 where c3_def:"c3 = c1+2"
	+ define c4 where c4_def:"c4 = c1+3"
	+ from \<open>L>4\<close> c1_def c2_def c3_def c4_def L_def have "c4<(2/3 + sqrt (8*N/3 - 8))" by linarith
	+
	+ define Nn where "Nn = int N"
	+ have "c4<(2/3 + sqrt (8Nn/3 - 8))" using Nn_def \<open>c4<(2/3 + sqrt (8N/3 - 8))\<close> by simp
	+ have Nn3:"(Nn-3)^2 - (sqrt (8Nn/3 - 8))^2 = Nn^2-3Nn-3Nn+9 - (sqrt (8Nn/3 - 8))^2"
	+ using assms Nn_def power2_diff by (simp add: power2_eq_square algebra_simps)
	+ have "(Nn-3)^2 - (sqrt (8Nn/3 - 8))^2 = Nn^2-3Nn-3Nn+9 - (8Nn/3 - 8)" using assms Nn_def Nn3 by fastforce
	+ hence "(Nn-3)^2 - (sqrt (8Nn/3 - 8))^2 = Nn^2-6Nn+9-8*Nn/3 +8" by linarith
	+ hence Nn4:"(Nn-3)^2 - (sqrt (8Nn/3 - 8))^2 = Nn(Nn-26/3)+17" by (simp add: Rings.ring_distribs(4) power2_eq_square)
	+ have "Nn*(Nn-26/3)+17>17" using assms Nn_def by auto
	+ hence "(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 > 0" using Nn4 by auto
	+ hence "Nn-3 > sqrt (8*Nn/3 - 8)" using assms Nn_def by (simp add: real_less_lsqrt)
	+ hence "Nn-2 > sqrt (8*Nn/3 - 8)+2/3" by linarith
	+ hence "N > c4" using Nn_def \<open>c4<(2/3 + sqrt (8*Nn/3 - 8))\<close> by simp
	+
	+ have "N/3 \<ge> 108" using assms using le_divide_eq by fastforce
	+ hence "sqrt(6*N/3 - 3)\<ge>1" by simp
	+ hence "1/2 + sqrt(6*N/3 - 3) \<ge>1" by linarith
	+ hence "c1 \<ge>1" using c1_def by simp
	+
	+ obtain b1 b2 where bproperties:"{b1, b2} \<subseteq> {c1, c2, c3, c4} \<and> (b2 = b1+2) \<and> odd b1"
	+ using two_consec_odd c1_def c2_def c3_def c4_def by (metis (no_types, opaque_lifting) Groups.add_ac(2)
	+empty_subsetI even_plus_one_iff insert_commute insert_mono nat_arith.add1 numeral.simps(2) numeral.simps(3))
	+ have b1andb2:"odd b1 \<and> b2 = b1+2" using bproperties by auto
	+ have b1pos:"b1 \<ge>1" using \<open>c1\<ge>1\<close> c2_def c3_def c4_def bproperties by auto
	+ hence b2pos:"b2 \<ge>3" using bproperties by simp
	+ have b2odd:"odd b2" using bproperties by simp
	+ define b1n where "b1n = nat b1"
	+ define b2n where "b2n = nat b2"
	+
	+ from b1n_def b1pos have "b1n mod 3 = b1 mod 3" using int_ops(9) by force
	+ from b2n_def b2pos have "b2n mod 3 = b2 mod 3" using int_ops(9) by force
	+
	+ have b_and_r:"\<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ proof -
	+ have case1:"\<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm1:"b1 mod 3 = 0"
	+ proof -
	+ have "b1n mod 3 = 0" using asm1 \<open>b1n mod 3 = b1 mod 3\<close> by simp
	+ hence "b2n mod 3 = 2" using \<open>b2n mod 3 = b2 mod 3\<close> bproperties asm1 by fastforce
	+ have case1_1:"[0 = b1n+0] (mod 3)" using \<open>b1n mod 3 = 0\<close>
	+ by (metis mod_0 nat_arith.rule0 unique_euclidean_semiring_class.cong_def)
	+ have case1_2:"[1 = b1n+1] (mod 3)" using \<open>b1n mod 3 = 0\<close>
	+ by (metis \<open>[0 = b1n + 0] (mod 3)\<close> add.commute cong_add_lcancel_0_nat cong_sym)
	+ have case1_3:"[2 = b2n+0] (mod 3)" using \<open>b2n mod 3 = 2\<close>
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ have "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 \<ge> 1" by linarith
	+ hence "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 = 1 \<or> N mod 3 = 2" by linarith
	+ hence "\<forall>N. \<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm1:"b1 mod 3 = 0" using case1_1 case1_2 case1_3 by (metis cong_mod_left)
	+ thus ?thesis using asm1 by auto
	+ qed
	+
	+ have case2:"\<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm2:"b1 mod 3 = 1"
	+ proof -
	+ have "b1n mod 3 = 1" using asm2 \<open>b1n mod 3 = b1 mod 3\<close> by simp
	+ hence "b2n mod 3 = 0" using \<open>b2n mod 3 = b2 mod 3\<close> bproperties asm2
	+ by (smt (verit, best) Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign
	+ int_ops(1) mod_diff_eq mod_pos_pos_trivial of_nat_eq_iff)
	+ have case2_1:"[0 = b2n+0] (mod 3)" using \<open>b2n mod 3 = 0\<close>
	+ by (metis mod_0 nat_arith.rule0 unique_euclidean_semiring_class.cong_def)
	+ have case2_2:"[1 = b1n+0] (mod 3)" using \<open>b1n mod 3 = 1\<close>
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ have case2_3:"[2 = b1n+1] (mod 3)" using \<open>b1n mod 3 = 1\<close>
	+ by (metis case2_2 cong_add_rcancel_nat nat_1_add_1 nat_arith.rule0)
	+ have "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 \<ge> 1" by linarith
	+ hence "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 = 1 \<or> N mod 3 = 2" by linarith
	+ hence "\<forall>N. \<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm2:"b1 mod 3 = 1" using case2_1 case2_2 case2_3 by (metis cong_mod_left)
	+ thus ?thesis using asm2 by auto
	+ qed
	+
	+ have case3:"\<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm3:"b1 mod 3 = 2"
	+ proof -
	+ have "b1n mod 3 = 2" using asm3 \<open>b1n mod 3 = b1 mod 3\<close> by simp
	+ have "(b1+2) mod 3 = (2+2) mod 3" using asm3 by (metis Groups.add_ac(2) mod_add_right_eq)
	+ hence "b2n mod 3 = 1" using \<open>b2n mod 3 = b2 mod 3\<close> bproperties by simp
	+ have case3_1:"[0 = b1n+1] (mod 3)" using \<open>b1n mod 3 = 2\<close>
	+ by (metis One_nat_def add.commute mod_0 mod_add_right_eq mod_self nat_1_add_1 numeral_3_eq_3
	+ plus_1_eq_Suc unique_euclidean_semiring_class.cong_def)
	+ have case3_2:"[1 = b2n+0] (mod 3)" using \<open>b2n mod 3 = 1\<close>
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ have case3_3:"[2 = b1n+0] (mod 3)" using \<open>b1n mod 3 = 2\<close>
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ have "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 \<ge> 1" by linarith
	+ hence "\<forall>N::nat. N mod 3 = 0 \<or> N mod 3 = 1 \<or> N mod 3 = 2" by linarith
	+ hence "\<forall>N. \<exists>b r::nat. [N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ if asm3:"b1 mod 3 = 2" using case3_1 case3_2 case3_3 by (metis cong_mod_left)
	+ thus ?thesis using asm3 by auto
	+ qed
	+
	+ have "b1 mod 3 = 0 \<or> b1 mod 3 = 1 \<or> b1 mod 3 = 2" by auto
	+ thus ?thesis using case1 case2 case3 by auto
	+ qed
	+
	+ obtain b r where b_r:"[N = b+r] (mod 3) \<and> (b = b1n \<or> b = b2n) \<and> (r = 0 \<or> r = 1)"
	+ using b_and_r by auto
	+
	+ have bpos:"b\<ge>1" using b1pos b2pos b_r b1n_def b2n_def by auto
	+ have bodd:"odd b"
	+ using b_r bproperties by (metis b1n_def b2n_def b2odd bpos even_nat_iff nat_eq_iff2 rel_simps(45))
	+
	+ define a where a_def:"a = b+2*(N-b-r) div 3"
	+ have "int b1n = b1" using b1n_def b1pos by linarith
	+ have "int b2n = b2" using b2n_def b2pos by linarith
	+ have m_div_num:"3 dvd (N-b-r)" using b_r
	+ by (metis cong_altdef_nat diff_diff_left diff_is_0_eq' dvd_0_right nat_le_linear)
	+ hence a_def1:"a = b+2*(N-b-r)/3" using a_def m_div_num real_of_nat_div by auto
	+ from \<open>N>c4\<close> have "N>b" using b_r bproperties b1n_def b2n_def
	+ by (smt (verit, del_insts) \<open>int b1n = b1\<close> \<open>int b2n = b2\<close> c2_def c3_def c4_def empty_iff insert_iff insert_subset of_nat_less_imp_less)
	+ hence "(N-b-r)/3 = (N-r)/3 - b/3" using \<open>b < N\<close> b_r by force
	+ hence "a = b- b2/3+2(N-r)/3" using a_def1 by linarith
	+ hence a_def3:"a = b(1-2/3) + 2(N-r)/3" by simp
	+
	+ have size1:"b*(1-2/3)>0" using bpos by simp
	+ have "N-r>0" using b_r assms by auto
	+ hence size2:"2*(N-r)/3>0" using assms(1) by simp
	+ have apos:"a\<ge>1" using size1 size2 a_def3 by simp
	+
	+ have "odd (b+2*(N-b-r) div 3)" using m_div_num b_r b2odd bproperties by (simp add: bodd mult_2)
	+ hence aodd:"odd a" using a_def by simp
	+ from a_def1 have "a-b = 2*(N-b-r)/3" by simp
	+ hence "(a-b)/2 = (N-b-r)/3" by simp
	+ hence "3*(a-b)/2 = N-b-r" by simp
	+ have "N-b-r\<ge>0" using b_r by simp
	+ hence N_expr:"N = r+b+3(a-b)/2" using \<open>N-b-r\<ge>0\<close> \<open>b < N\<close> b_r \<open>real (3 (a - b)) / 2 = real (N - b - r)\<close> by linarith
	+ from a_def \<open>N-b-r\<ge>0\<close> have "a\<ge>b" using a_def le_add1 by blast
	+
	+ have "b1 \<ge> c1" using bproperties c2_def c3_def c4_def by force
	+ hence "b1 \<ge> 1/2 + sqrt (6*N/3 - 3)" using c1_def using ceiling_le_iff by blast
	+ hence b1ngreater:"b1n \<ge> 1/2 + sqrt (6*N/3 - 3)" using b1n_def by simp
	+ hence b2ngreater:"b2n \<ge> 1/2 + sqrt (6*N/3 - 3)" using bproperties b1n_def b2n_def by linarith
	+ hence b_ineq1:"b \<ge> 1/2 + sqrt (6*N/3 - 3)" using b_r b1ngreater by auto
	+
	+ have "b2 \<le> c4" using bproperties c1_def c2_def c3_def c4_def by fastforce
	+ hence "b2 \<le> (2/3 + sqrt (8*N/3 - 8))"
	+ using \<open>real_of_int c4 < 2 / 3 + sqrt (real (8 * N) / 3 - 8)\<close> by linarith
	+ hence b2nsmaller:"b2n \<le> (2/3 + sqrt (8*N/3 - 8))" using b2n_def by (metis \<open>int b2n = b2\<close> of_int_of_nat_eq)
	+ hence "b1n \<le> (2/3 + sqrt (8*N/3 - 8))" using b1n_def bproperties using \<open>int b2n = b2\<close> by linarith
	+ hence b_ineq2:"b\<le>(2/3 + sqrt (8*N/3 - 8))" using b_r b2nsmaller by auto
	+
	+ define Nr where "Nr = real_of_nat N"
	+ define ar where "ar = real_of_int a"
	+ define br where "br = real_of_int b"
	+ define rr where "rr = real_of_nat r"
	+ define m where "m = real_of_nat 3"
	+ from assms have "N\<ge>2*m" using m_def by simp
	+ then have "Nr \<ge> 2m" using Nr_def \<open>N \<ge> 2 m\<close> by auto
	+ moreover have "br\<ge>0" using br_def bpos by auto
	+ moreover have "ar\<ge>0" using ar_def apos by auto
	+ moreover have "rr\<ge>0" using rr_def b_r by auto
	+ moreover have "m\<ge>3" using m_def by auto
	+ moreover have "m>rr" using m_def rr_def b_r by auto
	+ moreover have "Nr = m*(ar-br)/2+br+rr" using Nr_def ar_def br_def N_expr rr_def m_def \<open>a\<ge>b\<close> by auto
	+ moreover have "1/2+sqrt(6Nr/m-3)\<le>br \<and> br\<le>2/3+sqrt(8Nr/m-8)" using Nr_def br_def b_ineq1 b_ineq2 m_def by auto
	+ ultimately have "br^2<4ar \<and> 3ar<br^2+2*br+4" using Cauchy_lemma by auto
	+ hence real_ineq:"(real_of_int b)^2 < 4(real_of_int a) \<and> 3(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
	+ using br_def ar_def by auto
	+ hence nat_ineq1: "b^2<4*a" using br_def by (smt (verit, del_insts) Num.of_nat_simps(4)
	+mult.commute mult_2_right nat_distrib(1) numeral_Bit0 of_int_of_nat_eq of_nat_less_of_nat_power_cancel_iff)
	+ from real_ineq have nat_ineq2: "3a<b^2+2b+4" using ar_def br_def of_nat_less_iff by fastforce
	+
	+ obtain s t u v where stuv:"s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> int a = s^2 + t^2 + u^2 + v^2 \<and>
	+ int b = s+t+u+v" using apos bpos aodd bodd nat_ineq1 nat_ineq2 four_nonneg_int_sum by presburger
	+ have a_expr:"a = s^2 + t^2 + u^2 + v^2" using apos stuv by linarith
	+ have b_expr:"b = s+t+u+v" using bpos stuv by linarith
	+
	+ have "N = r + (s+t+u+v)+ 3*(a-(s+t+u+v))/2" using b_expr N_expr
	+ by (metis Num.of_nat_simps(4) Num.of_nat_simps(5) \<open>b \<le> a\<close> of_int_of_nat_eq of_nat_diff of_nat_numeral)
	+ hence "N = 3/2*(s^2-s+t^2-t+u^2-u+v^2-v)+r+(s+t+u+v)" using a_expr by simp
	+ hence N_expr2:"N = 3/2(s^2-s)+ 3/2(t^2-t)+ 3/2(u^2-u)+ 3/2(v^2-v)+ r+(s+t+u+v)"
	+ by (metis (no_types, opaque_lifting) add_diff_eq nat_distrib(2) of_int_add)
	+
	+ have s_div2:"3/2(s^2-s) = 3(s^2-s) div 2" using real_of_int_div by auto
	+ have t_div2:"3/2(t^2-t) = 3(t^2-t) div 2" using real_of_int_div by auto
	+ have u_div2:"3/2(u^2-u) = 3(u^2-u) div 2" using real_of_int_div by auto
	+ have v_div2:"3/2(v^2-v) = 3(v^2-v) div 2" using real_of_int_div by auto
	+ have N_expr3:"N = (3(s^2-s) div 2+s)+(3(t^2-t) div 2+t)+(3(u^2-u) div 2+u)+(3(v^2-v) div 2+v)+r"
	+ using N_expr2 s_div2 t_div2 u_div2 v_div2 by simp
	+
	+ define sn where "sn = nat s"
	+ define tn where "tn = nat t"
	+ define un where "un = nat u"
	+ define vn where "vn = nat v"
	+ have seqsn:"s^2-s = sn^2 - sn" using stuv sn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have teqtn:"t^2-t = tn^2 - tn" using stuv tn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have uequn:"u^2-u = un^2 - un" using stuv un_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+ have veqvn:"v^2-v = vn^2 - vn" using stuv vn_def
	+ by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
	+
	+ from N_expr3 have
	+ "N = (3(sn^2-sn) div 2+s)+(3(tn^2-tn) div 2+t)+(3(un^2-un) div 2+u)+(3(vn^2-vn) div 2+ v)+r"
	+ using seqsn teqtn uequn veqvn
	+ by (metis (mono_tags, lifting) Num.of_nat_simps(5) of_nat_numeral zdiv_int)
	+ hence "N = (3(sn^2-sn) div 2+sn)+(3(tn^2-tn) div 2+tn)+(3(un^2-un) div 2+un)+(3(vn^2-vn) div 2+ v)+r"
	+ using sn_def tn_def un_def stuv int_nat_eq int_ops(5) by presburger
	+ hence "N = (3(sn^2-sn) div 2+sn)+(3(tn^2-tn) div 2+tn)+(3(un^2-un) div 2+un)+(3(vn^2-vn) div 2+ vn)+r"
	+ using vn_def stuv by (smt (verit, del_insts) Num.of_nat_simps(4) int_nat_eq of_nat_eq_iff)
	+ hence "N = (3* sn(sn-1) div 2+sn)+(3tn(tn-1) div 2+tn)+(3un(un-1) div 2+un)+(3 vn*(vn-1) div 2+ vn)+r"
	+ by (smt (verit, ccfv_threshold) more_arith_simps(11) mult.right_neutral power2_eq_square right_diff_distrib')
	+ hence N_expr4:"N = polygonal_number 3 sn + polygonal_number 3 tn + polygonal_number 3 un + polygonal_number 3 vn +r"
	+ using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger
	+
	+ define p1 where "p1 = polygonal_number 3 sn"
	+ define p2 where "p2 = polygonal_number 3 tn"
	+ define p3 where "p3 = polygonal_number 3 un"
	+ define p4 where "p4 = polygonal_number 3 vn"
	+ have N_expr5:"N = p1 + p2 + p3 + p4 + r" using N_expr4 p1_def p2_def p3_def p4_def by auto
	+ thus ?thesis using p1_def p2_def p3_def p4_def b_r by blast
	+qed
	+end
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Gauss.thy b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Gauss.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Gauss.thy
	@@ -0,0 +1,44 @@
	+section \<open> Polygonal Number Theorem \<close>
	+subsection \<open> Gauss's Theorem on Triangular Numbers\<close>
	+text \<open>We show Gauss's theorem which states that every non-negative integer is the sum of three triangles, using the Three Squares Theorem AFP entry \cite{Three_Squares-AFP}. This corresponds to Theorem 1.8 in \cite{nathanson1996}.\<close>
	+
	+theory Polygonal_Number_Theorem_Gauss
	+ imports Polygonal_Number_Theorem_Lemmas
	+begin
	+
	+text\<open>The following is the formula for the $k$-th polygonal number of order $m+2$.\<close>
	+
	+definition polygonal_number :: "nat \<Rightarrow> nat \<Rightarrow> nat"
	+ where "polygonal_number m k = mk(k-1) div 2 + k"
	+
	+text\<open>When $m = 1$, the polygonal numbers have order 3 and the formula represents triangular numbers.
	+Gauss showed that all natural numbers can be written as the sum of three triangular numbers.
	+In other words, the triangular numbers form an additive basis of order 3 of the natural numbers.\<close>
	+
	+theorem Gauss_Sum_of_Three_Triangles:
	+ fixes n :: nat
	+ shows "\<exists> x y z. n = polygonal_number 1 x + polygonal_number 1 y + polygonal_number 1 z"
	+
	+proof -
	+ have "(8 * n + 3) mod 8 = 3" by auto
	+ then obtain a b c where 0: "odd a \<and> odd b \<and> odd c \<and> 8 * n + 3 = a^2 + b^2 + c^2"
	+ using odd_three_squares_using_mod_eight by presburger
	+ then obtain x y z where "a = 2 * x + 1 \<and> b = 2 * y + 1 \<and> c = 2 * z + 1" by (meson oddE)
	+ hence "8 * n + 3 = (2 * x + 1)^2 + (2 * y + 1)^2 + (2 * z + 1)^2"
	+ using 0 by auto
	+ hence "n = (x * x + x + y * y + y + z * z + z) div 2"
	+ by (auto simp add: power2_eq_square)
	+ hence n_expr:"n = (x * (x + 1) + y * (y + 1) + z * (z + 1)) div 2"
	+ by (metis (no_types, lifting) arithmetic_simps(79) nat_arith.add1 nat_distrib(2))
	+
	+ have triangle_identity: "polygonal_number 1 k = k*(k+1) div 2" for k
	+ proof -
	+ have "k(k-1)+2k = k*k+k" by (simp add: right_diff_distrib')
	+ hence "k(k-1) div 2 + k = (kk+k) div 2"
	+ by (metis Groups.add_ac(2) bot_nat_0.not_eq_extremum div_mult_self2 pos2)
	+ thus ?thesis using polygonal_number_def by simp
	+ qed
	+ from n_expr triangle_identity show ?thesis
	+ by (metis div_plus_div_distrib_dvd_right even_mult_iff odd_even_add odd_one)
	+qed
	+end
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Legendre.thy b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Legendre.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Legendre.thy
	@@ -0,0 +1,613 @@
	+subsection \<open> Legendre's Polygonal Number Theorem \<close>
	+text\<open>We will use the definition of the polygonal numbers from the Gauss Theorem theory file
	+which also imports the Three Squares Theorem AFP entry \cite{Three_Squares-AFP}.\<close>
	+
	+theory Polygonal_Number_Theorem_Legendre
	+ imports Polygonal_Number_Theorem_Gauss
	+begin
	+
	+text \<open>This lemma shows that under certain conditions, an integer $N$ can be written as the sum of four polygonal numbers.\<close>
	+
	+lemma sum_of_four_polygonal_numbers:
	+ fixes N m :: nat
	+ fixes b :: int
	+ assumes "m \<ge> 3"
	+ assumes "N \<ge> 2*m"
	+ assumes "[N = b] (mod m)"
	+ assumes odd_b: "odd b"
	+ assumes "b \<in> {1/2 + sqrt (6N/m - 3) .. 2/3 + sqrt (8N/m - 8)}"
	+ assumes "N \<ge> 9"
	+ shows "\<exists>k1 k2 k3 k4. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4"
	+
	+proof -
	+ define I where "I = {1/2 + sqrt (6N/m - 3) .. 2/3 + sqrt (8N/m - 8)}"
	+ from assms(5) I_def have "b \<in> I" by auto
	+ define a::int where a_def: "a = 2*(N-b) div m + b"
	+ have "m dvd (N-b)" using assms(3)
	+ by (smt (verit, ccfv_threshold) cong_iff_dvd_diff zdvd_zdiffD)
	+ hence "even (2*(N-b) div m)"
	+ by (metis div_mult_swap dvd_triv_left)
	+ hence "odd a" using a_def assms(3) odd_b by auto
	+ from assms(1) have "m^3 \<ge> m"
	+ by (simp add: power3_eq_cube)
	+ hence "N \<ge> 2 * m" using assms(1,2) by simp
	+ from assms(1) have m_pos: "m > 0" by auto
	+ have "N \<ge> b"
	+ proof -
	+ from assms(1) have "m \<ge> 1" by auto
	+ hence "1/m \<le> 1" using m_pos by auto
	+ moreover have "N > 0" using \<open>N \<ge> 2 * m\<close> m_pos by auto
	+ ultimately have "N/m \<le> N"
	+ using divide_less_eq_1 less_eq_real_def by fastforce
	+ hence "sqrt(8N/m - 8) \<le> sqrt(8(N-1))" by auto
	+ from assms(1) have "m^3 \<ge> 33(3::real)"
	+ by (metis numeral_le_real_of_nat_iff numeral_times_numeral power3_eq_cube power_mono zero_le_numeral)
	+ from \<open>N \<ge> 9\<close> have "N-1 \<ge> 8" by auto
	+ hence "(N-1)^2 \<ge> 8*(N-1)" using \<open>N > 0\<close> by (simp add: power2_eq_square)
	+ hence "(N-1) \<ge> sqrt (8*(N-1))" using \<open>N > 0\<close>
	+ by (metis of_nat_0_le_iff of_nat_mono of_nat_power real_sqrt_le_mono real_sqrt_pow2 real_sqrt_power)
	+ hence "N - (1::real) - sqrt(8*N/m - 8) \<ge> 0"
	+ using \<open>sqrt (real (8 * N) / real m - 8) \<le> sqrt (real (8 * (N - 1)))\<close> \<open>9 \<le> N\<close> by linarith
	+ hence expr_pos: "N - (2/3::real) - sqrt(8*N/m - 8) \<ge> 0" by auto
	+ have "b \<le> 2/3 + sqrt(8*N/m - 8)" using \<open>b \<in> I\<close> I_def by auto
	+ hence "N - b \<ge> N - (2/3 + sqrt(8*N/m-8))" by auto
	+ hence "N - b \<ge> 0"
	+ using expr_pos of_int_0_le_iff by auto
	+ thus ?thesis by auto
	+ qed
	+ from \<open>N \<ge> 2 * m\<close> m_pos have "6*N/m - 3 \<ge> 0" by (simp add: mult_imp_le_div_pos)
	+ hence "1/2 + sqrt (6*N/m - 3) > 0"
	+ by (smt (verit, del_insts) divide_le_0_1_iff real_sqrt_ge_zero)
	+ with \<open>b \<in> I\<close> assms(3) I_def have "b \<ge> 1" by auto
	+ hence b_pos: "b \<ge> 0" by auto
	+ from \<open>b \<in> I\<close> have b_in_I: "(1/2::real) + sqrt (6* real N / real m - 3) \<le> b \<and> b \<le> (2/3::real) + sqrt (8 * real N/real m - 8)" unfolding I_def by auto
	+ from b_pos \<open>N \<ge> b\<close> a_def have a_pos: "a \<ge> 0"
	+ by (smt (verit) m_pos of_nat_0_less_iff pos_imp_zdiv_neg_iff)
	+ hence "a \<ge> 1"
	+ by (smt (verit) \<open>odd a\<close> dvd_0_right)
	+ have "a - b = 2*(N-b) div m" using a_def by auto
	+ from \<open>int m dvd (int N - b)\<close> have "m dvd 2*(N-b)" by fastforce
	+ have "a = 2*(N-b)/m + b" using a_def m_pos
	+ using \<open>int m dvd 2 * (int N - b)\<close> by fastforce
	+ hence "a = 2N/m - 2b/m + b"
	+ by (simp add: assms diff_divide_distrib of_nat_diff)
	+ hence "(2::real)N/m = a + 2b/m - b" by auto
	+ hence "(2::real)N = (a + 2b/m - b)*m"
	+ using m_pos by (simp add: divide_eq_eq)
	+ hence "(2::real)N = m(a-b) + 2*b"
	+ using \<open>int m dvd 2 * (int N - b)\<close> a_def by auto
	+ hence "N = m*(a-b)/2 + b" by auto
	+ hence N_expr: "real N = real m * (of_int a - of_int b) / 2 + of_int b" by auto
	+ have "even (a-b)" using \<open>odd a\<close> \<open>odd b\<close> by auto
	+ hence "2 dvd m*(a-b)" by auto
	+ hence N_expr2: "N = m(a-b) div 2 + b" using \<open>N = m(a-b)/2 + b\<close> by linarith
	+ define Nr where "Nr = real_of_nat N"
	+ define mr where "mr = real m"
	+ define ar where "ar = real_of_int a"
	+ define br where "br = real_of_int b"
	+ from assms(1) have "mr \<ge> 3" using mr_def by auto
	+ moreover have "Nr \<ge> 2mr" using Nr_def mr_def \<open>N \<ge> 2 m\<close> by auto
	+ moreover have "br \<ge> 0" using br_def b_pos by auto
	+ moreover have "mr > 0" using mr_def m_pos by auto
	+ moreover have "ar \<ge> 0" using ar_def \<open>a \<ge> 0\<close> by auto
	+ moreover have "Nr = mr*(ar-br)/2 + br" using Nr_def mr_def ar_def br_def N_expr by auto
	+ moreover have "1/2 + sqrt (6Nr/mr-3) \<le> br \<and> br \<le> 2/3 + sqrt (8Nr/mr-8)" using Nr_def mr_def br_def b_in_I by auto
	+ ultimately have "br^2 < 4ar \<and> 3ar < br^2+2*br+4" using Cauchy_lemma_r_eq_zero
	+ by auto
	+ hence real_ineq:"(real_of_int b)^2 < 4(real_of_int a) \<and> 3(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
	+ using br_def ar_def by auto
	+ hence int_ineq1: "b^2<4*a" using of_int_less_iff by fastforce
	+ from real_ineq have int_ineq2: "3a < b^2+2b+4" using of_int_less_iff by fastforce
	+
	+ define an:: nat where "an = nat a"
	+ from a_pos have "an = a" unfolding an_def by auto
	+ define bn:: nat where "bn = nat b"
	+ from b_pos have "bn = b" unfolding bn_def by auto
	+ have "an \<ge> 1" using \<open>int an = a\<close> \<open>a \<ge> 1\<close> by auto
	+ moreover have "bn \<ge> 1" using \<open>int bn = b\<close> \<open>b \<ge> 1\<close> by auto
	+ moreover have "odd an" using \<open>odd a\<close> \<open>int an = a\<close> by auto
	+ moreover have "odd bn" using \<open>odd b\<close> \<open>int bn = b\<close> by auto
	+ moreover have "bn ^ 2 < 4 * an" using int_ineq1 \<open>int an = a\<close> \<open>int bn = b\<close>
	+ using of_nat_less_iff by fastforce
	+ moreover have "3 * an < bn ^ 2 + 2 * bn + 4" using int_ineq2 \<open>int an = a\<close> \<open>int bn = b\<close>
	+ using of_nat_less_iff by fastforce
	+ ultimately have "\<exists>s t u v :: int. s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> an = s^2 + t^2 + u^2 + v^2 \<and>
	+bn = s+t+u+v" using four_nonneg_int_sum by auto
	+ hence "\<exists>s t u v :: int. s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> a = s^2 + t^2 + u^2 + v^2 \<and>
	+b = s+t+u+v" using \<open>int an = a\<close> \<open>int bn = b\<close> by auto
	+ then obtain s t u v :: int where stuv: "s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> a = s^2 + t^2 + u^2 + v^2 \<and>
	+b = s+t+u+v" by auto
	+ hence "N = (m*(s^2+t^2+u^2+v^2-s-t-u-v) div 2) + s+t+u+v" using N_expr2 by (smt (verit, ccfv_threshold))
	+ hence "N = (m*(s^2-s+t^2-t+u^2-u+v^2-v) div 2) + s+t+u+v" by (smt (verit, ccfv_SIG))
	+ hence "N = (m * (s * (s-1) + t * (t-1) + u * (u-1) + v * (v-1)) div 2) +s+t+u+v" by (simp add: power2_eq_square algebra_simps)
	+ hence previous_step: "N = (m * s * (s-1) + m * t * (t-1) + m * u * (u-1) + m * v * (v-1)) div 2 + s+t+u+v" by (simp add: algebra_simps)
	+ moreover have "2 dvd m * s * (s-1)" by simp
	+ moreover have "2 dvd m * t * (t-1)" by simp
	+ moreover have "2 dvd m * u * (u-1)" by simp
	+ moreover have "2 dvd m * v * (v-1)" by simp
	+ ultimately have "N = m * s * (s-1) div 2 + m * t * (t-1) div 2 + m * u * (u-1) div 2 + m * v *(v-1) div 2 + s+t+u+v" by fastforce
	+ hence N_expr3: "N = m * s * (s-1) div 2 + s + m * t * (t-1) div 2 + t + m * u * (u-1) div 2 + u + m * v * (v-1) div 2 + v" by auto
	+ define sn::nat where "sn = nat s"
	+ define tn::nat where "tn = nat t"
	+ define un::nat where "un = nat u"
	+ define vn::nat where "vn = nat v"
	+ have "sn = s" using stuv sn_def by auto
	+ hence "m * sn * (sn-1) = m * s * (s-1)" by fastforce
	+ hence "m * sn * (sn-1) div 2 = m * s * (s-1) div 2" by linarith
	+ have "tn = t" using stuv tn_def by auto
	+ hence "m * tn * (tn-1) = m * t * (t-1)" by fastforce
	+ hence "m * tn * (tn-1) div 2 = m * t * (t-1) div 2" by linarith
	+ have "un = u" using stuv un_def by auto
	+ hence "m * un * (un-1) = m * u * (u-1)" by fastforce
	+ hence "m * un * (un-1) div 2 = m * u * (u-1) div 2" by linarith
	+ have "vn = v" using stuv vn_def by auto
	+ hence "m * vn * (vn-1) = m * v * (v-1)" by fastforce
	+ hence "m * vn * (vn-1) div 2 = m * v * (v-1) div 2" by linarith
	+ have "N = m * sn * (sn-1) div 2 + sn + m * tn * (tn-1) div 2 + tn + m * un * (un-1) div 2 + un + m * vn * (vn-1) div 2 + vn"
	+ using N_expr3 \<open>sn = s\<close> \<open>tn = t\<close> \<open>un = u\<close> \<open>vn = v\<close> \<open>m * sn * (sn-1) div 2 = m * s * (s-1) div 2\<close> \<open>m * tn * (tn-1) div 2 = m * t * (t-1) div 2\<close> \<open>m * un * (un-1) div 2 = m * u * (u-1) div 2\<close> \<open>m * vn * (vn-1) div 2 = m * v * (v-1) div 2\<close> by linarith
	+ hence "N = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn"
	+ using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger
	+ thus ?thesis by blast
	+qed
	+
	+text \<open>We show Legendre's polygonal number theorem which corresponds to Theorem 1.10 in \cite{nathanson1996}.\<close>
	+
	+theorem Legendre_Polygonal_Number_Theorem:
	+ fixes m N :: nat
	+ assumes "m \<ge> 3"
	+ assumes "N \<ge> 28*m^3"
	+ shows "odd m \<Longrightarrow> \<exists>k1 k2 k3 k4::nat. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4"
	+and "even m \<Longrightarrow> \<exists>k1 k2 k3 k4 k5::nat. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4 + polygonal_number m k5 \<and> (k1 = 0 \<or> k1 = 1 \<or> k2 = 0 \<or> k2 = 1 \<or> k3 = 0 \<or> k3 = 1 \<or> k4 = 0 \<or> k4 = 1 \<or> k5 = 0 \<or> k5 = 1)"
	+
	+proof -
	+ define L :: real where "L = (2/3 + sqrt (8N/m - 8)) - (1/2 + sqrt (6N/m - 3))"
	+ define I where "I = {1/2 + sqrt (6N/m - 3) .. 2/3 + sqrt (8N/m - 8)}"
	+ from assms(1) have "m^3 \<ge> m"
	+ by (simp add: power3_eq_cube)
	+ hence "N \<ge> 2 * m" using assms by simp
	+ have m_pos: "m > 0" using assms(1) by simp
	+ have "L > 2 * of_nat m" using assms \<open>N \<ge> 2 * m\<close> m_pos L_def
	+ apply -
	+ apply (rule interval_length_greater_than_2m[where N="of_nat N" and m="of_nat m"])
	+ apply (simp_all)
	+ by (metis (no_types, opaque_lifting) of_nat_le_iff of_nat_mult of_nat_numeral power3_eq_cube)
	+ hence 2: "L > 2 * m" by simp
	+ show thm_odd_m: "odd m \<Longrightarrow> \<exists>k1 k2 k3 k4. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4"
	+ proof -
	+ assume odd_m: "odd m"
	+ from assms(1) have "m > 0" by auto
	+ define ce where "ce = \<lceil>1/2 + sqrt (6*N/m - 3)\<rceil>"
	+ have "\<forall> i\<in>{0..2*m-1}. ce + i \<ge> ce" by auto
	+ hence lower_bound: "\<forall> i\<in>{0..2m-1}. ce + i \<ge> 1/2 + sqrt (6N/m - 3)" using ceiling_le_iff ce_def by blast
	+ have "2m-1 + ce \<le> 2/3 + sqrt (8N/m - 8)" using 2 L_def assms(1) ce_def by linarith
	+ hence upper_bound: "\<forall> i\<in>{0..2m-1}. ce + i \<le> 2/3 + sqrt (8N/m - 8)" by auto
	+ from lower_bound upper_bound have in_interval: "\<forall> i\<in>{0..2*m-1}. ce + i \<in> I" unfolding ce_def I_def by auto
	+ have "\<exists> f::nat \<Rightarrow> int. (\<forall> i\<in>{0..m-1}. odd (f i)) \<and> (\<forall> i\<in>{1..m-1}. f i = f 0 + 2*i) \<and> (\<forall> i\<in>{0..m-1}. f i \<in> I)"
	+ proof -
	+ have ?thesis if odd_f0: "odd ce"
	+ proof -
	+ define g::"nat \<Rightarrow> int" where "g i = ce + 2*i"
	+ have "odd (g 0)" using odd_f0 \<open>g \<equiv> \<lambda>i. ce + int (2 * i)\<close> by auto
	+ hence "\<forall> i\<in>{0..m-1}. odd (g i)" using \<open>g \<equiv> \<lambda>i. ce + int (2 * i)\<close> by auto
	+ have "\<forall> i\<in>{1..m-1}. g i = g 0 + 2i" using \<open>g \<equiv> \<lambda>i. ce + int (2 i)\<close> by auto
	+ have "\<forall> i\<in>{0..m-1}. 2i < 2m-1" using m_pos by auto
	+ hence "\<forall> i\<in>{0..m-1}. g i \<in> I" using \<open>g \<equiv> \<lambda>i. ce + int (2 * i)\<close> in_interval by fastforce
	+ show ?thesis using \<open>\<forall>i\<in>{0..m - 1}. odd (g i)\<close> \<open>\<forall>i\<in>{0..m - 1}. real_of_int (g i) \<in> I\<close> \<open>\<forall>i\<in>{1..m - 1}. g i = g 0 + int(2 * i)\<close> by blast
	+ qed
	+ moreover have ?thesis if "even ce"
	+ proof -
	+ from \<open>even ce\<close> have odd_f1: "odd (ce + 1)" by auto
	+ define g::"nat \<Rightarrow> int" where "g i = ce + (2*i + 1)"
	+ have "odd (g 0)" using odd_f1 \<open>g \<equiv> \<lambda>i. ce + int (2 * i + 1)\<close> by auto
	+ hence "\<forall> i\<in>{0..m-1}. odd (g i)" using \<open>g \<equiv> \<lambda>i. ce + int (2 * i + 1)\<close> by auto
	+ have "\<forall> i\<in>{1..m-1}. g i = g 0 + 2i" using \<open>g \<equiv> \<lambda>i. ce + int (2 i + 1)\<close> by auto
	+ have "\<forall> i\<in>{0..m-1}. 2i + 1 \<le> 2m-1" using m_pos by auto
	+ hence "\<forall> i\<in>{0..m-1}. g i \<in> I" using \<open>g \<equiv> \<lambda>i. ce + int (2 * i + 1)\<close> in_interval by fastforce
	+ show ?thesis using \<open>\<forall>i\<in>{0..m - 1}. odd (g i)\<close> \<open>\<forall>i\<in>{0..m - 1}. real_of_int (g i) \<in> I\<close> \<open>\<forall>i\<in>{1..m - 1}. g i = g 0 + int(2 *i)\<close> by blast
	+ qed
	+ ultimately show ?thesis by blast
	+ qed
	+ then obtain f::"nat \<Rightarrow> int" where f_def: "(\<forall> i\<in>{0..m-1}. odd (f i)) \<and> (\<forall> i\<in>{1..m-1}. f i = f 0 + 2*i) \<and> (\<forall> i\<in>{0..m-1}. f i \<in> I)" by auto
	+
	+ have inj_lemma: "\<lbrakk>i \<in> {0..m-1}; j \<in> {0..m-1}; [f i = f j] (mod m)\<rbrakk> \<Longrightarrow> i = j" for i j
	+ proof -
	+ assume asm1: "i \<in> {0..m-1}"
	+ assume asm2: "j \<in> {0..m-1}"
	+ assume asm3: "[f i = f j] (mod m)"
	+ from f_def have "odd (f 0)" by auto
	+ hence "\<exists> k::int. f 0 = 2*k + 1" by (metis oddE)
	+ then obtain k::int where k_def: "f 0 = 2*k + 1" by auto
	+ have False if case2: "i = 0 \<and> j > 0"
	+ proof -
	+ have "f j = f 0 + 2*j" using f_def case2 asm2 by auto
	+ hence "[2k + 1 = 2k + 1 + 2*j] (mod m)" using asm3 case2 k_def by auto
	+ hence "[2*j = 0] (mod m)"
	+ by (metis cong_add_lcancel_0 cong_int_iff cong_sym_eq int_ops(1))
	+ have "coprime 2 m" using odd_m by simp
	+ hence "[j = 0] (mod m)" using \<open>[2*j = 0] (mod m)\<close> by (simp add: cong_0_iff coprime_dvd_mult_right_iff)
	+ thus False using asm2 case2 cong_less_modulus_unique_nat by fastforce
	+ qed
	+ moreover have False if case3: "i > 0 \<and> j = 0"
	+ proof -
	+ have "f i = f 0 + 2*i" using f_def case3 asm1 by auto
	+ hence "[2k + 1 + 2i = 2*k + 1] (mod m)" using asm3 case3 k_def by auto
	+ hence "[2*i = 0] (mod m)"
	+ by (metis cong_add_lcancel_0 cong_int_iff cong_sym_eq int_ops(1))
	+ have "coprime 2 m" using odd_m by simp
	+ hence "[i = 0] (mod m)" using \<open>[2*i = 0] (mod m)\<close> by (simp add: cong_0_iff coprime_dvd_mult_right_iff)
	+ thus False using asm1 case3 cong_less_modulus_unique_nat by fastforce
	+ qed
	+ moreover have ?thesis if case4: "i > 0 \<and> j > 0"
	+ proof -
	+ have "i > 0" and "j > 0" using case4 by auto
	+ have "f i = f 0 + 2*i" using f_def case4 asm1 by auto
	+ moreover have "f j = f 0 + 2*j" using f_def case4 asm2 by auto
	+ ultimately have "[2k + 1 + 2i = 2k + 1 + 2j] (mod m)" using case4 k_def asm3 by fastforce
	+ hence "[2i = 2j] (mod m)"
	+ using cong_add_lcancel cong_int_iff by blast
	+ have "coprime 2 m" using odd_m by simp
	+ hence "[i = j] (mod m)"
	+ using \<open>[2 * i = 2 * j] (mod m)\<close> cong_mult_lcancel_nat by auto
	+ thus ?thesis using asm1 asm2 case4 cong_less_modulus_unique_nat by auto
	+ qed
	+ ultimately show ?thesis by fastforce
	+ qed
	+ have complete_cong_class: "\<exists>i \<in> {0..m-1}. [f i = S] (mod m)" for S
	+ proof -
	+ have "(f i) mod m = (f j) mod m \<Longrightarrow> [f i = f j] (mod m)" for i j
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ hence inj2: "\<lbrakk>i \<in> {0..m-1}; j \<in> {0..m-1}; (f i) mod m = (f j) mod m\<rbrakk> \<Longrightarrow> i = j" for i j
	+ using inj_lemma by auto
	+ hence injective: "\<forall>i \<in> {0..m-1}. \<forall>j \<in> {0..m-1}. (f i) mod m = (f j) mod m \<longrightarrow> i = j"
	+ by auto
	+ define g :: "nat \<Rightarrow> int" where "g i = (f i) mod m"
	+ then have g_inj2: "\<forall>i \<in> {0..m-1}. \<forall> j \<in> {0..m-1}. g i = g j \<longrightarrow> i = j"
	+ using \<open>g \<equiv> \<lambda>i. f i mod int m\<close> injective by fastforce
	+ then have g_inj: "inj_on g {0..m-1}"
	+ by (meson inj_onI)
	+ have g_range2: "\<forall> i \<in> {0..m-1}. g i \<in> {0..m-1}" using \<open>g \<equiv> \<lambda>i. f i mod int m\<close>
	+ by (metis m_pos Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign atLeastAtMost_iff mod_by_1 mod_if not_gr0 of_nat_0_less_iff of_nat_1 of_nat_diff verit_comp_simplify1(3) zle_diff1_eq)
	+ hence image_subset: "g ` {0..m-1} \<subseteq> {0..m-1}" by blast
	+ have g_range: "i \<in> {0..m-1} \<Longrightarrow> g i \<in> {0..m-1}" using \<open>g \<equiv> \<lambda>i. f i mod int m\<close>
	+ by (metis m_pos Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign atLeastAtMost_iff mod_by_1 mod_if not_gr0 of_nat_0_less_iff of_nat_1 of_nat_diff verit_comp_simplify1(3) zle_diff1_eq)
	+ have card_ge_m: "card (g ` {0..m-1}) \<ge> m" using g_inj
	+ by (metis m_pos Suc_diff_1 card_atLeastAtMost card_image minus_nat.diff_0 verit_comp_simplify1(2))
	+ have "card {0..m-1} = m" using m_pos by force
	+ hence card_le_m: "card (g ` {0..m-1}) \<le> m" using m_pos
	+ by (metis card_image g_inj le_refl)
	+ from card_ge_m card_le_m have image_size: "card (g ` {0..m-1}) = m" by auto
	+ with \<open>card {0..m-1} = m\<close> have equal_card: "card (g ` {0..m-1}) = card {0..m-1}" by auto
	+ have "finite (g ` {0..m-1})" using image_size by auto
	+ with equal_card image_subset have "g ` {0..m-1} = {0..m-1}"
	+ by (metis card_image card_subset_eq finite_atLeastAtMost_int image_int_atLeastAtMost inj_on_of_nat of_nat_0)
	+ hence "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. i = g j" for i by auto
	+ hence "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. i = (f j) mod m" for i
	+ using \<open>g \<equiv> \<lambda>i. f i mod int m\<close> by auto
	+ hence surj: "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. [i = f j] (mod m)" for i
	+ by (metis mod_mod_trivial unique_euclidean_semiring_class.cong_def)
	+ have "S mod m \<ge> 0" using m_pos by simp
	+ moreover have "S mod m \<le> m-1"
	+ using m_pos by (simp add: of_nat_diff)
	+ ultimately have "S mod m \<in> {0..m-1}" by auto
	+ with surj m_pos have "\<exists> j \<in> {0..m-1}. [S mod m = f j] (mod m)"
	+ by (metis atLeastAtMost_iff less_eq_nat.simps(1) nonneg_int_cases of_nat_less_iff verit_comp_simplify(3))
	+ thus ?thesis using cong_mod_right cong_sym by blast
	+ qed
	+ have "\<exists> b::int. [N = b] (mod m) \<and> odd b \<and> b \<in> I"
	+ proof -
	+ have "N mod m \<ge> 0" by auto
	+ moreover have "N mod m \<le> m-1"
	+ using m_pos less_Suc_eq_le by fastforce
	+ ultimately have "N mod m \<in> {0..m-1}" by auto
	+ with complete_cong_class have "\<exists> i. i \<in> {0..m-1} \<and> [f i = N] (mod m)" by blast
	+ then obtain c::nat where c_def: "c \<in> {0..m-1} \<and> [f c = N] (mod m)" by auto
	+ define b::int where "b = f c"
	+ have "[N = b] (mod m)" using b_def c_def by (metis cong_sym)
	+ moreover have "odd b" using b_def f_def c_def by auto
	+ moreover have "b \<in> I" using b_def f_def c_def by auto
	+ ultimately show ?thesis by auto
	+ qed
	+ then obtain b::int where b_def: "[N = b] (mod m) \<and> odd b \<and> b \<in> I" by auto
	+ have "m^3 \<ge> m" using m_pos by (auto simp add: power3_eq_cube)
	+ hence "N \<ge> 28*m" using assms(1,2) by linarith
	+ hence "N \<ge> 2*m" by simp
	+ have "m^3 \<ge> 33(3::nat)" using assms(1)
	+ by (metis power3_eq_cube power_mono zero_le_numeral)
	+ hence "N \<ge> 2833*(3::nat)" using assms(2) by auto
	+ hence "N \<ge> 9" by simp
	+ show ?thesis using sum_of_four_polygonal_numbers assms(1) b_def I_def \<open>N \<ge> 2 * m\<close> \<open>N \<ge> 9\<close> by blast
	+ qed
	+ show thm_even_m: "even m \<Longrightarrow> \<exists>k1 k2 k3 k4 k5. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4 + polygonal_number m k5 \<and> (k1 = 0 \<or> k1 = 1 \<or> k2 = 0 \<or> k2 = 1 \<or> k3 = 0 \<or> k3 = 1 \<or> k4 = 0 \<or> k4 = 1 \<or> k5 = 0 \<or> k5 = 1)"
	+ proof -
	+ assume even_m: "even m"
	+ from assms(1) have "m > 0" by auto
	+ define ce where "ce = \<lceil>1/2 + sqrt (6*N/m - 3)\<rceil>"
	+ have "\<forall> i\<in>{0..m-1}. ce + i \<ge> ce" by auto
	+ hence lower_bound: "\<forall> i\<in>{0..m-1}. ce + i \<ge> 1/2 + sqrt (6*N/m - 3)" using ceiling_le_iff ce_def by blast
	+ have "m-1 + ce \<le> 2/3 + sqrt (8*N/m - 8)" using 2 L_def assms(1) ce_def by linarith
	+ hence upper_bound: "\<forall> i\<in>{0..m-1}. ce + i \<le> 2/3 + sqrt (8*N/m - 8)" by auto
	+ from lower_bound upper_bound have in_interval: "\<forall> i\<in>{0..m-1}. ce + i \<in> I" unfolding ce_def I_def by auto
	+ have "\<exists> f::nat \<Rightarrow> int. (\<forall> i\<in>{1..m-1}. f i = f 0 + i) \<and> (\<forall> i\<in>{0..m-1}. f i \<in> I)"
	+ proof -
	+ define g::"nat \<Rightarrow> int" where "g i = ce + i"
	+ have "\<forall> i\<in>{1..m-1}. g i = g 0 + i" using \<open>g \<equiv> \<lambda>i. ce + int i\<close> by auto
	+ have "\<forall> i\<in>{0..m-1}. i < m" using m_pos by auto
	+ hence "\<forall> i\<in>{0..m-1}. g i \<in> I" using \<open>g \<equiv> \<lambda>i. ce + int i\<close> in_interval by fastforce
	+ show ?thesis by (metis Num.of_nat_simps(1) \<open>\<forall>i\<in>{0..m - 1}. real_of_int (g i) \<in> I\<close> \<open>g \<equiv> \<lambda>i. ce + int i\<close> arith_extra_simps(6))
	+ qed
	+ then obtain f::"nat \<Rightarrow> int" where f_def: "(\<forall> i\<in>{1..m-1}. f i = f 0 + i) \<and> (\<forall> i\<in>{0..m-1}. f i \<in> I)" by auto
	+ have inj_lemma: "\<lbrakk>i \<in> {0..m-1}; j \<in> {0..m-1}; [f i = f j] (mod m)\<rbrakk> \<Longrightarrow> i = j" for i j
	+ proof -
	+ assume asm1: "i \<in> {0..m-1}"
	+ assume asm2: "j \<in> {0..m-1}"
	+ assume asm3: "[f i = f j] (mod m)"
	+ have False if case2: "i = 0 \<and> j > 0"
	+ proof -
	+ have "f j = f 0 + j" using f_def case2 asm2 by auto
	+ hence "[f 0 = f 0 + j] (mod m)" using asm3 case2 by auto
	+ hence "[j = 0] (mod m)"
	+ by (metis cong_add_lcancel_0 cong_int_iff cong_sym_eq int_ops(1))
	+ thus False using asm2 case2 cong_less_modulus_unique_nat by fastforce
	+ qed
	+ moreover have False if case3: "i > 0 \<and> j = 0"
	+ proof -
	+ have "f i = f 0 + i" using f_def case3 asm1 by auto
	+ hence "[f 0 + i = f 0] (mod m)" using asm3 case3 by auto
	+ hence "[i = 0] (mod m)"
	+ by (metis cong_add_lcancel_0 cong_int_iff cong_sym_eq int_ops(1))
	+ thus False using asm1 case3 cong_less_modulus_unique_nat by fastforce
	+ qed
	+ moreover have ?thesis if case4: "i > 0 \<and> j > 0"
	+ proof -
	+ have "i > 0" and "j > 0" using case4 by auto
	+ have "f i = f 0 + i" using f_def case4 asm1 by auto
	+ moreover have "f j = f 0 + j" using f_def case4 asm2 by auto
	+ ultimately have "[f 0 + i = f 0 + j] (mod m)" using case4 asm3 by fastforce
	+ hence "[i = j] (mod m)"
	+ using cong_add_lcancel cong_int_iff by blast
	+ thus ?thesis using asm1 asm2 case4 cong_less_modulus_unique_nat by auto
	+ qed
	+ ultimately show ?thesis by fastforce
	+ qed
	+ have complete_cong_class: "\<exists>i \<in> {0..m-1}. [f i = S] (mod m)" for S
	+ proof -
	+ have "(f i) mod m = (f j) mod m \<Longrightarrow> [f i = f j] (mod m)" for i j
	+ by (simp add: unique_euclidean_semiring_class.cong_def)
	+ hence inj2: "\<lbrakk>i \<in> {0..m-1}; j \<in> {0..m-1}; (f i) mod m = (f j) mod m\<rbrakk> \<Longrightarrow> i = j" for i j
	+ using inj_lemma by auto
	+ hence injective: "\<forall>i \<in> {0..m-1}. \<forall>j \<in> {0..m-1}. (f i) mod m = (f j) mod m \<longrightarrow> i = j"
	+ by auto
	+ define g :: "nat \<Rightarrow> int" where "g i = (f i) mod m"
	+ then have g_inj2: "\<forall>i \<in> {0..m-1}. \<forall> j \<in> {0..m-1}. g i = g j \<longrightarrow> i = j"
	+ using \<open>g \<equiv> \<lambda>i. f i mod int m\<close> injective by fastforce
	+ then have g_inj: "inj_on g {0..m-1}"
	+ by (meson inj_onI)
	+ have g_range2: "\<forall> i \<in> {0..m-1}. g i \<in> {0..m-1}" using \<open>g \<equiv> \<lambda>i. f i mod int m\<close>
	+ by (metis m_pos Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign atLeastAtMost_iff mod_by_1 mod_if not_gr0 of_nat_0_less_iff of_nat_1 of_nat_diff verit_comp_simplify1(3) zle_diff1_eq)
	+ hence image_subset: "g ` {0..m-1} \<subseteq> {0..m-1}" by blast
	+ have g_range: "i \<in> {0..m-1} \<Longrightarrow> g i \<in> {0..m-1}" using \<open>g \<equiv> \<lambda>i. f i mod int m\<close>
	+ by (metis m_pos Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign atLeastAtMost_iff mod_by_1 mod_if not_gr0 of_nat_0_less_iff of_nat_1 of_nat_diff verit_comp_simplify1(3) zle_diff1_eq)
	+ have card_ge_m: "card (g ` {0..m-1}) \<ge> m" using g_inj
	+ by (metis m_pos Suc_diff_1 card_atLeastAtMost card_image minus_nat.diff_0 verit_comp_simplify1(2))
	+ have "card {0..m-1} = m" using m_pos by force
	+ hence card_le_m: "card (g ` {0..m-1}) \<le> m" using m_pos
	+ by (metis card_image g_inj le_refl)
	+ from card_ge_m card_le_m have image_size: "card (g ` {0..m-1}) = m" by auto
	+ with \<open>card {0..m-1} = m\<close> have equal_card: "card (g ` {0..m-1}) = card {0..m-1}" by auto
	+ have "finite (g ` {0..m-1})" using image_size by auto
	+ with equal_card image_subset have "g ` {0..m-1} = {0..m-1}"
	+ by (metis card_image card_subset_eq finite_atLeastAtMost_int image_int_atLeastAtMost inj_on_of_nat of_nat_0)
	+ hence "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. i = g j" for i by auto
	+ hence "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. i = (f j) mod m" for i
	+ using \<open>g \<equiv> \<lambda>i. f i mod int m\<close> by auto
	+ hence surj: "i \<in> {0..m-1} \<Longrightarrow> \<exists> j \<in> {0..m-1}. [i = f j] (mod m)" for i
	+ by (metis mod_mod_trivial unique_euclidean_semiring_class.cong_def)
	+ have "S mod m \<ge> 0" using m_pos by simp
	+ moreover have "S mod m \<le> m-1"
	+ using m_pos by (simp add: of_nat_diff)
	+ ultimately have "S mod m \<in> {0..m-1}" by auto
	+ with surj m_pos have "\<exists> j \<in> {0..m-1}. [S mod m = f j] (mod m)"
	+ by (metis atLeastAtMost_iff less_eq_nat.simps(1) nonneg_int_cases of_nat_less_iff verit_comp_simplify(3))
	+ thus ?thesis using cong_mod_right cong_sym by blast
	+ qed
	+ have thm_odd_n: ?thesis if "odd N"
	+ proof -
	+ have "\<exists> b::int. [N = b] (mod m) \<and> odd b \<and> b \<in> I"
	+ proof -
	+ from complete_cong_class have "\<exists> i. i \<in> {0..m-1} \<and> [f i = N] (mod m)" by blast
	+ then obtain c::nat where c_def: "c \<in> {0..m-1} \<and> [f c = N] (mod m)" by auto
	+ define b::int where "b = f c"
	+ have "[N = b] (mod m)" using b_def c_def by (metis cong_sym)
	+ moreover have "odd b"
	+ proof
	+ assume "even b"
	+ have "\<exists>k::int. N = b + k*m" using \<open>[N = b] (mod m)\<close>
	+ by (metis cong_iff_lin cong_sym_eq mult.commute)
	+ then obtain k::int where k_def: "N = b + k*m" by auto
	+ have "even (k*m)" using even_m by auto
	+ hence "even N" using k_def \<open>even b\<close> by presburger
	+ thus False using \<open>odd N\<close> by blast
	+ qed
	+ moreover have "b \<in> I" using b_def f_def c_def by auto
	+ ultimately show ?thesis by auto
	+ qed
	+ then obtain b::int where b_def: "[N = b] (mod m) \<and> odd b \<and> b \<in> I" by auto
	+ have "m^3 \<ge> m" using m_pos by (auto simp add: power3_eq_cube)
	+ hence "N \<ge> 28*m" using assms(1,2) by linarith
	+ hence "N \<ge> 2*m" by simp
	+ have "m^3 \<ge> 33(3::nat)" using assms(1)
	+ by (metis power3_eq_cube power_mono zero_le_numeral)
	+ hence "N \<ge> 2833*(3::nat)" using assms(2) by auto
	+ hence "N \<ge> 9" by simp
	+ hence "\<exists>k1 k2 k3 k4. N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4"
	+ using sum_of_four_polygonal_numbers assms(1) b_def I_def \<open>N \<ge> 2 * m\<close> \<open>N \<ge> 9\<close> by blast
	+ then obtain k1 k2 k3 k4 where "N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4" by blast
	+ moreover have "polygonal_number m 0 = 0" using Polygonal_Number_Theorem_Gauss.polygonal_number_def by auto
	+ ultimately have "N = polygonal_number m k1 + polygonal_number m k2 + polygonal_number m k3 + polygonal_number m k4 + polygonal_number m 0" by linarith
	+ thus ?thesis by blast
	+ qed
	+ have thm_even_n: ?thesis if "even N"
	+ proof -
	+ have "\<exists> b::int. [N-1 = b] (mod m) \<and> odd b \<and> b \<in> I"
	+ proof -
	+ from complete_cong_class have "\<exists> i. i \<in> {0..m-1} \<and> [f i = N-1] (mod m)" by blast
	+ then obtain c::nat where c_def: "c \<in> {0..m-1} \<and> [f c = N-1] (mod m)" by auto
	+ define b::int where "b = f c"
	+ have "[N-1 = b] (mod m)" using b_def c_def by (metis cong_sym)
	+ moreover have "odd b"
	+ proof
	+ assume "even b"
	+ have "\<exists>k::int. N-1 = b + k*m" using \<open>[N-1 = b] (mod m)\<close>
	+ by (metis (full_types) cong_iff_lin cong_sym_eq mult.commute)
	+ then obtain k::int where k_def: "N-1 = b + k*m" by auto
	+ have "even (k*m)" using even_m by auto
	+ hence "even (N-1)" using k_def \<open>even b\<close> by presburger
	+ hence "odd N"
	+ by (metis Groups.mult_ac(2) \<open>2 * m \<le> N\<close> add_eq_self_zero add_leD1 assms(1) dvd_diffD1 le_trans mult_2_right nat_1_add_1 nat_dvd_1_iff_1 rel_simps(25) zero_neq_numeral)
	+ thus False using \<open>even N\<close> by blast
	+ qed
	+ moreover have "b \<in> I" using b_def f_def c_def by auto
	+ ultimately show ?thesis by auto
	+ qed
	+ then obtain b::int where b_def: "[N-1 = b] (mod m) \<and> odd b \<and> b \<in> I" by auto
	+ from b_def have "b \<in> I" by auto
	+ define a::int where a_def: "a = 2*(N-1-b) div m + b"
	+ have "m dvd (N-1-b)" using b_def
	+ by (smt (verit, ccfv_threshold) cong_iff_dvd_diff zdvd_zdiffD)
	+ hence "even (2*(N-1-b) div m)"
	+ by (metis div_mult_swap dvd_triv_left)
	+ hence "odd a" using a_def b_def by auto
	+ from assms(1) have "m^3 \<ge> m"
	+ by (simp add: power3_eq_cube)
	+ hence "N \<ge> 2 * m" using assms(1,2) by simp
	+ from assms(1) have m_pos: "m > 0" by auto
	+ have "N-1 \<ge> b"
	+ proof -
	+ from assms(1) have "m \<ge> 1" by auto
	+ hence "1/m \<le> 1" using m_pos by auto
	+ moreover have "N > 0" using \<open>N \<ge> 2 * m\<close> m_pos by auto
	+ ultimately have "N/m \<le> N"
	+ using divide_less_eq_1 less_eq_real_def by fastforce
	+ hence "sqrt(8N/m - 8) \<le> sqrt(8(N-1))" by auto
	+ from assms(1) have "m^3 \<ge> 33(3::real)"
	+ by (metis numeral_le_real_of_nat_iff numeral_times_numeral power3_eq_cube power_mono zero_le_numeral)
	+ hence "N \<ge> 2833*(3::real)" using assms(2) by linarith
	+ hence "N-6 \<ge> 6" by simp
	+ hence "N-6 \<ge> 0" by simp
	+ with \<open>N-6 \<ge> 6\<close> have "(N-6)^2 \<ge> 6^2"
	+ using power2_nat_le_eq_le by blast
	+ hence "(N-6)^2 \<ge> 24" by simp
	+ hence "(N-2)^2 \<ge> 8*(N-1)" by (simp add: power2_eq_square algebra_simps)
	+ hence "(N-2) \<ge> sqrt (8*(N-1))" using \<open>N > 0\<close>
	+ by (metis of_nat_0_le_iff of_nat_mono of_nat_power real_sqrt_le_mono real_sqrt_pow2 real_sqrt_power)
	+ hence "N - (2::real) - sqrt(8*N/m - 8) \<ge> 0"
	+ using \<open>sqrt (real (8 * N) / real m - 8) \<le> sqrt (real (8 * (N - 1)))\<close> \<open>N-6 \<ge> 6\<close> by linarith
	+ hence expr_pos: "N - 1 - (2/3::real) - sqrt(8*N/m - 8) \<ge> 0" by auto
	+ have "b \<le> 2/3 + sqrt(8*N/m - 8)" using \<open>b \<in> I\<close> I_def by auto
	+ hence "N - 1 - b \<ge> N - 1 - (2/3 + sqrt(8*N/m-8))" by auto
	+ hence "N - 1 - b \<ge> 0"
	+ using expr_pos of_int_0_le_iff by auto
	+ thus ?thesis by auto
	+ qed
	+ from \<open>N \<ge> 2 * m\<close> m_pos have "6*N/m - 3 \<ge> 0" by (simp add: mult_imp_le_div_pos)
	+ hence "1/2 + sqrt (6*N/m - 3) > 0"
	+ by (smt (verit, del_insts) divide_le_0_1_iff real_sqrt_ge_zero)
	+ with \<open>b \<in> I\<close> b_def I_def have "b \<ge> 1" by auto
	+ hence b_pos: "b \<ge> 0" by auto
	+ from \<open>b \<in> I\<close> have b_in_I: "(1/2::real) + sqrt (6* real N / real m - 3) \<le> b \<and> b \<le> (2/3::real) + sqrt (8 * real N/real m - 8)" unfolding I_def by auto
	+ from b_pos \<open>N-1 \<ge> b\<close> a_def have a_pos: "a \<ge> 0"
	+ by (smt (verit) m_pos of_nat_0_less_iff pos_imp_zdiv_neg_iff)
	+ hence "a \<ge> 1"
	+ by (smt (verit) \<open>odd a\<close> dvd_0_right)
	+ have "a - b = 2*(N-1-b) div m" using a_def by auto
	+ from \<open>int m dvd (N - 1 - b)\<close> have "m dvd 2*(N-1-b)" by fastforce
	+ have "a = 2*(N-1-b)/m + b" using a_def m_pos
	+ using \<open>int m dvd 2 * (N - 1 - b)\<close> by fastforce
	+ hence "a = 2(N-1)/m - 2b/m + b"
	+ by (simp add: assms diff_divide_distrib of_nat_diff)
	+ hence "(2::real)(N-1)/m = a + 2b/m - b" by auto
	+ hence "(2::real)(N-1) = (a + 2b/m - b)*m"
	+ using m_pos by (simp add: divide_eq_eq)
	+ hence "(2::real)(N-1) = m(a-b) + 2*b"
	+ using \<open>int m dvd 2 * (N - 1 - b)\<close> a_def by auto
	+ hence "N-1 = m*(a-b)/2 + b" by auto
	+ hence "N = m*(a-b)/2 + b + 1"
	+ using \<open>2 * m \<le> N\<close> assms(1) by linarith
	+ hence N_expr: "N = real m * (of_int a - of_int b) / 2 + of_int b + 1" by auto
	+ have "even (a-b)" using \<open>odd a\<close> b_def by auto
	+ hence "2 dvd m*(a-b)" by auto
	+ hence N_expr2: "N = m(a-b) div 2 + b + 1" using \<open>N = m(a-b)/2 + b + 1\<close> by linarith
	+ define Nr where "Nr = real_of_nat N"
	+ define mr where "mr = real m"
	+ define ar where "ar = real_of_int a"
	+ define br where "br = real_of_int b"
	+ from assms(1) have "mr \<ge> 3" using mr_def by auto
	+ moreover have "Nr \<ge> 2mr" using Nr_def mr_def \<open>N \<ge> 2 m\<close> by auto
	+ moreover have "br \<ge> 0" using br_def b_pos by auto
	+ moreover have "mr > 0" using mr_def m_pos by auto
	+ moreover have "ar \<ge> 0" using ar_def \<open>a \<ge> 0\<close> by auto
	+ moreover have "Nr = mr*(ar-br)/2 + br + 1" using Nr_def mr_def ar_def br_def N_expr by auto
	+ moreover have "1/2 + sqrt (6Nr/mr-3) \<le> br \<and> br \<le> 2/3 + sqrt (8Nr/mr-8)" using Nr_def mr_def br_def b_in_I by auto
	+ ultimately have "br^2 < 4ar \<and> 3ar < br^2+2*br+4" using Cauchy_lemma
	+ by auto
	+ hence real_ineq:"(real_of_int b)^2 < 4(real_of_int a) \<and> 3(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
	+ using br_def ar_def by auto
	+ hence int_ineq1: "b^2<4*a" using of_int_less_iff by fastforce
	+ from real_ineq have int_ineq2: "3a < b^2+2b+4" using of_int_less_iff by fastforce
	+
	+ define an:: nat where "an = nat a"
	+ from a_pos have "an = a" unfolding an_def by auto
	+ define bn:: nat where "bn = nat b"
	+ from b_pos have "bn = b" unfolding bn_def by auto
	+ have "an \<ge> 1" using \<open>int an = a\<close> \<open>a \<ge> 1\<close> by auto
	+ moreover have "bn \<ge> 1" using \<open>int bn = b\<close> \<open>b \<ge> 1\<close> by auto
	+ moreover have "odd an" using \<open>odd a\<close> \<open>int an = a\<close> by auto
	+ moreover have "odd bn" using b_def \<open>int bn = b\<close> by auto
	+ moreover have "bn ^ 2 < 4 * an" using int_ineq1 \<open>int an = a\<close> \<open>int bn = b\<close>
	+ using of_nat_less_iff by fastforce
	+ moreover have "3 * an < bn ^ 2 + 2 * bn + 4" using int_ineq2 \<open>int an = a\<close> \<open>int bn = b\<close>
	+ using of_nat_less_iff by fastforce
	+ ultimately have "\<exists>s t u v :: int. s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> an = s^2 + t^2 + u^2 + v^2 \<and>
	+ bn = s+t+u+v" using four_nonneg_int_sum by auto
	+ hence "\<exists>s t u v :: int. s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> a = s^2 + t^2 + u^2 + v^2 \<and>
	+ b = s+t+u+v" using \<open>int an = a\<close> \<open>int bn = b\<close> by auto
	+ then obtain s t u v :: int where stuv: "s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> a = s^2 + t^2 + u^2 + v^2 \<and>
	+ b = s+t+u+v" by auto
	+ hence "N = (m*(s^2+t^2+u^2+v^2-s-t-u-v) div 2) + s+t+u+v + 1" using N_expr2 by (smt (verit, ccfv_threshold))
	+ hence "N = (m*(s^2-s+t^2-t+u^2-u+v^2-v) div 2) + s+t+u+v + 1" by (smt (verit, ccfv_SIG))
	+ hence "N = (m * (s * (s-1) + t * (t-1) + u * (u-1) + v * (v-1)) div 2) +s+t+u+v + 1" by (simp add: power2_eq_square algebra_simps)
	+ hence previous_step: "N = (m * s * (s-1) + m * t * (t-1) + m * u * (u-1) + m * v * (v-1)) div 2 + s+t+u+v + 1" by (simp add: algebra_simps)
	+ moreover have "2 dvd m * s * (s-1)" by simp
	+ moreover have "2 dvd m * t * (t-1)" by simp
	+ moreover have "2 dvd m * u * (u-1)" by simp
	+ moreover have "2 dvd m * v * (v-1)" by simp
	+ ultimately have "N = m * s * (s-1) div 2 + m * t * (t-1) div 2 + m * u * (u-1) div 2 + m * v *(v-1) div 2 + s+t+u+v + 1" by fastforce
	+ hence N_expr3: "N = m * s * (s-1) div 2 + s + m * t * (t-1) div 2 + t + m * u * (u-1) div 2 + u + m * v * (v-1) div 2 + v + 1" by auto
	+ define sn::nat where "sn = nat s"
	+ define tn::nat where "tn = nat t"
	+ define un::nat where "un = nat u"
	+ define vn::nat where "vn = nat v"
	+ have "sn = s" using stuv sn_def by auto
	+ hence "m * sn * (sn-1) = m * s * (s-1)" by fastforce
	+ hence "m * sn * (sn-1) div 2 = m * s * (s-1) div 2" by linarith
	+ have "tn = t" using stuv tn_def by auto
	+ hence "m * tn * (tn-1) = m * t * (t-1)" by fastforce
	+ hence "m * tn * (tn-1) div 2 = m * t * (t-1) div 2" by linarith
	+ have "un = u" using stuv un_def by auto
	+ hence "m * un * (un-1) = m * u * (u-1)" by fastforce
	+ hence "m * un * (un-1) div 2 = m * u * (u-1) div 2" by linarith
	+ have "vn = v" using stuv vn_def by auto
	+ hence "m * vn * (vn-1) = m * v * (v-1)" by fastforce
	+ hence "m * vn * (vn-1) div 2 = m * v * (v-1) div 2" by linarith
	+ have "N = m * sn * (sn-1) div 2 + sn + m * tn * (tn-1) div 2 + tn + m * un * (un-1) div 2 + un + m * vn * (vn-1) div 2 + vn + 1"
	+ using N_expr3 \<open>sn = s\<close> \<open>tn = t\<close> \<open>un = u\<close> \<open>vn = v\<close> \<open>m * sn * (sn-1) div 2 = m * s * (s-1) div 2\<close> \<open>m * tn * (tn-1) div 2 = m * t * (t-1) div 2\<close> \<open>m * un * (un-1) div 2 = m * u * (u-1) div 2\<close> \<open>m * vn * (vn-1) div 2 = m * v * (v-1) div 2\<close> by linarith
	+ hence "N = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn + 1"
	+ using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger
	+ also have "polygonal_number m 1 = 1" using Polygonal_Number_Theorem_Gauss.polygonal_number_def by auto
	+ ultimately have "N = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn + polygonal_number m 1" by auto
	+ thus ?thesis by blast
	+ qed
	+ show ?thesis using thm_odd_n thm_even_n by blast
	+ qed
	+qed
	+end
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Lemmas.thy b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Lemmas.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/Polygonal_Number_Theorem_Lemmas.thy
	@@ -0,0 +1,724 @@
	+section \<open> Technical Lemmas \<close>
	+text \<open>We show three lemmas used in the proof of both main theorems.\<close>
	+
	+theory Polygonal_Number_Theorem_Lemmas
	+ imports "Three_Squares.Three_Squares"
	+
	+begin
	+
	+subsection \<open> Lemma 1.10 in \cite{nathanson1996} \<close>
	+text \<open>This lemma is split into two parts. We modify the proof given in \cite{nathanson1996} slightly as we require the second result to hold for $l=2$ in the proof of Legendre's polygonal number theorem.\<close>
	+
	+theorem interval_length_greater_than_four:
	+ fixes m N L :: real
	+ assumes "m \<ge> 3"
	+ assumes "N \<ge> 2*m"
	+ assumes "L = (2/3 + sqrt (8N/m - 8)) - (1/2 + sqrt (6N/m - 3))"
	+ shows "N \<ge> 108*m \<Longrightarrow> L > 4"
	+
	+proof -
	+ assume asm: "N \<ge> 108*m"
	+ show "L > 4"
	+ proof -
	+ define x :: real where "x = N / m"
	+ define l :: real where "l = 4"
	+ define l_0 :: real where "l_0 = 4 - 1/6"
	+ have 0: "x \<ge> 2" unfolding x_def using assms(2)
	+ by (metis assms(1) divide_right_mono dual_order.trans linorder_le_cases mult.commute mult_1 nonzero_mult_div_cancel_left numeral_le_one_iff semiring_norm(70) zero_le_square)
	+ have 1: "L = sqrt (8x - 8) - sqrt (6x - 3) + 1/6" by (auto simp add: x_def assms(3))
	+ hence 2: "L > l \<longleftrightarrow> sqrt (8x - 8) > sqrt (6x - 3) + l_0" unfolding l_0_def l_def by auto
	+ have 3: "sqrt (8x - 8) > sqrt (6x - 3) + l_0 \<longleftrightarrow> 2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)"
	+ proof
	+ assume "sqrt (8x - 8) > sqrt (6x - 3) + l_0"
	+ hence "(sqrt (8x - 8))^2 > (sqrt (6x - 3) + l_0)^2"
	+ using l_0_def asm by (smt (verit, ccfv_SIG) "0" divide_less_eq_1_pos one_power2 pos2 power_mono_iff real_less_rsqrt)
	+ hence "8x - 8 > 6x - 3 + l_0^2 + 2l_0 sqrt (6*x - 3)"
	+ by (smt (verit, del_insts) "0" power2_sum real_sqrt_pow2_iff)
	+ thus "2x - l_0^2 - 5 > 2l_0* sqrt (6*x - 3)" by auto
	+ next
	+ assume "2x - l_0^2 - 5 > 2l_0* sqrt (6*x - 3)"
	+ hence "8x - 8 > 6x - 3 + l_0^2 + 2l_0 sqrt (6*x - 3)" by auto
	+ hence "(sqrt (8x - 8))^2 > (sqrt (6x - 3) + l_0)^2"
	+ by (smt (verit, best) 0 power2_sum real_sqrt_pow2_iff)
	+ thus "sqrt (8x - 8) > sqrt (6x - 3) + l_0"
	+ using 0 real_less_rsqrt by force
	+ qed
	+ have "2x - l_0^2 - 5 > 2l_0* sqrt (6x - 3) \<longleftrightarrow> 4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12*l_0^2 > 0"
	+ proof
	+ assume "2x - l_0^2 - 5 > 2l_0* sqrt (6*x - 3)"
	+ hence "(2x - l_0^2 - 5)^2 > (2l_0* sqrt (6*x - 3))^2"
	+ by (smt (verit, del_insts) "0" asm l_0_def le_divide_eq_1_pos less_1_mult one_power2 pos2 power_mono_iff sqrt_le_D)
	+ thus "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ using 0 by (simp add: algebra_simps power2_eq_square power4_eq_xxxx)
	+ next
	+ assume "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ hence "(2x - l_0^2 - 5)^2 > (2l_0* sqrt (6*x - 3))^2"
	+ using 0 by (simp add: algebra_simps power2_eq_square power4_eq_xxxx)
	+ from assms(1) have "m > 0" by auto
	+ hence "2x \<ge> 2108"
	+ using x_def asm by (simp add: le_divide_eq)
	+ hence "2x - l_0^2 - 5 \<ge> 2108 - (4-1/6)*(4-1/6) - 5" unfolding l_0_def by (auto simp add: power2_eq_square)
	+ hence "2*x - l_0^2 - 5 > 0" by auto
	+ thus "2x - l_0^2 - 5 > 2l_0* sqrt (6*x - 3)"
	+ using \<open>(2x - l_0^2 - 5)^2 > (2l_0* sqrt (6*x - 3))^2\<close> using power2_less_imp_less by fastforce
	+ qed
	+ from assms(1) have "m > 0" by auto
	+ hence "x \<ge> 108" using x_def asm by (simp add: le_divide_eq)
	+ have "7(4-1/6)(4-1/6) + 5 < (108::real)" by simp
	+ hence "7*l_0^2 + 5 < 108" unfolding l_0_def by (auto simp add: power2_eq_square)
	+ hence "x \<ge> 7*l_0^2 + 5" using \<open>108 \<le> x\<close> by auto
	+ hence "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ by (smt (verit) mult_nonneg_nonneg power2_less_eq_zero_iff zero_le_power2)
	+ thus ?thesis
	+ using "2" "3" \<open>(2 * l_0 * sqrt (6 * x - 3) < 2 * x - l_0\<^sup>2 - 5) = (0 < 4 * x * (x - (7 * l_0\<^sup>2 + 5)) + (l_0\<^sup>2 + 5)\<^sup>2 + 12 * l_0\<^sup>2)\<close> l_def by blast
	+ qed
	+ qed
	+
	+theorem interval_length_greater_than_lm:
	+ fixes m N :: real
	+ fixes L l :: real
	+ assumes "m \<ge> 3"
	+ assumes "N \<ge> 2*m"
	+ assumes "L = (2/3 + sqrt (8N/m - 8)) - (1/2 + sqrt (6N/m - 3))"
	+ shows "l \<ge> 2 \<and> N \<ge> 7l^2m^3 \<Longrightarrow> L > l*m"
	+
	+proof -
	+ assume asm: "l \<ge> 2 \<and> N \<ge> 7l^2m^3"
	+ show "L > l*m"
	+ proof -
	+ from asm have asm1: "l \<ge> 2" and asm2: "N \<ge> 7l^2m^3" by auto
	+ define x :: real where "x = N / m"
	+ define l_0 :: real where "l_0 = l*m - 1/6"
	+ have "l_0 > 0" using l_0_def asm1 assms(1)
	+ by (smt (verit, ccfv_threshold) le_divide_eq_1 mult_le_cancel_left2 of_int_le_1_iff)
	+ have 0: "x \<ge> 2" using x_def assms(1,2) by (simp add: pos_le_divide_eq)
	+ have 1: "L = sqrt (8x - 8) - sqrt (6x - 3) + 1/6" by (auto simp add: x_def assms(3))
	+ hence 2: "L > lm \<longleftrightarrow> sqrt (8x - 8) > sqrt (6*x - 3) + l_0" by (auto simp add: l_0_def)
	+ have 3: "sqrt (8x - 8) > sqrt (6x - 3) + l_0 \<longleftrightarrow> 2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)"
	+ proof
	+ assume "sqrt (8x - 8) > sqrt (6x - 3) + l_0"
	+ hence "(sqrt (8x - 8))^2 > (sqrt (6x - 3) + l_0)^2"
	+ using l_0_def asm1 by (smt (verit, best) \<open>0 < l_0\<close> real_le_lsqrt real_sqrt_four real_sqrt_less_iff real_sqrt_pow2_iff)
	+ hence "8x - 8 > 6x - 3 + l_0^2 + 2* l_0 * sqrt (6*x - 3)"
	+ by (smt (verit, del_insts) "0" power2_sum real_sqrt_pow2_iff)
	+ thus "2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)" by auto
	+ next
	+ assume "2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)"
	+ hence "8x - 8 > 6x - 3 + l_0^2 + 2l_0 sqrt (6*x - 3)" by auto
	+ hence "(sqrt (8x - 8))^2 > (sqrt (6x - 3) + l_0)^2"
	+ by (smt (verit, del_insts) "0" power2_sum real_sqrt_pow2_iff)
	+ thus "sqrt (8x - 8) > sqrt (6x - 3) + l_0"
	+ using 0 real_less_rsqrt by force
	+ qed
	+ have "2x - l_0^2 - 5 > 2l_0 * sqrt (6x - 3) \<longleftrightarrow> 4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12*l_0^2 > 0"
	+ proof
	+ assume "2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)"
	+ have "(2x - l_0^2 - 5)^2 > (2l_0 * sqrt (6*x - 3))^2"
	+ using \<open>0 < l_0\<close> by (smt (verit, ccfv_SIG) 0 \<open>2 * l_0 * sqrt (6 * x - 3) < 2 * x - l_0\<^sup>2 - 5\<close> pos2 power_strict_mono real_sqrt_ge_zero zero_le_mult_iff)
	+ thus "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ using 0 by (simp add: algebra_simps power2_eq_square power4_eq_xxxx)
	+ next
	+ assume "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ hence "(2x - l_0^2 - 5)^2 > (2l_0 * sqrt (6*x - 3))^2"
	+ using 0 by (simp add: algebra_simps power2_eq_square power4_eq_xxxx)
	+ have "m > 0" using assms(1) by simp
	+ hence "x \<ge> 7l^2m^2"
	+ unfolding x_def using asm2 assms(1)
	+ by (simp add: mult_imp_le_div_pos power2_eq_square power3_eq_cube)
	+ hence 4: "2x - l_0^2 - 5 \<ge> 14l^2m^2 - (lm-1/6)^2 - 5"
	+ by (simp add: x_def l_0_def power2_eq_square)
	+ have "(lm-(1/6::real))^2 = (lm)^2 - l*m/3 + (1/36::real)"
	+ apply (simp add: power2_eq_square)
	+ by argo
	+ hence "14l^2m^2 - (lm-1/6)^2 - 5 = 14l^2m^2 - l^2m^2 + l*m/3 - 1/36 - 5"
	+ using 4 by (auto simp add: power2_eq_square)
	+ hence "14l^2m^2 - l^2m^2 + lm/3 - 1/36 - 5 = 13l^2m^2 + l*m/3 - 1/36 - 5" by argo
	+ from asm1 assms(1) have 5: "l*m/3 > 0" by simp
	+ have "l > 0" using asm1 by auto
	+ hence "ll \<ge> 22" using asm1 mult_mono' zero_le_numeral by blast
	+ have "m > 0" using assms(1) by auto
	+ hence "mm \<ge> 33"
	+ by (metis assms(1) less_eq_real_def mult_le_less_imp_less zero_less_numeral)
	+ hence "13mm - 1 \<ge> 1333-1" by simp
	+ have "3*3 > (0::real)" by auto
	+ hence "13llmm \<ge>(13::real)2233" using \<open>ll \<ge> 22\<close> asm1
	+ by (meson \<open>0 < l\<close> \<open>0 < m\<close> assms(1) less_eq_real_def mult_mono split_mult_pos_le zero_le_numeral)
	+ hence "13l^2m^2 + lm/3 - 1/36 - 5 \<ge> 13223*3-1/36-(5::real)"
	+ using 5 by (auto simp add: power2_eq_square)
	+ have "133333-1/36-(5::real) > 0" by auto
	+ hence "2*x - l_0^2 - 5 > 0"
	+ using "4" \<open>13 * 2 * 2 * 3 * 3 - 1 / 36 - 5 \<le> 13 * l\<^sup>2 * m\<^sup>2 + l * m / 3 - 1 / 36 - 5\<close> \<open>14 * l\<^sup>2 * m\<^sup>2 - (l * m - 1 / 6)\<^sup>2 - 5 = 14 * l\<^sup>2 * m\<^sup>2 - l\<^sup>2 * m\<^sup>2 + l * m / 3 - 1 / 36 - 5\<close> by force
	+ thus "2x - l_0^2 - 5 > 2l_0 * sqrt (6*x - 3)"
	+ by (smt (verit) \<open>(2 * l_0 * sqrt (6 * x - 3))\<^sup>2 < (2 * x - l_0\<^sup>2 - 5)\<^sup>2\<close> power_mono)
	+ qed
	+ have "(1/6)^2 * (36::real) = 1" by (auto simp add: power2_eq_square)
	+ from assms(1) have "m > 0" by auto
	+ hence "x \<ge> 7l^2m^2" unfolding x_def using asm2
	+ by (simp add: pos_le_divide_eq power2_eq_square power3_eq_cube)
	+ from asm1 have "l > 0" by auto
	+ from assms(1) asm1 \<open>m > 0\<close> \<open>l > 0\<close> have "lm \<ge> 2(3::real)"
	+ by (metis mult_less_cancel_right mult_mono verit_comp_simplify1(1) verit_comp_simplify1(3) zero_le_numeral)
	+ hence "-27lm/6 + 7(1/6)*(1/6) + 5 < (0::real)" by simp
	+ hence "7l^2m^2 > 7*l_0^2 + (5::real)" unfolding l_0_def
	+ apply (auto simp add: power2_eq_square)
	+ by argo
	+ hence "x \<ge> 7*l_0^2 + 5"
	+ using \<open>7 * l\<^sup>2 * m\<^sup>2 \<le> x\<close> by linarith
	+ hence "4x(x - (7l_0^2 + 5)) + (l_0^2 + 5)^2 + 12l_0^2 > 0"
	+ by (smt (verit) mult_nonneg_nonneg power2_less_eq_zero_iff zero_le_power2)
	+ thus ?thesis
	+ using "2" "3" \<open>(2 * l_0 * sqrt (6 * x - 3) < 2 * x - l_0\<^sup>2 - 5) = (0 < 4 * x * (x - (7 * l_0\<^sup>2 + 5)) + (l_0\<^sup>2 + 5)\<^sup>2 + 12 * l_0\<^sup>2)\<close> by fastforce
	+ qed
	+qed
	+
	+lemmas interval_length_greater_than_2m [simp] = interval_length_greater_than_lm [where l=2, simplified]
	+
	+subsection \<open> Lemma 1.11 in \cite{nathanson1996} \<close>
	+
	+text \<open> We show Lemma 1.11 in \cite{nathanson1996} which is also known as Cauchy's Lemma.\<close>
	+
	+theorem Cauchy_lemma:
	+ fixes m N a b r :: real
	+ assumes "m \<ge> 3" "N \<ge> 2*m"
	+ and "0 \<le> a" "0 \<le> b" "0 \<le> r" "r < m"
	+ and "N = m*(a - b)/2 + b + r"
	+ and "1/2 + sqrt (6N/m - 3) \<le> b \<and> b \<le> 2/3 + sqrt (8N/m - 8)"
	+ shows "b^2 < 4a \<and> 3a < b^2 + 2*b + 4"
	+
	+proof-
	+ from assms have asm1: "1/2 + sqrt (6N/m - 3) \<le> b" and asm2: "b \<le> 2/3 + sqrt (8N/m - 8)" by auto
	+ have "N - b - r = m*(a - b)/2" using assms(7) by simp
	+ hence "a = (N - b - r)*2/m + b" using assms(1) by simp
	+ hence "a = b - 2/m * b + 2 * (N - r)/m"
	+ apply (simp add: algebra_simps)
	+ by (smt (verit, del_insts) add_divide_distrib)
	+ hence a: "a = b(1 - 2/m) + 2(N - r)/m"
	+ by (simp add: right_diff_distrib')
	+ have "b^2 < 4*a"
	+ proof -
	+ from a have 0: "b^2 - 4a = b^2 - 4(1 - 2/m)b - 8(N-r)/m" by simp
	+ have "3/m \<le> 1" using assms(1) by simp
	+ hence 1: "2/3 \<le> 2*(1 - 2/m)" by simp
	+ have "N/m - 1 < N/m - r/m" using assms(1,6) by simp
	+ hence "sqrt(8(N/m - 1)) < sqrt (8((N - r)/m))" by (simp add: diff_divide_distrib)
	+ hence 2: "sqrt(8N/m - 8) < sqrt (8((N - r)/m))" by simp
	+ have "2/3 + sqrt (8N/m - 8) < 2(1 - 2/m) + sqrt (8*((N-r)/m))"
	+ using 1 2 by linarith
	+ hence "b < 2(1-2/m) + sqrt (8(N - r)/m)" using asm2 by simp
	+ hence 3: "b < 2(1-2/m) + sqrt (4(1-2/m)^2 + 8*(N - r)/m)"
	+ by (smt (verit, best) power2_less_0 real_sqrt_le_iff)
	+ define r1 where "r1 = 2(1-2/m) - sqrt (4(1-2/m)^2 + 8*(N - r)/m)"
	+ define r2 where "r2 = 2(1-2/m) + sqrt (4(1-2/m)^2 + 8*(N - r)/m)"
	+
	+ have "r1r2 = (2(1-2/m) - sqrt (4(1-2/m)^2 + 8(N - r)/m))(2(1-2/m) + sqrt (4(1-2/m)^2 + 8(N - r)/m))"
	+ using r1_def r2_def by simp
	+ hence "r1r2 = 2(1-2/m)(2(1-2/m) + sqrt (4(1-2/m)^2 + 8(N - r)/m)) -
	+sqrt (4(1-2/m)^2 + 8(N - r)/m)(2(1-2/m) + sqrt (4(1-2/m)^2 + 8(N - r)/m))"
	+ by (simp add: Rings.ring_distribs(3))
	+ hence "r1r2 = (2(1-2/m))^2+2(1-2/m) sqrt (4(1-2/m)^2 + 8(N - r)/m) - 2(1-2/m) sqrt (4(1-2/m)^2 + 8(N - r)/m)
	+- (sqrt (4(1-2/m)^2 + 8(N - r)/m))^2" by (simp add: distrib_left power2_eq_square)
	+ hence "r1r2 = (2(1-2/m))^2 -(sqrt (4(1-2/m)^2 + 8(N - r)/m))^2" by simp
	+
	+ hence "r1 * r2 = 4(1-2/m)^2 - 4(1-2/m)^2 - 8*(N - r)/m"
	+ using assms(1) assms(2) assms(6) four_x_squared
	+ by (smt (verit) divide_nonneg_nonneg real_sqrt_pow2_iff zero_compare_simps(12))
	+ hence r1_times_r2:"r1r2 = -8(N-r)/m" by linarith
	+
	+ have "(b-r1)(b-r2) = b(b-r2) - r1*(b-r2)" using cross3_simps(28) by blast
	+ hence "(b-r1)(b-r2) = b^2-br2-br1+r1r2" by (simp add: power2_eq_square right_diff_distrib)
	+ hence "(b-r1)(b-r2) = b^2-b(2(1-2/m) + sqrt (4(1-2/m)^2 + 8(N - r)/m))-b(2(1-2/m) - sqrt (4(1-2/m)^2 + 8(N - r)/m))+r1r2"
	+ using r1_def r2_def by simp
	+ hence "(b-r1)(b-r2) = b^2-b2(1-2/m)-b sqrt (4(1-2/m)^2 + 8(N - r)/m)-b(2(1-2/m) - sqrt (4(1-2/m)^2 + 8(N - r)/m)) +r1*r2"
	+ by (simp add: distrib_left)
	+ hence "(b-r1)(b-r2) = b^2-b2(1-2/m)-b sqrt (4(1-2/m)^2 + 8(N - r)/m)-b2(1-2/m)+b* sqrt (4(1-2/m)^2 + 8(N - r)/m)+r1*r2"
	+ by (simp add: Rings.ring_distribs(4))
	+ hence "(b-r1)(b-r2) = b^2-b4(1-2/m)+r1r2" by simp
	+ hence "(b-r1)(b-r2) = b^2 - 4(1 - 2/m)b - 8(N-r)/m" using r1_times_r2
	+ by (simp add: \<open>r1 * r2 = 4 * (1 - 2 / m)\<^sup>2 - 4 * (1 - 2 / m)\<^sup>2 - 8 * (N - r) / m\<close>)
	+ hence "b^2 - 4(1 - 2/m)b - 8*(N-r)/m < 0" using 3 assms(4)
	+ by (smt (verit, del_insts) \<open>r1 * r2 = 4 * (1 - 2 / m)\<^sup>2 - 4 * (1 - 2 / m)\<^sup>2 - 8 * (N - r) / m\<close> assms(1) assms(2) assms(6) divide_pos_pos mult_nonneg_nonneg mult_pos_neg r2_def)
	+ thus ?thesis using 0 by simp
	+ qed
	+ have "3a < b^2 + 2b + 4"
	+ proof -
	+ from a have 4: "b^2 + 2b + 4 - 3a = b^2 - (1-6/m)b - (6(N-r)/m - 4)" by argo
	+ have 5: "1/2 > 1/2 - 3/m" using assms(1) by simp
	+ hence "1/2 - 3/m < 1" by linarith
	+ also have "1/2 - 3/m > -1" using assms(1)
	+ by (smt (verit) divide_le_0_1_iff less_divide_eq_1_pos)
	+ hence "(1/2 - 3/m)^2 < 1"
	+ by (metis (no_types, opaque_lifting) calculation less_eq_real_def power2_eq_1_iff square_le_1 verit_comp_simplify1(3))
	+ hence 6: "sqrt (6N/m - 3) > sqrt ((1/2 - 3/m)^2 + 6N/m - 4)" using assms(1) by simp
	+ from asm1 5 6 have "b > (1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6*N/m - 4)" by linarith
	+ hence 7: "b > (1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4)"
	+ by (smt (verit, ccfv_SIG) assms(1) assms(5) divide_right_mono real_sqrt_le_mono)
	+ define s1 where "s1 = (1/2 - 3/m) - sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4)"
	+ define s2 where "s2 = (1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4)"
	+ have "s1* s2=(1/2-3/m)((1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4))-
	+sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4)((1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4))"
	+ using s1_def s2_def Rings.ring_distribs(3) by blast
	+ hence "s1* s2= (1/2-3/m)^2+(1/2-3/m)* sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4)-
	+sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4)((1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4))"
	+ by (simp add: nat_distrib(2) power2_eq_square)
	+ hence "s1* s2= (1/2-3/m)^2+(1/2-3/m)* sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4)-
	+(1/2-3/m)* sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4) - (sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4))^2"
	+ by (smt (verit, ccfv_SIG) Groups.mult_ac(2) Rings.ring_distribs(3) power2_eq_square)
	+ hence 8:"s1* s2=(1/2-3/m)^2- (sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4))^2" by simp
	+ from assms(1,6) have "-r/m > -1" by simp
	+ hence "-6*r/m > -6" by simp
	+ hence "12 - 4 - 6*r/m > 0" by simp
	+ hence "12m/m - 6r/m - 4 > 0" using assms(1) by simp
	+ hence "6(2m - r)/m - 4 > 0" by argo
	+ hence "6*(N - r)/m - 4 > 0" using assms(1,2)
	+ by (smt (verit, best) divide_right_mono)
	+ hence "s1 * s2 = (1/2 - 3/m)^2 - (1/2 - 3/m)^2 - 6*(N - r)/m + 4" using 8
	+ by (smt (verit) real_sqrt_pow2_iff zero_le_power2)
	+
	+ have "(b-s1)(b-s2) = b(b-s2) - s1*(b-s2)" using cross3_simps(28) by blast
	+ hence "(b-s1)(b-s2) = b^2-b s2-b* s1+s1* s2" by (simp add: power2_eq_square right_diff_distrib)
	+ hence "(b-s1)(b-s2) = b^2-b((1/2 - 3/m) + sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4))-b((1/2 - 3/m) - sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4))
	++ s1* s2" using s1_def s2_def by simp
	+ hence "(b-s1)(b-s2) = b^2-b(1/2 - 3/m)-b* sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4)-b((1/2 - 3/m) - sqrt ((1/2 - 3/m)^2 + 6*(N - r)/m - 4))
	++ s1* s2" by (simp add: nat_distrib(2))
	+ hence "(b-s1)(b-s2) = b^2-b(1/2 - 3/m)-b* sqrt ((1/2 - 3/m)^2 + 6(N - r)/m - 4)-b(1/2 - 3/m)+b* sqrt ((1/2 - 3/m)^2 + 6(N - r)/m-4)+s1 s2"
	+ by (smt (verit, ccfv_SIG) nat_distrib(2))
	+ hence "(b-s1)(b-s2) = b^2-2b(1/2 - 3/m)+s1 s2" by simp
	+ hence "(b-s1)(b-s2) = b^2-2b(1/2 - 3/m)+ (1/2 - 3/m)^2 - (1/2 - 3/m)^2 - 6(N - r)/m + 4"
	+ using \<open>s1 * s2 = (1 / 2 - 3 / m)\<^sup>2 - (1 / 2 - 3 / m)\<^sup>2 - 6 * (N - r) / m + 4\<close> by fastforce
	+ hence "(b-s1)(b-s2) = b^2-2b(1/2-3/m)- 6(N - r)/m + 4" by simp
	+ hence "(b-s1)(b-s2) = b^2-b(1-6/m)- 6*(N - r)/m + 4" by simp
	+ hence "(b-s1)(b-s2) = b^2-b(1-6/m)- (6*(N - r)/m - 4)" by simp
	+ hence "b^2 - (1-6/m)b - (6(N-r)/m - 4) > 0" using 7 by (smt (verit, del_insts) "8" Groups.mult_ac(2)
	+\<open>s1 * s2 = (1 / 2 - 3 / m)\<^sup>2 - (1 / 2 - 3 / m)\<^sup>2 - 6 * (N - r) / m + 4\<close> real_sqrt_ge_0_iff s1_def s2_def zero_compare_simps(8) zero_le_power2)
	+ thus ?thesis using 4 by simp
	+ qed
	+ show ?thesis by (simp add: \<open>3 * a < b\<^sup>2 + 2 * b + 4\<close> \<open>b\<^sup>2 < 4 * a\<close>)
	+qed
	+
	+lemmas Cauchy_lemma_r_eq_zero = Cauchy_lemma [where r=0, simplified]
	+
	+subsection \<open> Lemma 1.12 in \cite{nathanson1996} \<close>
	+
	+lemma not_one:
	+ fixes a b :: nat
	+ assumes "a\<ge>1"
	+ assumes "b\<ge>1"
	+ assumes "\<exists>k1 :: nat. a = 2*k1+1"
	+ assumes "\<exists>k2 :: nat. b = 2*k2+1"
	+ assumes "b^2 < 4*a"
	+ shows "4*a-b^2 \<noteq> 1"
	+
	+proof
	+ assume "4*a-b^2 = 1"
	+ hence "b^2 = 4*a-1" by auto
	+ hence "b^2 mod 4 = (4*a-1) mod 4" by auto
	+ have "(4*a-1) mod 4 = 3 mod 4" using assms(1) by (simp add: mod_diff_eq_nat)
	+ hence "b^2 mod 4 = 3" using \<open>b^2 = 4*a-1\<close> mod_less by presburger
	+ thus "False" using assms by (metis One_nat_def eq_numeral_Suc insert_iff nat.simps(3)
	+ power_two_mod_four pred_numeral_simps(3) singletonD)
	+qed
	+
	+
	+lemma not_two:
	+ fixes a b :: nat
	+ assumes "a\<ge>1"
	+ assumes "b\<ge>1"
	+ assumes "\<exists>k1 :: nat. a = 2*k1+1"
	+ assumes 1:"\<exists>k2 :: nat. b = 2*k2+1"
	+ assumes "b^2 < 4*a"
	+ shows "4*a-b^2 \<noteq> 2"
	+
	+proof
	+ assume "4*a-b^2=2"
	+ hence "b^2=4*a-2" by auto
	+ from 1 have 2: "\<not> 2 dvd b^2" by auto
	+ have "2 dvd (4*a-2)" by auto
	+ thus "False" using \<open>b^2=4*a-2\<close> 2 by auto
	+qed
	+
	+text \<open>The following lemma shows that given odd positive integers $x,y,z$ and $b$, where $x \geq y \geq z$, we may pick
	+a suitable integer $u$ where $u = z$ or $u = -z$, such that $b+x+y+u \equiv 0 \pmod{4}$.\<close>
	+
	+lemma suit_z:
	+ fixes b x y z :: nat
	+ assumes "odd b \<and> odd x \<and> odd y \<and> odd z"
	+ assumes "x\<ge>y \<and> y\<ge>z"
	+ shows "\<exists> u :: int. (u=z \<or> u=-z) \<and> (b+x+y+u) mod 4 = 0"
	+
	+proof -
	+ from assms have 0: "(b+x+y) mod 4 = 1 \<or> (b+x+y) mod 4 = 3" by (metis dvd_refl even_add
	+even_even_mod_4_iff landau_product_preprocess(53) mod_exhaust_less_4)
	+ from assms have 1: "z mod 4 = 1 \<or> z mod 4 = 3" by (metis dvd_0_right dvd_refl
	+ even_even_mod_4_iff mod_exhaust_less_4)
	+
	+ have c1:"\<exists>u1::int. (u1=z \<or> u1=-z) \<and> (b+x+y+u1) mod 4 = 0"
	+ if asm1:"(b+x+y) mod 4 = 1 \<and> z mod 4 = 3"
	+ proof -
	+ from asm1 have 2:"(b+x+y+z) mod 4 = 0" by (metis add_num_simps(1) add_num_simps(7)
	+ mod_add_eq mod_self numeral_plus_one one_plus_numeral_commute)
	+ define u1 :: int where "u1=z"
	+ show "\<exists>u1::int. (u1=z \<or> u1=-z) \<and> (b+x+y+u1) mod 4 = 0" using "2" u1_def
	+ by (metis Num.of_nat_simps(4) of_nat_0 of_nat_numeral zmod_int)
	+ qed
	+
	+ have c2:"\<exists>u2::int.(u2=z \<or> u2=-z) \<and> (b+x+y+u2) mod 4 = 0"
	+ if asm2:"(b+x+y) mod 4 = 1 \<and> z mod 4 = 1"
	+ proof -
	+ from asm2 have 3:"(b+x+y-z) mod 4 = 0"
	+ by (metis assms(2) mod_eq_0_iff_dvd mod_eq_dvd_iff_nat trans_le_add2)
	+ define u2::int where "u2=-z"
	+ show "\<exists>u2::int.(u2=z \<or> u2=-z) \<and> (b+x+y+u2) mod 4 = 0" using "3" u2_def
	+ by (metis Num.of_nat_simps(2) asm2 mod_0
	+ mod_add_cong more_arith_simps(4) of_nat_numeral zmod_int)
	+ qed
	+
	+ have c3:"\<exists>u3::int.(u3=z \<or> u3=-z) \<and> (b+x+y+u3) mod 4 = 0"
	+ if asm3:"(b+x+y) mod 4 = 3 \<and> z mod 4 = 1"
	+ proof -
	+ from asm3 have 4: "(b+x+y+z) mod 4 = 0" by (metis add_num_simps(1) add_num_simps(7)
	+ mod_add_eq mod_self numeral_plus_one)
	+ define u3::int where "u3=z"
	+ show "\<exists>u3::int.(u3=z \<or> u3=-z) \<and> (b+x+y+u3) mod 4 = 0" using "4" u3_def
	+ by (metis Num.of_nat_simps(4) of_nat_0 of_nat_numeral zmod_int)
	+ qed
	+
	+ have c4:"\<exists>u4::int.(u4=z \<or> u4=-z) \<and> (b+x+y+u4) mod 4 = 0"
	+ if asm4:"(b+x+y) mod 4 = 3 \<and> z mod 4 = 3"
	+ proof -
	+ from asm4 have 5: "(b+x+y-z) mod 4 = 0"
	+ by (metis assms(2) mod_eq_0_iff_dvd mod_eq_dvd_iff_nat trans_le_add2)
	+ define u4::int where "u4=-z"
	+ show "\<exists>u4::int.(u4=z \<or> u4=-z) \<and> (b+x+y+u4) mod 4 = 0" using "5" u4_def by (metis asm4 mod_0
	+ mod_add_cong more_arith_simps(4) of_nat_numeral zmod_int)
	+ qed
	+
	+ show ?thesis using assms 0 1 c1 c2 c3 c4 by auto
	+qed
	+
	+lemma four_terms_bin_exp_allsum:
	+ fixes b s t u v :: int
	+ assumes "b = s+t+u+v"
	+ shows "b^2 = t^2+u^2+s^2+v^2+2tu+2 * s * v + 2t s + 2t v +2u s +2u v"
	+
	+proof -
	+ from assms have "b^2 = (t+u)^2+(s+v)^2+2(t+u)(s+v)" by (smt (verit, best) power2_sum)
	+ hence b_simp1:"b^2 = (t^2+u^2+2tu) + (s^2+v^2+2 * s * v)+2(t+u)(s+v)"
	+ by (simp add: power2_sum)
	+ have "2(t+u)(s+v) = 2t s + 2t v +2u s +2u v"
	+ using int_distrib(1) int_distrib(2) by force
	+ from this b_simp1 have b_expression:"b^2 = t^2+u^2+s^2+v^2+2tu+2 * s * v +
	+2t s + 2t v +2u s +2u v" by auto
	+ thus ?thesis by auto
	+qed
	+
	+lemma four_terms_bin_exp_twodiff:
	+ fixes b s t u v :: int
	+ assumes "b = s+t-u-v"
	+ shows "b^2 = t^2+u^2+s^2+v^2-2tu-2 * s * v + 2t s - 2t v -2u s +2u v"
	+
	+proof -
	+ from assms have "b^2 = (s-u)^2+(t-v)^2+2(s-u)(t-v)" by (smt (verit, best) power2_sum)
	+ hence b_simp1:"b^2 = s^2+u^2-2 * s u + t^2+v^2 - 2 t * v + 2(s-u)(t-v)"
	+ by (simp add: power2_diff)
	+ have "2(s-u)(t-v) = 2* s * t - 2* s * v - 2ut+2u v"
	+ by (simp add: Rings.ring_distribs(3) Rings.ring_distribs(4))
	+ from this b_simp1 have b_expression: "b^2 = t^2+u^2+s^2+v^2-2tu-2 * s * v +
	+2t s - 2t v -2u s +2u v" by auto
	+ thus ?thesis by auto
	+qed
	+
	+text\<open>If a quadratic with positive leading coefficient is always non-negative, its discriminant is
	+non-positive.\<close>
	+
	+lemma qua_disc:
	+ fixes a b c :: real
	+ assumes "a>0"
	+ assumes "\<forall>x::real. ax^2+bx+c \<ge>0"
	+ shows "b^2 - 4ac \<le> 0"
	+
	+proof -
	+ from assms have 0:"\<forall>x::real. (ax^2+bx+c)/a \<ge>0" by simp
	+ from assms have 1:"\<forall>x::real.(bx+c)/a = b/ax+c/a" by (simp add: add_divide_distrib)
	+ from assms have "\<forall>x::real.(ax^2+bx+c)/a = x^2+(b*x+c)/a" by (simp add: is_num_normalize(1))
	+ from 1 this have "\<forall>x::real.(ax^2+bx+c)/a = x^2+b/a*x+c/a" by simp
	+ hence atleastzero:"\<forall>x::real. x^2+b/a*x+c/a \<ge>0" using 0 by simp
	+
	+ from assms have 2:"\<forall>x::real. x^2+b/ax+c/a = x^2+2b/(2a)x+c/a+b^2/(4a^2)-b^2/(4a^2)" by simp
	+ have simp1:"\<forall>x::real.(x+b/(2a))^2 = x^2+2b/(2a)x+(b/(2*a))^2" by (simp add: power2_sum)
	+ have "(b/(2a))^2 = b^2/(4a^2)" by (metis four_x_squared power_divide)
	+ hence "\<forall>x::real. x^2+b/ax+c/a = (x+b/(2a))^2+c/a-b^2/(4*a^2)" using 2 simp1 by auto
	+ hence "\<forall>x::real. (x+b/(2a))^2+c/a-b^2/(4a^2) \<ge>0" using atleastzero by presburger
	+ hence 3:"\<forall>x::real. b^2/(4a^2)-c/a\<le>(x+b/(2a))^2" by (smt (verit, del_insts))
	+ have "\<exists>x::real. (x+b/(2*a))^2=0" by (metis diff_add_cancel power_zero_numeral)
	+ hence "b^2/(4*a^2)-c/a\<le>0" using 3 by metis
	+ hence 4:"4a^2(b^2/(4*a^2)-c/a)\<le>0" using assms by (simp add: mult_nonneg_nonpos)
	+ have 5:"4a^2b^2/(4*a^2) = b^2" using assms by simp
	+ have 6:"4a^2c/a = 4ac" using assms by (simp add: power2_eq_square)
	+ show ?thesis using 4 5 6 assms by (simp add: Rings.ring_distribs(4))
	+qed
	+
	+text\<open>The following lemma shows for any point on a 3D sphere with radius $a$, the sum of its coordinates
	+lies between $\sqrt{3a}$ and $-\sqrt{3a}$.\<close>
	+
	+lemma three_terms_Cauchy_Schwarz:
	+ fixes x y z a :: real
	+ assumes "a > 0"
	+ assumes "x^2+y^2+z^2 = a"
	+ shows "(x+y+z)\<ge>-sqrt(3a) \<and> (x+y+z)\<le>sqrt(3a)"
	+
	+proof -
	+ have 1:"\<forall>t::real. (tx+1)^2 = t^2x^2+1+2tx" by (simp add: power2_sum power_mult_distrib)
	+ have 2:"\<forall>t::real. (ty+1)^2 = t^2y^2+1+2ty" by (simp add: power2_sum power_mult_distrib)
	+ have 3:"\<forall>t::real. (tz+1)^2 = t^2z^2+1+2tz" by (simp add: power2_sum power_mult_distrib)
	+ from 1 2 3 have 4:"\<forall>t::real.(tx+1)^2+(ty+1)^2+(tz+1)^2 = t^2x^2+1+2tx + t^2y^2+1+2t*y +
	+t^2z^2+1+2t*z" by auto
	+
	+ have "\<forall>t::real. t^2x^2+t^2y^2=t^2*(x^2+y^2)" by (simp add: nat_distrib(2))
	+ hence 5:"\<forall>t::real. t^2x^2+t^2y^2+t^2z^2=t^2(x^2+y^2+z^2)" by (metis nat_distrib(2))
	+ have 6:"\<forall>t::real. 2tx+2ty+2tz = t2(x+y+z)" by (simp add: Groups.mult_ac(2) distrib_right)
	+ from 4 5 6 have "\<forall>t::real.(tx+1)^2+(ty+1)^2+(tz+1)^2 = t^2(x^2+y^2+z^2)+ t2(x+y+z)+3"
	+ by (smt (verit, best))
	+ hence "\<forall>t::real. t^2(x^2+y^2+z^2)+ t2*(x+y+z)+3 \<ge>0" by (metis add_nonneg_nonneg zero_le_power2)
	+ hence "(2(x+y+z))^2 - 12(x^2+y^2+z^2)\<le>0" using qua_disc
	+ by (smt (z3) power2_diff power2_sum power_zero_numeral sum_squares_bound)
	+ hence "12(x^2+y^2+z^2)\<ge>4(x+y+z)^2" by (simp add: four_x_squared)
	+ hence "3*a\<ge>(x+y+z)^2" using assms by auto
	+ thus ?thesis by (smt (verit, del_insts) real_sqrt_abs real_sqrt_le_iff)
	+qed
	+
	+text\<open>We adapt the lemma above through changing the types for the convenience of our proof.\<close>
	+
	+lemma three_terms_Cauchy_Schwarz_nat_ver:
	+ fixes x y z a :: nat
	+ assumes "a>0"
	+ assumes "x^2+y^2+z^2 = a"
	+ shows "(x+y+z)\<ge>-sqrt(3a) \<and> (x+y+z)\<le>sqrt(3a)"
	+
	+proof -
	+ have fac1:"real(x+y+z) = real x + real y + real z" by auto
	+ have fac2: "3(real a) = real(3a)" by auto
	+ thus ?thesis using fac1 three_terms_Cauchy_Schwarz fac2 by (smt (verit) assms(1) assms(2) nat_less_real_le of_nat_0_le_iff of_nat_add of_nat_power)
	+qed
	+
	+text\<open>This theorem is Lemma 1.12 in \cite{nathanson1996}, which shows for odd positive integers $a$ and $b$
	+satisfying certain properties, there exist four non-negative integers $s,t,u$ and $v$ such that
	+$a = s^2+t^2+u^2+v^2$ and $b = s+t+u+v$. We use the Three Squares Theorem AFP entry \cite{Three_Squares-AFP}.\<close>
	+
	+theorem four_nonneg_int_sum:
	+ fixes a b :: nat
	+ assumes "a\<ge>1"
	+ assumes "b\<ge>1"
	+ assumes "odd a"
	+ assumes "odd b"
	+ assumes 3:"b^2 < 4*a"
	+ assumes "3a < b^2+2b+4"
	+ shows "\<exists>s t u v :: int. s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> a = s^2 + t^2 + u^2 + v^2 \<and>
	+ b = s+t+u+v"
	+
	+proof -
	+ from assms have 0:"\<exists>k1 :: nat. a = 2*k1+1" by (meson oddE)
	+ from assms have 1:"\<exists>k2 :: nat. b = 2*k2+1" by (meson oddE)
	+ from 0 have "4*a mod 8 = 4" by auto
	+ hence 2:"8 dvd (4*a-4)" by (metis dvd_minus_mod)
	+
	+ obtain k2 where "b = 2*k2+1" using 1 by auto
	+ have "2 dvd k2*(k2+1)" by auto
	+ hence "8 dvd 4k2(k2+1)" by (metis ab_semigroup_mult_class.mult_ac(1)
	+ mult_2_right nat_mult_dvd_cancel_disj numeral_Bit0)
	+ hence "b^2 mod 8 = 1" using 1 by (metis One_nat_def Suc_0_mod_numeral(2) assms(4)
	+ square_mod_8_eq_1_iff unique_euclidean_semiring_class.cong_def)
	+ hence "8 dvd (b^2-1)" by (metis dvd_minus_mod)
	+ from 2 this have "8 dvd ((4*a-4)-(b^2-1))" using dvd_diff_nat by blast
	+ from assms 0 1 and this have 7:"8 dvd ((4*a-b^2)-3)" by auto
	+ from assms 0 1 have 5:"4*a-b^2\<noteq>1" using not_one by auto
	+ from assms 0 1 have 6:"4*a-b^2\<noteq>2" using not_two by auto
	+ from 3 5 6 have "4*a-b^2 \<ge> 3" by auto
	+ from this 7 have 8:"(4*a-b^2) mod 8 = 3" using mod_nat_eqI by presburger
	+
	+ obtain j k l where ints:"odd j \<and> odd k \<and> odd l \<and> (4*a-b^2) = j^2+k^2+l^2"
	+ using 8 odd_three_squares_using_mod_eight by presburger
	+ define x where "x = sort[j,k,l] ! 2"
	+ define y where "y = sort[j,k,l] ! 1"
	+ define z where "z = sort[j,k,l] ! 0"
	+
	+ have "x^2+y^2+z^2 = sum_list (map (\<lambda>x. x^2) [j,k,l])" using x_def y_def z_def by auto
	+ from this ints have a_and_b:"(4*a-b^2) = x^2+y^2+z^2" by auto
	+
	+ have size:"x\<ge>y \<and> y\<ge>z" using x_def y_def z_def by auto
	+ have x_par:"x = j \<or> x = k \<or> x = l" using x_def by auto
	+ have y_par:"y = j \<or> y = k \<or> y = l" using y_def by auto
	+ have z_par:"z = j \<or> z = k \<or> z = l" using z_def by auto
	+ hence parity:"odd x \<and> odd y \<and> odd z" using ints x_par y_par z_par by fastforce
	+ from "1" have b_par:"odd b" by auto
	+
	+ obtain w::int where w_def:"(w=z \<or> w=-z) \<and> (b+x+y+w) mod 4 = 0" (This is the \pm z in proof)
	+ using suit_z size parity b_par by presburger
	+
	+ from parity have fac1:"(int z) mod 4 = 3 \<or> (int z) mod 4 = 1" by presburger
	+ from parity have fac2:"-z mod 4 = 3 \<or> -z mod 4 = 1" by presburger
	+ from w_def have fac3:"w mod 4 = 3 \<or> w mod 4 = 1" using fac1 fac2 by auto
	+
	+ have s_int:"4 dvd (b+x+y+w)" using b_par parity fac3 w_def by presburger
	+ have b_x_int:"2 dvd (b+x)" using b_par parity by presburger
	+ have b_y_int:"2 dvd (b+y)" using b_par parity by presburger
	+ have b_w_int:"2 dvd (b+w)" using b_par fac3 by presburger
	+
	+ obtain s::int where s_def:"s = (b+x+y+w) div 4" using s_int by fastforce
	+ obtain t::int where t_def:"t = (b+x) div 2 - s" using s_int b_x_int by blast
	+ obtain u::int where u_def:"u = (b+y) div 2 - s" using s_int b_y_int by blast
	+ obtain v::int where v_def:"v = (b+w) div 2 - s" using s_int b_w_int by blast
	+
	+ from t_def s_def have t_simp1:"t = (2b+2x) div 4 - (b+x+y+w) div 4" by auto
	+ have t_simp2:"(2* b+2* x) - (b+x+y+w) = b+x-y-w" using size by auto
	+ hence t_expre:"t = (b+x-y-w) div 4" using t_simp1 by (smt (verit, ccfv_SIG) add_num_simps(1)
	+div_plus_div_distrib_dvd_right numeral_Bit0 of_nat_numeral one_plus_numeral s_int
	+unique_euclidean_semiring_with_nat_class.of_nat_div)
	+ from b_x_int have "4 dvd (2b+2x)"
	+ by (metis distrib_left_numeral mult_2_right nat_mult_dvd_cancel_disj numeral_Bit0)
	+ hence four_div_tn:"4 dvd (b+x-y-w)" using s_int t_simp2 by presburger
	+
	+ have " (b+x) div 2 + (b+y) div 2 = (2*b+x+y) div 2"
	+ by (smt (verit, best) Groups.add_ac(2) b_y_int div_plus_div_distrib_dvd_right
	+left_add_twice nat_arith.add2)
	+ hence threesum:"t + u + s = (2*b+x+y) div 2 - s" using t_def u_def by auto
	+
	+ have "2 dvd (x+y)" using parity by auto
	+ hence "(2b+x+y) div 2 + (b+w) div 2 = (2b+b+x+y+w) div 2"
	+ by (smt (verit, ccfv_threshold) Num.of_nat_simps(4) b_w_int div_plus_div_distrib_dvd_right
	+ landau_product_preprocess(4) numerals(1) of_nat_1 one_plus_numeral
	+ unique_euclidean_semiring_with_nat_class.of_nat_div)
	+
	+ hence "t+u+s+v = (2*b+b+x+y+w) div 2 -s -s" using v_def threesum by auto
	+ hence foursum0:"t+u+s+v = (2*b+b+x+y+w) div 2 - (b+x+y+w) div 4 - (b+x+y+w) div 4"
	+ using s_def by auto
	+ have foursum1:"(b+x+y+w) div 4 + (b+x+y+w) div 4 = (b+x+y+w) div 2"
	+ using div_mult_swap s_int by auto
	+ have "(2b+b+x+y+w) div 2 - (b+x+y+w) div 2 = (2b) div 2" by auto
	+ hence "t+u+s+v = (2*b) div 2" using foursum0 foursum1 by linarith
	+ hence second:"t+u+s+v = b" by auto
	+
	+ from a_and_b have "4*a = x^2+y^2+z^2+b^2"
	+ by (metis Nat.add_diff_assoc2 add_diff_cancel_right' assms(5) less_or_eq_imp_le)
	+ hence "a = (x^2+y^2+z^2+b^2) div 4" using parity b_par by auto
	+
	+ from second have b_expresion:"b^2 = t^2+u^2+s^2+v^2+2tu+2* s * v +
	+2t s + 2t v +2u s +2u v" using four_terms_bin_exp_allsum
	+ by (metis is_num_normalize(1) nat_arith.add2 of_nat_power)
	+
	+ define sn where sn_def:"sn = b+x+y+w" (Numerator of s)
	+ from sn_def s_def have sn_nume: "4* s = sn" by (metis dvd_div_mult_self mult.commute s_int)
	+ from sn_def have sn_sqr:"sn^2 = b^2+x^2+y^2+w^2+2* b * x+2* b * y+2bw+2xy+2xw+2yw"
	+ using four_terms_bin_exp_allsum w_def by auto
	+ hence s_pen:"16* s^2 = b^2+x^2+y^2+w^2+2bx+2by+2bw+2xy+2xw+2yw" using sn_nume by auto
	+ have "4 dvd sn" using s_int sn_def by auto
	+ hence "16 dvd sn^2" by auto
	+ hence s_sqr_expression:"s^2=(b^2+x^2+y^2+w^2+2bx+2by+2bw+2xy+2xw+2yw) div 16"
	+ using sn_sqr s_pen by auto
	+
	+ define tn where tn_def:"tn = b+x-y-w" (Numerator of t)
	+ from tn_def t_expre size four_div_tn have tn_nume: "4* t = tn"
	+ by (metis dvd_div_mult_self mult.commute)
	+ from size assms have "b+x-y > 0" by auto
	+ hence "tn = int b + int x - int y -w" using tn_def by auto
	+ from this have tn_sqr:"tn^2 = b^2+x^2+y^2+w^2+2bx-2by-2bw-2xy-2xw+2yw"
	+ using four_terms_bin_exp_twodiff w_def by auto
	+ hence t_pen:"16t^2 = b^2+x^2+y^2+w^2+2bx-2by-2bw-2xy-2xw+2y*w" using tn_nume by auto
	+ have "16 dvd tn^2" using tn_def four_div_tn by auto
	+ hence t_sqr_expression:"t^2=(b^2+x^2+y^2+w^2+2bx-2by-2bw-2xy-2xw+2yw) div 16"
	+ using tn_sqr t_pen by auto
	+
	+ from size s_def t_expre w_def have sgeqt:"s\<ge>t" by auto
	+ from size s_def t_def u_def have tgequ:"t\<ge>u" by auto
	+ from size s_def u_def v_def w_def have ugeqv:"u\<ge>v" by auto
	+
	+ from assms(6) have "12a < 4b^2+8*b+16" by auto
	+ hence "12a-3b^2 < b^2+8*b+16" by auto
	+ hence "12a-3b^2 < (b+4)^2"
	+ by (smt (z3) add.commute add.left_commute mult_2 numeral_Bit0 power2_eq_square power2_sum)
	+ hence mid_ineq:"sqrt(12a-3b^2) < b+4"
	+ by (meson of_nat_0_le_iff of_nat_power_less_of_nat_cancel_iff real_less_lsqrt)
	+
	+ define ab::nat where ab_def:"ab = 4*a-b^2"
	+ from assms ab_def have nonneg_ab:"ab>0" by auto
	+ from a_and_b ab_def have sum_of_sqrs:"x^2+y^2+z^2 = ab" by auto
	+ from this nonneg_ab have "x^2+y^2+z^2>0" by auto
	+ from this sum_of_sqrs three_terms_Cauchy_Schwarz_nat_ver have "x+y+z \<le> sqrt(3*ab)" by auto
	+ hence left_ineq:"x+y+z \<le> sqrt(3(4a-b^2))" using ab_def by auto
	+ have "sqrt(3(4a-b^2)) = sqrt(12a-3b^2)" by (simp add: diff_mult_distrib2)
	+ from left_ineq mid_ineq this have "x+y+z < b+4" by auto
	+ hence num_bound:"int b- x- y- z > -4" by auto
	+
	+ define vn where vn_def:"vn = int b+w-x-y"
	+ from num_bound vn_def w_def have vn_bound:"vn > -4" by auto
	+ from w_def have four_div_sn:"4 dvd (int b +x+y+w)" by auto
	+ from parity have "4 dvd (int 2x+2y)"
	+ by (metis Num.of_nat_simps(5) \<open>even (x + y)\<close> distrib_left int_dvd_int_iff
	+nat_mult_dvd_cancel_disj num_double numeral_mult of_nat_add of_nat_numeral)
	+ hence "4 dvd (int b +x+y+w - 2x - 2y)" using four_div_sn
	+ by (smt (verit) Num.of_nat_simps(5) dvd_add_left_iff)
	+ hence "4 dvd vn" using vn_def by presburger
	+
	+ from v_def s_def have "v = (int 2b + 2w) div 4 - (int b + x + y + w) div 4" by auto
	+ hence v_expre:"v = (int b-x-y+w) div 4" using four_div_sn by fastforce
	+ hence "v = vn div 4" using vn_def by auto
	+ hence "v \<ge> 0" using vn_bound four_div_sn using \<open>4 dvd vn\<close> by fastforce
	+ hence stuv_nonneg:" s \<ge> 0 \<and> t \<ge> 0 \<and> u \<ge> 0 \<and> v \<ge> 0 " using sgeqt tgequ ugeqv by linarith
	+
	+ from vn_def have vn_sqr:"vn^2 = b^2+x^2+y^2+w^2 - 2bx-2by+2bw+2xy-2xw-2yw"
	+ using four_terms_bin_exp_twodiff w_def by auto
	+ from \<open>v = vn div 4\<close> have vn_is_num:"v^2 = vn^2 div 16" using \<open>4 dvd vn\<close> by fastforce
	+ hence "16 dvd vn^2" using v_def using \<open>4 dvd vn\<close> by fastforce
	+ from vn_is_num vn_sqr have
	+ v_sqr_expression:"v^2=(b^2+x^2+y^2+w^2-2bx-2by+2bw+2xy-2xw-2yw) div 16" by auto
	+
	+ define un where un_def:"un = int b+y-x-w"
	+ from parity w_def have "even (x+w)" by auto
	+ from this parity have "4 dvd (int 2x+2w)"
	+ by (metis distrib_left even_numeral mult_2_right mult_dvd_mono numeral_Bit0 of_nat_numeral)
	+ hence "4 dvd (int b +x+y+w - 2x-2w)" using four_div_sn
	+ by (smt (verit) Num.of_nat_simps(5) dvd_add_left_iff)
	+ hence "4 dvd un" using un_def by presburger
	+
	+ from u_def s_def have "u = (int 2b+2y) div 4 - (int b + x + y + w) div 4" by auto
	+ hence u_expre:"u = (int b-x+y-w) div 4" using four_div_sn by fastforce
	+ hence "u = un div 4" using un_def by auto
	+ from un_def have un_sqr:"un^2 = b^2+x^2+y^2+w^2+2by-2bx-2bw-2yx-2yw+2xw"
	+ using four_terms_bin_exp_twodiff w_def by auto
	+ from \<open>u = un div 4\<close> have un_is_num:"u^2 = un^2 div 16" using \<open>4 dvd un\<close> by fastforce
	+ hence "16 dvd un^2" using u_def using \<open>4 dvd un\<close> by fastforce
	+ from un_is_num un_sqr have
	+ u_sqr_expression:"u^2 = (b^2+x^2+y^2+w^2+2by-2bx-2bw-2yx-2yw+2xw) div 16" by auto
	+
	+ from u_sqr_expression v_sqr_expression have
	+ uv_simp1:"u^2+v^2 = (int b^2+x^2+y^2+w^2-2bx-2by+2bw+2xy-2xw-2yw) div 16 +
	+ (int b^2+x^2+y^2+w^2+2by-2bx-2bw-2yx-2yw+2xw) div 16" by auto
	+ have uv_simp2:"(int b^2+x^2+y^2+w^2-2bx-2by+2bw+2xy-2xw-2yw)+
	+ (int b^2+x^2+y^2+w^2+2by-2bx-2bw-2yx-2yw+2xw)=
	+ (int 2b^2+2x^2+2y^2+2w^2-4bx-4yw)" by auto
	+ hence "16 dvd (int 2b^2+2x^2+2y^2+2w^2-4bx-4yw)" by (smt (verit) \<open>16 dvd un\<^sup>2\<close> \<open>16 dvd vn\<^sup>2\<close>
	+ dvd_add_right_iff of_nat_power un_sqr vn_sqr zadd_int_left)
	+ hence usqr_plus_vsqr:"u^2+v^2 = (int 2b^2+2x^2+2y^2+2w^2-4bx-4yw) div 16"
	+ using uv_simp1 uv_simp2 by (smt (verit, ccfv_threshold) Num.of_nat_simps(4) Num.of_nat_simps(5)
	+ \<open>16 dvd vn\<^sup>2\<close> div_plus_div_distrib_dvd_right power2_eq_square vn_sqr)
	+
	+ have allsum0:"s^2+t^2+u^2+v^2 = (sn^2+tn^2+un^2+vn^2) div 16" using \<open>16 dvd vn\<^sup>2\<close> \<open>16 dvd sn\<^sup>2\<close>
	+\<open>16 dvd un\<^sup>2\<close> \<open>16 dvd tn\<^sup>2\<close> s_sqr_expression t_sqr_expression u_sqr_expression v_sqr_expression
	+sn_sqr tn_sqr un_sqr vn_sqr by (metis add.commute div_plus_div_distrib_dvd_left)
	+ have allsum1:"(sn^2+tn^2+un^2+vn^2) = (int 4b^2+4x^2+4y^2+4w^2)"
	+ using sn_sqr tn_sqr un_sqr vn_sqr by auto
	+ have "16 dvd (sn^2+tn^2+un^2+vn^2)"
	+ by (simp add: \<open>16 dvd sn\<^sup>2\<close> \<open>16 dvd tn\<^sup>2\<close> \<open>16 dvd un\<^sup>2\<close> \<open>16 dvd vn\<^sup>2\<close>)
	+ hence "16 dvd 4*(int b^2+x^2+y^2+w^2)" using allsum1 by auto
	+ hence "4 dvd (int b^2+x^2+y^2+w^2)" by presburger
	+ from allsum1 have "s^2+t^2+u^2+v^2 = (int 4b^2+4x^2+4y^2+4w^2) div 16"
	+ using allsum0 by presburger
	+ hence "s^2+t^2+u^2+v^2 = 4*(int b^2+x^2+y^2+w^2) div 16" by simp
	+ hence allsum2:"s^2+t^2+u^2+v^2 = (int b^2+x^2+y^2+w^2) div 4" by simp
	+
	+ from a_and_b have "4*a = int b^2+x^2+y^2+w^2" using w_def
	+ using \<open>4 * a = x\<^sup>2 + y\<^sup>2 + z\<^sup>2 + b\<^sup>2\<close> by fastforce
	+ hence first:"a = s^2+t^2+u^2+v^2" using allsum2 by linarith
	+
	+ show ?thesis using first second stuv_nonneg by (smt (verit, best))
	+qed
	+end
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/ROOT b/thys/Polygonal_Number_Theorem/ROOT
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/ROOT
	@@ -0,0 +1,17 @@
	+chapter AFP
	+
	+session "Polygonal_Number_Theorem" (AFP) = "HOL-Analysis" +
	+ options [timeout = 600]
	+
	+ sessions
	+ "Three_Squares"
	+
	+ theories
	+ "Polygonal_Number_Theorem_Lemmas"
	+ "Polygonal_Number_Theorem_Gauss"
	+ "Polygonal_Number_Theorem_Cauchy"
	+ "Polygonal_Number_Theorem_Legendre"
	+
	+ document_files
	+ "root.tex"
	+ "root.bib"
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/document/root.bib b/thys/Polygonal_Number_Theorem/document/root.bib
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/document/root.bib
	@@ -0,0 +1,19 @@
	+@book{nathanson1996,
	+ title={Additive Number Theory: The Classical Bases},
	+ author={Nathanson, Melvyn Bernard},
	+ series={Graduate Texts in Mathematics},
	+ volume={164},
	+ year={1996},
	+ publisher={Springer},
	+ address={New York}
	+}
	+@article{Three_Squares-AFP,
	+ author = {Anton Danilkin and Loïc Chevalier},
	+ title = {Three Squares Theorem},
	+ journal = {Archive of Formal Proofs},
	+ month = {May},
	+ year = {2023},
	+ note = {\url{https://isa-afp.org/entries/Three_Squares.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}
	\ No newline at end of file
	diff --git a/thys/Polygonal_Number_Theorem/document/root.tex b/thys/Polygonal_Number_Theorem/document/root.tex
	new file mode 100644
	--- /dev/null
	+++ b/thys/Polygonal_Number_Theorem/document/root.tex
	@@ -0,0 +1,78 @@
	+\documentclass[11pt,a4paper]{article}
	+\usepackage[T1]{fontenc}
	+\usepackage{isabelle,isabellesym}
	+
	+% further packages required for unusual symbols (see also
	+% isabellesym.sty), use only when needed
	+
	+%\usepackage{amssymb}
	+ %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
	+ %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
	+ %\<triangleq>, \<yen>, \<lozenge>
	+
	+%\usepackage{eurosym}
	+ %for \<euro>
	+
	+%\usepackage[only,bigsqcap,bigparallel,fatsemi,interleave,sslash]{stmaryrd}
	+ %for \<Sqinter>, \<Parallel>, \<Zsemi>, \<Parallel>, \<sslash>
	+
	+%\usepackage{eufrak}
	+ %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
	+
	+%\usepackage{textcomp}
	+ %for \<onequarter>, \<onehalf>, \<threequarters>, \<degree>, \<cent>,
	+ %\<currency>
	+
	+% this should be the last package used
	+\usepackage{pdfsetup}
	+
	+% urls in roman style, theory text in math-similar italics
	+\urlstyle{rm}
	+\isabellestyle{it}
	+
	+% for uniform font size
	+%\renewcommand{\isastyle}{\isastyleminor}
	+
	+
	+\begin{document}
	+
	+\title{Polygonal Number Theorem}
	+\author{Kevin Lee, Zhengkun Ye and Angeliki Koutsoukou-Argyraki}
	+\maketitle
	+
	+\begin{abstract}
	+ We formalize the proofs of Cauchy's and Legendre's Polygonal Number Theorems given in Melvyn B. Nathanson's book `Additive Number Theory: The Classical Bases' \cite{nathanson1996}.
	+
	+ For $m \geq 1$, the $k$-th polygonal number of order $m+2$ is defined to be $p_m(k)=\frac{mk(k-1)}{2}+k$. The theorems state that:
	+ \begin{itemize}
	+ \item If $m \ge 4$ and $N \geq 108m$, then $N$ can be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four of which are different from $0$ or $1$. If $N \geq 324$, then $N$ can be written as the sum of five pentagonal numbers, at least one of which is $0$ or $1$.
	+ \item Let $m \geq 3$ and $N \geq 28m^3$. If $m$ is odd, then $N$ is the sum of four polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.
	+ \end{itemize}
	+ We also formalize the proof of Gauss's theorem which states that every non-negative integer is the sum of three triangular numbers.
	+\end{abstract}
	+\newpage
	+\tableofcontents
	+
	+\subsection*{Acknowledgements}
	+The project was completed during the 2023 summer internship of the first two authors within the Cambridge
	+Mathematics Placements (CMP) Programme, supervised by the third author and hosted at the Department of Computer Science and Technology, University of Cambridge. All three authors were funded by the ERC Advanced Grant ALEXANDRIA (Project GA 742178)
	+led by Lawrence C. Paulson.
	+
	+Kevin Lee and Zhengkun Ye wish to thank the Zulip community for help with beginners' questions.
	+\newpage
	+% sane default for proof documents
	+\parindent 0pt\parskip 0.5ex
	+
	+% generated text of all theories
	+\input{session}
	+
	+% bibliography
	+\bibliographystyle{abbrv}
	+\bibliography{root}
	+
	+\end{document}
	+
	+%%% Local Variables:
	+%%% mode: latex
	+%%% TeX-master: t
	+%%% End:
	diff --git a/thys/Quantales_Converse/Kleene_Algebra_Converse.thy b/thys/Quantales_Converse/Kleene_Algebra_Converse.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/Kleene_Algebra_Converse.thy
	@@ -0,0 +1,159 @@
	+(*
	+Title: Kleene algebra with converse
	+Author: Cameron Calk, Georg Struth
	+Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
	+*)
	+
	+section \<open>Kleene algebra with converse\<close>
	+
	+theory "Kleene_Algebra_Converse"
	+ imports Kleene_Algebra.Kleene_Algebra
	+
	+begin
	+
	+text \<open>We start from involutive dioids and Kleene algebra and then add a so-called strong Gelfand property
	+to obtain an operation of converse that is closer to algebras of paths and relations.\<close>
	+
	+subsection \<open>Involutive Kleene algebra\<close>
	+
	+class invol_op =
	+ fixes invol :: "'a \<Rightarrow> 'a" ("_\<^sup>\<circ>" [101] 100)
	+
	+class involutive_dioid = dioid_one_zero + invol_op +
	+ assumes inv_invol [simp]: "(x\<^sup>\<circ>)\<^sup>\<circ> = x"
	+ and inv_contrav [simp]: "(x \<cdot> y)\<^sup>\<circ> = y\<^sup>\<circ> \<cdot> x\<^sup>\<circ>"
	+ and inv_sup [simp]: "(x + y)\<^sup>\<circ> = x\<^sup>\<circ> + y\<^sup>\<circ>"
	+
	+begin
	+
	+lemma inv_zero [simp]: "0\<^sup>\<circ> = 0"
	+proof-
	+ have "0\<^sup>\<circ> = (0\<^sup>\<circ> \<cdot> 0)\<^sup>\<circ>"
	+ by simp
	+ also have "\<dots> = 0\<^sup>\<circ> \<cdot> (0\<^sup>\<circ>)\<^sup>\<circ>"
	+ using local.inv_contrav by blast
	+ also have "\<dots> = 0\<^sup>\<circ> \<cdot> 0"
	+ by simp
	+ also have "\<dots> = 0"
	+ by simp
	+ finally show ?thesis.
	+qed
	+
	+lemma inv_one [simp]: "1\<^sup>\<circ> = 1"
	+proof-
	+ have "1\<^sup>\<circ> = 1\<^sup>\<circ> \<cdot> (1\<^sup>\<circ>)\<^sup>\<circ>"
	+ by simp
	+ also have "\<dots> = (1\<^sup>\<circ> \<cdot> 1)\<^sup>\<circ>"
	+ using local.inv_contrav by presburger
	+ also have "\<dots> = (1\<^sup>\<circ>)\<^sup>\<circ>"
	+ by simp
	+ also have "\<dots> = 1"
	+ by simp
	+ finally show ?thesis.
	+qed
	+
	+lemma inv_iso: "x \<le> y \<Longrightarrow> x\<^sup>\<circ> \<le> y\<^sup>\<circ>"
	+ by (metis local.inv_sup local.less_eq_def)
	+
	+lemma inv_adj: "(x\<^sup>\<circ> \<le> y) = (x \<le> y\<^sup>\<circ>)"
	+ using inv_iso by fastforce
	+
	+end
	+
	+text \<open>Here is an equivalent axiomatisation from Doornbos, Backhouse and van der Woude's paper
	+on a calculational approach to mathematical induction.\<close>
	+
	+class involutive_dioid_alt = dioid_one_zero +
	+ fixes inv_alt :: "'a \<Rightarrow> 'a"
	+ assumes inv_alt: "(inv_alt x \<le> y) = (x \<le> inv_alt y)"
	+ and inv_alt_contrav [simp]: "inv_alt (x \<cdot> y) = inv_alt y \<cdot> inv_alt x"
	+
	+begin
	+
	+lemma inv_alt_invol [simp]: "inv_alt (inv_alt x) = x"
	+proof-
	+ have "inv_alt (inv_alt x) \<le> x"
	+ by (simp add: inv_alt)
	+ thus ?thesis
	+ by (meson inv_alt order_antisym)
	+qed
	+
	+lemma inv_alt_add: "inv_alt (x + y) = inv_alt x + inv_alt y"
	+proof-
	+ {fix z
	+ have "(inv_alt (x + y) \<le> z) = (x + y \<le> inv_alt z)"
	+ by (simp add: inv_alt)
	+ also have "\<dots> = (x \<le> inv_alt z \<and> y \<le> inv_alt z)"
	+ by simp
	+ also have "\<dots> = (inv_alt x \<le> z \<and> inv_alt y \<le> z)"
	+ by (simp add: inv_alt)
	+ also have "\<dots> = (inv_alt x + inv_alt y \<le> z)"
	+ by force
	+ finally have "(inv_alt (x + y) \<le> z) = (inv_alt x + inv_alt y \<le> z)".}
	+ thus ?thesis
	+ using order_antisym by blast
	+qed
	+
	+sublocale altinv: involutive_dioid _ _ _ _ _ _ inv_alt
	+ by unfold_locales (simp_all add: inv_alt_add)
	+
	+end
	+
	+sublocale involutive_dioid \<subseteq> altinv: involutive_dioid_alt _ _ _ _ _ _ invol
	+ by unfold_locales (simp_all add: local.inv_adj)
	+
	+class involutive_kleene_algebra = involutive_dioid + kleene_algebra
	+
	+begin
	+
	+lemma inv_star: "(x\<^sup>\<star>)\<^sup>\<circ> = (x\<^sup>\<circ>)\<^sup>\<star>"
	+proof (rule order.antisym)
	+ have "((x\<^sup>\<circ>)\<^sup>\<star>)\<^sup>\<circ> = (1 + (x\<^sup>\<circ>)\<^sup>\<star> \<cdot> x\<^sup>\<circ>)\<^sup>\<circ>"
	+ by simp
	+ also have "\<dots> = 1 + (x\<^sup>\<circ>)\<^sup>\<circ> \<cdot> ((x\<^sup>\<circ>)\<^sup>\<star>)\<^sup>\<circ>"
	+ using local.inv_contrav local.inv_one local.inv_sup by presburger
	+ finally have "1 + x \<cdot> ((x\<^sup>\<circ>)\<^sup>\<star>)\<^sup>\<circ> \<le> ((x\<^sup>\<circ>)\<^sup>\<star>)\<^sup>\<circ>"
	+ by simp
	+ hence "x\<^sup>\<star> \<le> ((x\<^sup>\<circ>)\<^sup>\<star>)\<^sup>\<circ>"
	+ using local.star_inductl by force
	+ thus "(x\<^sup>\<star>)\<^sup>\<circ> \<le> (x\<^sup>\<circ>)\<^sup>\<star>"
	+ by (simp add: local.inv_adj)
	+next
	+ have "(x\<^sup>\<star>)\<^sup>\<circ> = (1 + x\<^sup>\<star> \<cdot> x)\<^sup>\<circ>"
	+ by simp
	+ also have "\<dots> = 1 + x\<^sup>\<circ> \<cdot> (x\<^sup>\<star>)\<^sup>\<circ>"
	+ using local.inv_contrav local.inv_one local.inv_sup by presburger
	+ finally have "1 + x\<^sup>\<circ> \<cdot> (x\<^sup>\<star>)\<^sup>\<circ> \<le> (x\<^sup>\<star>)\<^sup>\<circ>"
	+ by simp
	+ thus "(x\<^sup>\<circ>)\<^sup>\<star> \<le> (x\<^sup>\<star>)\<^sup>\<circ>"
	+ using local.star_inductl by force
	+qed
	+
	+end
	+
	+
	+subsection \<open>Kleene algebra with converse\<close>
	+
	+text \<open>The name "strong Gelfand property" has been borrowed from Palmigiano and Re.\<close>
	+
	+class dioid_converse = involutive_dioid +
	+ assumes strong_gelfand: "x \<le> x \<cdot> x\<^sup>\<circ> \<cdot> x"
	+
	+lemma (in dioid_converse) subid_conv: "x \<le> 1 \<Longrightarrow> x\<^sup>\<circ> = x"
	+proof (rule order.antisym)
	+ assume h: "x \<le> 1"
	+ have "x \<le> x \<cdot> x\<^sup>\<circ> \<cdot> x"
	+ by (simp add: local.strong_gelfand)
	+ also have "\<dots> \<le> 1 \<cdot> x\<^sup>\<circ> \<cdot> 1"
	+ using h local.mult_isol_var by blast
	+ also have "\<dots> = x\<^sup>\<circ>"
	+ by simp
	+ finally show "x \<le> x\<^sup>\<circ>"
	+ by simp
	+ thus "x\<^sup>\<circ> \<le> x"
	+ by (simp add: local.inv_adj)
	+qed
	+
	+class kleene_algebra_converse = involutive_kleene_algebra + dioid_converse
	+
	+end
	diff --git a/thys/Quantales_Converse/Modal_Kleene_Algebra_Converse.thy b/thys/Quantales_Converse/Modal_Kleene_Algebra_Converse.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/Modal_Kleene_Algebra_Converse.thy
	@@ -0,0 +1,131 @@
	+(*
	+Title: Modal Kleene Algebra with converse
	+Author: Cameron Calk, Georg Struth
	+Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
	+*)
	+
	+section \<open>Modal Kleene algebra with converse\<close>
	+
	+theory "Modal_Kleene_Algebra_Converse"
	+ imports Modal_Kleene_Algebra_Var Kleene_Algebra_Converse
	+
	+begin
	+
	+text \<open>Here we mainly study the interaction of converse with domain and codomain.\<close>
	+
	+subsection \<open>Involutive modal Kleene algebras\<close>
	+
	+class involutive_domain_semiring = domain_semiring + involutive_dioid
	+
	+begin
	+
	+notation domain_op ("dom")
	+
	+lemma strong_conv_conv: "dom x \<le> x \<cdot> x\<^sup>\<circ> \<Longrightarrow> x \<le> x \<cdot> x\<^sup>\<circ> \<cdot> x"
	+proof-
	+ assume h: "dom x \<le> x \<cdot> x\<^sup>\<circ>"
	+ have "x = dom x \<cdot> x"
	+ by simp
	+ also have "\<dots> \<le> x \<cdot> x\<^sup>\<circ> \<cdot> x"
	+ using h local.mult_isor by presburger
	+ finally show ?thesis.
	+qed
	+
	+end
	+
	+class involutive_dr_modal_semiring = dr_modal_semiring + involutive_dioid
	+
	+class involutive_dr_modal_kleene_algebra = involutive_dr_modal_semiring + kleene_algebra
	+
	+
	+subsection \<open>Modal semirings algebras with converse\<close>
	+
	+class dr_modal_semiring_converse = dr_modal_semiring + dioid_converse
	+
	+begin
	+
	+lemma d_conv [simp]: "(dom x)\<^sup>\<circ> = dom x"
	+proof-
	+ have "dom x \<le> dom x \<cdot> (dom x)\<^sup>\<circ> \<cdot> dom x"
	+ by (simp add: local.strong_gelfand)
	+ also have "\<dots> \<le> 1 \<cdot> (dom x)\<^sup>\<circ> \<cdot> 1"
	+ by (simp add: local.subid_conv)
	+ finally have a: "dom x \<le> (dom x)\<^sup>\<circ>"
	+ by simp
	+ hence "(dom x)\<^sup>\<circ> \<le> dom x"
	+ by (simp add: local.inv_adj)
	+ thus ?thesis
	+ using a by auto
	+qed
	+
	+lemma cod_conv: "(cod x)\<^sup>\<circ> = cod x"
	+ by (metis d_conv local.dc_compat)
	+
	+lemma d_conv_cod [simp]: "dom (x\<^sup>\<circ>) = cod x"
	+proof-
	+ have "dom (x\<^sup>\<circ>) = dom ((x \<cdot> cod x)\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = dom ((cod x)\<^sup>\<circ> \<cdot> x\<^sup>\<circ>)"
	+ using local.inv_contrav by presburger
	+ also have "\<dots> = dom (cod x \<cdot> x\<^sup>\<circ>)"
	+ by (simp add: cod_conv)
	+ also have "\<dots> = dom (dom (cod x) \<cdot> x\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = dom (cod x) \<cdot> dom (x\<^sup>\<circ>)"
	+ using local.dsg3 by blast
	+ also have "\<dots> = cod x \<cdot> dom (x\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = cod x \<cdot> cod (dom (x\<^sup>\<circ>))"
	+ by simp
	+ also have "\<dots> = cod (x \<cdot> cod (dom (x\<^sup>\<circ>)))"
	+ using local.rdual.dsg3 by presburger
	+ also have "\<dots> = cod (x \<cdot> dom (x\<^sup>\<circ>))"
	+ by simp
	+ also have "\<dots> = cod ((x\<^sup>\<circ>)\<^sup>\<circ> \<cdot> (dom (x\<^sup>\<circ>))\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = cod ((dom (x\<^sup>\<circ>) \<cdot> x\<^sup>\<circ>)\<^sup>\<circ>)"
	+ using local.inv_contrav by presburger
	+ also have "\<dots> = cod ((x\<^sup>\<circ>)\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = cod x"
	+ by simp
	+ finally show ?thesis.
	+qed
	+
	+lemma cod_conv_d: "cod (x\<^sup>\<circ>) = dom x"
	+ by (metis d_conv_cod local.inv_invol)
	+
	+lemma "dom y = y \<Longrightarrow> fd (x\<^sup>\<circ>) y = bd x y"
	+proof-
	+ assume h: "dom y = y"
	+ have "fd (x\<^sup>\<circ>) y = dom (x\<^sup>\<circ> \<cdot> dom y)"
	+ by (simp add: local.fd_def)
	+ also have "\<dots> = dom ((dom y \<cdot> x)\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = cod (dom y \<cdot> x)"
	+ using d_conv_cod by blast
	+ also have "\<dots> = bd x y"
	+ by (simp add: h local.bd_def)
	+ finally show ?thesis.
	+qed
	+
	+lemma "dom y = y \<Longrightarrow> bd (x\<^sup>\<circ>) y = fd x y"
	+ by (metis cod_conv_d d_conv local.bd_def local.fd_def local.inv_contrav)
	+
	+end
	+
	+
	+subsection \<open>Modal Kleene algebras with converse\<close>
	+
	+class dr_modal_kleene_algebra_converse = dr_modal_semiring_converse + kleene_algebra
	+
	+class dr_modal_semiring_strong_converse = involutive_dr_modal_semiring +
	+ assumes weak_dom_def: "dom x \<le> x \<cdot> x\<^sup>\<circ>"
	+ and weak_cod_def: "cod x \<le> x\<^sup>\<circ> \<cdot> x"
	+
	+subclass (in dr_modal_semiring_strong_converse) dr_modal_semiring_converse
	+ by unfold_locales (metis local.ddual.mult_isol_var local.dsg1 local.eq_refl local.weak_dom_def)
	+
	+class dr_modal_kleene_algebra_strong_converse = dr_modal_semiring_strong_converse + kleene_algebra
	+
	+end
	diff --git a/thys/Quantales_Converse/Modal_Kleene_Algebra_Var.thy b/thys/Quantales_Converse/Modal_Kleene_Algebra_Var.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/Modal_Kleene_Algebra_Var.thy
	@@ -0,0 +1,51 @@
	+(*
	+Title: Modal Kleene algebra based on domain and range semirings
	+Author: Georg Struth
	+Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
	+*)
	+
	+section \<open>Modal Kleene algebra based on domain and range semirings\<close>
	+
	+theory "Modal_Kleene_Algebra_Var"
	+ imports KAD.Domain_Semiring KAD.Range_Semiring
	+
	+begin
	+
	+notation domain_op ("dom")
	+notation range_op ("cod")
	+
	+subclass (in domain_semiring) dioid_one_zero..
	+
	+subclass (in range_semiring) dioid_one_zero
	+ by unfold_locales simp
	+
	+subsection \<open>Modal semirings\<close>
	+
	+text \<open>The following modal semirings are based on domain and range semirings instead of antidomain and antirange semirings,
	+as in the AFP entry for Kleene algebra with domain.\<close>
	+
	+class dr_modal_semiring = domain_semiring + range_semiring +
	+ assumes dc_compat [simp]: "dom (cod x) = cod x"
	+ and cd_compat [simp]: "cod (dom x) = dom x"
	+
	+begin
	+
	+sublocale msrdual: dr_modal_semiring "(+)" "\<lambda>x y. y \<cdot> x" 1 0 cod "(\<le>)" "(<)" dom
	+ by unfold_locales simp_all
	+
	+lemma d_cod_fix: "(dom x = x) = (x = cod x)"
	+ by (metis local.cd_compat local.dc_compat)
	+
	+lemma local_var: "(x \<cdot> y = 0) = (cod x \<cdot> dom y = 0)"
	+ using local.dom_weakly_local local.rdual.dom_weakly_local by force
	+
	+lemma fbdia_conjugation: "(fd x (dom p) \<cdot> dom q = 0) = (dom p \<cdot> bd x (dom q) = 0)"
	+ by (metis local.bd_def local.cd_compat local.ddual.mult_assoc local.dom_weakly_local local.fd_def local.rdual.dom_weakly_local local.rdual.dsg4)
	+
	+end
	+
	+subsection \<open>Modal Kleene algebra\<close>
	+
	+class dr_modal_kleene_algebra = dr_modal_semiring + kleene_algebra
	+
	+end
	diff --git a/thys/Quantales_Converse/Modal_Quantale.thy b/thys/Quantales_Converse/Modal_Quantale.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/Modal_Quantale.thy
	@@ -0,0 +1,329 @@
	+(*
	+Title: Modal quantales
	+Author: Georg Struth
	+Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
	+*)
	+
	+section \<open>Modal quantales\<close>
	+
	+theory Modal_Quantale
	+ imports Quantales.Quantale_Star Modal_Kleene_Algebra_Var KAD.Modal_Kleene_Algebra
	+
	+begin
	+
	+subsection \<open>Simplified modal semirings and Kleene algebras\<close>
	+
	+text \<open>The previous formalisation of modal Kleene algebra in the AFP adds two compatibility axioms between domain and codomain
	+when combining an antidomain semiring with an antirange semiring. But these are unnecessary. They are derivable from the other
	+axioms. Thus I provide a simpler axiomatisation that should eventually replace the one in the AFP.\<close>
	+
	+class modal_semiring_simp = antidomain_semiring + antirange_semiring
	+
	+lemma (in modal_semiring_simp) dr_compat [simp]: "d (r x) = r x"
	+proof-
	+ have a: "ar x \<cdot> d (r x) = 0"
	+ using local.ads_d_def local.ars_r_def local.dpdz.dom_weakly_local by auto
	+ have "r x \<cdot> d (r x) \<cdot> ar x \<le> r x \<cdot> ar x"
	+ by (simp add: local.a_subid_aux2 local.ads_d_def local.mult_isor)
	+ hence b: "r x \<cdot> d (r x) \<cdot> ar x = 0"
	+ by (simp add: local.ardual.am2 local.ars_r_def local.join.bot_unique)
	+ have "d (r x) = (ar x + r x) \<cdot> d (r x)"
	+ using local.add_comm local.ardual.ans3 local.ars_r_def local.mult_1_left by presburger
	+ also have "\<dots> = ar x \<cdot> d (r x) + r x \<cdot> d (r x)"
	+ by simp
	+ also have "\<dots> = r x \<cdot> d (r x)"
	+ by (simp add: a)
	+ also have "\<dots> = r x \<cdot> d (r x) \<cdot> (ar x + r x)"
	+ using local.add_comm local.ardual.ans3 local.ars_r_def by auto
	+ also have "\<dots> = r x \<cdot> d (r x) \<cdot> ar x + r x \<cdot> d (r x) \<cdot> r x"
	+ by simp
	+ also have "\<dots> = r x \<cdot> d (r x) \<cdot> r x"
	+ using b by auto
	+ also have "\<dots> = r x"
	+ by (metis local.ads_d_def local.am3 local.ardual.a_mult_idem local.ars_r_def local.ds.ddual.mult_assoc)
	+ finally show ?thesis
	+ by simp
	+qed
	+
	+lemma (in modal_semiring_simp) rd_compat [simp]: "r (d x) = d x"
	+ by (smt (verit) local.a_mult_idem local.ads_d_def local.am2 local.ardual.dpdz.dom_weakly_local local.ars_r_def local.dr_compat local.kat_3_equiv')
	+
	+subclass (in modal_semiring_simp) modal_semiring
	+ apply unfold_locales by simp_all
	+
	+class modal_kleene_algebra_simp = modal_semiring_simp + kleene_algebra
	+
	+subclass (in modal_kleene_algebra_simp) modal_kleene_algebra..
	+
	+
	+subsection \<open>Domain quantales\<close>
	+
	+class domain_quantale = unital_quantale + domain_op +
	+ assumes dom_absorb: "x \<le> dom x \<cdot> x"
	+ and dom_local: "dom (x \<cdot> dom y) = dom (x \<cdot> y)"
	+ and dom_add: "dom (x \<squnion> y) = dom x \<squnion> dom y"
	+ and dom_subid: "dom x \<le> 1"
	+ and dom_zero [simp]: "dom \<bottom> = \<bottom>"
	+
	+text \<open>The definition is that of a domain semiring. I cannot extend the quantale class with respect to domain semirings
	+because of different operations are used for addition/sup. The following sublocale statement brings all those properties into scope.\<close>
	+
	+sublocale domain_quantale \<subseteq> dqmsr: domain_semiring "(\<squnion>)" "(\<cdot>)" 1 \<bottom> dom "(\<le>)" "(<)"
	+ by unfold_locales (simp_all add: dom_add dom_local dom_absorb sup.absorb2 dom_subid)
	+
	+sublocale domain_quantale \<subseteq> dqmka: domain_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> dom "(\<le>)" "(<)" qstar..
	+
	+typedef (overloaded) 'a d_element = "{x :: 'a :: domain_quantale. dom x = x}"
	+ using dqmsr.dom_one by blast
	+
	+setup_lifting type_definition_d_element
	+
	+instantiation d_element :: (domain_quantale) bounded_lattice
	+
	+begin
	+
	+lift_definition less_eq_d_element :: "'a d_element \<Rightarrow> 'a d_element \<Rightarrow> bool" is "(\<le>)" .
	+
	+lift_definition less_d_element :: "'a d_element \<Rightarrow> 'a d_element \<Rightarrow> bool" is "(<)" .
	+
	+lift_definition bot_d_element :: "'a d_element" is \<bottom>
	+ by simp
	+
	+lift_definition top_d_element :: "'a d_element" is 1
	+ by simp
	+
	+lift_definition inf_d_element :: "'a d_element \<Rightarrow> 'a d_element \<Rightarrow> 'a d_element" is "(\<cdot>)"
	+ by (metis dqmsr.dom_mult_closed)
	+
	+lift_definition sup_d_element :: "'a d_element \<Rightarrow> 'a d_element \<Rightarrow> 'a d_element" is "(\<squnion>)"
	+ by simp
	+
	+instance
	+ apply (standard; transfer)
	+ apply (simp_all add: less_le_not_le)
	+ apply (metis dqmsr.dom_subid_aux2'')
	+ apply (metis dqmsr.dom_subid_aux1'')
	+ apply (metis dqmsr.dom_glb_eq)
	+ by (metis dqmsr.dom_subid)
	+
	+end
	+
	+instance d_element :: (domain_quantale) distrib_lattice
	+ by (standard, transfer, metis dqmsr.dom_distrib)
	+
	+context domain_quantale
	+begin
	+
	+lemma dom_top [simp]: "dom \<top> = 1"
	+proof-
	+ have "1 \<le> \<top>"
	+ by simp
	+ hence "dom 1 \<le> dom \<top>"
	+ using local.dqmsr.d_iso by blast
	+ thus ?thesis
	+ by (simp add: local.dual_order.antisym)
	+qed
	+
	+lemma dom_top2: "x \<cdot> \<top> \<le> dom x \<cdot> \<top>"
	+proof-
	+ have "x \<cdot> \<top> = dom x \<cdot> x \<cdot> \<top>"
	+ by simp
	+ also have "\<dots> \<le> dom x \<cdot> \<top> \<cdot> \<top>"
	+ using local.dqmsr.d_restrict_iff_1 local.top_greatest local.top_times_top mult_assoc by presburger
	+ finally show ?thesis
	+ by (simp add: local.mult.semigroup_axioms semigroup.assoc)
	+qed
	+
	+lemma weak_twisted: "x \<cdot> dom y \<le> dom (x \<cdot> y) \<cdot> x"
	+ using local.dqmsr.fd_def local.dqmsr.fdemodalisation2 local.eq_refl by blast
	+
	+lemma dom_meet: "dom x \<cdot> dom y = dom x \<sqinter> dom y"
	+ apply (rule order.antisym)
	+ apply (simp add: local.dqmsr.dom_subid_aux2 local.dqmsr.dom_subid_aux2'')
	+ by (smt (z3) local.dom_absorb local.dqmsr.dom_iso local.dqmsr.dom_subid_aux2 local.dqmsr.dsg3 local.dqmsr.dsg4 local.dual_order.antisym local.inf.cobounded1 local.inf.cobounded2 local.psrpq.mult_isol_var)
	+
	+lemma dom_meet_pres: "dom (dom x \<sqinter> dom y) = dom x \<sqinter> dom y"
	+ using dom_meet local.dqmsr.dom_mult_closed by presburger
	+
	+lemma dom_meet_distl: "dom x \<cdot> (y \<sqinter> z) = (dom x \<cdot> y) \<sqinter> (dom x \<cdot> z)"
	+proof-
	+ have a: "dom x \<cdot> (y \<sqinter> z) \<le> (dom x \<cdot> y) \<sqinter> (dom x \<cdot> z)"
	+ using local.mult_isol by force
	+ have "(dom x \<cdot> y) \<sqinter> (dom x \<cdot> z) = dom ((dom x \<cdot> y) \<sqinter> (dom x \<cdot> z)) \<cdot> ((dom x \<cdot> y) \<sqinter> (dom x \<cdot> z))"
	+ by simp
	+ also have "\<dots> \<le> dom ((dom x \<cdot> y)) \<cdot> ((dom x \<cdot> y) \<sqinter> (dom x \<cdot> z))"
	+ using calculation local.dqmsr.dom_iso local.dqmsr.dom_llp2 local.inf.cobounded1 by presburger
	+ also have "\<dots> \<le> dom x \<cdot> ((dom x \<cdot> y) \<sqinter> (dom x \<cdot> z))"
	+ by (metis local.dqmsr.domain_1'' local.dqmsr.domain_invol local.mult_isor)
	+ also have "\<dots> \<le> dom x \<cdot> (y \<sqinter> z)"
	+ by (meson local.dqmsr.dom_subid_aux2 local.inf_mono local.order_refl local.psrpq.mult_isol_var)
	+ finally show ?thesis
	+ using a local.dual_order.antisym by blast
	+qed
	+
	+lemma dom_meet_approx: "dom ((dom x \<cdot> y) \<sqinter> (dom x \<cdot> z)) \<le> dom x"
	+ by (metis dom_meet_distl local.dqmsr.domain_1'' local.dqmsr.domain_invol)
	+
	+lemma dom_inf_pres_aux: "Y \<noteq> {} \<Longrightarrow> dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> dom x"
	+proof-
	+ assume "Y \<noteq> {}"
	+ have "\<forall>z\<in>Y. dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> dom (dom x \<cdot> z)"
	+ by (meson local.INF_lower local.dqmsr.dom_iso)
	+ hence "\<forall>z\<in>Y. dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> dom x \<cdot> dom z"
	+ by fastforce
	+ hence "\<forall>z\<in>Y. dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> dom x"
	+ using dom_meet by fastforce
	+ thus "dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> dom x"
	+ using \<open>Y \<noteq> {}\<close> by blast
	+qed
	+
	+lemma dom_inf_pres_aux2: "(\<Sqinter>y \<in> Y. dom x \<cdot> y) \<le> \<Sqinter>Y"
	+ by (simp add: local.INF_lower2 local.dqmsr.dom_subid_aux2 local.le_Inf_iff)
	+
	+lemma dom_inf_pres: "Y \<noteq> {} \<Longrightarrow> dom x \<cdot> (\<Sqinter>Y) = (\<Sqinter>y \<in> Y. dom x \<cdot> y)"
	+proof-
	+ assume hyp: "Y \<noteq> {}"
	+ have a: "dom x \<cdot> (\<Sqinter>Y) \<le> (\<Sqinter>y \<in> Y. dom x \<cdot> y)"
	+ by (simp add: Setcompr_eq_image local.Inf_subdistl)
	+ have "(\<Sqinter>y \<in> Y. dom x \<cdot> y) = dom (\<Sqinter>y \<in> Y. dom x \<cdot> y) \<cdot> (\<Sqinter>y \<in> Y. dom x \<cdot> y)"
	+ by simp
	+ also have "\<dots> \<le> dom x \<cdot> (\<Sqinter>y \<in> Y. dom x \<cdot> y)"
	+ using dom_inf_pres_aux hyp local.mult_isor by blast
	+ also have "\<dots> \<le> dom x \<cdot> \<Sqinter>Y"
	+ by (simp add: dom_inf_pres_aux2 local.psrpq.mult_isol_var)
	+ finally show ?thesis
	+ using a order.antisym by blast
	+qed
	+
	+lemma "dom (\<Sqinter>X) \<le> \<Sqinter> (dom ` X)"
	+ by (simp add: local.INF_greatest local.Inf_lower local.dqmsr.dom_iso)
	+
	+text \<open>The domain operation need not preserve arbitrary sups, though this property holds, for instance,
	+in quantales of binary relations. I do not aim at a stronger axiomatisation in this theory.\<close>
	+
	+lemma dom_top_pres: "(x \<le> dom y \<cdot> x) = (x \<le> dom y \<cdot> \<top>)"
	+ apply standard
	+ apply (meson local.dqmsr.ddual.mult_isol_var local.dual_order.eq_iff local.dual_order.trans local.top_greatest)
	+ using local.dqmsr.dom_iso local.dqmsr.dom_llp by fastforce
	+
	+lemma dom_lla_var: "(dom x \<le> dom y) = (x \<le> dom y \<cdot> \<top>)"
	+ using dom_top_pres local.dqmsr.dom_llp by force
	+
	+lemma "dom (1 \<sqinter> x) = 1 \<sqinter> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> dom x = x"
	+ using local.inf_absorb2 by force
	+
	+lemma dom_meet_sub: "dom (x \<sqinter> y) \<le> dom x \<sqinter> dom y"
	+ by (simp add: local.dqmsr.d_iso)
	+
	+lemma dom_dist1: "dom x \<squnion> (dom y \<sqinter> dom z) = (dom x \<squnion> dom y) \<sqinter> (dom x \<squnion> dom z)"
	+ by (metis dom_meet local.dom_add local.dqmsr.dom_distrib)
	+
	+lemma dom_dist2: "dom x \<sqinter> (dom y \<squnion> dom z) = (dom x \<sqinter> dom y) \<squnion> (dom x \<sqinter> dom z)"
	+ by (metis dom_meet local.dom_add local.sup_distl)
	+
	+abbreviation "fd' \<equiv> dqmsr.fd"
	+
	+definition "bb x y = \<Squnion>{dom z \|z. fd' x z \<le> dom y}"
	+
	+lemma fd'_bb_galois_aux: "fd' x (dom p) \<le> dom q \<Longrightarrow> dom p \<le> bb x (dom q)"
	+ by (simp add: bb_def local.SUP_upper setcompr_eq_image)
	+
	+lemma dom_iso_var: "(\<Squnion>x \<in> X. dom x) \<le> dom (\<Squnion>x \<in> X. dom x)"
	+ by (meson local.SUP_le_iff local.dom_subid local.dqmsr.domain_subid)
	+
	+lemma dom_iso_var2: "(\<Squnion>x \<in> X. dom x) \<le> dom (\<Squnion>x \<in> X. x)"
	+ by (simp add: local.SUP_le_iff local.Sup_upper local.dqmsr.dom_iso)
	+
	+end
	+
	+
	+subsection \<open>Codomain quantales\<close>
	+
	+class codomain_quantale = unital_quantale + range_op +
	+ assumes cod_absorb: "x \<le> x \<cdot> cod x"
	+ and cod_local: "cod (cod x \<cdot> y) = cod (x \<cdot> y)"
	+ and cod_add: "cod (x \<squnion> y) = cod x \<squnion> cod y"
	+ and cod_subid: "cod x \<le> 1"
	+ and cod_zero: "cod \<bottom> = \<bottom>"
	+
	+sublocale codomain_quantale \<subseteq> coddual: domain_quantale range_op _ "\<lambda>x y. y \<cdot> x" _ _ _ _ _ _ _ _
	+ by unfold_locales (auto simp: mult_assoc cod_subid cod_zero cod_add cod_local cod_absorb Sup_distr Sup_distl)
	+
	+abbreviation (in codomain_quantale) "bd' \<equiv> coddual.fd'"
	+
	+definition (in codomain_quantale) "fb x y = \<Squnion>{cod z \|z. bd' x z \<le> cod y}"
	+
	+lemma (in codomain_quantale) bd'_fb_galois_aux: "bd' x (cod p) \<le> cod q \<Longrightarrow> cod p \<le> fb x (cod q)"
	+ using local.coddual.bb_def local.coddual.fd'_bb_galois_aux local.fb_def by auto
	+
	+
	+subsection \<open>Modal quantales\<close>
	+
	+class dc_modal_quantale = domain_quantale + codomain_quantale +
	+ assumes dc_compat [simp]: "dom (cod x) = cod x"
	+ and cd_compat [simp]: "cod (dom x) = dom x"
	+
	+sublocale dc_modal_quantale \<subseteq> mqs: dr_modal_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar dom cod
	+ by unfold_locales simp_all
	+
	+sublocale dc_modal_quantale \<subseteq> mqdual: dc_modal_quantale _ "\<lambda>x y. y \<cdot> x" _ _ _ _ _ _ _ _ dom cod
	+ by unfold_locales simp_all
	+
	+lemma (in dc_modal_quantale) "x \<cdot> \<top> = dom x \<cdot> \<top>"
	+(* nitpick[expect=genuine] *)
	+ oops
	+
	+lemma (in dc_modal_quantale) "\<top> \<cdot> x = \<top> \<cdot> cod x"
	+(* nitpick[expect=genuine] *)
	+ oops
	+
	+
	+subsection \<open>Antidomain and anticodomain quantales\<close>
	+
	+notation antidomain_op ("adom")
	+
	+class antidomain_quantale = unital_quantale + antidomain_op +
	+ assumes as1 [simp]: "adom x \<cdot> x = \<bottom>"
	+ and as2 [simp]: "adom (x \<cdot> y) \<le> adom (x \<cdot> adom (adom y))"
	+ and as3 [simp]: "adom (adom x) \<squnion> adom x = 1"
	+
	+definition (in antidomain_quantale) "ddom = adom \<circ> adom"
	+
	+sublocale antidomain_quantale \<subseteq> adqmsr: antidomain_semiring adom "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)"
	+ by unfold_locales (simp_all add: local.sup.absorb2)
	+
	+sublocale antidomain_quantale \<subseteq> adqmka: antidomain_kleene_algebra adom "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar..
	+
	+sublocale antidomain_quantale \<subseteq> addq: domain_quantale ddom
	+ by unfold_locales (simp_all add: ddom_def local.adqmsr.ans_d_def)
	+
	+notation antirange_op ("acod")
	+
	+class anticodomain_quantale = unital_quantale + antirange_op +
	+ assumes ars1 [simp]: "x \<cdot> acod x = \<bottom>"
	+ and ars2 [simp]: "acod (x \<cdot> y) \<le> acod (acod (acod x) \<cdot> y)"
	+ and ars3 [simp]: "acod (acod x) \<squnion> acod x = 1"
	+
	+sublocale anticodomain_quantale \<subseteq> acoddual: antidomain_quantale acod _ "\<lambda>x y. y \<cdot> x" _ _ _ _ _ _ _ _
	+ by unfold_locales (auto simp: mult_assoc Sup_distl Sup_distr)
	+
	+definition (in anticodomain_quantale) "ccod = acod \<circ> acod"
	+
	+sublocale anticodomain_quantale \<subseteq> acdqmsr: antirange_semiring "(\<squnion>)" "(\<cdot>)" 1 \<bottom> acod "(\<le>)" "(<)"..
	+
	+sublocale anticodomain_quantale \<subseteq> acdqmka: antirange_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar acod..
	+
	+sublocale anticodomain_quantale \<subseteq> acddq: codomain_quantale _ _ _ _ _ _ _ _ _ _ "\<lambda> x. acod (acod x)"
	+ by unfold_locales (simp_all add: local.acoddual.adqmsr.ans_d_def)
	+
	+class modal_quantale = antidomain_quantale + anticodomain_quantale
	+
	+sublocale modal_quantale \<subseteq> mmqs: modal_kleene_algebra_simp "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar adom acod..
	+
	+sublocale modal_quantale \<subseteq> mmqdual: modal_quantale _ "\<lambda>x y. y \<cdot> x" _ _ _ _ _ _ _ _ adom acod
	+ by unfold_locales simp_all
	+
	+end
	+
	+
	+
	diff --git a/thys/Quantales_Converse/Quantale_Converse.thy b/thys/Quantales_Converse/Quantale_Converse.thy
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/Quantale_Converse.thy
	@@ -0,0 +1,1583 @@
	+(*
	+Title: Quantales with converse
	+Author: Cameron Calk, Georg Struth
	+Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
	+*)
	+
	+section \<open>Quantales with converse\<close>
	+
	+theory Quantale_Converse
	+ imports Modal_Quantale Modal_Kleene_Algebra_Converse
	+
	+begin
	+
	+subsection \<open>Properties of unital quantales\<close>
	+
	+text \<open>These properties should eventually added to the quantales AFP entry.\<close>
	+
	+lemma (in quantale) bres_bot_top [simp]: "\<bottom> \<rightarrow> \<top> = \<top>"
	+ by (simp add: local.bres_galois_imp local.order.antisym)
	+
	+lemma (in quantale) fres_top_bot [simp]: "\<top> \<leftarrow> \<bottom> = \<top>"
	+ by (meson local.fres_galois local.order_antisym local.top_greatest)
	+
	+lemma (in unital_quantale) bres_top_top2 [simp]: "(x \<rightarrow> y \<cdot> \<top>) \<cdot> \<top> = x \<rightarrow> y \<cdot> \<top>"
	+proof-
	+ have "(x \<rightarrow> y \<cdot> \<top>) \<cdot> \<top> \<le> x \<rightarrow> y \<cdot> \<top> \<cdot> \<top>"
	+ by (simp add: local.bres_interchange)
	+ hence "(x \<rightarrow> y \<cdot> \<top>) \<cdot> \<top> \<le> x \<rightarrow> y \<cdot> \<top>"
	+ by (simp add: mult_assoc)
	+ thus ?thesis
	+ by (metis local.mult_1_right local.order_eq_iff local.psrpq.subdistl local.sup_top_right)
	+qed
	+
	+lemma (in unital_quantale) fres_top_top2 [simp]: "\<top> \<cdot> (\<top> \<cdot> y \<leftarrow> x) = \<top> \<cdot> y \<leftarrow> x"
	+ by (metis local.dual_order.antisym local.fres_interchange local.le_top local.top_greatest mult_assoc)
	+
	+lemma (in unital_quantale) bres_top_bot [simp]: "\<top> \<rightarrow> \<bottom> = \<bottom>"
	+ by (metis local.bot_least local.bres_canc1 local.le_top local.order.antisym)
	+
	+lemma (in unital_quantale) fres_bot_top [simp]: "\<bottom> \<leftarrow> \<top> = \<bottom>"
	+ by (metis local.bot_unique local.fres_canc1 local.top_le local.uqka.independence2 local.uwqlka.star_ext)
	+
	+lemma (in unital_quantale) top_bot_iff: "(x \<cdot> \<top> = \<bottom>) = (x = \<bottom>)"
	+ by (metis local.fres_bot_top local.fres_canc2 local.le_bot local.mult_botl)
	+
	+
	+subsection \<open>Involutive quantales\<close>
	+
	+text \<open>The following axioms for involutive quantales are standard.\<close>
	+
	+class involutive_quantale = unital_quantale + invol_op +
	+ assumes inv_invol [simp]: "(x\<^sup>\<circ>)\<^sup>\<circ> = x"
	+ and inv_contrav: "(x \<cdot> y)\<^sup>\<circ> = y\<^sup>\<circ> \<cdot> x\<^sup>\<circ>"
	+ and inv_sup [simp]: "(\<Squnion>X)\<^sup>\<circ> = (\<Squnion>x \<in> X. x\<^sup>\<circ>)"
	+
	+context involutive_quantale
	+begin
	+
	+lemma inv_binsup [simp]: "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> \<squnion> y\<^sup>\<circ>"
	+proof-
	+ have "(x \<squnion> y)\<^sup>\<circ> = (\<Squnion>z \<in> {x,y}. z\<^sup>\<circ>)"
	+ using local.inv_sup local.sup_Sup by presburger
	+ also have "\<dots> = (\<Squnion>z \<in> {x\<^sup>\<circ>,y\<^sup>\<circ>}. z)"
	+ by simp
	+ also have "\<dots> = x\<^sup>\<circ> \<squnion> y\<^sup>\<circ>"
	+ by fastforce
	+ finally show ?thesis.
	+qed
	+
	+lemma inv_iso: "x \<le> y \<Longrightarrow> x\<^sup>\<circ> \<le> y\<^sup>\<circ>"
	+ by (metis inv_binsup local.sup.absorb_iff1)
	+
	+lemma inv_galois: "(x\<^sup>\<circ> \<le> y) = (x \<le> y\<^sup>\<circ>)"
	+ using local.inv_iso by fastforce
	+
	+lemma bres_fres_conv: "(y\<^sup>\<circ> \<leftarrow> x\<^sup>\<circ>)\<^sup>\<circ> = x \<rightarrow> y"
	+proof-
	+ have "(y\<^sup>\<circ> \<leftarrow> x\<^sup>\<circ>)\<^sup>\<circ> = (\<Squnion>{z. z \<cdot> x\<^sup>\<circ> \<le> y\<^sup>\<circ>})\<^sup>\<circ>"
	+ by (simp add: local.fres_def)
	+ also have "\<dots> = \<Squnion>{z\<^sup>\<circ> \|z. z \<cdot> x\<^sup>\<circ> \<le> y\<^sup>\<circ>}"
	+ by (simp add: image_Collect)
	+ also have "\<dots> = \<Squnion>{z. z\<^sup>\<circ> \<cdot> x\<^sup>\<circ> \<le> y\<^sup>\<circ>}"
	+ by (metis local.inv_invol)
	+ also have "\<dots> = \<Squnion>{z. (x \<cdot> z)\<^sup>\<circ> \<le> y\<^sup>\<circ>}"
	+ by (simp add: local.inv_contrav)
	+ also have "\<dots> = \<Squnion>{z. x \<cdot> z \<le> y}"
	+ by (metis local.inv_invol local.inv_iso)
	+ also have "\<dots> = x \<rightarrow> y"
	+ by (simp add: local.bres_def)
	+ finally show ?thesis.
	+qed
	+
	+lemma fres_bres_conv: "(y\<^sup>\<circ> \<rightarrow> x\<^sup>\<circ>)\<^sup>\<circ> = x \<leftarrow> y"
	+ by (metis bres_fres_conv local.inv_invol)
	+
	+sublocale invqka: involutive_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar invol
	+ by unfold_locales (simp_all add: local.inv_contrav)
	+
	+lemma inv_binf [simp]: "(x \<sqinter> y)\<^sup>\<circ> = x\<^sup>\<circ> \<sqinter> y\<^sup>\<circ>"
	+proof-
	+ {fix z
	+ have "(z \<le> (x \<sqinter> y)\<^sup>\<circ>) = (z\<^sup>\<circ> \<le> x \<sqinter> y)"
	+ using invqka.inv_adj by blast
	+ also have "\<dots> = (z\<^sup>\<circ> \<le> x \<and> z\<^sup>\<circ> \<le> y)"
	+ by simp
	+ also have "\<dots> = (z \<le> x\<^sup>\<circ> \<and> z \<le> y\<^sup>\<circ>)"
	+ by (simp add: invqka.inv_adj)
	+ also have "\<dots> = (z \<le> x\<^sup>\<circ> \<sqinter> y\<^sup>\<circ>)"
	+ by simp
	+ finally have "(z \<le> (x \<sqinter> y)\<^sup>\<circ>) = (z \<le> x\<^sup>\<circ> \<sqinter> y\<^sup>\<circ>)".}
	+ thus ?thesis
	+ using local.dual_order.antisym by blast
	+qed
	+
	+lemma inv_inf [simp]: "(\<Sqinter>X)\<^sup>\<circ> = (\<Sqinter>x \<in> X. x\<^sup>\<circ>)"
	+ by (metis invqka.inv_adj local.INF_eqI local.Inf_greatest local.Inf_lower local.inv_invol)
	+
	+lemma inv_top [simp]: "\<top>\<^sup>\<circ> = \<top>"
	+proof-
	+ have a: "\<top>\<^sup>\<circ> \<le> \<top>"
	+ by simp
	+ hence "(\<top>\<^sup>\<circ>)\<^sup>\<circ> \<le> \<top>\<^sup>\<circ>"
	+ using local.inv_iso by blast
	+ hence "\<top> \<le> \<top>\<^sup>\<circ>"
	+ by simp
	+ thus ?thesis
	+ by (simp add: local.top_le)
	+qed
	+
	+lemma inv_qstar_aux [simp]: "(x ^ i)\<^sup>\<circ> = (x\<^sup>\<circ>) ^ i"
	+ by (induct i, simp_all add: local.power_commutes)
	+
	+lemma inv_conjugate: "(x\<^sup>\<circ> \<sqinter> y = \<bottom>) = (x \<sqinter> y\<^sup>\<circ> = \<bottom>)"
	+ using inv_binf invqka.inv_zero by fastforce
	+
	+text \<open>We define domain and codomain as in relation algebra and compare with the domain
	+and codomain axioms above.\<close>
	+
	+definition do :: "'a \<Rightarrow> 'a" where
	+ "do x = 1 \<sqinter> (x \<cdot> x\<^sup>\<circ>)"
	+
	+definition cd :: "'a \<Rightarrow> 'a" where
	+ "cd x = 1 \<sqinter> (x\<^sup>\<circ> \<cdot> x)"
	+
	+lemma do_inv: "do (x\<^sup>\<circ>) = cd x"
	+proof-
	+ have "do (x\<^sup>\<circ>) = 1 \<sqinter> (x\<^sup>\<circ> \<cdot> (x\<^sup>\<circ>)\<^sup>\<circ>)"
	+ by (simp add: do_def)
	+ also have "\<dots> = 1 \<sqinter> (x\<^sup>\<circ> \<cdot> x)"
	+ by simp
	+ also have "\<dots> = cd x"
	+ by (simp add: cd_def)
	+ finally show ?thesis.
	+qed
	+
	+lemma cd_inv: "cd (x\<^sup>\<circ>) = do x"
	+ by (simp add: cd_def do_def)
	+
	+lemma do_le_top: "do x \<le> 1 \<sqinter> (x \<cdot> \<top>)"
	+ by (simp add: do_def local.inf.coboundedI2 local.mult_isol)
	+
	+lemma do_subid: "do x \<le> 1"
	+ by (simp add: do_def)
	+
	+lemma cd_subid: "cd x \<le> 1"
	+ by (simp add: cd_def)
	+
	+lemma do_bot [simp]: "do \<bottom> = \<bottom>"
	+ by (simp add: do_def)
	+
	+lemma cd_bot [simp]: "cd \<bottom> = \<bottom>"
	+ by (simp add: cd_def)
	+
	+lemma do_iso: "x \<le> y \<Longrightarrow> do x \<le> do y"
	+ by (simp add: do_def local.inf.coboundedI2 local.inv_iso local.psrpq.mult_isol_var)
	+
	+lemma cd_iso: "x \<le> y \<Longrightarrow> cd x \<le> cd y"
	+ using cd_def local.eq_refl local.inf_mono local.inv_iso local.psrpq.mult_isol_var by presburger
	+
	+lemma do_subdist: "do x \<squnion> do y \<le> do (x \<squnion> y)"
	+proof-
	+ have "do x \<le> do (x \<squnion> y)" and "do y \<le> do (x \<squnion> y)"
	+ by (simp_all add: do_iso)
	+ thus ?thesis
	+ by simp
	+qed
	+
	+lemma cd_subdist: "cd x \<squnion> cd y \<le> cd (x \<squnion> y)"
	+ by (simp add: cd_iso)
	+
	+lemma inv_do [simp]: "(do x)\<^sup>\<circ> = do x"
	+ by (simp add: do_def)
	+
	+lemma inv_cd [simp]: "(cd x)\<^sup>\<circ> = cd x"
	+ by (metis do_inv inv_do)
	+
	+lemma dedekind_modular:
	+ assumes "(x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+ shows "(x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y"
	+ using assms local.inf.cobounded1 local.mult_isol local.order_trans by blast
	+
	+lemma modular_eq1:
	+ assumes "\<forall>x y z w. (y \<sqinter> (z \<cdot> x\<^sup>\<circ>) \<le> w \<longrightarrow> (y \<cdot> x) \<sqinter> z \<le> w \<cdot> x)"
	+ shows "\<forall>x y z. (x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y"
	+ using assms by blast
	+
	+lemma "do x \<cdot> do y = do x \<sqinter> do y" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> p \<cdot> q = p \<sqinter> q" (* nitpick[expect=genuine]*)
	+ oops
	+
	+end
	+
	+sublocale ab_unital_quantale \<subseteq> ciq: involutive_quantale id _ _ _ _ _ _ _ _ _ _
	+ by unfold_locales (simp_all add: mult_commute)
	+
	+class distributive_involutive_quantale = involutive_quantale + distrib_unital_quantale
	+
	+class boolean_involutive_quantale = involutive_quantale + bool_unital_quantale
	+
	+begin
	+
	+lemma res_peirce:
	+ assumes "\<forall>x y. x\<^sup>\<circ> \<cdot> -(x \<cdot> y) \<le> -y"
	+ shows "((x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>) = ((y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>)"
	+proof
	+ assume "(x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>"
	+ hence "z\<^sup>\<circ> \<le> -(x \<cdot> y)"
	+ by (simp add: local.inf.commute local.inf_shunt)
	+ thus "(y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>"
	+ by (metis assms local.inf_shunt local.inv_conjugate local.inv_contrav local.inv_invol local.mult_isol local.order.trans)
	+next
	+ assume "(y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>"
	+ hence "x\<^sup>\<circ> \<le> -(y \<cdot> z)"
	+ using local.compl_le_swap1 local.inf_shunt by blast
	+ thus "(x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>"
	+ by (metis assms local.dual_order.trans local.inf_shunt local.inv_conjugate local.inv_contrav local.mult_isol)
	+qed
	+
	+lemma res_schroeder1:
	+ assumes "\<forall>x y. x\<^sup>\<circ> \<cdot> -(x \<cdot> y) \<le> -y"
	+ shows "((x \<cdot> y) \<sqinter> z = \<bottom>) = (y \<sqinter> (x\<^sup>\<circ> \<cdot> z) = \<bottom>)"
	+proof
	+ assume h: "(x \<cdot> y) \<sqinter> z = \<bottom>"
	+ hence "z \<le> -(x \<cdot> y)"
	+ by (simp add: local.inf.commute local.inf_shunt)
	+ thus "y \<sqinter> (x\<^sup>\<circ> \<cdot> z) = \<bottom>"
	+ by (metis assms local.dual_order.trans local.inf.commute local.inf_shunt local.mult_isol)
	+next
	+ assume "y \<sqinter> (x\<^sup>\<circ> \<cdot> z) = \<bottom>"
	+ hence "y \<le> -(x\<^sup>\<circ> \<cdot> z)"
	+ by (simp add: local.inf_shunt)
	+ thus "(x \<cdot> y) \<sqinter> z = \<bottom>"
	+ by (metis assms local.inf_shunt local.inv_invol local.order_trans mult_isol)
	+qed
	+
	+lemma res_schroeder2:
	+ assumes "\<forall>x y. x\<^sup>\<circ> \<cdot> -(x \<cdot> y) \<le> -y"
	+ shows "((x \<cdot> y) \<sqinter> z = \<bottom>) = (x \<sqinter> (z \<cdot> y\<^sup>\<circ>) = \<bottom>)"
	+ by (metis assms local.inv_invol local.res_peirce local.res_schroeder1)
	+
	+lemma res_mod:
	+ assumes "\<forall>x y. x\<^sup>\<circ> \<cdot> -(x \<cdot> y) \<le> -y"
	+ shows "(x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y"
	+proof-
	+ have "(x \<cdot> y) \<sqinter> z = ((x \<sqinter> ((z \<cdot> y\<^sup>\<circ>) \<squnion> -(z \<cdot> y\<^sup>\<circ>))) \<cdot> y) \<sqinter> z"
	+ by simp
	+ also have "\<dots> = (((x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y) \<sqinter> z) \<squnion> (((x \<sqinter> -(z \<cdot> y\<^sup>\<circ>)) \<cdot> y) \<sqinter> z)"
	+ using local.chaq.wswq.distrib_left local.inf.commute local.sup_distr by presburger
	+ also have "\<dots> \<le> ((x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y) \<squnion> ((x \<cdot> y) \<sqinter> (-(z \<cdot> y\<^sup>\<circ>)) \<cdot> y \<sqinter> z)"
	+ by (metis assms local.inf.commute local.inf_compl_bot_right local.sup.orderI local.sup_inf_absorb res_schroeder2)
	+ also have "\<dots> \<le> ((x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y) \<squnion> ((x \<cdot> y) \<sqinter> -z \<sqinter> z)"
	+ by (metis assms local.dual_order.eq_iff local.inf.commute local.inf_compl_bot_right res_schroeder2)
	+ also have "\<dots> \<le> ((x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y)"
	+ by (simp add: local.inf.commute)
	+ finally show ?thesis.
	+qed
	+
	+end
	+
	+text \<open>The strong Gelfand property (name by Palmigiano and Re) is important for dioids and Kleene algebras.
	+The modular law is a convenient axiom for relational quantales, in a setting where the underlying
	+lattice is not boolean.\<close>
	+
	+class quantale_converse = involutive_quantale +
	+ assumes strong_gelfand: "x \<le> x \<cdot> x\<^sup>\<circ> \<cdot> x"
	+
	+begin
	+
	+lemma do_gelfand [simp]: "do x \<cdot> do x \<cdot> do x = do x"
	+ apply (rule order.antisym)
	+ using local.do_subid local.h_seq local.mult_isol apply fastforce
	+ by (metis local.inv_do local.strong_gelfand)
	+
	+lemma cd_gelfand [simp]: "cd x \<cdot> cd x \<cdot> cd x = cd x"
	+ by (metis do_gelfand local.do_inv)
	+
	+lemma do_idem [simp]: "do x \<cdot> do x = do x"
	+ apply (rule order.antisym)
	+ using local.do_subid local.mult_isol apply fastforce
	+ by (metis do_gelfand local.do_subid local.eq_refl local.nsrnqo.mult_oner local.psrpq.mult_isol_var)
	+
	+lemma cd_idem [simp]: "cd x \<cdot> cd x = cd x"
	+ by (metis do_idem local.do_inv)
	+
	+lemma dodo [simp]: "do (do x) = do x"
	+proof-
	+ have "do (do x) = 1 \<sqinter> (do x \<cdot> do x)"
	+ using local.do_def local.inv_do by force
	+ also have "\<dots> = 1 \<sqinter> do x"
	+ by simp
	+ also have "\<dots> = do x"
	+ by (simp add: local.do_def)
	+ finally show ?thesis.
	+qed
	+
	+lemma cdcd [simp]: "cd (cd x) = cd x"
	+ using cd_idem local.cd_def local.inv_cd by force
	+
	+lemma docd_compat [simp]: "do (cd x) = cd x"
	+proof-
	+ have "do (cd x) = do (do (x\<^sup>\<circ>))"
	+ by (simp add: local.do_inv)
	+ also have "\<dots> = do (x\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = cd x"
	+ by (simp add: local.do_inv)
	+ finally show ?thesis.
	+qed
	+
	+lemma cddo_compat [simp]: "cd (do x) = do x"
	+ by (metis docd_compat local.cd_inv local.inv_do)
	+
	+end
	+
	+sublocale quantale_converse \<subseteq> convqka: kleene_algebra_converse "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" invol qstar
	+ by unfold_locales (simp add: local.strong_gelfand)
	+
	+
	+subsection \<open>Dedekind quantales\<close>
	+
	+class dedekind_quantale = involutive_quantale +
	+ assumes modular_law: "(x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y"
	+
	+begin
	+
	+sublocale convdqka: kleene_algebra_converse "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" invol qstar
	+ by unfold_locales (metis local.inf.absorb2 local.le_top local.modular_law local.top_greatest)
	+
	+subclass quantale_converse
	+ by unfold_locales (simp add: local.convdqka.strong_gelfand)
	+
	+lemma modular_2 [simp]: "((x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> y) \<sqinter> z = (x \<cdot> y) \<sqinter> z"
	+ apply (rule order.antisym)
	+ using local.inf.cobounded1 local.inf_mono local.mult_isor local.order_refl apply presburger
	+ by (simp add: local.modular_law)
	+
	+lemma modular_1 [simp]: "(x \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))) \<sqinter> z = (x \<cdot> y) \<sqinter> z"
	+ by (metis local.inv_binf local.inv_contrav local.inv_invol modular_2)
	+
	+lemma modular3: "(x \<cdot> y) \<sqinter> z \<le> x \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+ by (metis local.inf.cobounded1 modular_1)
	+
	+text \<open>The name Dedekind quantale owes to the following formula, which is equivalent to the modular law. Dedekind quantales
	+are called modular quantales in Rosenthal's book on quantaloids (to be precise: he discusses modular quantaloids, but the
	+notion of modular quantale is then obvious).\<close>
	+
	+lemma dedekind: "(x \<cdot> y) \<sqinter> z \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+proof-
	+ have "(x \<cdot> y) \<sqinter> z = (x \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))) \<sqinter> z"
	+ by simp
	+ also have "\<dots> \<le> (x \<sqinter> (z \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))\<^sup>\<circ>)) \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+ using local.modular_law by presburger
	+ also have "\<dots> = (x \<sqinter> (z \<cdot> (y\<^sup>\<circ> \<sqinter> (z\<^sup>\<circ> \<cdot> x)))) \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+ by simp
	+ also have "\<dots> \<le> (x \<sqinter> (z \<cdot> y\<^sup>\<circ>)) \<cdot> (y \<sqinter> (x\<^sup>\<circ> \<cdot> z))"
	+ using local.inf.commute modular_1 by fastforce
	+ finally show ?thesis.
	+qed
	+
	+lemma peirce: "((x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>) = ((y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>)"
	+proof
	+ assume "(x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>"
	+ hence "((x \<cdot> y) \<sqinter> z\<^sup>\<circ>) \<cdot> y\<^sup>\<circ> = \<bottom>"
	+ by simp
	+ hence "(z\<^sup>\<circ> \<cdot> y\<^sup>\<circ>) \<sqinter> x = \<bottom>"
	+ by (metis local.inf.commute local.inv_invol local.le_bot local.modular_law)
	+ hence "((y \<cdot> z) \<sqinter> x\<^sup>\<circ>)\<^sup>\<circ> = \<bottom>\<^sup>\<circ>"
	+ by simp
	+ thus "(y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>"
	+ by (metis local.inv_invol)
	+next
	+ assume h: "(y \<cdot> z) \<sqinter> x\<^sup>\<circ> = \<bottom>"
	+ hence "z\<^sup>\<circ> \<cdot> ((y \<cdot> z) \<sqinter> x\<^sup>\<circ>) = \<bottom>"
	+ by simp
	+ hence "(y\<^sup>\<circ> \<cdot> x\<^sup>\<circ>) \<sqinter> z = \<bottom>"
	+ by (metis h local.inf.commute local.inv_invol local.le_bot local.mult_botr modular3)
	+ hence "((x \<cdot> y) \<sqinter> z\<^sup>\<circ>)\<^sup>\<circ> = \<bottom>\<^sup>\<circ>"
	+ by simp
	+ thus "(x \<cdot> y) \<sqinter> z\<^sup>\<circ> = \<bottom>"
	+ by (metis local.inv_invol)
	+qed
	+
	+lemma schroeder_1: "((x \<cdot> y) \<sqinter> z = \<bottom>) = (y \<sqinter> (x\<^sup>\<circ> \<cdot> z) = \<bottom>)"
	+ by (metis local.inf.commute local.inf_bot_right local.inv_invol local.mult_botr modular_1)
	+
	+lemma schroeder_2: "((x \<cdot> y) \<sqinter> z = \<bottom>) = (x \<sqinter> (z \<cdot> y\<^sup>\<circ>) = \<bottom>)"
	+ by (metis local.inv_invol peirce schroeder_1)
	+
	+lemma modular_eq2: "y \<sqinter> (z \<cdot> x\<^sup>\<circ>) \<le> w \<Longrightarrow> (y \<cdot> x) \<sqinter> z \<le> w \<cdot> x"
	+ by (meson local.dual_order.trans local.eq_refl local.h_w1 local.modular_law)
	+
	+lemma lla_top_aux: "p \<le> 1 \<Longrightarrow> ((x \<le> p \<cdot> x) = (x \<le> p \<cdot> \<top>))"
	+proof
	+ assume h: "p \<le> 1"
	+ and h1: "x \<le> p \<cdot> x"
	+ thus "x \<le> p \<cdot> \<top>"
	+ by (meson local.mult_isol local.order_trans local.top_greatest)
	+next
	+ assume h: "p \<le> 1"
	+ and "x \<le> p \<cdot> \<top>"
	+ hence "x = (p \<cdot> \<top>) \<sqinter> x"
	+ using local.inf.absorb_iff2 by auto
	+ also have "\<dots> \<le> p \<cdot> (\<top> \<sqinter> (p\<^sup>\<circ> \<cdot> x))"
	+ using modular3 by blast
	+ also have "\<dots> = p \<cdot> p \<cdot> x"
	+ by (simp add: h local.convdqka.subid_conv mult_assoc)
	+ finally show "x \<le> p \<cdot> x"
	+ by (metis h local.dual_order.trans local.mult_isor local.nsrnqo.mult_onel)
	+qed
	+
	+text \<open>Next we turn to properties of domain and codomain in Dedekind quantales.\<close>
	+
	+lemma lra_top_aux: "p \<le> 1 \<Longrightarrow> ((x \<le> x \<cdot> p ) = (x \<le> \<top> \<cdot> p))"
	+ by (metis convdqka.subid_conv local.inf.absorb_iff2 local.mult_1_right local.psrpq.subdistl local.sup.absorb_iff2 local.top_greatest modular_eq2)
	+
	+lemma lla: "p \<le> 1 \<Longrightarrow> ((do x \<le> p) = (x \<le> p \<cdot> \<top>))"
	+proof
	+ assume a1: "x \<le> p \<cdot> \<top>"
	+ assume a2: "p \<le> 1"
	+ have f3: "x \<cdot> \<top> \<le> p \<cdot> \<top> \<cdot> \<top>"
	+ by (simp add: a1 local.mult_isor)
	+ have f4: "p \<cdot> do x \<le> p"
	+ by (simp add: local.do_subid local.uqka.star_inductr_var_equiv local.uwqlka.star_subid)
	+ have "x \<cdot> \<top> \<le> p \<cdot> \<top>"
	+ using f3 by (simp add: local.mult.semigroup_axioms semigroup.assoc)
	+ thus "do x \<le> p"
	+ using f4 a2 lla_top_aux local.do_le_top local.inf.bounded_iff local.order_trans by blast
	+next
	+ assume a1: "do x \<le> p"
	+ assume a2: "p \<le> 1"
	+ hence "do x \<cdot> x \<le> p \<cdot> x"
	+ by (simp add: a1 local.mult_isor)
	+ hence "x \<le> p \<cdot> x"
	+ using a1 local.do_def modular_eq2 by fastforce
	+ thus "x \<le> p \<cdot> \<top>"
	+ by (simp add: a2 lla_top_aux)
	+qed
	+
	+lemma lla_Inf: "do x = \<Sqinter>{p. x \<le> p \<cdot> \<top> \<and> p \<le> 1}"
	+ apply (rule order.antisym)
	+ using lla local.Inf_greatest apply fastforce
	+ by (metis CollectI lla local.Inf_lower local.do_subid local.order.refl)
	+
	+lemma lra: "p \<le> 1 \<Longrightarrow> ((cd x \<le> p) = (x \<le> \<top> \<cdot> p))"
	+ by (metis invqka.inv_adj lla local.convdqka.subid_conv local.do_inv local.inv_contrav local.inv_top)
	+
	+lemma lra_Inf: "cd x = \<Sqinter>{p. x \<le> \<top> \<cdot> p \<and> p \<le> 1}"
	+ apply (rule order.antisym)
	+ using local.Inf_greatest lra apply fastforce
	+ by (metis CollectI local.Inf_lower local.cd_subid local.order.refl lra)
	+
	+lemma lla_var: "p \<le> 1 \<Longrightarrow> ((do x \<le> p) = (x \<le> p \<cdot> x))"
	+ by (simp add: lla lla_top_aux)
	+
	+lemma lla_Inf_var: "do x = \<Sqinter>{p. x \<le> p \<cdot> x \<and> p \<le> 1}"
	+ apply (rule order.antisym)
	+ using lla_var local.Inf_greatest apply fastforce
	+ by (metis CollectI lla_var local.Inf_lower local.do_subid local.order.refl)
	+
	+lemma lra_var: "p \<le> 1 \<Longrightarrow> ((cd x \<le> p) = (x \<le> x \<cdot> p))"
	+ by (simp add: lra lra_top_aux)
	+
	+lemma lra_Inf_var: "cd x = \<Sqinter>{p. x \<le> x \<cdot> p \<and> p \<le> 1}"
	+ apply (rule order.antisym)
	+ using local.Inf_greatest lra_var apply fastforce
	+ by (metis CollectI local.Inf_lower local.cd_subid local.order.refl lra_var)
	+
	+lemma do_top: "do x = 1 \<sqinter> (x \<cdot> \<top>)"
	+proof-
	+ have "1 \<sqinter> (x \<cdot> \<top>) = 1 \<sqinter> (x \<cdot> (\<top> \<sqinter> x\<^sup>\<circ> \<cdot> 1))"
	+ by (metis local.inf.commute local.inf_top.left_neutral modular_1)
	+ also have "\<dots> = do x"
	+ by (simp add: local.do_def)
	+ finally show ?thesis..
	+qed
	+
	+lemma cd_top: "cd x = 1 \<sqinter> (\<top> \<cdot> x)"
	+ by (metis do_top invqka.inv_one local.do_inv local.inv_binf local.inv_cd local.inv_contrav local.inv_invol local.inv_top)
	+
	+text \<open>We start deriving the axioms of modal semirings and modal quantales.\<close>
	+
	+lemma do_absorp: "x \<le> do x \<cdot> x"
	+ using lla_var local.do_subid by blast
	+
	+lemma cd_absorp: "x \<le> x \<cdot> cd x"
	+ using local.cd_subid lra_var by blast
	+
	+lemma do_absorp_eq [simp]: "do x \<cdot> x = x"
	+ using do_absorp local.do_subid local.dual_order.antisym local.h_w1 by fastforce
	+
	+lemma cd_absorp_eq [simp]: "x \<cdot> cd x = x"
	+ by (metis do_top local.do_inv local.inf_top.right_neutral local.nsrnqo.mult_oner modular_1)
	+
	+lemma do_top2: "x \<cdot> \<top> = do x \<cdot> \<top>"
	+proof (rule order.antisym)
	+ have "x \<cdot> \<top> = do x \<cdot> x \<cdot> \<top>"
	+ by simp
	+ also have "\<dots> \<le> do x \<cdot> \<top> \<cdot> \<top>"
	+ using local.psrpq.mult_double_iso local.top_greatest by presburger
	+ also have "\<dots> = do x \<cdot> \<top>"
	+ by (simp add: local.mult.semigroup_axioms semigroup.assoc)
	+ finally show "x \<cdot> \<top> \<le> do x \<cdot> \<top>".
	+ have "do x \<cdot> \<top> = (1 \<sqinter> (x \<cdot> \<top>)) \<cdot> \<top>"
	+ by (simp add: do_top)
	+ also have "\<dots> \<le> (1 \<cdot> \<top>) \<sqinter> (x \<cdot> \<top> \<cdot> \<top>)"
	+ by (simp add: local.mult_isor)
	+ also have "\<dots> = x \<cdot> \<top>"
	+ by (simp add: mult_assoc)
	+ finally show "do x \<cdot> \<top> \<le> x \<cdot> \<top>".
	+qed
	+
	+lemma cd_top2: "\<top> \<cdot> x = \<top> \<cdot> cd x"
	+ by (metis do_top2 local.do_inv local.inv_cd local.inv_contrav local.inv_invol local.inv_top)
	+
	+lemma do_local [simp]: "do (x \<cdot> do y) = do (x \<cdot> y)"
	+proof-
	+ have "do (x \<cdot> do y) = 1 \<sqinter> (x \<cdot> do y \<cdot> \<top>)"
	+ using do_top by force
	+ also have "\<dots> = 1 \<sqinter> (x \<cdot> y \<cdot> \<top>)"
	+ using do_top2 mult_assoc by force
	+ also have "\<dots> = do (x \<cdot> y)"
	+ by (simp add: do_top)
	+ finally show ?thesis
	+ by force
	+qed
	+
	+lemma cd_local [simp]: "cd (cd x \<cdot> y) = cd (x \<cdot> y)"
	+ by (metis cd_top cd_top2 mult_assoc)
	+
	+lemma do_fix_subid: "(do x = x) = (x \<le> 1)"
	+proof
	+ assume "do x = x"
	+ thus "x \<le> 1"
	+ by (metis local.do_subid)
	+next
	+ assume a: "x \<le> 1"
	+ hence "x = do x \<cdot> x"
	+ by simp
	+ hence b: "x \<le> do x"
	+ by (metis a local.mult_isol local.nsrnqo.mult_oner)
	+ have "x \<cdot> x \<le> x"
	+ using a local.mult_isor by fastforce
	+ hence "do x \<le> x"
	+ by (simp add: a lla_var local.le_top lra_top_aux)
	+ thus "do x = x"
	+ by (simp add: b local.dual_order.antisym)
	+qed
	+
	+lemma cd_fix_subid: "(cd x = x) = (x \<le> 1)"
	+ by (metis local.convdqka.subid_conv local.do_inv local.do_fix_subid local.docd_compat)
	+
	+lemma do_inf2: "do (do x \<sqinter> do y) = do x \<sqinter> do y"
	+ using do_top local.do_fix_subid local.inf.assoc by auto
	+
	+lemma do_inf_comp: "do x \<cdot> do y = do x \<sqinter> do y"
	+ by (smt (verit, best) local.do_def local.do_idem local.do_fix_subid local.dodo local.dual_order.trans local.inf_commute local.inf_idem local.inv_contrav local.inv_do local.mult_1_right local.mult_isol modular_1 mult_assoc)
	+
	+lemma cd_inf_comp: "cd x \<cdot> cd y = cd x \<sqinter> cd y"
	+ by (metis do_inf_comp local.docd_compat)
	+
	+lemma subid_mult_meet: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> p \<cdot> q = p \<sqinter> q"
	+ by (metis do_inf_comp local.do_fix_subid)
	+
	+lemma dodo_sup: "do (do x \<squnion> do y) = do x \<squnion> do y"
	+proof-
	+ have "do (do x \<squnion> do y) = 1 \<sqinter> ((do x \<squnion> do y) \<cdot> (do x \<squnion> do y)\<^sup>\<circ>)"
	+ using local.do_def by blast
	+ also have "\<dots> = 1 \<sqinter> ((do x \<squnion> do y) \<cdot> (do x \<squnion> do y))"
	+ by simp
	+ also have "\<dots> = 1 \<sqinter> (do x \<squnion> do y)"
	+ using local.do_subid local.dodo local.inf.idem local.le_supI subid_mult_meet by presburger
	+ also have "\<dots> = do x \<squnion> do y"
	+ by (simp add: local.do_def local.inf_absorb2)
	+ finally show ?thesis.
	+qed
	+
	+lemma do_sup [simp]: "do (x \<squnion> y) = do x \<squnion> do y"
	+proof-
	+ have "do (x \<squnion> y) = 1 \<sqinter> ((x \<squnion> y) \<cdot> \<top>)"
	+ by (simp add: do_top)
	+ also have "\<dots> = 1 \<sqinter> (x \<cdot> \<top> \<squnion> y \<cdot> \<top>)"
	+ by simp
	+ also have "\<dots> = 1 \<sqinter> (do x \<cdot> \<top> \<squnion> do y \<cdot> \<top>)"
	+ using do_top2 by force
	+ also have "\<dots> = 1 \<sqinter> ((do x \<squnion> do y) \<cdot> \<top>)"
	+ by simp
	+ also have "\<dots> = do (do x \<squnion> do y)"
	+ by (simp add: do_top)
	+ finally show ?thesis
	+ by (simp add: dodo_sup)
	+qed
	+
	+lemma cdcd_sup: "cd (cd x \<squnion> cd y) = cd x \<squnion> cd y"
	+ using local.cd_fix_subid by fastforce
	+
	+lemma cd_sup [simp]: "cd (x \<squnion> y) = cd x \<squnion> cd y"
	+ by (metis do_sup local.do_inv local.inv_binsup)
	+
	+text \<open>Next we show that Dedekind quantales are modal quantales, hence also modal semirings.\<close>
	+
	+sublocale dmq: dc_modal_quantale 1 "(\<cdot>)" Inf Sup "(\<sqinter>)" "(\<le>)" "(<)" "(\<squnion>)" "\<bottom>" "\<top>" cd do
	+proof
	+ show "\<And>x. cd x \<le> 1"
	+ by (simp add: cd_top)
	+ show "\<And>x. do x \<le> 1"
	+ by (simp add: do_top)
	+qed simp_all
	+
	+lemma do_top3 [simp]: "do (x \<cdot> \<top>) = do x"
	+ using dmq.coddual.le_top dmq.dqmsr.domain_1'' local.do_iso local.order.antisym by presburger
	+
	+lemma cd_top3 [simp]: "cd (\<top> \<cdot> x) = cd x"
	+ by (simp add: cd_top dmq.coddual.mult_assoc)
	+
	+lemma dodo_Sup_pres: "do (\<Squnion>x \<in> X. do x) = (\<Squnion>x \<in> X. do x)"
	+ by (simp add: local.SUP_least local.do_fix_subid)
	+
	+text \<open>The domain elements form a complete Heyting algebra.\<close>
	+
	+lemma do_complete_heyting: "do x \<sqinter> (\<Squnion>y \<in> Y. do y) = (\<Squnion>y \<in> Y. do x \<sqinter> do y)"
	+proof-
	+ have "do x \<sqinter> (\<Squnion>y \<in> Y. do y) = do x \<cdot> (\<Squnion>y \<in> Y. do y)"
	+ by (metis do_inf_comp dodo_Sup_pres)
	+ also have "\<dots> = (\<Squnion>y \<in> Y. do x \<cdot> do y)"
	+ by (simp add: dmq.coddual.Sup_distr image_image)
	+ also have "\<dots> = (\<Squnion>y \<in> Y. do x \<sqinter> do y)"
	+ using do_inf_comp by force
	+ finally show ?thesis.
	+qed
	+
	+lemma cdcd_Sup_pres: "cd (\<Squnion>x \<in> X. cd x) = (\<Squnion>x \<in> X. cd x)"
	+ by (simp add: local.SUP_least local.cd_fix_subid)
	+
	+lemma cd_complete_heyting: "cd x \<sqinter> (\<Squnion>y \<in> Y. cd y) = (\<Squnion>y \<in> Y. cd x \<sqinter> cd y)"
	+proof-
	+ have "cd x \<sqinter> (\<Squnion>y \<in> Y. cd y) = cd x \<cdot> (\<Squnion>y \<in> Y. cd y)"
	+ by (metis cd_inf_comp cdcd_Sup_pres)
	+ also have "\<dots> = (\<Squnion>y \<in> Y. cd x \<cdot> cd y)"
	+ by (simp add: dmq.coddual.Sup_distr image_image)
	+ also have "\<dots> = (\<Squnion>y \<in> Y. cd x \<sqinter> cd y)"
	+ using cd_inf_comp by force
	+ finally show ?thesis.
	+qed
	+
	+lemma subid_complete_heyting:
	+ assumes "p \<le> 1"
	+ and "\<forall>q \<in> Q. q \<le> 1"
	+ shows "p \<sqinter> (\<Squnion>Q) = (\<Squnion>q \<in> Q. p \<sqinter> q)"
	+proof-
	+ have a: "do p = p"
	+ by (simp add: assms(1) local.do_fix_subid)
	+ have b: "\<forall>q \<in> Q. do q = q"
	+ using assms(2) local.do_fix_subid by presburger
	+ hence "p \<sqinter> (\<Squnion>Q) = do p \<sqinter> (\<Squnion>q \<in> Q. do q)"
	+ by (simp add: a)
	+ also have "\<dots> = (\<Squnion>q \<in> Q. do p \<sqinter> do q)"
	+ using do_complete_heyting by blast
	+ also have "\<dots> = (\<Squnion>q \<in> Q. p \<sqinter> q)"
	+ using a b by force
	+ finally show ?thesis.
	+qed
	+
	+text \<open>Next we show that domain and codomain preserve arbitrary Sups.\<close>
	+
	+lemma do_Sup_pres_aux: "(\<Squnion>x \<in> X. do x \<cdot> \<top>) = (\<Squnion>x \<in> do ` X. x \<cdot> \<top>)"
	+ by (smt (verit, best) Sup.SUP_cong image_image)
	+
	+lemma do_Sup_pres: "do (\<Squnion>x \<in> X. x) = (\<Squnion>x \<in> X. do x)"
	+proof-
	+ have "do (\<Squnion>x \<in> X. x) = 1 \<sqinter> ((\<Squnion>x \<in> X. x) \<cdot> \<top>)"
	+ by (simp add: do_top)
	+ also have "\<dots> = 1 \<sqinter> (\<Squnion>x \<in> X. x \<cdot> \<top>)"
	+ by (simp add: dmq.coddual.Sup_distl)
	+ also have "\<dots> = 1 \<sqinter> (\<Squnion>x \<in> X. do x \<cdot> \<top>)"
	+ using do_top2 by force
	+ also have "\<dots> = 1 \<sqinter> (\<Squnion>x \<in> do ` X. x \<cdot> \<top>)"
	+ using do_Sup_pres_aux by presburger
	+ also have "\<dots> = 1 \<sqinter> ((\<Squnion>x \<in> X. do x) \<cdot> \<top>)"
	+ using dmq.coddual.Sup_distl by presburger
	+ also have "\<dots> = do (\<Squnion>x \<in> X. do x)"
	+ by (simp add: do_top)
	+ finally show ?thesis
	+ using dodo_Sup_pres by presburger
	+qed
	+
	+lemma cd_Sup_pres: "cd (\<Squnion>x \<in> X. x) = (\<Squnion>x \<in> X. cd x)"
	+proof-
	+ have "cd (\<Squnion>x \<in> X. x) = do ((\<Squnion>x \<in> X. x)\<^sup>\<circ>)"
	+ using local.do_inv by presburger
	+ also have "\<dots> = do (\<Squnion>x \<in> X. x\<^sup>\<circ>)"
	+ by simp
	+ also have "\<dots> = (\<Squnion>x \<in> X. do (x\<^sup>\<circ>))"
	+ by (metis do_Sup_pres image_ident image_image)
	+ also have "\<dots> = (\<Squnion>x \<in> X. cd x)"
	+ using local.do_inv by presburger
	+ finally show ?thesis.
	+qed
	+
	+lemma do_inf: "do (x \<sqinter> y) = 1 \<sqinter> (y \<cdot> x\<^sup>\<circ>)"
	+ by (smt (z3) do_absorp_eq invqka.inv_one local.do_def local.inf_commute local.inv_binf local.inv_contrav local.inv_invol local.mult_1_right modular_1 modular_2 mult_assoc)
	+
	+lemma cd_inf: "cd (x \<sqinter> y) = 1 \<sqinter> (y\<^sup>\<circ> \<cdot> x)"
	+ by (metis do_inf local.do_inv local.inv_binf local.inv_invol)
	+
	+lemma do_bres_prop: "p \<le> 1 \<Longrightarrow> do (x \<rightarrow> p \<cdot> \<top>) = 1 \<sqinter> (x \<rightarrow> p \<cdot> \<top>)"
	+ by (simp add: do_top)
	+
	+lemma cd_fres_prop: "p \<le> 1 \<Longrightarrow> cd (\<top> \<cdot> p \<leftarrow> x) = 1 \<sqinter> (\<top> \<cdot> p \<leftarrow> x)"
	+ by (simp add: cd_top)
	+
	+lemma do_meet_prop: "(do p \<cdot> x) \<sqinter> (x \<cdot> do q) = do p \<cdot> x \<cdot> do q"
	+proof (rule order.antisym)
	+ have "x \<sqinter> (do p \<cdot> x \<cdot> do q) \<le> do p \<cdot> x"
	+ by (simp add: dmq.dqmsr.dom_subid_aux2'' local.inf.coboundedI2)
	+ thus "(do p \<cdot> x) \<sqinter> (x \<cdot> do q) \<le> do p \<cdot> x \<cdot> do q"
	+ by (simp add: local.inf.commute modular_eq2)
	+next
	+ have "do p \<cdot> x \<cdot> do q = (do p \<cdot> x \<cdot> do q) \<sqinter> (do p \<cdot> x \<cdot> do q)"
	+ by simp
	+ also have "\<dots> \<le> (do p \<cdot> x) \<sqinter> (x \<cdot> do q)"
	+ using dmq.dqmsr.dom_subid_aux2 dmq.dqmsr.dom_subid_aux2'' local.psrpq.mult_isol_var by auto
	+ finally show "do p \<cdot> x \<cdot> do q \<le> (do p \<cdot> x) \<sqinter> (x \<cdot> do q)".
	+qed
	+
	+lemma subid_meet_prop: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> (p \<cdot> x) \<sqinter> (x \<cdot> q) = p \<cdot> x \<cdot> q"
	+ by (metis do_fix_subid do_meet_prop)
	+
	+text \<open>Next we consider box and diamond operators like in modal semirings and modal quantales.
	+These are inherited from domain quantales. Diamonds are defined with respect to domain and codomain.
	+The box operators are defined as Sups and hence right adjoints of diamonds.\<close>
	+
	+abbreviation "do_dia \<equiv> dmq.fd'"
	+
	+abbreviation "cd_dia \<equiv> dmq.bd'"
	+
	+abbreviation "do_box \<equiv> dmq.bb"
	+
	+abbreviation "cd_box \<equiv> dmq.fb"
	+
	+text \<open>In the sense of modal logic, the domain-based diamond is a backward operator, the codomain-based
	+one a forward operator. These are related by opposition/converse.\<close>
	+
	+lemma do_dia_cd_dia_conv: "p \<le> 1 \<Longrightarrow> do_dia (x\<^sup>\<circ>) p = cd_dia x p"
	+ by (simp add: convdqka.subid_conv dmq.coddual.dqmsr.fd_def dmq.dqmsr.fd_def local.cd_def local.do_def)
	+
	+lemma cd_dia_do_dia_conv: "p \<le> 1 \<Longrightarrow> cd_dia (x\<^sup>\<circ>) p = do_dia x p"
	+ by (metis do_dia_cd_dia_conv local.inv_invol)
	+
	+text \<open>Diamonds preserve sups in both arguments.\<close>
	+
	+lemma do_dia_Sup: "do_dia (\<Squnion>X) p = (\<Squnion>x \<in> X. do_dia x p)"
	+proof-
	+ have "do_dia (\<Squnion>X) p = do ((\<Squnion>X) \<cdot> p)"
	+ by (simp add: dmq.dqmsr.fd_def)
	+ also have "\<dots> = do (\<Squnion>x \<in> X. x \<cdot> p)"
	+ using local.Sup_distr by fastforce
	+ also have "\<dots> = (\<Squnion>x \<in> X. do (x \<cdot> p))"
	+ by (metis do_Sup_pres image_ident image_image)
	+ also have "\<dots> = (\<Squnion>x \<in> X. do_dia x p)"
	+ using dmq.dqmsr.fd_def by fastforce
	+ finally show ?thesis.
	+qed
	+
	+lemma do_dia_Sup2: "do_dia x (\<Squnion>P) = (\<Squnion>p \<in> P. do_dia x p)"
	+proof-
	+ have "do_dia x (\<Squnion>P) = do (x \<cdot> (\<Squnion>P))"
	+ by (simp add: dmq.dqmsr.fd_def)
	+ also have "\<dots> = (\<Squnion>p \<in> P. do (x \<cdot> p))"
	+ by (metis dmq.coddual.Sup_distr do_Sup_pres image_ident image_image)
	+ also have "\<dots> = (\<Squnion>p \<in> P. do_dia x p)"
	+ using dmq.dqmsr.fd_def by auto
	+ finally show ?thesis.
	+qed
	+
	+lemma cd_dia_Sup: "cd_dia (\<Squnion>X) p = (\<Squnion>x \<in> X. cd_dia x p)"
	+proof-
	+ have "cd_dia (\<Squnion>X) p = cd (p \<cdot> (\<Squnion>X))"
	+ by (simp add: dmq.coddual.dqmsr.fd_def)
	+ also have "\<dots> = cd (\<Squnion>x \<in> X. p \<cdot> x)"
	+ using dmq.coddual.Sup_distr by auto
	+ also have "\<dots> = (\<Squnion>x \<in> X. cd (p \<cdot> x))"
	+ by (metis cd_Sup_pres image_ident image_image)
	+ also have "\<dots> = (\<Squnion>x \<in> X. cd_dia x p)"
	+ using dmq.coddual.dqmsr.fd_def by force
	+ finally show ?thesis.
	+qed
	+
	+lemma cd_dia_Sup2: "cd_dia x (\<Squnion>P) = (\<Squnion>p \<in> P. cd_dia x p)"
	+proof-
	+ have "cd_dia x (\<Squnion>P) = cd ((\<Squnion>P) \<cdot> x)"
	+ by (simp add: dmq.coddual.dqmsr.fd_def)
	+ also have "\<dots> = (\<Squnion>p \<in> P. cd (p \<cdot> x))"
	+ by (metis cd_Sup_pres dmq.coddual.Sup_distl image_ident image_image)
	+ also have "\<dots> = (\<Squnion>p \<in> P. cd_dia x p)"
	+ using dmq.coddual.dqmsr.fd_def by auto
	+ finally show ?thesis.
	+qed
	+
	+text \<open>The domain-based box is a forward operator, the codomain-based on a backward one. These interact
	+again with respect to converse.\<close>
	+
	+lemma do_box_var: "p \<le> 1 \<Longrightarrow> do_box x p = \<Squnion>{q. do_dia x q \<le> p \<and> q \<le> 1}"
	+ by (smt (verit, best) Collect_cong dmq.bb_def dmq.dqmsr.fdia_d_simp local.do_fix_subid local.dodo)
	+
	+lemma cd_box_var: "p \<le> 1 \<Longrightarrow> cd_box x p = \<Squnion>{q. cd_dia x q \<le> p \<and> q \<le> 1}"
	+ by (smt (verit, best) Collect_cong dmq.coddual.dqmsr.fdia_d_simp dmq.fb_def local.cd_fix_subid local.cdcd)
	+
	+lemma do_box_cd_box_conv: "p \<le> 1 \<Longrightarrow> do_box (x\<^sup>\<circ>) p = cd_box x p"
	+proof-
	+ assume a: "p \<le> 1"
	+ hence "do_box (x\<^sup>\<circ>) p = \<Squnion>{q. do_dia (x\<^sup>\<circ>) q \<le> p \<and> q \<le> 1}"
	+ by (simp add: do_box_var)
	+ also have "\<dots> = \<Squnion>{q. cd_dia x q \<le> p \<and> q \<le> 1}"
	+ by (metis do_dia_cd_dia_conv)
	+ also have "\<dots> = cd_box x p"
	+ using a cd_box_var by auto
	+ finally show ?thesis.
	+qed
	+
	+lemma cd_box_do_box_conv: "p \<le> 1 \<Longrightarrow> cd_box (x\<^sup>\<circ>) p = do_box x p"
	+ by (metis do_box_cd_box_conv local.inv_invol)
	+
	+lemma do_box_subid: "do_box x p \<le> 1"
	+ using dmq.bb_def local.Sup_le_iff by force
	+
	+lemma cd_box_subid: "p \<le> 1 \<Longrightarrow> cd_box x p \<le> 1"
	+ by (metis do_box_subid local.do_box_cd_box_conv)
	+
	+text \<open>Next we prove that boxes and diamonds are adjoints, and then demodalisation laws known
	+from modal semirings.\<close>
	+
	+lemma do_dia_do_box_galois:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(do_dia x p \<le> q) = (p \<le> do_box x q)"
	+proof
	+ show "do_dia x p \<le> q \<Longrightarrow> p \<le> do_box x q"
	+ by (simp add: assms do_box_var local.Sup_upper)
	+next
	+ assume "p \<le> do_box x q"
	+ hence "do_dia x p \<le> do (x \<cdot> \<Squnion>{t. do_dia x t \<le> q \<and> t \<le> 1})"
	+ by (simp add: assms(2) local.dmq.dqmsr.fd_def local.do_box_var local.do_iso local.mult_isol)
	+ also have "\<dots> = \<Squnion>{do (x \<cdot> t) \|t. do_dia x t \<le> q \<and> t \<le> 1}"
	+ by (metis do_Sup_pres image_ident image_image local.Sup_distl setcompr_eq_image)
	+ also have "\<dots> = \<Squnion>{do_dia x t \|t. do_dia x t \<le> q \<and> t \<le> 1}"
	+ using local.dmq.dqmsr.fd_def by fastforce
	+ also have "\<dots> \<le> q"
	+ using local.Sup_le_iff by blast
	+ finally have "do_dia x p \<le> q".
	+ thus "p \<le> do_box x q \<Longrightarrow> do_dia x p \<le> q".
	+qed
	+
	+lemma cd_dia_cd_box_galois:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(cd_dia x p \<le> q) = (p \<le> cd_box x q)"
	+ by (metis assms do_box_cd_box_conv do_dia_cd_dia_conv do_dia_do_box_galois)
	+
	+lemma do_dia_demod_subid:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(do_dia x p \<le> q) = (x \<cdot> p \<le> q \<cdot> x)"
	+ by (metis assms dmq.dqmsr.fdemodalisation2 local.do_fix_subid)
	+
	+text \<open>The demodalisation laws have variants based on residuals.\<close>
	+
	+lemma do_dia_demod_subid_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(do_dia x p \<le> q) = (p \<le> x \<rightarrow> q \<cdot> x)"
	+ by (simp add: assms do_dia_demod_subid local.bres_galois)
	+
	+lemma do_dia_demod_subid_var:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(do_dia x p \<le> q) = (x \<cdot> p \<le> q \<cdot> \<top>)"
	+ by (simp add: assms(2) dmq.dqmsr.fd_def lla)
	+
	+lemma do_dia_demod_subid_var_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(do_dia x p \<le> q) = (p \<le> x \<rightarrow> q \<cdot> \<top>)"
	+ by (simp add: assms do_dia_demod_subid_var local.bres_galois)
	+
	+lemma cd_dia_demod_subid:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(cd_dia x p \<le> q) = (p \<cdot> x \<le> x \<cdot> q)"
	+ by (metis assms dmq.coddual.dqmsr.fdemodalisation2 local.cd_fix_subid)
	+
	+lemma cd_dia_demod_subid_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(cd_dia x p \<le> q) = (p \<le> x \<cdot> q \<leftarrow> x)"
	+ by (simp add: assms cd_dia_demod_subid local.fres_galois)
	+
	+lemma cd_dia_demod_subid_var:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(cd_dia x p \<le> q) = (p \<cdot> x \<le> \<top> \<cdot> q)"
	+ by (simp add: assms(2) dmq.coddual.dqmsr.fd_def lra)
	+
	+lemma cd_dia_demod_subid_var_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+shows "(cd_dia x p \<le> q) = (p \<le> \<top> \<cdot> q \<leftarrow> x)"
	+ by (simp add: assms cd_dia_demod_subid_var local.fres_galois)
	+
	+lemma do_box_iso:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ and "p \<le> q"
	+shows "do_box x p \<le> do_box x q"
	+ by (meson assms do_box_subid do_dia_do_box_galois local.order.refl local.order.trans)
	+
	+lemma cd_box_iso:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ and "p \<le> q"
	+shows "cd_box x p \<le> cd_box x q"
	+ by (metis assms do_box_cd_box_conv do_box_iso)
	+
	+lemma do_box_demod_subid:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> do_box x q) = (x \<cdot> p \<le> q \<cdot> x)"
	+ using assms do_dia_do_box_galois local.do_dia_demod_subid by force
	+
	+lemma do_box_demod_subid_bres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> do_box x q) = (p \<le> x \<rightarrow> q \<cdot> x)"
	+ by (simp add: assms do_box_demod_subid local.bres_galois)
	+
	+lemma do_box_demod_subid_var:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> do_box x q) = (x \<cdot> p \<le> q \<cdot> \<top>)"
	+ using assms do_dia_demod_subid_var do_dia_do_box_galois by auto
	+
	+lemma do_box_demod_subid_var_bres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> do_box x q) = (p \<le> x \<rightarrow> q \<cdot> \<top>)"
	+ by (simp add: assms do_box_demod_subid_var local.bres_galois)
	+
	+lemma do_box_demod_subid_var_bres_do:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> do_box x q) = (p \<le> do (x \<rightarrow> q \<cdot> \<top>))"
	+ by (simp add: assms do_box_demod_subid_var_bres do_top)
	+
	+lemma cd_box_demod_subid:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> cd_box x q) = (p \<cdot> x \<le> x \<cdot> q)"
	+ using assms local.cd_dia_cd_box_galois local.cd_dia_demod_subid by force
	+
	+lemma cd_box_demod_subid_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> cd_box x q) = (p \<le> x \<cdot> q \<leftarrow> x)"
	+ by (simp add: assms cd_box_demod_subid local.fres_galois)
	+
	+lemma cd_box_demod_subid_var:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> cd_box x q) = (p \<cdot> x \<le> \<top> \<cdot> q)"
	+ using assms cd_dia_cd_box_galois cd_dia_demod_subid_var by force
	+
	+lemma cd_box_demod_subid_var_fres:
	+ assumes "p \<le> 1"
	+ and "q \<le> 1"
	+ shows "(p \<le> cd_box x q) = (p \<le> \<top> \<cdot> q \<leftarrow> x)"
	+ by (simp add: assms cd_box_demod_subid_var local.fres_galois)
	+
	+text \<open>We substitute demodalisation inequalities for diamonds in the definitions of boxes.\<close>
	+
	+lemma do_box_var2: "p \<le> 1 \<Longrightarrow> do_box x p = \<Squnion>{q. x \<cdot> q \<le> p \<cdot> x \<and> q \<le> 1}"
	+ unfolding do_box_var by (meson do_dia_demod_subid)
	+
	+lemma do_box_bres1: "p \<le> 1 \<Longrightarrow> do_box x p = \<Squnion>{q. q \<le> x \<rightarrow> p \<cdot> x \<and> q \<le> 1}"
	+ unfolding do_box_var by (meson do_dia_demod_subid_fres)
	+
	+lemma do_box_bres2: "p \<le> 1 \<Longrightarrow> do_box x p = \<Squnion>{q. q \<le> x \<rightarrow> p \<cdot> \<top> \<and> q \<le> 1}"
	+ unfolding do_box_var by (simp add: dmq.dqmsr.fd_def lla local.bres_galois)
	+
	+lemma do_box_var3: "p \<le> 1 \<Longrightarrow> do_box x p = \<Squnion>{q. x \<cdot> q \<le> p \<cdot> \<top> \<and> q \<le> 1}"
	+ unfolding do_box_var using dmq.dqmsr.fd_def lla by force
	+
	+lemma cd_box_var2: "p \<le> 1 \<Longrightarrow> cd_box x p = \<Squnion>{q. q \<cdot> x \<le> x \<cdot> p \<and> q \<le> 1}"
	+ unfolding cd_box_var by (metis cd_dia_demod_subid)
	+
	+lemma cd_box_var3: "p \<le> 1 \<Longrightarrow> cd_box x p = \<Squnion>{q. q \<cdot> x \<le> \<top> \<cdot> p \<and> q \<le> 1}"
	+ unfolding cd_box_var by (simp add: dmq.coddual.dqmsr.fd_def lra)
	+
	+text \<open>Using these results we get a simple characterisation of boxes via domain and codomain.
	+Similar laws can be found implicitly in Doornbos, Backhouse and van der Woude's paper on a calculational
	+approach to mathematical induction, which speaks about wlp operators instead modal operators.\<close>
	+
	+lemma bres_do_box: "p \<le> 1 \<Longrightarrow> do_box x p = do (x \<rightarrow> p \<cdot> \<top>)"
	+ by (meson do_box_demod_subid_var_bres_do do_box_subid local.cd_fix_subid local.cddo_compat local.dual_order.antisym local.eq_refl)
	+
	+lemma bres_do_box_var: "p \<le> 1 \<Longrightarrow> do_box x p = 1 \<sqinter> (x \<rightarrow> p \<cdot> \<top>)"
	+ using bres_do_box do_bres_prop by force
	+
	+lemma bres_do_box_top: "p \<le> 1 \<Longrightarrow> (do_box x p) \<cdot> \<top> = x \<rightarrow> p \<cdot> \<top>"
	+ using bres_do_box do_top2 by auto
	+
	+lemma fres_cd_box: "p \<le> 1 \<Longrightarrow> cd_box x p = cd (\<top> \<cdot> p \<leftarrow> x)"
	+proof-
	+ assume h0: "p \<le> 1"
	+ {fix q
	+ assume h1: "q \<le> 1"
	+ have "(q \<le> cd_box x p) = (q \<cdot> x \<le> \<top> \<cdot> p)"
	+ by (simp add: cd_box_demod_subid_var h0 h1)
	+ also have "\<dots> = (q \<le> \<top> \<cdot> p \<leftarrow> x)"
	+ by (simp add: local.fres_galois)
	+ also have "\<dots> = (q \<le> 1 \<sqinter> (\<top> \<cdot> p \<leftarrow> x))"
	+ by (simp add: h1)
	+ also have "\<dots> = (q \<le> cd (\<top> \<cdot> p \<leftarrow> x))"
	+ by (simp add: cd_fres_prop h0)
	+ finally have "(q \<le> cd_box x p) = (q \<le> cd (\<top> \<cdot> p \<leftarrow> x))".}
	+ hence "\<forall>y. y \<le> cd_box x p \<longleftrightarrow> y \<le> cd (\<top> \<cdot> p \<leftarrow> x)"
	+ by (meson cd_box_subid dmq.coddual.dqmsr.dpd3 h0 local.dual_order.trans)
	+ thus ?thesis
	+ using local.dual_order.antisym by blast
	+qed
	+
	+lemma fres_cd_box_var: "p \<le> 1 \<Longrightarrow> cd_box x p = 1 \<sqinter> (\<top> \<cdot> p \<leftarrow> x)"
	+ by (simp add: cd_fres_prop fres_cd_box)
	+
	+lemma fres_cd_box_top: "p \<le> 1 \<Longrightarrow> \<top> \<cdot> cd_box x p = \<top> \<cdot> p \<leftarrow> x"
	+ using cd_top2 fres_cd_box by auto
	+
	+
	+text \<open>Next we show that the box operators act on the complete Heyting algebra of subidentities.\<close>
	+
	+lemma cd_box_act:
	+ assumes "p \<le> 1"
	+ shows "cd_box (x \<cdot> y) p = cd_box x (cd_box y p)"
	+proof-
	+ {fix q
	+ assume a: "q \<le> 1"
	+ hence "(q \<le> cd_box (x \<cdot> y) p) = (cd_dia (x \<cdot> y) q \<le> p)"
	+ by (simp add: assms cd_dia_cd_box_galois)
	+ also have "\<dots> = (cd_dia y (cd_dia x q) \<le> p)"
	+ by (simp add: local.dmq.coddual.dqmsr.fdia_mult)
	+ also have "\<dots> = (cd_dia x q \<le> cd_box y p)"
	+ using assms cd_dia_cd_box_galois cd_top dmq.coddual.dqmsr.fd_def by force
	+ also have "\<dots> = (q \<le> cd_box x (cd_box y p))"
	+ by (simp add: a assms cd_dia_cd_box_galois local.cd_box_subid)
	+ finally have "(q \<le> cd_box (x \<cdot> y) p) = (q \<le> cd_box x (cd_box y p))".}
	+ thus ?thesis
	+ by (meson assms local.cd_box_subid local.dual_order.eq_iff)
	+qed
	+
	+lemma do_box_act:
	+ assumes "p \<le> 1"
	+ shows "do_box (x \<cdot> y) p = do_box y (do_box x p)"
	+ by (smt (verit) assms cd_box_act local.cd_box_do_box_conv local.do_box_subid local.inv_contrav)
	+
	+text \<open>Next we show that the box operators are Sup reversing in the first and Inf preserving
	+in the second argument.\<close>
	+
	+lemma do_box_sup_inf: "p \<le> 1 \<Longrightarrow> do_box (x \<squnion> y) p = do_box x p \<cdot> do_box y p"
	+proof-
	+ assume h1: "p \<le> 1"
	+ {fix q
	+ assume h2: "q \<le> 1"
	+ hence "(q \<le> do_box (x \<squnion> y) p) = (do_dia (x \<squnion> y) q \<le> p)"
	+ by (simp add: do_dia_do_box_galois h1)
	+ also have "\<dots> = (do_dia x q \<le> p \<and> do_dia y q \<le> p)"
	+ by (simp add: dmq.dqmsr.fdia_add2)
	+ also have "\<dots> = (q \<le> do_box x p \<and> q \<le> do_box y p)"
	+ by (simp add: do_dia_do_box_galois h1 h2)
	+ also have "\<dots> = (q \<le> do_box x p \<cdot> do_box y p)"
	+ by (simp add: do_box_subid subid_mult_meet)
	+ finally have "(q \<le> do_box (x \<squnion> y) p) = (q \<le> do_box x p \<cdot> do_box y p)".}
	+ hence "\<forall>z. z \<le> do_box (x \<squnion> y) p \<longleftrightarrow> z \<le> do_box x p \<cdot> do_box y p"
	+ by (metis do_box_subid local.dual_order.trans local.inf.boundedE subid_mult_meet)
	+ thus ?thesis
	+ using local.dual_order.antisym by blast
	+qed
	+
	+lemma do_box_sup_inf_var: "p \<le> 1 \<Longrightarrow> do_box (x \<squnion> y) p = do_box x p \<sqinter> do_box y p"
	+ by (simp add: do_box_subid do_box_sup_inf subid_mult_meet)
	+
	+lemma do_box_Sup_Inf:
	+ assumes "X \<noteq> {}"
	+ and "p \<le> 1"
	+ shows "do_box (\<Squnion>X) p = (\<Sqinter>x \<in> X. do_box x p)"
	+proof-
	+ {fix q
	+ assume h: "q \<le> 1"
	+ hence "(q \<le> do_box (\<Squnion>X) p) = (do_dia (\<Squnion>X) q \<le> p)"
	+ by (simp add: do_dia_do_box_galois assms(2))
	+ also have "\<dots> = ((\<Squnion>x \<in> X. do_dia x q) \<le> p)"
	+ using do_dia_Sup by force
	+ also have "\<dots> = (\<forall>x \<in> X. do_dia x q \<le> p)"
	+ by (simp add: local.SUP_le_iff)
	+ also have "\<dots> = (\<forall>x \<in> X. q \<le> do_box x p)"
	+ using do_dia_do_box_galois assms(2) h by blast
	+ also have "\<dots> = (q \<le> (\<Sqinter>x \<in> X. do_box x p))"
	+ by (simp add: local.le_INF_iff)
	+ finally have "(q \<le> do_box (\<Squnion>X) p) = (q \<le> (\<Sqinter>x \<in> X. do_box x p))".}
	+ hence "\<forall>y. y \<le> do_box (\<Squnion>X) p \<longleftrightarrow> y \<le> (\<Sqinter>x \<in> X. do_box x p)"
	+ by (smt (verit, ccfv_threshold) assms(1) do_box_subid local.INF_le_SUP local.SUP_least local.order_trans)
	+ thus ?thesis
	+ using local.dual_order.antisym by blast
	+qed
	+
	+lemma do_box_Sup_Inf2:
	+ assumes "P \<noteq> {}"
	+ and "\<forall>p \<in> P. p \<le> 1"
	+ shows "do_box x (\<Sqinter>P) = (\<Sqinter>p \<in> P. do_box x p)"
	+proof-
	+ have h0: "(\<Sqinter>p \<in> P. do_box x p) \<le> 1"
	+ by (meson all_not_in_conv assms(1) do_box_subid local.INF_lower2)
	+ {fix q
	+ assume h1: "q \<le> 1"
	+ hence "(q \<le> do_box x (\<Sqinter>P)) = (do_dia x q \<le> \<Sqinter>P)"
	+ by (simp add: assms do_dia_do_box_galois local.Inf_less_eq)
	+ also have "\<dots> = (\<forall>p \<in> P. do_dia x q \<le> p)"
	+ using local.le_Inf_iff by blast
	+ also have "\<dots> = (\<forall>p \<in> P. q \<le> do_box x p)"
	+ using assms(2) do_dia_do_box_galois h1 by auto
	+ also have "\<dots> = (q \<le> (\<Sqinter>p \<in> P. do_box x p))"
	+ by (simp add: local.le_INF_iff)
	+ finally have "(q \<le> do_box x (\<Sqinter>P)) = (q \<le> (\<Sqinter>p \<in> P. do_box x p))".}
	+ thus ?thesis
	+ using do_box_subid h0 local.dual_order.antisym by blast
	+qed
	+
	+lemma cd_box_sup_inf: "p \<le> 1 \<Longrightarrow> cd_box (x \<squnion> y) p = cd_box x p \<cdot> cd_box y p"
	+ by (metis do_box_cd_box_conv do_box_sup_inf local.inv_binsup)
	+
	+lemma cd_box_sup_inf_var: "p \<le> 1 \<Longrightarrow> cd_box (x \<squnion> y) p = cd_box x p \<sqinter> cd_box y p"
	+ by (simp add: cd_box_subid cd_box_sup_inf subid_mult_meet)
	+
	+lemma cd_box_Sup_Inf:
	+ assumes "X \<noteq> {}"
	+ and "p \<le> 1"
	+shows "cd_box (\<Squnion>X) p = (\<Sqinter>x \<in> X. cd_box x p)"
	+proof-
	+ have "cd_box (\<Squnion>X) p = do_box ((\<Squnion>X)\<^sup>\<circ>) p"
	+ using assms(2) do_box_cd_box_conv by presburger
	+ also have "\<dots> = (\<Sqinter>y \<in> {x\<^sup>\<circ>\|x. x \<in> X}. do_box y p)"
	+ by (simp add: Setcompr_eq_image assms do_box_Sup_Inf)
	+ also have "\<dots> = (\<Sqinter>x \<in> X. do_box (x\<^sup>\<circ>) p)"
	+ by (simp add: Setcompr_eq_image image_image)
	+ also have "\<dots> = (\<Sqinter>x \<in> X. cd_box x p)"
	+ using assms(2) do_box_cd_box_conv by force
	+ finally show ?thesis.
	+qed
	+
	+lemma cd_box_Sup_Inf2:
	+ assumes "P \<noteq> {}"
	+ and "\<forall>p \<in> P. p \<le> 1"
	+shows "cd_box x (\<Sqinter>P) = (\<Sqinter>p \<in> P. cd_box x p)"
	+proof-
	+ have "cd_box x (\<Sqinter>P) = do_box (x\<^sup>\<circ>) (\<Sqinter>P)"
	+ by (simp add: assms do_box_cd_box_conv local.Inf_less_eq)
	+ also have "\<dots> = (\<Sqinter>p \<in> P. do_box (x\<^sup>\<circ>) p)"
	+ using assms do_box_Sup_Inf2 by presburger
	+ also have "\<dots> = (\<Sqinter>p \<in> P. cd_box x p)"
	+ using assms(2) do_box_cd_box_conv by force
	+ finally show ?thesis.
	+qed
	+
	+text \<open>Next we define an antidomain operation in the style of modal semirings. A natural condition is
	+that the antidomain of an element is the greatest test that cannot be left-composed with that elements,
	+and hence a greatest left annihilator. The definition of anticodomain is similar. As we are not in a
	+boolean domain algebra, we cannot expect that the antidomain of the antidomain yields the domain or
	+that the union of a domain element with the corresponding antidomain element equals one.\<close>
	+
	+definition "ado x = \<Squnion>{p. p \<cdot> x = \<bottom> \<and> p \<le> 1}"
	+
	+definition "acd x = \<Squnion>{p. x \<cdot> p = \<bottom> \<and> p \<le> 1}"
	+
	+lemma ado_acd: "ado (x\<^sup>\<circ>) = acd x"
	+ unfolding ado_def acd_def
	+ by (metis convdqka.subid_conv invqka.inv_zero local.inv_contrav local.inv_invol)
	+
	+lemma acd_ado: "acd (x\<^sup>\<circ>) = ado x"
	+ unfolding ado_def acd_def
	+ by (metis acd_def ado_acd ado_def local.inv_invol)
	+
	+lemma ado_left_zero [simp]: "ado x \<cdot> x = \<bottom>"
	+ unfolding ado_def
	+ using dmq.coddual.Sup_distl by auto
	+
	+lemma acd_right_zero [simp]: "x \<cdot> acd x = \<bottom>"
	+ unfolding acd_def
	+ by (simp add: dmq.coddual.h_Sup local.dual_order.antisym)
	+
	+lemma ado_greatest: "p \<le> 1 \<Longrightarrow> p \<cdot> x = \<bottom> \<Longrightarrow> p \<le> ado x"
	+ by (simp add: ado_def local.Sup_upper)
	+
	+lemma acd_greatest: "p \<le> 1 \<Longrightarrow> x \<cdot> p = \<bottom> \<Longrightarrow> p \<le> acd x"
	+ by (simp add: acd_def local.Sup_upper)
	+
	+lemma ado_subid: "ado x \<le> 1"
	+ using ado_def local.Sup_le_iff by force
	+
	+lemma acd_subid: "acd x \<le> 1"
	+ by (simp add: acd_def local.Sup_le_iff)
	+
	+lemma ado_left_zero_iff: "p \<le> 1 \<Longrightarrow> (p \<le> ado x) = (p \<cdot> x = \<bottom>)"
	+ by (metis ado_greatest ado_left_zero local.bot_unique local.mult_isor)
	+
	+lemma acd_right_zero_iff: "p \<le> 1 \<Longrightarrow> (p \<le> acd x) = (x \<cdot> p = \<bottom>)"
	+ by (metis acd_greatest acd_right_zero local.bot_unique local.mult_isol)
	+
	+text \<open>This gives an eqational characterisation of antidomain and anticodomain.\<close>
	+
	+lemma ado_cd_bot: "ado x = cd (\<bottom> \<leftarrow> x)"
	+proof-
	+ {fix p
	+ assume h: "p \<le> 1"
	+ hence "(p \<le> ado x) = (p \<cdot> x = \<bottom>)"
	+ by (simp add: ado_left_zero_iff)
	+ also have "\<dots> = (p \<le> \<bottom> \<leftarrow> x)"
	+ using local.bot_unique local.fres_galois by blast
	+ also have "\<dots> = (p \<le> 1 \<sqinter> (\<bottom> \<leftarrow> x))"
	+ by (simp add: h)
	+ also have "\<dots> = (p \<le> cd (\<bottom> \<leftarrow> x))"
	+ by (metis cd_fres_prop local.bot_least local.mult_botr)
	+ finally have "(p \<le> ado x) = (p \<le> cd (\<bottom> \<leftarrow> x))".}
	+ hence "\<forall>y. y \<le> ado x \<longleftrightarrow> y \<le> cd (\<bottom> \<leftarrow> x)"
	+ using ado_subid local.cd_subid local.dual_order.trans by blast
	+ thus ?thesis
	+ using local.dual_order.antisym by blast
	+qed
	+
	+lemma acd_do_bot: "acd x = do (x \<rightarrow> \<bottom>)"
	+ by (metis ado_acd ado_cd_bot invqka.inv_zero local.bres_fres_conv local.cd_inv local.inv_invol)
	+
	+lemma ado_cd_bot_id: "ado x = 1 \<sqinter> (\<bottom> \<leftarrow> x)"
	+ by (metis ado_cd_bot cd_fres_prop local.cd_bot local.cd_subid local.mult_botr)
	+
	+lemma acd_do_bot_id: "acd x = 1 \<sqinter> (x \<rightarrow> \<bottom>)"
	+ by (metis acd_do_bot do_bres_prop local.bot_least local.mult_botl)
	+
	+lemma ado_cd_bot_var: "ado x = cd (\<bottom> \<leftarrow> do x)"
	+ by (metis ado_cd_bot do_top2 local.fres_bot_top local.fres_curry)
	+
	+lemma acd_do_bot_var: "acd x = do (cd x \<rightarrow> \<bottom>)"
	+ by (metis acd_do_bot cd_top2 local.bres_curry local.bres_top_bot)
	+
	+lemma ado_do_bot: "ado x = do (do x \<rightarrow> \<bottom>)"
	+ using acd_ado acd_do_bot_var local.cd_inv by auto
	+
	+lemma "do x = ado (ado x)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma acd_cd_bot: "acd x = cd (\<bottom> \<leftarrow> cd x)"
	+ by (metis ado_acd ado_cd_bot_var local.cd_inv local.inv_invol)
	+
	+lemma ado_do_bot_var: "ado x = 1 \<sqinter> (do x \<rightarrow> \<bottom>)"
	+ by (metis ado_do_bot do_bres_prop local.bot_unique local.bres_bot_bot local.bres_canc1 local.do_bot local.do_subid)
	+
	+ lemma acd_cd_bot_var: "acd x = 1 \<sqinter> (\<bottom> \<leftarrow> cd x)"
	+ by (metis acd_cd_bot acd_right_zero cd_top local.fres_top_top2)
	+
	+
	+text \<open>Domain and codomain are compatible with the boxes.\<close>
	+
	+lemma cd_box_ado: "cd_box x \<bottom> = ado x"
	+ by (simp add: ado_cd_bot fres_cd_box)
	+
	+lemma do_box_acd: "do_box x \<bottom> = acd x"
	+ by (simp add: acd_do_bot bres_do_box)
	+
	+lemma ado_subid_prop: "p \<le> 1 \<Longrightarrow> ado p = 1 \<sqinter> (p \<rightarrow> \<bottom>)"
	+ by (metis ado_do_bot_var do_fix_subid)
	+
	+lemma ado_do: "p \<le> 1 \<Longrightarrow> ado p = do (p \<rightarrow> \<bottom>)"
	+ using ado_do_bot do_fix_subid by force
	+
	+lemma ado_do_compl: "ado x \<cdot> do x = \<bottom>"
	+ using dmq.dqmsr.dom_weakly_local by force
	+
	+lemma "ado x \<squnion> do x = \<top>" (* nitpick[expect = genuine]*)
	+ oops
	+
	+lemma "\<forall>x p. \<exists>f. 1 \<sqinter> (\<top> \<cdot> p \<leftarrow> x) = 1 \<sqinter> (\<bottom> \<leftarrow> (x \<rightarrow> p \<cdot> \<top>))" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "cd_box x p = ado (x \<cdot> ado p)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma ad_do_bot [simp]: "(1 \<sqinter> (do x \<rightarrow> \<bottom>)) \<cdot> do x = \<bottom>"
	+ using ado_do_bot_var ado_left_zero dmq.dqmsr.dom_weakly_local by presburger
	+
	+lemma do_heyting_galois: "(do x \<cdot> do y \<le> do z) = (do x \<le> 1 \<sqinter> (do y \<rightarrow> do z))"
	+ by (metis dmq.dqmsr.dsg4 dmq.mqdual.cod_subid local.bres_galois local.le_inf_iff)
	+
	+lemma do_heyting_galois_var: "(do x \<cdot> do y \<le> do z) = (do x \<le> cd_box (do y) (do z))"
	+ by (metis cd_dia_cd_box_galois cd_fix_subid dmq.coddual.dqmsr.fd_def dmq.dqmsr.dom_mult_closed local.do_subid)
	+
	+text \<open>Antidomain is therefore Heyting negation.\<close>
	+
	+lemma ado_heyting_negation: "ado (do x) = cd_box (do x) \<bottom>"
	+ by (simp add: cd_box_ado)
	+
	+lemma ad_ax1 [simp]: "(1 \<sqinter> (do x \<rightarrow> \<bottom>)) \<cdot> x = \<bottom>"
	+ by (simp add: local.dmq.dqmsr.dom_weakly_local)
	+
	+lemma "1 \<sqinter> (do (1 \<sqinter> (do x \<rightarrow> \<bottom>)) \<rightarrow> \<bottom>) = do x" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "p \<le> 1 \<Longrightarrow> do_dia x p = 1 \<sqinter> (cd_box x (1 \<sqinter> (p \<rightarrow> \<bottom>)) \<rightarrow> \<bottom>)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "p \<le> 1 \<Longrightarrow> cd_box x p = 1 \<sqinter> (do_dia x (1 \<sqinter> (p \<rightarrow> \<bottom>)) \<rightarrow> \<bottom>)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "p \<le> 1 \<Longrightarrow> cd_dia x p = 1 \<sqinter> (do_box x (1 \<sqinter> (p \<rightarrow> \<bottom>)) \<rightarrow> \<bottom>)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+lemma "p \<le> 1 \<Longrightarrow> do_box x p = 1 \<sqinter> (cd_dia x (1 \<sqinter> (p \<rightarrow> \<bottom>)) \<rightarrow> \<bottom>)" (* nitpick[expect=genuine]*)
	+ oops
	+
	+end
	+
	+subsection \<open>Boolean Dedekind quantales\<close>
	+
	+class distributive_dedekind_quantale = distrib_unital_quantale + dedekind_quantale
	+
	+class boolean_dedekind_quantale = bool_unital_quantale + distributive_dedekind_quantale
	+
	+begin
	+
	+lemma ad_do_bot [simp]: "(1 - do x) \<cdot> do x = \<bottom>"
	+ by (simp add: local.diff_eq local.inf_shunt local.subid_mult_meet)
	+
	+lemma ad_ax1 [simp]: "(1 - do x) \<cdot> x = \<bottom>"
	+ by (simp add: local.dmq.dqmsr.dom_weakly_local)
	+
	+lemma ad_do [simp]: "1 - do (1 - do x) = do x"
	+proof-
	+ have "1 - do (1 - do x) = 1 - (1 - do x)"
	+ by (metis local.diff_eq local.do_fix_subid local.inf.cobounded1)
	+ also have "\<dots> = 1 \<sqinter> -(1 \<sqinter> - do x)"
	+ by (simp add: local.diff_eq)
	+ also have "\<dots> = do x"
	+ by (simp add: local.chaq.wswq.distrib_left local.do_top)
	+ finally show ?thesis.
	+qed
	+
	+lemma ad_ax2: "1 - do (x \<cdot> y) \<squnion> (1 - do (x \<cdot> (1 - do (1 - do y)))) = 1 - do (x \<cdot> (1 - do (1 - do y)))"
	+ by simp
	+
	+lemma ad_ax3 [simp]: "do x \<squnion> (1 - do x) = 1"
	+proof-
	+ have "do x \<squnion> (1 - do x) = do x \<squnion> (1 \<sqinter> -do x)"
	+ using local.diff_eq by force
	+ also have "\<dots> = 1 \<sqinter> (do x \<squnion> -do x)"
	+ by (simp add: local.chaq.wswq.distrib_left local.do_top)
	+ also have "\<dots> = 1"
	+ by simp
	+ finally show ?thesis.
	+qed
	+
	+sublocale bdad: antidomain_semiring "\<lambda>x. 1 - do x" "(\<squnion>)" "(\<cdot>)" 1 \<bottom> _ _
	+ by unfold_locales simp_all
	+
	+sublocale bdadka: antidomain_kleene_algebra "\<lambda>x. 1 - do x" "(\<squnion>)" "(\<cdot>)" 1 \<bottom> _ _ qstar..
	+
	+sublocale bdar: antirange_semiring "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "\<lambda>x. 1 - cd x" _ _
	+ apply unfold_locales
	+ apply (metis ad_ax1 ad_do dmq.mqs.local_var local.docd_compat)
	+ apply (metis ad_do local.cddo_compat local.dmq.coddual.dqmsr.dsr2 local.docd_compat local.sup.idem)
	+ by (metis bdad.a_subid' bdad.as3 local.convdqka.subid_conv local.do_inv)
	+
	+sublocale bdaka: antirange_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> _ _ qstar "\<lambda>x. 1 - cd x"..
	+
	+sublocale bmod: modal_semiring_simp "\<lambda>x. 1 - do x" "(\<squnion>)" "(\<cdot>)" 1 \<bottom> _ _ "\<lambda>x. 1 - cd x"..
	+
	+sublocale bmod: modal_kleene_algebra_simp "(\<squnion>)" "(\<cdot>)" 1 \<bottom> _ _ qstar "\<lambda>x. 1 - do x" "\<lambda>x. 1 - cd x"..
	+
	+lemma inv_neg: "(-x)\<^sup>\<circ> = -(x\<^sup>\<circ>)"
	+ by (metis local.diff_eq local.diff_shunt_var local.dual_order.eq_iff local.inf.commute local.inv_conjugate local.inv_galois)
	+
	+lemma residuation: "x\<^sup>\<circ> \<cdot> -(x \<cdot> y) \<le> -y"
	+ by (metis local.inf.commute local.inf_compl_bot local.inf_shunt local.schroeder_1)
	+
	+lemma bres_prop: "x \<rightarrow> y = -(x\<^sup>\<circ> \<cdot> -y)"
	+ by (metis local.ba_dual.dual_iff local.bres_canc1 local.bres_galois_imp local.compl_le_swap1 local.dmq.coddual.h_w1 local.dual_order.antisym local.inv_invol residuation)
	+
	+lemma fres_prop: "x \<leftarrow> y = -(-x \<cdot> y\<^sup>\<circ>)"
	+ using bres_prop inv_neg local.fres_bres_conv by auto
	+
	+lemma do_dia_fdia: "do_dia x p = bdad.fdia x p"
	+ by (simp add: local.bdad.fdia_def local.dmq.dqmsr.fd_def)
	+
	+lemma cd_dia_bdia: "cd_dia x p = bdar.bdia x p"
	+ by (metis ad_do bdar.ardual.a_subid' bdar.bdia_def local.cd_fix_subid local.dmq.coddual.dqmsr.fd_def local.docd_compat)
	+
	+lemma do_dia_fbox_de_morgan: "p \<le> 1 \<Longrightarrow> do_dia x p = 1 - bdad.fbox x (1 - p)"
	+ by (smt (verit, ccfv_SIG) ad_do bdad.a_closure bdad.am_d_def bdad.fbox_def local.dmq.dqmsr.fd_def local.do_fix_subid)
	+
	+lemma fbox_do_dia_de_morgan: "p \<le> 1 \<Longrightarrow> bdad.fbox x p = 1 - do_dia x (1 - p)"
	+ using bdad.fbox_def local.dmq.dqmsr.fd_def local.do_fix_subid by force
	+
	+lemma cd_dia_bbox_de_morgan: "p \<le> 1 \<Longrightarrow> cd_dia x p = 1 - bdar.bbox x (1 - p)"
	+ by (smt (verit, best) ad_do bdar.bbox_def bdar.bdia_def cd_dia_bdia local.cd_fix_subid local.do_subid local.docd_compat)
	+
	+lemma bbox_cd_dia_de_morgan: "p \<le> 1 \<Longrightarrow> bdar.bbox x p = 1 - cd_dia x (1 - p)"
	+ using bdar.bbox_def local.cd_fix_subid local.dmq.coddual.dqmsr.fd_def by force
	+
	+lemma do_box_bbox: "p \<le> 1 \<Longrightarrow> do_box x p = bdar.bbox x p"
	+proof-
	+ assume a: "p \<le> 1"
	+ {fix q
	+ assume b: "q \<le> 1"
	+ hence "(q \<le> do_box x p) = (x \<cdot> q \<le> p \<cdot> x)"
	+ by (simp add: a local.do_box_demod_subid)
	+ also have "\<dots> = (x \<cdot> cd q \<le> cd p \<cdot> x)"
	+ by (metis a b local.cd_fix_subid)
	+ also have "\<dots> = (cd q \<le> bdar.bbox x p)"
	+ by (metis bdar.bbox_def bdar.bdia_def cd_dia_bdia local.bdar.ardual.a_closure' local.bdar.ardual.ans_d_def local.bdar.ardual.dnsz.dsg1 local.bdar.ardual.fbox_demodalisation3 local.bdar.ardual.fbox_one local.dmq.coddual.dqmsr.fd_def local.nsrnqo.mult_oner)
	+ also have "\<dots> = (q \<le> bdar.bbox x p)"
	+ using b local.cd_fix_subid by force
	+ finally have "(q \<le> do_box x p) = (q \<le> bdar.bbox x p)".}
	+ thus ?thesis
	+ by (metis bdar.bbox_def local.bdar.ardual.a_subid' local.do_box_subid local.dual_order.antisym local.eq_refl)
	+qed
	+
	+lemma cd_box_fbox: "p \<le> 1 \<Longrightarrow> cd_box x p = bdad.fbox x p"
	+ by (smt (verit, ccfv_SIG) bdar.bbox_def do_box_bbox local.bdad.fbox_def local.cd_dia_do_dia_conv local.cd_inv local.cd_fix_subid local.cd_top local.diff_eq local.dmq.bb_def local.dmq.coddual.dqmsr.fd_def local.dmq.coddual.mult_assoc local.dmq.dqmsr.fd_def local.dmq.fb_def local.do_box_cd_box_conv local.do_fix_subid local.do_top local.inf.cobounded1)
	+
	+lemma do_dia_cd_box_de_morgan: "p \<le> 1 \<Longrightarrow> do_dia x p = 1 - cd_box x (1 - p)"
	+ by (simp add: cd_box_fbox local.diff_eq local.do_dia_fbox_de_morgan)
	+
	+lemma cd_box_do_dia_de_morgan: "p \<le> 1 \<Longrightarrow> cd_box x p = 1 - do_dia x (1 - p)"
	+ by (simp add: cd_box_fbox local.fbox_do_dia_de_morgan)
	+
	+lemma cd_dia_do_box_de_morgan: "p \<le> 1 \<Longrightarrow> cd_dia x p = 1 - do_box x (1 - p)"
	+ by (simp add: do_box_bbox local.cd_dia_bbox_de_morgan local.diff_eq)
	+
	+lemma do_box_cd_dia_de_morgan: "p \<le> 1 \<Longrightarrow> do_box x p = 1 - cd_dia x (1 - p)"
	+ by (simp add: do_box_bbox local.bbox_cd_dia_de_morgan)
	+
	+end
	+
	+class dc_involutive_modal_quantale = dc_modal_quantale + involutive_quantale
	+
	+begin
	+
	+sublocale invmqmka: involutive_dr_modal_kleene_algebra "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar invol dom cod..
	+
	+lemma do_approx_dom: "do x \<le> dom x"
	+proof -
	+ have "do x = dom (do x) \<cdot> do x"
	+ by simp
	+ also have "\<dots> \<le> dom (1 \<sqinter> (x \<cdot> x\<^sup>\<circ>))"
	+ by (simp add: local.do_def local.dqmsr.domain_subid)
	+ also have "\<dots> \<le> dom 1 \<sqinter> dom (x \<cdot> x\<^sup>\<circ>)"
	+ using local.dom_meet_sub by presburger
	+ also have "\<dots> \<le> dom (x \<cdot> dom (x\<^sup>\<circ>))"
	+ by simp
	+ also have "\<dots> \<le> dom x"
	+ by (simp add: local.dqmsr.domain_1'')
	+ finally show ?thesis.
	+qed
	+
	+end
	+
	+class dc_modal_quantale_converse = dc_involutive_modal_quantale + quantale_converse
	+
	+sublocale dc_modal_quantale_converse \<subseteq> invmqmka: dr_modal_kleene_algebra_converse "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar invol dom cod..
	+
	+class dc_modal_quantale_strong_converse = dc_involutive_modal_quantale +
	+ assumes weak_dom_def: "dom x \<le> x \<cdot> x\<^sup>\<circ>"
	+ and weak_cod_def: "cod x \<le> x\<^sup>\<circ> \<cdot> x"
	+
	+begin
	+
	+sublocale invmqmka: dr_modal_kleene_algebra_strong_converse "(\<squnion>)" "(\<cdot>)" 1 \<bottom> "(\<le>)" "(<)" qstar invol dom cod
	+ by (unfold_locales, simp_all add: local.weak_dom_def local.weak_cod_def)
	+
	+lemma dom_def: "dom x = 1 \<sqinter> (x \<cdot> x\<^sup>\<circ>)"
	+ using local.do_approx_dom local.do_def local.dual_order.eq_iff local.weak_dom_def by force
	+
	+lemma cod_def: "cod x = 1 \<sqinter> (x\<^sup>\<circ> \<cdot> x)"
	+ using local.dom_def local.invmqmka.d_conv_cod by auto
	+
	+lemma do_dom: "do x = dom x"
	+ by (simp add: local.do_def local.dom_def)
	+
	+lemma cd_cod: "cd x = cod x"
	+ by (simp add: local.cd_def local.cod_def)
	+
	+end
	+
	+class dc_modal_dedekind_quantale = dc_involutive_modal_quantale + dedekind_quantale
	+
	+class cd_distributive_modal_dedekind_quantale = dc_modal_dedekind_quantale + distrib_unital_quantale
	+
	+class dc_boolean_modal_dedekind_quantale = dc_modal_dedekind_quantale + bool_unital_quantale
	+
	+begin
	+
	+lemma subid_idem: "p \<le> 1 \<Longrightarrow> p \<cdot> p = p"
	+ by (simp add: local.subid_mult_meet)
	+
	+lemma subid_comm: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> p \<cdot> q = q \<cdot> p"
	+ using local.inf.commute local.subid_mult_meet by force
	+
	+lemma subid_meet_comp: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> p \<sqinter> q = p \<cdot> q"
	+ by (simp add: local.subid_mult_meet)
	+
	+lemma subid_dom: "p \<le> 1 \<Longrightarrow> dom p = p"
	+proof-
	+ assume h: "p \<le> 1"
	+ have a: "p \<squnion> (1 \<sqinter> -p) = 1"
	+ by (metis h local.inf_sup_absorb local.sup.commute local.sup.orderE local.sup_compl_top local.sup_inf_distrib1)
	+ have b: "(1 \<sqinter> -p) \<sqinter> p = \<bottom>"
	+ by (simp add: local.inf.commute)
	+ hence "dom p = (p \<squnion> (1 \<sqinter> -p)) \<cdot> dom p"
	+ by (simp add: a)
	+ also have "\<dots> = p \<cdot> dom p \<squnion> dom ((1 \<sqinter> -p) \<cdot> dom p) \<cdot> (1 \<sqinter> -p) \<cdot> dom p"
	+ using local.dqmsr.dsg1 local.wswq.distrib_right mult_assoc by presburger
	+ also have "\<dots> \<le> p \<squnion> dom ((1 \<sqinter> -p) \<cdot> dom p)"
	+ by (metis b h local.dom_subid local.dom_zero local.inf.cobounded1 local.mqdual.cod_local local.mult_botr local.sup.mono subid_comm subid_meet_comp)
	+ also have "\<dots> = p \<squnion> dom ((1 \<sqinter> -p) \<cdot> p)"
	+ by simp
	+ also have "\<dots> = p \<squnion> dom \<bottom>"
	+ using b h subid_meet_comp by fastforce
	+ also have "\<dots> = p"
	+ by simp
	+ finally have "dom p \<le> p".
	+ thus ?thesis
	+ using h local.dqmsr.domain_subid local.dual_order.antisym by presburger
	+qed
	+
	+lemma do_prop: "(do x \<le> do y) = (x \<le> do y \<cdot> \<top>)"
	+ by (simp add: local.lla)
	+
	+lemma do_lla: "(do x \<le> do y) = (x \<le> do y \<cdot> x)"
	+ by (simp add: local.lla_var)
	+
	+lemma lla_subid: "p \<le> 1 \<Longrightarrow> ((dom x \<le> p) = (x \<le> p \<cdot> x))"
	+ by (metis local.dqmsr.dom_llp subid_dom)
	+
	+lemma dom_do: "dom x = do x"
	+proof-
	+ have "x \<le> do x \<cdot> x"
	+ by simp
	+ hence "dom x \<le> do x"
	+ using lla_subid local.do_subid local.dodo by presburger
	+ thus ?thesis
	+ by (simp add: local.antisym_conv local.do_approx_dom)
	+qed
	+
	+end
	+
	+end
	+
	diff --git a/thys/Quantales_Converse/ROOT b/thys/Quantales_Converse/ROOT
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/ROOT
	@@ -0,0 +1,20 @@
	+chapter AFP
	+
	+session "Quantales_Converse" (AFP) = HOL +
	+ options [timeout = 600]
	+
	+ sessions
	+ Kleene_Algebra
	+ KAD
	+ Quantales
	+
	+ theories
	+ Modal_Kleene_Algebra_Var
	+ Kleene_Algebra_Converse
	+ Modal_Kleene_Algebra_Converse
	+ Modal_Quantale
	+ Quantale_Converse
	+
	+ document_files
	+ "root.tex"
	+ "root.bib"
	diff --git a/thys/Quantales_Converse/document/root.bib b/thys/Quantales_Converse/document/root.bib
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/document/root.bib
	@@ -0,0 +1,126 @@
	+@article{ArmstrongSW13,
	+ author = {Alasdair Armstrong and
	+ Georg Struth and
	+ Tjark Weber},
	+ title = {Kleene Algebra},
	+ journal = {Archive of Formal Proofs},
	+ year = {2013},
	+ url = {https://www.isa-afp.org/entries/Kleene\_Algebra.shtml},
	+ timestamp = {Mon, 25 May 2020 09:13:27 +0200},
	+ biburl = {https://dblp.org/rec/journals/afp/ArmstrongSW13.bib},
	+ bibsource = {dblp computer science bibliography, https://dblp.org}
	+}
	+
	+@article{CalkGMS22,
	+ author = {Cameron Calk and
	+ Eric Goubault and
	+ Philippe Malbos and
	+ Georg Struth},
	+ title = {Algebraic coherent confluence and higher globular {K}leene algebras},
	+ journal = {Logical Methods in Computer Science},
	+ volume = {18},
	+ number = {4},
	+ year = {2022},
	+ url = {https://doi.org/10.46298/lmcs-18(4:9)2022},
	+ doi = {10.46298/lmcs-18(4:9)2022},
	+ timestamp = {Tue, 24 Jan 2023 10:48:24 +0100},
	+ biburl = {https://dblp.org/rec/journals/lmcs/CalkGMS22.bib},
	+ bibsource = {dblp computer science bibliography, https://dblp.org}
	+}
	+
	+@article{CalkMPS23,
	+Author = {Cameron Calk and Philippe Malbos and Damien Pous and Georg Struth},
	+Title = {Higher Catoids, Higher Quantales and their Correspondences},
	+Year = {2023},
	+journal = {arXiv},
	+volume = {2307.09253},
	+}
	+
	+@article{FahrenbergJSZ23,
	+ author = {Fahrenberg, Uli and
	+ Johansen, Christian and
	+ Struth, Georg and
	+ Ziemia{\'n}ski, Krzysztof},
	+ title = {Catoids and Modal Convolution Algebras},
	+ journal = {Algebra Universalis},
	+ volume = {84},
	+ pages = {10},
	+ year = {2023}
	+}
	+
	+@article{GomesGHSW16,
	+ author = {Victor B. F. Gomes and
	+ Walter Guttmann and
	+ Peter H{\"{o}}fner and
	+ Georg Struth and
	+ Tjark Weber},
	+ title = {Kleene Algebras with Domain},
	+ journal = {Archive of Formal Proofs},
	+ year = {2016},
	+ url = {https://www.isa-afp.org/entries/KAD.shtml},
	+ timestamp = {Mon, 25 May 2020 09:13:24 +0200},
	+ biburl = {https://dblp.org/rec/journals/afp/GomesGHSW16.bib},
	+ bibsource = {dblp computer science bibliography, https://dblp.org}
	+}
	+
	+@inproceedings{GomesS16,
	+ author = {Victor B. F. Gomes and Georg Struth},
	+ title = {Modal {K}leene Algebra Applied to Program Correctness},
	+ booktitle = {{FM} 2016},
	+ series = {LNCS},
	+ volume = {9995},
	+ pages = {310--325},
	+ year = {2016},
	+ url = {https://doi.org/10.1007/978-3-319-48989-6\_19},
	+ doi = {10.1007/978-3-319-48989-6\_19},
	+ timestamp = {Tue, 14 May 2019 10:00:46 +0200},
	+ biburl = {https://dblp.org/rec/conf/fm/GomesS16.bib},
	+ bibsource = {dblp computer science bibliography, https://dblp.org}
	+}
	+
	+@Unpublished{PousS23,
	+ author = {Damien Pous and Georg Struth},
	+ title = {Dedekind Quantaloids as Intuitionistic Modal Algebras},
	+ note = {In preparation},
	+ year = 2023}
	+
	+@Article{Resende18,
	+ author = {Pedro Resende},
	+ title = {Open Maps of Involutive Quantales},
	+ journal = {Applied Categorical Structures},
	+ year = 2018,
	+ volume = 26,
	+ pages = {631-644}}
	+
	+@Book{Rosenthal90,
	+ author = {Rosenthal, Kimmo. I.},
	+ title = {Quantales and Their Applications},
	+ publisher = {Longman Scientific $\&$ Technical},
	+ year = 1990}
	+
	+@Book{Rosenthal96,
	+ author = {Rosenthal, K. I.},
	+ title = {The Theory of Quantaloids},
	+ publisher = {Addison Wesley Longman Limited},
	+ year = 1996}
	+
	+@article{Struth18,
	+ author = {Georg Struth},
	+ title = {Quantales},
	+ journal = {Archive of Formal Proofs},
	+ year = {2018},
	+ url = {https://www.isa-afp.org/entries/Quantales.html},
	+ timestamp = {Mon, 25 May 2020 09:13:16 +0200},
	+ biburl = {https://dblp.org/rec/journals/afp/Struth18a.bib},
	+ bibsource = {dblp computer science bibliography, https://dblp.org}
	+}
	+
	+@Article{Tarski41,
	+ author = {Alfred Tarski},
	+ title = {On the Calculus of Relations},
	+ journal = {Journal of Symbolic Logic},
	+ year = 1941,
	+ volume = 6,
	+ number = 3,
	+ pages = {73--89}}
	+
	diff --git a/thys/Quantales_Converse/document/root.tex b/thys/Quantales_Converse/document/root.tex
	new file mode 100644
	--- /dev/null
	+++ b/thys/Quantales_Converse/document/root.tex
	@@ -0,0 +1,121 @@
	+\documentclass[11pt,a4paper]{article}
	+\usepackage[T1]{fontenc}
	+\usepackage{isabelle,isabellesym}
	+
	+% further packages required for unusual symbols (see also
	+% isabellesym.sty), use only when needed
	+
	+%\usepackage{amssymb}
	+ %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
	+ %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
	+ %\<triangleq>, \<yen>, \<lozenge>
	+
	+%\usepackage{eurosym}
	+ %for \<euro>
	+
	+\usepackage[only,bigsqcap,bigparallel,fatsemi,interleave,sslash]{stmaryrd}
	+ %for \<Sqinter>, \<Parallel>, \<Zsemi>, \<Parallel>, \<sslash>
	+
	+%\usepackage{eufrak}
	+ %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
	+
	+%\usepackage{textcomp}
	+ %for \<onequarter>, \<onehalf>, \<threequarters>, \<degree>, \<cent>,
	+ %\<currency>
	+
	+% this should be the last package used
	+\usepackage{pdfsetup}
	+
	+% urls in roman style, theory text in math-similar italics
	+\urlstyle{rm}
	+\isabellestyle{it}
	+
	+% for uniform font size
	+%\renewcommand{\isastyle}{\isastyleminor}
	+
	+\begin{document}
	+
	+\title{Modal quantales, involutive Quantales, Dedekind Quantales}
	+\author{Cameron Calk and Georg Struth}
	+\maketitle
	+
	+\begin{abstract}
	+ This AFP entry provides mathematical components for modal quantales,
	+ involutive quantales and Dedekind quantales. Modal quantales are
	+ simple extensions of modal Kleene algebras useful for the
	+ verification of recursive programs. Involutive quantales appear in
	+ the study of $C^\ast$-algebras. Dedekind quantales are relatives of
	+ Tarski's relation algebras, hence relevant to program verification
	+ and beyond that to higher rewriting. We also provide
	+ components for weaker variants such as Kleene algebras with converse
	+ and modal Kleene algebras with converse.
	+\end{abstract}
	+
	+\tableofcontents
	+
	+% sane default for proof documents
	+\parindent 0pt\parskip 0.5ex
	+
	+% generated text of all theories
	+
	+\section{Introductory Remarks}
	+
	+In this AFP entry we provide mathematical components for modal
	+quantales, involutive quantales and Dedekind quantales. Modal
	+quantales are simple extensions of modal Kleene algebras that can be
	+used in the verification of recursive
	+programs~\cite{GomesS16}. Involutive quantales appear in the study of
	+$C^\ast$-algebras~\cite{Resende18}. Dedekind quantales,
	+categorifications of which are known as \emph{modular
	+ quantaloids}~\cite{Rosenthal96}, are relatives of Tarski's relation
	+algebras~\cite{Tarski41}, and hence relevant to program verification
	+as well. We also provide components for weaker variants such as Kleene
	+algebras and modal Kleene algebras with converse.
	+
	+Our main interest in these structures comes from recent applications
	+in higher-dimensional rewriting~\cite{CalkGMS22,CalkMPS23}, where they
	+are used in coherence proofs for rewriting systems based on computads
	+or polygraphs. This includes proofs of coherent Church-Rosser theorems
	+and coherent Newman's lemmas. A more long-term programme considers
	+the formalisation of algebraic aspects of higher rewriting with proof
	+assistants.
	+
	+ Modal quantales have previously been studied
	+ in~\cite{FahrenbergJSZ23}, where it is shown, for instance, that any
	+ category can be lifted to a modal quantale at powerset level. Such
	+ lifting results will be formalised in a companion AFP entry.
	+
	+ Dedekind quantales give also rise to intuitionistic modal algebras,
	+ as the results in this AFP entry show. In particular, the set of all
	+ subidentities or coreflexives of a Dedekind quantale forms a
	+ complete Heyting algebra (aka frame or locale), on which modal box
	+ and diamond operators can be defined. A paper explaining these
	+ results is in preparation~\cite{PousS23}. A further application of
	+ Dedekind quantales lies once again in higher-dimensional
	+ rewriting~\cite{CalkGMS22,CalkMPS23}. Any groupoid, in particular,
	+ can be lifted to a Dedekind quantale at powerset level, a result
	+ which will once again be formalised in a companion AFP entry.
	+
	+ Our components build on extant AFP components for Kleene
	+ algebras~\cite{ArmstrongSW13}, modal Kleene
	+ algebras~\cite{GomesGHSW16} and quantales~\cite{Struth18}.
	+
	+ \vspace{\baselineskip}
	+
	+ Georg Struth is grateful for an invited professorship at École
	+ polytechnique and a fellowship at the Collegium de Lyon, Institute
	+ of Advanced Study, during which most of this formalisation work
	+ has been done.
	+
	+\input{session}
	+
	+% optional bibliography
	+\bibliographystyle{abbrv}
	+\bibliography{root}
	+
	+\end{document}
	+
	+%%% Local Variables:
	+%%% mode: latex
	+%%% TeX-master: t
	+%%% End:
	diff --git a/thys/ROOTS b/thys/ROOTS
	--- a/thys/ROOTS
	+++ b/thys/ROOTS
	@@ -1,757 +1,760 @@
	ABY3_Protocols
	ADS_Functor
	AI_Planning_Languages_Semantics
	AODV
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	AVL-Trees
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	Architectural_Design_Patterns
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	ArrowImpossibilityGS
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	Pi_Calculus
	Pi_Transcendental
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	+Polygonal_Number_Theorem
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	+Quantales_Converse
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	Sort_Encodings
	Source_Coding_Theorem
	SpecCheck
	Special_Function_Bounds
	Splay_Tree
	Sqrt_Babylonian
	Stable_Matching
	Stalnaker_Logic
	Statecharts
	Stateful_Protocol_Composition_and_Typing
	Stellar_Quorums
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	Stochastic_Matrices
	Stone_Algebras
	Stone_Kleene_Relation_Algebras
	Stone_Relation_Algebras
	Store_Buffer_Reduction
	Stream-Fusion
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	StrictOmegaCategories
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	Sturm_Tarski
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	Subresultants
	Subset_Boolean_Algebras
	SumSquares
	Sunflowers
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	Surprise_Paradox
	Symmetric_Polynomials
	Syntax_Independent_Logic
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	Szemeredi_Regularity
	Szpilrajn
	TESL_Language
	TLA
	Tail_Recursive_Functions
	Tarskis_Geometry
	Taylor_Models
	Three_Circles
	Three_Squares
	Timed_Automata
	Topological_Semantics
	Topology
	TortoiseHare
	TsirelsonBound
	Transcendence_Series_Hancl_Rucki
	Transformer_Semantics
	Transition_Systems_and_Automata
	Transitive-Closure
	Transitive-Closure-II
	Transitive_Models
	Treaps
	Tree-Automata
	Tree_Decomposition
	Tree_Enumeration
	Triangle
	Trie
	Turans_Graph_Theorem
	Twelvefold_Way
	Two_Generated_Word_Monoids_Intersection
	Tycon
	Types_Tableaus_and_Goedels_God
	Types_To_Sets_Extension
	UPF
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	UTP
	Undirected_Graph_Theory
	Universal_Hash_Families
	Universal_Turing_Machine
	UpDown_Scheme
	VYDRA_MDL
	Valuation
	Van_Emde_Boas_Trees
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	VeriComp
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	pGCL
	diff --git a/tools/afp_submit.scala b/tools/afp_submit.scala
	--- a/tools/afp_submit.scala
	+++ b/tools/afp_submit.scala
	@@ -1,1616 +1,1617 @@
	/* Author: Fabian Huch, TU Muenchen

	AFP submission system backend.
	*/
	package afp


	import isabelle.*
	import isabelle.HTML.*

	import afp.Web_App.{ACTION, API, FILE, Params}
	import afp.Web_App.Params.{List_Key, Nest_Key, empty}
	import afp.Web_App.HTML.*
	import afp.Metadata.{Affiliation, Author, DOI, Email, Entry, Formatted, Homepage, License, Orcid, Reference, Release, Topic, Unaffiliated}

	import java.net.URL
	import java.text.Normalizer
	import java.time.LocalDate

	import scala.collection.immutable.StringView
	import scala.util.matching.Regex


	object AFP_Submit {

	case class Val_Opt[A] private(opt: Option[A], err: Option[String]) {
	def is_empty: Boolean = opt.isEmpty
	}
	object Val_Opt {
	def ok[A](value: A): Val_Opt[A] = Val_Opt(Some(value), None)
	def user_err[A](msg: String): Val_Opt[A] = Val_Opt(None, Some(msg))
	def error[A]: Val_Opt[A] = Val_Opt(None, None)
	}

	case class Val[A] private(v: A, err: Option[String]) {
	def with_err(err: String): Val[A] = Val.err(v, err)
	def perhaps_err(opt: Val_Opt[_]): Val[A] = opt.err.map(with_err).getOrElse(this)
	}

	object Val {
	def ok[A](value: A): Val[A] = Val(value, None)
	def err[A](value: A, msg: String): Val[A] = Val(value, Some(msg))
	}


	/* data model */

	object Model {
	object Related extends Enumeration {
	val DOI, Plaintext = Value

	def from_string(s: String): Option[Value] = values.find(_.toString == s)
	def get(r: Reference): Value = r match {
	case afp.Metadata.DOI(_) => DOI
	case afp.Metadata.Formatted(_) => Plaintext
	}
	}

	case class Create_Entry(
	name: Val[String] = Val.ok(""),
	title: Val[String] = Val.ok(""),
	affils: Val[List[Affiliation]] = Val.ok(Nil),
	notifies: Val[List[Email]] = Val.ok(Nil),
	author_input: Option[Author] = None,
	notify_input: Option[Author] = None,
	topics: Val[List[Topic]] = Val.ok(Nil),
	topic_input: Option[Topic] = None,
	license: License,
	`abstract`: Val[String] = Val.ok(""),
	related: List[Reference] = Nil,
	related_kind: Option[Related.Value] = None,
	related_input: Val[String] = Val.ok("")
	) {
	val used_affils: Set[Affiliation] = (affils.v ++ notifies.v).toSet
	val used_authors: Set[Author.ID] = used_affils.map(_.author)
	}

	case class Create(
	entries: Val[List[Create_Entry]] = Val.ok(Nil),
	new_authors: Val[List[Author]] = Val.ok(Nil),
	new_author_input: String = "",
	new_author_orcid: String = "",
	new_affils: Val[List[Affiliation]] = Val.ok(Nil),
	new_affils_author: Option[Author] = None,
	new_affils_input: String = "",
	) extends T {
	def update_entry(num: Int, entry: Create_Entry): Create =
	this.copy(entries = Val.ok(entries.v.updated(num, entry)))

	def updated_authors(old_authors: Map[Author.ID, Author]): Map[Author.ID, Author] =
	(old_authors ++ new_authors.v.map(author => author.id -> author)).map {
	case (id, author) =>
	val emails =
	author.emails ++ new_affils.v.collect { case e: Email if e.author == id => e }
	val homepages =
	author.homepages ++ new_affils.v.collect { case h: Homepage if h.author == id => h }
	id -> author.copy(emails = emails.distinct, homepages = homepages.distinct)
	}

	val used_affils: Set[Affiliation] = entries.v.toSet.flatMap(_.used_affils)
	val used_authors: Set[Author.ID] =
	new_affils.v.map(_.author).toSet ++ entries.v.flatMap(_.used_authors)

	def create(authors: Map[Author.ID, Author]): (Map[Author.ID, Author], List[Entry]) =
	(updated_authors(authors), entries.v.map(entry =>
	Entry(
	name = entry.name.v,
	title = entry.title.v,
	authors = entry.affils.v,
	date = LocalDate.now(),
	topics = entry.topics.v,
	`abstract` = entry.`abstract`.v.trim,
	notifies = entry.notifies.v,
	license = entry.license,
	note = "",
	related = entry.related)))
	}

	object Build extends Enumeration {
	val Pending, Running, Aborted, Failed, Success = Value
	}

	object Status extends Enumeration {
	val Submitted, Review, Added, Rejected = Value

	def from_string(s: String): Option[Value] = values.find(_.toString == s)
	}

	case class Overview(id: String, date: LocalDate, name: String, status: Status.Value)
	case class Metadata(authors: Map[Author.ID, Author], entries: List[Entry]) {
	def new_authors(old_authors: Map[Author.ID, Author]): Set[Author] =
	entries.flatMap(_.authors).map(_.author).filterNot(old_authors.contains).toSet.map(authors)

	def new_affils(old_authors: Map[Author.ID, Author]): Set[Affiliation] =
	entries.flatMap(entry => entry.authors ++ entry.notifies).toSet.filter {
	case _: Unaffiliated => false
	case e: Email => !old_authors.get(e.author).exists(_.emails.contains(e))
	case h: Homepage => !old_authors.get(h.author).exists(_.homepages.contains(h))
	}
	}

	sealed trait T
	case object Invalid extends T
	case class Upload(meta: Metadata, message: String, error: String) extends T
	case class Created(id: String) extends T
	case class Submission(
	id: Handler.ID,
	meta: Metadata,
	message: String,
	build: Build.Value,
	status: Option[Status.Value],
	log: String) extends T
	case class Submission_List(submissions: List[Overview]) extends T
	}


	/* Physical submission handling */

	trait Handler {
	def create(
	date: Date,
	meta: Model.Metadata,
	message: String,
	archive: Bytes,
	ext: String
	): Handler.ID
	def list(): Model.Submission_List
	def get(id: Handler.ID,
	topics: Map[Topic.ID, Topic],
	licenses: Map[License.ID, License]
	): Option[Model.Submission]
	def submit(id: Handler.ID): Unit
	def set_status(id: Handler.ID, status: Model.Status.Value): Unit
	def abort_build(id: Handler.ID): Unit
	def get_patch(id: Handler.ID): Option[Path]
	def get_archive(id: Handler.ID): Option[Path]
	}
	object Handler {
	import Model._

	type ID = String

	object ID {
	private val format = Date.Format.make(
	List(
	Date.Formatter.pattern("uuuu-MM-dd_HH-mm-ss_SSS"),
	Date.Formatter.pattern("uuuu-MM-dd_HH-mm-ss_SSS_VV")),
	_ + "_" + Date.timezone().getId)

	def apply(submission_time: Date): ID = format(submission_time)
	def unapply(id: ID): Option[Date] = format.unapply(id)
	def check(id: ID): Option[ID] = unapply(id).map(apply)
	}


	/* Handler for local edits */

	class Edit(afp_structure: AFP_Structure) extends Handler {
	val authors = afp_structure.load_authors.map(author => author.id -> author).toMap
	val topics = afp_structure.load_topics.flatMap(_.all_topics)
	val all_topics = topics.map(topic => topic.id -> topic).toMap
	val licenses = afp_structure.load_licenses.map(license => license.id -> license).toMap
	val releases = afp_structure.load_releases.groupBy(_.entry)
	val dates = afp_structure.load().map(entry => entry.name -> entry.date).toMap

	override def create(
	date: Date,
	meta: Metadata,
	message: String,
	archive: Bytes,
	ext: String
	): ID = {
	val entry =
	meta.entries match {
	case e :: Nil => e
	case _ => isabelle.error("Must be a single entry")
	}

	val old = afp_structure.load_entry(entry.name, authors, all_topics, licenses, releases)
	val updated =
	old.copy(
	title = entry.title,
	authors = entry.authors,
	topics = entry.topics,
	`abstract` = entry.`abstract`,
	notifies = entry.notifies,
	license = entry.license,
	related = entry.related)

	afp_structure.save_entry(updated)
	// TODO what happens to the authors

	entry.name
	}

	override def list(): Submission_List =
	Submission_List(afp_structure.entries.sortBy(dates.get).reverse.map { entry =>
	Overview(entry, dates(entry), entry, Status.Added)
	})

	override def get(
	id: ID, topics: Map[ID, Topic], licenses: Map[ID, License]
	): Option[Submission] =
	if (!afp_structure.entries.contains(id)) None
	else {
	val entry = afp_structure.load_entry(id, authors, all_topics, licenses, releases)
	val meta = Metadata(authors, List(entry))
	Some(Submission(id, meta, "", Model.Build.Success, Some(Status.Added), ""))
	}

	override def submit(id: ID): Unit = ()
	override def set_status(id: ID, status: Model.Status.Value): Unit = ()
	override def abort_build(id: ID): Unit = ()
	override def get_patch(id: ID): Option[Path] = None
	override def get_archive(id: ID): Option[Path] = None
	}


	/* Adapter to existing submission system */

	class Adapter(submission_dir: Path, afp_structure: AFP_Structure) extends Handler {

	private val up: Path = submission_dir + Path.basic("up")
	private def up(id: ID): Path = up + Path.basic(id)
	private def down(id: ID): Path = submission_dir + Path.basic("down") + Path.basic(id)

	private def signal(id: ID, s: String): Unit =
	File.write(up(id) + Path.basic(s), s.toUpperCase)
	private def is_signal(id: ID, s: String): Boolean = (up(id) + Path.basic(s)).file.exists()

	private def read_build(id: ID): Build.Value = {
	val build = down(id) + Path.basic("result")
	if (!build.file.exists) Build.Pending
	else File.read(build).trim match {
	case "" => Build.Running
	case "NOT_FINISHED" => Build.Running
	case "FAILED" => if (is_signal(id, "kill")) Build.Aborted else Build.Failed
	case "SUCCESS" => Build.Success
	case s => isabelle.error("Unkown build status: " + quote(s))
	}
	}

	private def status_file(id: ID): Path = up(id) + Path.basic("AFP_STATUS")
	private def read_status(id: ID): Option[Status.Value] = {
	val status = status_file(id)
	if (!status.file.exists()) None
	else File.read(status).trim match {
	case "SUBMITTED" => Some(Status.Submitted)
	case "PROCESSING" => Some(Status.Review)
	case "REJECTED" => Some(Status.Rejected)
	case "ADDED" => Some(Status.Added)
	case s => isabelle.error("Unknown status: " + quote(s))
	}
	}

	private def info_file(id: ID): Path = up(id) + Path.basic("info.json")
	private def patch_file(id: ID): Path = up(id) + Path.basic("patch")

	private val archive_name: String = "archive"

	def make_partial_patch(base_dir: Path, src: Path, dst: Path): String = {
	val patch = Isabelle_System.make_patch(base_dir, src, dst, "--unidirectional-new-file")
	split_lines(patch).filterNot(_.startsWith("Only in")).mkString("\n")
	}

	def create(date: Date, meta: Metadata, message: String, archive: Bytes, ext: String): ID = {
	val id = ID(date)
	val dir = up(id)
	dir.file.mkdirs()

	val structure = AFP_Structure(dir)
	structure.save_authors(meta.authors.values.toList.sortBy(_.id))
	meta.entries.foreach(structure.save_entry)

	val archive_file = dir + Path.basic(archive_name + ext)
	Bytes.write(archive_file, archive)

	val metadata_rel =
	File.relative_path(afp_structure.base_dir, afp_structure.metadata_dir).getOrElse(
	afp_structure.metadata_dir)
	val metadata_patch =
	make_partial_patch(afp_structure.base_dir, metadata_rel, structure.metadata_dir)
	File.write(patch_file(id), metadata_patch)

	val info =
	JSON.Format(JSON.Object(
	"comment" -> message,
	"entries" -> meta.entries.map(_.name),
	"notify" -> meta.entries.flatMap(_.notifies).map(_.address).distinct))
	File.write(info_file(id), info)

	signal(id, "done")
	id
	}

	def list(): Submission_List =
	Submission_List(
	File.read_dir(up).flatMap(ID.unapply).reverse.flatMap { date =>
	val id = ID(date)
	val day = date.rep.toLocalDate
	read_status(id).map(Overview(id, day, AFP_Structure(up(id)).entries_unchecked.head, _))
	})

	def get(
	id: ID,
	topics: Map[Topic.ID, Topic],
	licenses: Map[License.ID, License]
	): Option[Submission] =
	ID.check(id).filter(up(_).file.exists).map { id =>
	val structure = AFP_Structure(up(id))
	val authors = structure.load_authors.map(author => author.id -> author).toMap
	val entries = structure.entries_unchecked.map(
	structure.load_entry(_, authors, topics, licenses, Map.empty))

	val log_file = down(id) + Path.basic("isabelle.log")
	val log = if (log_file.file.exists) File.read(log_file) else ""

	JSON.parse(File.read(info_file(id))) match {
	case JSON.Object(m) if m.contains("comment") =>
	val meta = Metadata(authors, entries)
	Submission(id, meta, m("comment").toString, read_build(id), read_status(id), log)
	case _ => isabelle.error("Could not read info")
	}
	}

	private def check(id: ID): Option[ID] = ID.check(id).filter(up(_).file.exists)

	def submit(id: ID): Unit = check(id).foreach(signal(_, "mail"))

	def set_status(id: ID, status: Status.Value): Unit =
	check(id).foreach { id =>
	val content =
	status match {
	case Status.Submitted => "SUBMITTED"
	case Status.Review => "PROCESSING"
	case Status.Added => "ADDED"
	case Status.Rejected => "REJECTED"
	}
	File.write(status_file(id), content)
	}

	def abort_build(id: ID): Unit = check(id).foreach(signal(_, "kill"))

	def get_patch(id: ID): Option[Path] = check(id).map(patch_file)
	def get_archive(id: ID): Option[Path] = check(id).flatMap { id =>
	val dir = up(id)
	File.read_dir(dir).find(_.startsWith(archive_name + ".")).map(dir + Path.basic(_))
	}
	}
	}


	/* server */

	object Mode extends Enumeration {
	val EDIT, SUBMISSION = Value
	}

	class Server(
	api: API,
	afp_structure: AFP_Structure,
	mode: Mode.Value,
	handler: Handler,
	devel: Boolean,
	verbose: Boolean,
	progress: Progress,
	port: Int = 0
	) extends Web_App.Server[Model.T](api, port, verbose, progress) {
	val repo = Mercurial.the_repository(afp_structure.base_dir)

	var authors: Map[Author.ID, Metadata.Author] = Map.empty
	var topics: List[Topic] = Nil
	var all_topics: Map[Topic.ID, Topic] = Map.empty
	var licenses: Map[License.ID, License] = Map.empty
	var releases: Map[Entry.Name, List[Release]] = Map.empty
	var entries: Set[Entry.Name] = Set.empty

	private def load(): Unit = synchronized {
	authors = afp_structure.load_authors.map(author => author.id -> author).toMap
	topics = afp_structure.load_topics.flatMap(_.all_topics)
	all_topics = topics.map(topic => topic.id -> topic).toMap
	licenses = afp_structure.load_licenses.map(license => license.id -> license).toMap
	releases = afp_structure.load_releases.groupBy(_.entry)
	entries = afp_structure.entries.toSet
	}

	load()


	/* endpoints */

	val SUBMIT = Path.explode("submit")
	val SUBMISSION = Path.explode("submission")
	val SUBMISSIONS = Path.explode("submissions")
	val API_SUBMISSION = Path.explode("api/submission")
	val API_SUBMISSION_UPLOAD = Path.explode("api/submission/upload")
	val API_SUBMISSION_CREATE = Path.explode("api/submission/create")
	val API_SUBMISSION_STATUS = Path.explode("api/submission/status")
	val API_RESUBMIT = Path.explode("api/resubmit")
	val API_BUILD_ABORT = Path.explode("api/build/abort")
	val API_SUBMIT = Path.explode("api/submit")
	val API_SUBMISSION_AUTHORS_ADD = Path.explode("api/submission/authors/add")
	val API_SUBMISSION_AUTHORS_REMOVE = Path.explode("api/submission/authors/remove")
	val API_SUBMISSION_AFFILIATIONS_ADD = Path.explode("api/submission/affiliations/add")
	val API_SUBMISSION_AFFILIATIONS_REMOVE = Path.explode("api/submission/affiliations/remove")
	val API_SUBMISSION_ENTRIES_ADD = Path.explode("api/submission/entries/add")
	val API_SUBMISSION_ENTRIES_REMOVE = Path.explode("api/submission/entries/remove")
	val API_SUBMISSION_ENTRY_TOPICS_ADD = Path.explode("api/submission/entry/topics/add")
	val API_SUBMISSION_ENTRY_TOPICS_REMOVE = Path.explode("api/submission/entry/topics/remove")
	val API_SUBMISSION_ENTRY_AUTHORS_ADD = Path.explode("api/submission/entry/authors/add")
	val API_SUBMISSION_ENTRY_AUTHORS_REMOVE = Path.explode("api/submission/entry/authors/remove")
	val API_SUBMISSION_ENTRY_NOTIFIES_ADD = Path.explode("api/submission/entry/notifies/add")
	val API_SUBMISSION_ENTRY_NOTIFIES_REMOVE = Path.explode("api/submission/entry/notifies/remove")
	val API_SUBMISSION_ENTRY_RELATED_ADD = Path.explode("api/submission/entry/related/add")
	val API_SUBMISSION_ENTRY_RELATED_REMOVE = Path.explode("api/submission/entry/related/remove")
	val API_SUBMISSION_DOWNLOAD = Path.explode("api/download/patch")
	val API_SUBMISSION_DOWNLOAD_ZIP = Path.explode("api/download/archive.zip")
	val API_SUBMISSION_DOWNLOAD_TGZ = Path.explode("api/download/archive.tar.gz")
	val API_CSS = Path.explode("api/main.css")


	/* fields */

	val ABSTRACT = "abstract"
	val ADDRESS = "address"
	val AFFILIATION = "affiliation"
	val ARCHIVE = "archive"
	val AUTHOR = "author"
	val DATE = "date"
	val ENTRY = "entry"
	val ID = "id"
	val INPUT = "input"
	val KIND = "kind"
	val LICENSE = "license"
	val MESSAGE = "message"
	val NAME = "name"
	val NOTIFY = "notify"
	val ORCID = "orcid"
	val RELATED = "related"
	val STATUS = "status"
	val TITLE = "title"
	val TOPIC = "topic"


	/* utils */

	def keyed(id: String, value: String): String = "[" + id + "] " + value

	def author_string(author: Author): String = {
	val orcid =
	author.orcid.map(orcid => " (ORCID id: " + orcid.identifier + ")").getOrElse("")
	keyed(author.id, author.name) + orcid
	}

	def author_option(author: Author): XML.Elem = option(author.id, author_string(author))

	def affil_id(affil: Affiliation): String =
	affil match {
	case Unaffiliated(_) => ""
	case Email(_, id, _) => id
	case Homepage(_, id, _) => id
	}

	def affil_address(affil: Affiliation): String =
	affil match {
	case Unaffiliated(_) => ""
	case Email(_, _, address) => address
	case Homepage(_, _, url) => url.toString
	}

	def affil_string(affil: Affiliation): String =
	affil match {
	case Unaffiliated(_) => "No email or homepage"
	case Email(_, id, address) => keyed(id, address)
	case Homepage(_, id, url) => keyed(id, url.toString)
	}

	def related_string(related: Reference): String = related match {
	case Metadata.DOI(identifier) => identifier
	case Metadata.Formatted(rep) => rep
	}

	def indexed[A, B](l: List[A], key: Params.Key, field: String, f: (A, Params.Key) => B) =
	l.zipWithIndex map {
	case (a, i) => f(a, List_Key(key, field, i))
	}

	def fold[A](it: List[Params.Data], a: A)(f: (Params.Data, A) => Option[A]): Option[A] =
	it.foldLeft(Option(a)) {
	case (None, _) => None
	case (Some(a), param) => f(param, a)
	}

	def download_link(href: String, body: XML.Body): XML.Elem =
	class_("download")(link(href, body)) + ("target" -> "_blank")
	def frontend_link(path: Path, params: Properties.T, body: XML.Body): XML.Elem =
	link(api.app_url(path, params), body) + ("target" -> "_parent")

	def render_if(cond: Boolean, body: XML.Body): XML.Body = if (cond) body else Nil
	def render_if(cond: Boolean, elem: XML.Elem): XML.Body = if (cond) List(elem) else Nil
	def render_error(for_elem: String, validated: Val[_]): XML.Body =
	validated.err.map(error =>
	break ::: List(css("color: red")(label(for_elem, error)))).getOrElse(Nil)


	/* view */

	def render_create(model: Model.Create): XML.Body = {
	val updated_authors = model.updated_authors(authors)
	val authors_list = updated_authors.values.toList.sortBy(_.id)
	val (entry_authors, other_authors) =
	updated_authors.values.toList.sortBy(_.id).partition(author =>
	model.used_authors.contains(author.id))
	val email_authors = authors_list.filter(_.emails.nonEmpty)

	def render_topic(topic: Topic, key: Params.Key): XML.Elem =
	par(
	hidden(Nest_Key(key, ID), topic.id) ::
	text(topic.id) :::
	action_button(api.api_url(API_SUBMISSION_ENTRY_TOPICS_REMOVE), "x", key) :: Nil)

	def render_affil(affil: Affiliation, key: Params.Key): XML.Elem = {
	val author = updated_authors(affil.author)
	val affils = author.emails ::: author.homepages ::: Unaffiliated(author.id) :: Nil
	par(
	hidden(Nest_Key(key, ID), affil.author) ::
	text(author_string(updated_authors(affil.author))) :::
	selection(Nest_Key(key, AFFILIATION),
	Some(affil_id(affil)),
	affils.map(affil => option(affil_id(affil), affil_string(affil)))) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_AUTHORS_REMOVE), "x", key) :: Nil)
	}

	def render_notify(email: Email, key: Params.Key): XML.Elem = {
	val author = updated_authors(email.author)
	par(
	hidden(Nest_Key(key, ID), email.author) ::
	text(author_string(updated_authors(email.author))) :::
	selection(
	Nest_Key(key, AFFILIATION),
	Some(affil_id(email)),
	author.emails.map(affil => option(affil_id(affil), affil_string(affil)))) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_NOTIFIES_REMOVE), "x", key) :: Nil)
	}

	def render_related(related: Reference, key: Params.Key): XML.Elem =
	par(
	hidden(Nest_Key(key, KIND), Model.Related.get(related).toString) ::
	hidden(Nest_Key(key, INPUT), related_string(related)) ::
	text(related_string(related)) :::
	action_button(api.api_url(API_SUBMISSION_ENTRY_RELATED_REMOVE), "x", key) :: Nil)

	def render_entry(entry: Model.Create_Entry, key: Params.Key): XML.Elem =
	fieldset(legend("Entry") ::
	par(
	fieldlabel(Nest_Key(key, TITLE), "Title of article") ::
	textfield(Nest_Key(key, TITLE), "Example Submission", entry.title.v) ::
	render_error(Nest_Key(key, TITLE), entry.title)) ::
	par(
	fieldlabel(Nest_Key(key, NAME), "Short name") ::
	textfield(Nest_Key(key, NAME), "Example_Submission", entry.name.v) ::
	explanation(Nest_Key(key, NAME),
	"Name of the folder where your ROOT and theory files reside.") ::
	render_error(Nest_Key(key, NAME), entry.name)) ::
	fieldset(legend("Topics") ::
	indexed(entry.topics.v, key, TOPIC, render_topic) :::
	selection(Nest_Key(key, TOPIC),
	entry.topic_input.map(_.id),
	topics.map(topic => option(topic.id, topic.id))) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_TOPICS_ADD), "add", key) ::
	render_error("", entry.topics)) ::
	par(List(
	fieldlabel(Nest_Key(key, LICENSE), "License"),
	radio(Nest_Key(key, LICENSE),
	entry.license.id,
	licenses.values.toList.map(license => license.id -> license.name)),
	explanation(Nest_Key(key, LICENSE),
	"Note: For LGPL submissions, please include the header \"License: LGPL\" in each file"))) ::
	par(
	fieldlabel(Nest_Key(key, ABSTRACT), "Abstract") ::
	placeholder("HTML and MathJax, no LaTeX")(
	textarea(Nest_Key(key, ABSTRACT), entry.`abstract`.v) +
	("rows" -> "8") +
	("cols" -> "70")) ::
	explanation(Nest_Key(key, ABSTRACT),
	"Note: You can use HTML or MathJax (not LaTeX!) to format your abstract.") ::
	render_error(Nest_Key(key, ABSTRACT), entry.`abstract`)) ::
	fieldset(legend("Authors") ::
	indexed(entry.affils.v, key, AUTHOR, render_affil) :::
	selection(Nest_Key(key, AUTHOR),
	entry.author_input.map(_.id),
	authors_list.map(author => option(author.id, author_string(author)))) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_AUTHORS_ADD), "add", key) ::
	explanation(Nest_Key(key, AUTHOR),
	"Add an author from the list. Register new authors first below.") ::
	render_error(Nest_Key(key, AUTHOR), entry.affils)) ::
	fieldset(legend("Contact") ::
	indexed(entry.notifies.v, key, NOTIFY, render_notify) :::
	selection(Nest_Key(key, NOTIFY),
	entry.notify_input.map(_.id),
	optgroup("Suggested", email_authors.filter(author =>
	entry.used_authors.contains(author.id)).map(author_option)) ::
	email_authors.filter(author =>
	!entry.used_authors.contains(author.id)).map(author_option)) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_NOTIFIES_ADD), "add", key) ::
	explanation(Nest_Key(key, NOTIFY),
	"These addresses serve two purposes: " +
	"1. They are used to send you updates about the state of your submission. " +
	"2. They are the maintainers of the entry once it is accepted. " +
	"Typically this will be one or more of the authors.") ::
	render_error("", entry.notifies)) ::
	fieldset(legend("Related Publications") ::
	indexed(entry.related, key, RELATED, render_related) :::
	selection(Nest_Key(Nest_Key(key, RELATED), KIND),
	entry.related_kind.map(_.toString),
	Model.Related.values.toList.map(v => option(v.toString, v.toString))) ::
	textfield(Nest_Key(Nest_Key(key, RELATED), INPUT),
	"10.1109/5.771073", entry.related_input.v) ::
	action_button(api.api_url(API_SUBMISSION_ENTRY_RELATED_ADD), "add", key) ::
	explanation(Nest_Key(Nest_Key(key, RELATED), INPUT),
	"Publications related to the entry, as DOIs (10.1109/5.771073) or plaintext (HTML)." +
	"Typically a publication by the authors describing the entry," +
	" background literature (articles, books) or web resources. ") ::
	render_error(Nest_Key(Nest_Key(key, RELATED), INPUT), entry.related_input)) ::
	render_if(mode == Mode.SUBMISSION,
	action_button(api.api_url(API_SUBMISSION_ENTRIES_REMOVE), "remove entry", key)))

	def render_new_author(author: Author, key: Params.Key): XML.Elem =
	par(
	hidden(Nest_Key(key, ID), author.id) ::
	hidden(Nest_Key(key, NAME), author.name) ::
	hidden(Nest_Key(key, ORCID), author.orcid.map(_.identifier).getOrElse("")) ::
	text(author_string(author)) :::
	render_if(!model.used_authors.contains(author.id),
	action_button(api.api_url(API_SUBMISSION_AUTHORS_REMOVE), "x", key)))

	def render_new_affil(affil: Affiliation, key: Params.Key): XML.Elem =
	par(
	hidden(Nest_Key(key, AUTHOR), affil.author) ::
	hidden(Nest_Key(key, ID), affil_id(affil)) ::
	hidden(Nest_Key(key, AFFILIATION), affil_address(affil)) ::
	text(author_string(updated_authors(affil.author)) + ": " + affil_string(affil)) :::
	render_if(!model.used_affils.contains(affil),
	action_button(api.api_url(API_SUBMISSION_AFFILIATIONS_REMOVE), "x", key)))

	val (upload, preview) = mode match {
	case Mode.EDIT => ("Save", "preview and save >")
	case Mode.SUBMISSION => ("Upload", "preview and upload >")
	}

	List(submit_form(api.api_url(API_SUBMISSION),
	indexed(model.entries.v, Params.empty, ENTRY, render_entry) :::
	render_error("", model.entries) :::
	render_if(mode == Mode.SUBMISSION,
	par(List(
	explanation("",
	"You can submit multiple entries at once. " +
	"Put the corresponding folders in the archive " +
	"and use the button below to add more input fields for metadata. "),
	api_button(api.api_url(API_SUBMISSION_ENTRIES_ADD), "additional entry")))) :::
	break :::
	fieldset(legend("New Authors") ::
	explanation("", "If you are new to the AFP, add yourself here.") ::
	indexed(model.new_authors.v, Params.empty, AUTHOR, render_new_author) :::
	fieldlabel(Nest_Key(AUTHOR, NAME), "Name") ::
	textfield(Nest_Key(AUTHOR, NAME), "Gerwin Klein", model.new_author_input) ::
	fieldlabel(Nest_Key(AUTHOR, ORCID), "ORCID id (optional)") ::
	textfield(Nest_Key(AUTHOR, ORCID), "0000-0002-1825-0097", model.new_author_orcid) ::
	api_button(api.api_url(API_SUBMISSION_AUTHORS_ADD), "add") ::
	render_error("", model.new_authors)) ::
	fieldset(legend("New email or homepage") ::
	explanation("",
	"Add new email or homepages here. " +
	"If you would like to update an existing, " +
	"submit with the old one and write to the editors.") ::
	indexed(model.new_affils.v, Params.empty, AFFILIATION, render_new_affil) :::
	fieldlabel(AFFILIATION, "Author") ::
	selection(AFFILIATION,
	model.new_affils_author.map(_.id),
	optgroup("Entry authors", entry_authors.map(author_option)) ::
	other_authors.map(author_option)) ::
	fieldlabel(Nest_Key(AFFILIATION, ADDRESS), "Email or Homepage") ::
	textfield(Nest_Key(AFFILIATION, ADDRESS), "https://proofcraft.org",
	model.new_affils_input) ::
	api_button(api.api_url(API_SUBMISSION_AFFILIATIONS_ADD), "add") ::
	render_error("", model.new_affils)) ::
	break :::
	fieldset(List(legend(upload),
	api_button(api.api_url(API_SUBMISSION_UPLOAD), preview))) :: Nil))
	}

	def render_metadata(meta: Model.Metadata): XML.Body = {
	def render_topic(topic: Topic, key: Params.Key): XML.Elem =
	item(hidden(Nest_Key(key, ID), topic.id) :: text(topic.id))

	def render_affil(affil: Affiliation, key: Params.Key): XML.Elem =
	item(
	hidden(Nest_Key(key, ID), affil.author) ::
	hidden(Nest_Key(key, AFFILIATION), affil_id(affil)) ::
	text(author_string(meta.authors(affil.author)) + ", " + affil_string(affil)))

	def render_related(related: Reference, key: Params.Key): XML.Elem =
	item(
	hidden(Nest_Key(key, KIND), Model.Related.get(related).toString) ::
	hidden(Nest_Key(key, INPUT), related_string(related)) ::
	unescape(related_string(related)))

	def render_entry(entry: Entry, key: Params.Key): XML.Elem =
	fieldset(List(
	legend("Entry"),
	par(
	fieldlabel(Nest_Key(key, TITLE), "Title") ::
	hidden(Nest_Key(key, TITLE), entry.title) ::
	text(entry.title)),
	par(
	fieldlabel(Nest_Key(key, NAME), "Short Name") ::
	hidden(Nest_Key(key, NAME), entry.name) ::
	text(entry.name)),
	par(
	fieldlabel(Nest_Key(key, DATE), "Date") ::
	hidden(Nest_Key(key, DATE), entry.date.toString) ::
	text(entry.date.toString)),
	par(List(
	fieldlabel("", "Topics"),
	list(indexed(entry.topics, key, TOPIC, render_topic)))),
	par(
	fieldlabel(Nest_Key(key, LICENSE), "License") ::
	hidden(Nest_Key(key, LICENSE), entry.license.id) ::
	text(entry.license.name)),
	par(List(
	fieldlabel(Nest_Key(key, ABSTRACT), "Abstract"),
	hidden(Nest_Key(key, ABSTRACT), entry.`abstract`),
	class_("mathjax_process")(span(unescape(entry.`abstract`))))),
	par(List(
	fieldlabel("", "Authors"),
	list(indexed(entry.authors, key, AUTHOR, render_affil)))),
	par(List(
	fieldlabel("", "Contact"),
	list(indexed(entry.notifies, key, NOTIFY, render_affil)))),
	par(List(
	fieldlabel("", "Related Publications"),
	list(indexed(entry.related, key, RELATED, render_related))))))

	def render_new_author(author: Author, key: Params.Key): XML.Elem =
	par(List(
	hidden(Nest_Key(key, ID), author.id),
	hidden(Nest_Key(key, NAME), author.name),
	hidden(Nest_Key(key, ORCID), author.orcid.map(_.identifier).getOrElse(""))))

	def render_new_affil(affil: Affiliation, key: Params.Key): XML.Elem =
	par(List(
	hidden(Nest_Key(key, AUTHOR), affil.author),
	hidden(Nest_Key(key, ID), affil_id(affil)),
	hidden(Nest_Key(key, AFFILIATION), affil_address(affil))))

	indexed(meta.entries, Params.empty, ENTRY, render_entry) :::
	indexed(meta.new_authors(authors).toList, Params.empty, AUTHOR, render_new_author) :::
	indexed(meta.new_affils(authors).toList, Params.empty, AFFILIATION, render_new_affil)
	}

	def render_submission(submission: Model.Submission): XML.Body = {
	def status_text(status: Option[Model.Status.Value]): String =
	status.map {
	case Model.Status.Submitted => "AFP editors notified."
	case Model.Status.Review => "Review in progress."
	case Model.Status.Added => "Added to the AFP."
	case Model.Status.Rejected => "Submission rejected."
	} getOrElse "Draft saved. Submit to editors once successfully built."

	val archive_url =
	if (handler.get_archive(submission.id).exists(_.get_ext == "zip"))
	API_SUBMISSION_DOWNLOAD_ZIP
	else API_SUBMISSION_DOWNLOAD_TGZ

	val resubmit = mode match {
	case Mode.EDIT => "Update"
	case Mode.SUBMISSION => "Resubmit"
	}

	List(submit_form(api.api_url(SUBMISSION, List(ID -> submission.id)),
	render_if(mode == Mode.SUBMISSION,
	download_link(api.api_url(archive_url, List(ID -> submission.id)), text("archive")) ::
	download_link(api.api_url(API_SUBMISSION_DOWNLOAD, List(ID -> submission.id)),
	text("metadata patch")) ::
	text(" (apply with: 'patch -p0 < FILE')")) :::
	render_if(mode == Mode.SUBMISSION, par(
	hidden(MESSAGE, submission.message) ::
	text("Comment: " + submission.message))) :::
	section("Metadata") ::
	render_metadata(submission.meta) :::
	section("Status") ::
	span(text(status_text(submission.status))) ::
	render_if(submission.build != Model.Build.Running,
	action_button(api.api_url(API_RESUBMIT), resubmit, submission.id)) :::
	render_if(submission.build == Model.Build.Running,
	action_button(api.api_url(API_BUILD_ABORT), "Abort build", submission.id)) :::
	render_if(submission.build == Model.Build.Success && submission.status.isEmpty,
	action_button(api.api_url(API_SUBMIT), "Send submission to AFP editors", submission.id)) :::
	render_if(mode == Mode.SUBMISSION,
	fieldset(legend("Build") ::
	bold(text(submission.build.toString)) ::
	par(text("Isabelle log:") ::: source(submission.log) :: Nil) ::
	Nil))))
	}

	def render_upload(upload: Model.Upload): XML.Body = {
	val submit = mode match {
	case Mode.EDIT => "save"
	case Mode.SUBMISSION => "submit"
	}

	List(submit_form(api.api_url(API_SUBMISSION), List(
	fieldset(legend("Overview") :: render_metadata(upload.meta)),
	fieldset(legend("Submit") ::
	api_button(api.api_url(API_SUBMISSION), "< edit metadata") ::
	render_if(mode == Mode.SUBMISSION, List(
	par(List(
	fieldlabel(MESSAGE, "Message for the editors (optional)"),
	textfield(MESSAGE, "", upload.message),
	explanation(
	MESSAGE,
	"Note: Anything special or noteworthy about your submission can be covered here. " +
	"It will not become part of your entry. " +
	"You're also welcome to leave suggestions for our AFP submission service here. ")
	)),
	par(List(
	fieldlabel(ARCHIVE, "Archive file (.tar.gz or .zip)"),
	browse(ARCHIVE, List(".zip", ".tar.gz")),
	explanation(ARCHIVE,
	- "Note: Your zip or tar file should contain one folder with your theories, ROOT, etc. " +
	- "The folder name should be the short/folder name used in the submission form."))))) :::
	+ "Note: Your zip or tar file must contain exactly one folder for each entry with your theories, ROOT, etc. " +
	+ "The folder name must be the short/folder name used in the submission form. " +
	+ "Hidden files and folders (e.g., __MACOSX) are not allowed."))))) :::
	api_button(api.api_url(API_SUBMISSION_CREATE), submit) ::
	render_error(ARCHIVE, Val.err((), upload.error))))))
	}

	def render_submission_list(submission_list: Model.Submission_List): XML.Body = {
	def render_overview(overview: Model.Overview, key: Params.Key): XML.Elem =
	item(
	hidden(Nest_Key(key, ID), overview.id) ::
	hidden(Nest_Key(key, DATE), overview.date.toString) ::
	hidden(Nest_Key(key, NAME), overview.name) ::
	span(text(overview.date.toString)) ::
	span(List(frontend_link(SUBMISSION, List(ID -> overview.id),
	text(overview.name)))) ::
	render_if(mode == Mode.SUBMISSION,
	class_("right")(span(List(
	selection(Nest_Key(key, STATUS), Some(overview.status.toString),
	Model.Status.values.toList.map(v => option(v.toString, v.toString))),
	action_button(api.api_url(API_SUBMISSION_STATUS), "update", key))))))

	def list1(ls: List[XML.Elem]): XML.Elem = if (ls.isEmpty) par(Nil) else list(ls)

	val ls = indexed(submission_list.submissions, Params.empty, ENTRY, (o, k) => (o, k))
	val finished =
	ls.filter(t => Set(Model.Status.Added, Model.Status.Rejected).contains(t._1.status))

	List(submit_form(api.api_url(API_SUBMISSION_STATUS),
	render_if(mode == Mode.SUBMISSION,
	text("Open") :::
	list1(ls.filter(_._1.status == Model.Status.Submitted).map(render_overview)) ::
	text("In Progress") :::
	list1(ls.filter(_._1.status == Model.Status.Review).map(render_overview)) ::
	text("Finished")) :::
	list1(finished.map(render_overview)) :: Nil))
	}

	def render_created(created: Model.Created): XML.Body = {
	val status = mode match {
	case Mode.SUBMISSION => "View your submission status: "
	case Mode.EDIT => "View the entry at: "
	}

	List(div(
	span(text("Entry successfully saved. " + status)) ::
	break :::
	frontend_link(SUBMISSION, List(ID -> created.id),
	text(api.app_url(SUBMISSION, List(ID -> created.id)))) ::
	break :::
	render_if(mode == Mode.SUBMISSION, span(text("(keep that url!).")))))
	}

	def render_invalid: XML.Body =
	text("Invalid request")

	def render(model: Model.T): XML.Body =
	model match {
	case create: Model.Create => render_create(create)
	case upload: Model.Upload => render_upload(upload)
	case submission: Model.Submission => render_submission(submission)
	case submissions: Model.Submission_List => render_submission_list(submissions)
	case created: Model.Created => render_created(created)
	case Model.Invalid => render_invalid
	}


	/* validation */

	def validate_topic(id: String, selected: List[Topic]): Val_Opt[Topic] =
	if (id.isEmpty) Val_Opt.user_err("Select topic first")
	else {
	topics.find(_.id == id) match {
	case Some(topic) =>
	if (selected.contains(topic)) Val_Opt.user_err("Topic already selected")
	else Val_Opt.ok(topic)
	case _ => Val_Opt.error
	}
	}

	def validate_new_author(
	id: String,
	name: String,
	orcid_str: String,
	authors: Map[Author.ID, Author]
	): Val_Opt[Author] = {
	val Id = """[a-zA-Z][a-zA-Z0-9]*""".r
	id match {
	case Id() if !authors.contains(id) =>
	if (name.isEmpty \|\| name.trim != name)
	Val_Opt.user_err("Name must not be empty")
	else if (orcid_str.isEmpty)
	Val_Opt.ok(Author(id, name))
	else Exn.capture(Orcid(orcid_str)) match {
	case Exn.Res(orcid) =>
	if (authors.values.exists(_.orcid.exists(_ == orcid)))
	Val_Opt.user_err("Author with that orcid already exists")
	else Val_Opt.ok(Author(id, name, orcid = Some(orcid)))
	case _ => Val_Opt.user_err("Invalid orcid")
	}
	case _ => Val_Opt.error
	}
	}

	def validate_new_affil(id: String, address: String, author: Author): Val_Opt[Affiliation] = {
	if (address.isBlank) Val_Opt.user_err("Address must not be empty")
	else if (address.contains("@")) {
	val Id = (author.id + """_email\d*""").r
	id match {
	case Id() if !author.emails.map(_.id).contains(id) =>
	val Address = """([^@\s]+)@([^@\s]+)""".r
	address match {
	case Address(user, host) => Val_Opt.ok(Email(author.id, id, user, host))
	case _ => Val_Opt.user_err("Invalid email address")
	}
	case _ => Val_Opt.error
	}
	}
	else {
	val Id = (author.id + """_homepage\d*""").r
	id match {
	case Id() if !author.homepages.map(_.id).contains(id) =>
	Exn.capture(new URL(address)) match {
	case Exn.Res(url) => Val_Opt.ok(Homepage(author.id, id, url))
	case _ => Val_Opt.user_err("Invalid url")
	}
	case _ => Val_Opt.error
	}
	}
	}

	def validate_related(
	kind: Model.Related.Value,
	related: String,
	references: List[Reference]
	): Val_Opt[Reference] =
	if (related.isBlank) Val_Opt.user_err("Reference must not be empty")
	else {
	kind match {
	case Model.Related.DOI =>
	Exn.capture(DOI(related)) match {
	case Exn.Res(doi) =>
	if (references.contains(doi)) Val_Opt.user_err("Already present")
	else Val_Opt.ok(doi)
	case _ => Val_Opt.user_err("Invalid DOI format")
	}
	case Model.Related.Plaintext =>
	val formatted = Formatted(related)
	if (references.contains(formatted)) Val_Opt.user_err("Already present")
	else Val_Opt.ok(formatted)
	}
	}


	/* param parsing */

	def parse_create(params: Params.Data): Option[Model.Create] = {
	def parse_topic(topic: Params.Data, topics: List[Topic]): Option[Topic] =
	validate_topic(topic.get(ID).value, topics).opt

	def parse_email(email: Params.Data, authors: Map[Author.ID, Author]): Option[Email] =
	authors.get(email.get(ID).value).flatMap(
	_.emails.find(_.id == email.get(AFFILIATION).value))

	def parse_affil(affil: Params.Data, authors: Map[Author.ID, Author]): Option[Affiliation] =
	authors.get(affil.get(ID).value).flatMap { author =>
	val id = affil.get(AFFILIATION).value
	if (id.isEmpty) Some(Unaffiliated(author.id))
	else (author.emails ++ author.homepages).collectFirst {
	case e: Email if e.id == id => e
	case h: Homepage if h.id == id => h
	}
	}

	def parse_related(related: Params.Data, references: List[Reference]): Option[Reference] =
	Model.Related.from_string(related.get(KIND).value).flatMap(
	validate_related(_, related.get(INPUT).value, references).opt)

	def parse_new_author(author: Params.Data, authors: Map[Author.ID, Author]): Option[Author] =
	validate_new_author(
	author.get(ID).value, author.get(NAME).value, author.get(ORCID).value, authors).opt

	def parse_new_affil(affil: Params.Data, authors: Map[Author.ID, Author]): Option[Affiliation] =
	authors.get(affil.get(AUTHOR).value).flatMap(author =>
	validate_new_affil(affil.get(ID).value, affil.get(AFFILIATION).value, author).opt)

	def parse_entry(entry: Params.Data, authors: Map[Author.ID, Author]): Option[Model.Create_Entry] =
	for {
	topics <-
	fold(entry.list(TOPIC), List.empty[Topic]) {
	case (topic, topics) => parse_topic(topic, topics).map(topics :+ _)
	}
	affils <-
	fold(entry.list(AUTHOR), List.empty[Affiliation]) {
	case (affil, affils) => parse_affil(affil, authors).map(affils :+ _)
	}
	notifies <-
	fold(entry.list(NOTIFY), List.empty[Email]) {
	case (email, emails) => parse_email(email, authors).map(emails :+ _)
	}
	related <-
	fold(entry.list(RELATED), List.empty[Reference]) {
	case (related, references) => parse_related(related, references).map(references :+ _)
	}
	license <- licenses.get(entry.get(LICENSE).value)
	} yield Model.Create_Entry(
	name = Val.ok(entry.get(NAME).value),
	title = Val.ok(entry.get(TITLE).value),
	topics = Val.ok(topics),
	topic_input = parse_topic(entry.get(TOPIC), Nil),
	license = license,
	`abstract` = Val.ok(entry.get(ABSTRACT).value),
	author_input = authors.get(entry.get(AUTHOR).value),
	notify_input = authors.get(entry.get(NOTIFY).value),
	affils = Val.ok(affils),
	notifies = Val.ok(notifies),
	related = related,
	related_kind = Model.Related.from_string(entry.get(RELATED).get(KIND).value),
	related_input = Val.ok(entry.get(RELATED).get(INPUT).value))

	for {
	(new_author_ids, all_authors) <-
	fold(params.list(AUTHOR), (List.empty[Author.ID], authors)) {
	case (author, (new_authors, authors)) =>
	parse_new_author(author, authors).map(author =>
	(new_authors :+ author.id, authors.updated(author.id, author)))
	}
	(new_affils, all_authors) <-
	fold(params.list(AFFILIATION), (List.empty[Affiliation], all_authors)) {
	case (affil, (new_affils, authors)) =>
	parse_new_affil(affil, authors).map { affil =>
	val author = authors(affil.author)
	(new_affils :+ affil, affil match {
	case _: Unaffiliated => authors
	case e: Email =>
	authors.updated(author.id, author.copy(emails = author.emails :+ e))
	case h: Homepage =>
	authors.updated(author.id, author.copy(homepages = author.homepages :+ h))
	})
	}
	}
	new_authors = new_author_ids.map(all_authors)
	entries <- fold(params.list(ENTRY), List.empty[Model.Create_Entry]) {
	case (entry, entries) => parse_entry(entry, all_authors).map(entries :+ _)
	}
	} yield Model.Create(
	entries = Val.ok(entries),
	new_authors = Val.ok(new_authors),
	new_author_input = params.get(AUTHOR).get(NAME).value,
	new_author_orcid = params.get(AUTHOR).get(ORCID).value,
	new_affils = Val.ok(new_affils),
	new_affils_author = all_authors.get(params.get(AFFILIATION).value),
	new_affils_input = params.get(AFFILIATION).get(ADDRESS).value)
	}

	def parse_submission_list(params: Params.Data): Option[Model.Submission_List] = {
	def parse_overview(entry: Params.Data): Option[Model.Overview] =
	for {
	date <-
	Exn.capture(LocalDate.parse(entry.get(DATE).value)) match {
	case Exn.Res(date) => Some(date)
	case Exn.Exn(_) => None
	}
	status <- Model.Status.from_string(entry.get(STATUS).value)
	} yield Model.Overview(entry.get(ID).value, date, entry.get(NAME).value, status)

	val submissions =
	fold(params.list(ENTRY), List.empty[Model.Overview]) {
	case (entry, overviews) => parse_overview(entry).map(overviews :+ _)
	}
	submissions.map(Model.Submission_List.apply)
	}

	/* control */

	def add_entry(params: Params.Data): Option[Model.T] = parse_create(params).map { model =>
	model.copy(entries = Val.ok(model.entries.v :+ Model.Create_Entry(license = licenses.head._2)))
	}

	def remove_entry(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_entry <- List_Key.num(ENTRY, params.get(Web_App.ACTION).value)
	} yield model.copy(entries = Val.ok(Utils.remove_at(num_entry, model.entries.v)))

	def add_author(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_entry <- List_Key.num(ENTRY, params.get(Web_App.ACTION).value)
	entry <- model.entries.v.unapply(num_entry)
	} yield
	entry.author_input match {
	case None =>
	model.update_entry(num_entry, entry.copy(
	affils = entry.affils.with_err("Select author first")))
	case Some(author) =>
	val default_affil = author.emails.headOption.orElse(
	author.homepages.headOption).getOrElse(Unaffiliated(author.id))

	model.update_entry(num_entry, entry.copy(
	author_input = None, affils = Val.ok(entry.affils.v :+ default_affil)))
	}

	def remove_author(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	(action, num_affil) <- List_Key.split(AUTHOR, params.get(Web_App.ACTION).value)
	num_entry <- List_Key.num(ENTRY, action)
	entry <- model.entries.v.unapply(num_entry)
	} yield
	model.update_entry(num_entry, entry.copy(affils =
	Val.ok(Utils.remove_at(num_affil, entry.affils.v))))

	def add_notify(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_entry <- List_Key.num(ENTRY, params.get(Web_App.ACTION).value)
	entry <- model.entries.v.unapply(num_entry)
	entry1 <-
	entry.notify_input match {
	case Some(author) =>
	author.emails.headOption.map(email => entry.copy(
	notify_input = None, notifies = Val.ok(entry.notifies.v :+ email)))
	case None =>
	Some(entry.copy(notifies = entry.notifies.with_err("Select author first")))
	}
	} yield model.update_entry(num_entry, entry1)

	def remove_notify(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	(action, num_notify) <- List_Key.split(NOTIFY, params.get(Web_App.ACTION).value)
	num_entry <- List_Key.num(ENTRY, action)
	entry <- model.entries.v.unapply(num_entry)
	} yield
	model.update_entry(num_entry, entry.copy(notifies =
	Val.ok(Utils.remove_at(num_notify, entry.notifies.v))))

	def add_topic(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_entry <- List_Key.num(ENTRY, params.get(Web_App.ACTION).value)
	entry <- model.entries.v.unapply(num_entry)
	entry_params <- params.list(ENTRY).unapply(num_entry)
	} yield {
	val topic = validate_topic(entry_params.get(TOPIC).value, entry.topics.v)
	val topic_input = if (topic.is_empty) entry.topic_input else None
	model.update_entry(num_entry, entry.copy(
	topic_input = topic_input,
	topics = Val.ok(entry.topics.v ++ topic.opt).perhaps_err(topic)))
	}

	def remove_topic(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	(action, num_topic) <- List_Key.split(TOPIC, params.get(Web_App.ACTION).value)
	num_entry <- List_Key.num(ENTRY, action)
	entry <- model.entries.v.unapply(num_entry)
	} yield {
	val entry1 = entry.copy(topics = Val.ok(Utils.remove_at(num_topic, entry.topics.v)))
	model.copy(entries = Val.ok(model.entries.v.updated(num_entry, entry1)))
	}

	def add_related(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_entry <- List_Key.num(ENTRY, params.get(Web_App.ACTION).value)
	entry <- model.entries.v.unapply(num_entry)
	} yield {
	val entry1 =
	entry.related_kind match {
	case None =>
	entry.copy(related_input =
	entry.related_input.with_err("Select reference kind first"))
	case Some(kind) =>
	val reference = validate_related(kind, entry.related_input.v, entry.related)
	val related_input = if (reference.is_empty) entry.related_input.v else ""
	entry.copy(
	related = entry.related ++ reference.opt,
	related_input = Val.ok(related_input).perhaps_err(reference))
	}
	model.update_entry(num_entry, entry1)
	}

	def remove_related(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	(action, num_related) <- List_Key.split(RELATED, params.get(Web_App.ACTION).value)
	num_entry <- List_Key.num(ENTRY, action)
	entry <- model.entries.v.unapply(num_entry)
	} yield {
	val entry1 = entry.copy(related = Utils.remove_at(num_related, entry.related))
	model.copy(entries = Val.ok(model.entries.v.updated(num_entry, entry1)))
	}

	def add_new_author(params: Params.Data): Option[Model.T] = parse_create(params).map { model =>
	val name = model.new_author_input.trim
	if (name.isEmpty)
	model.copy(new_authors = model.new_authors.with_err("Name must not be empty"))
	else {
	def as_ascii(str: String) = {
	var res: String = str
	List("ö" -> "oe", "ü" -> "ue", "ä" -> "ae", "ß" -> "ss").foreach {
	case (c, rep) => res = res.replace(c, rep)
	}
	Normalizer.normalize(res, Normalizer.Form.NFD).replaceAll("[^\\x00-\\x7F]", "")
	}

	def make_author_id(name: String): String = {
	val normalized = as_ascii(name)
	val Suffix = """^.?([a-zA-Z])$""".r
	val suffix = normalized match {
	case Suffix(suffix) => suffix
	case _ => ""
	}
	val Prefix = """^([a-zA-Z]).$""".r
	val prefix = normalized.stripSuffix(suffix) match {
	case Prefix(prefix) => prefix
	case _ => ""
	}
	var ident = suffix.toLowerCase
	for (c <- prefix.toLowerCase) {
	if (model.updated_authors(authors).contains(ident)) ident += c.toString
	else return ident
	}
	Utils.make_unique(ident, model.updated_authors(authors).keySet)
	}

	val id = make_author_id(name)

	val author = validate_new_author(id, model.new_author_input, model.new_author_orcid,
	model.updated_authors(authors))
	val new_author_input = if (author.is_empty) model.new_author_input else ""
	val new_author_orcid = if (author.is_empty) model.new_author_orcid else ""

	model.copy(
	new_author_input = new_author_input,
	new_author_orcid = new_author_orcid,
	new_authors = Val.ok(model.new_authors.v ++ author.opt).perhaps_err(author))
	}
	}

	def remove_new_author(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_author <- List_Key.num(AUTHOR, params.get(Web_App.ACTION).value)
	author <- model.new_authors.v.unapply(num_author)
	if !model.used_authors.contains(author.id)
	} yield model.copy(new_authors =
	Val.ok(Utils.remove_at(num_author, model.new_authors.v)))

	def add_new_affil(params: Params.Data): Option[Model.T] =
	parse_create(params).map { model =>
	model.new_affils_author match {
	case Some(author) =>
	val address = model.new_affils_input.trim
	if (address.isEmpty)
	model.copy(new_affils = model.new_affils.with_err("Must not be empty"))
	else {
	val id =
	if (address.contains("@"))
	Utils.make_unique(author.id + "_email", author.emails.map(_.id).toSet)
	else
	Utils.make_unique(author.id + "_homepage", author.homepages.map(_.id).toSet)

	val affil = validate_new_affil(id, address, author)
	val new_affils_input = if (affil.is_empty) model.new_affils_input else ""
	model.copy(
	new_affils_input = new_affils_input,
	new_affils = Val.ok(model.new_affils.v ++ affil.opt).perhaps_err(affil))
	}
	case None => model.copy(new_affils = model.new_affils.with_err("Select author first"))
	}
	}

	def remove_new_affil(params: Params.Data): Option[Model.T] =
	for {
	model <- parse_create(params)
	num_affil <- List_Key.num(AFFILIATION, params.get(Web_App.ACTION).value)
	affil <- model.new_affils.v.unapply(num_affil)
	if !model.used_affils.contains(affil)
	} yield model.copy(new_affils = Val.ok(Utils.remove_at(num_affil, model.new_affils.v)))

	def upload(params: Params.Data): Option[Model.T] = parse_create(params).map { create =>
	var ok = true

	def validate[A](validator: A => Val[A], value: A): Val[A] = {
	val res = validator(value)
	if (res.err.nonEmpty) ok = false
	res
	}

	def title(title: String): Val[String] =
	if (title.isBlank) Val.err(title, "Title must not be blank")
	else if (title.trim != title) Val.err(title, "Title must not contain extra spaces")
	else Val.ok(title)

	def name(name: String): Val[String] =
	if (name.isBlank) Val.err(name, "Name must not be blank")
	else if (!"[a-zA-Z0-9_-]+".r.matches(name)) Val.err(name,
	"Invalid character in name")
	else mode match {
	case Mode.EDIT =>
	if (Server.this.entries.contains(name)) Val.ok(name)
	else Val.err(name, "Entry does not exist")
	case Mode.SUBMISSION =>
	if (Server.this.entries.contains(name)) Val.err(name, "Entry already exists")
	else Val.ok(name)
	}

	def entries(entries: List[Model.Create_Entry]): Val[List[Model.Create_Entry]] =
	if (entries.isEmpty) Val.err(entries, "Must contain at least one entry")
	else if (Library.duplicates(entries.map(_.name)).nonEmpty)
	Val.err(entries, "Entries must have different names")
	else Val.ok(entries)

	def new_authors(authors: List[Author]): Val[List[Author]] =
	if (!authors.forall(author => create.used_authors.contains(author.id)))
	Val.err(authors, "Unused authors")
	else Val.ok(authors)

	def new_affils(affils: List[Affiliation]): Val[List[Affiliation]] =
	if (!affils.forall(affil => create.used_affils.contains(affil)))
	Val.err(affils, "Unused affils")
	else Val.ok(affils)

	def entry_authors(authors: List[Affiliation]): Val[List[Affiliation]] =
	if (authors.isEmpty) Val.err(authors, "Must contain at least one author")
	else if (!Utils.is_distinct(authors)) Val.err(authors, "Duplicate affiliations")
	else Val.ok(authors)

	def notifies(notifies: List[Email]): Val[List[Email]] =
	if (notifies.isEmpty) Val.err(notifies, "Must contain at least one maintainer")
	else if (!Utils.is_distinct(notifies)) Val.err(notifies, "Duplicate emails")
	else Val.ok(notifies)

	def topics(topics: List[Topic]): Val[List[Topic]] =
	if (topics.isEmpty) Val.err(topics, "Must contain at least one topic") else Val.ok(topics)

	def `abstract`(txt: String): Val[String] =
	if (txt.isBlank) Val.err(txt, "Entry must contain an abstract")
	else if (List("\\cite", "\\emph", "\\texttt").exists(txt.contains(_)))
	Val.err(txt, "LaTeX not allowed, use MathJax for math symbols")
	else Val.ok(txt)

	val validated = create.copy(
	entries = validate(
	entries, create.entries.v.map(entry => entry.copy(
	name = validate(name, entry.name.v),
	title = validate(title, entry.title.v),
	affils = validate(entry_authors, entry.affils.v),
	notifies = validate(notifies, entry.notifies.v),
	topics = validate(topics, entry.topics.v),
	`abstract` = validate(`abstract`, entry.`abstract`.v)
	))),
	new_authors = validate(new_authors, create.new_authors.v),
	new_affils = validate(new_affils, create.new_affils.v))

	if (ok) {
	val (updated_authors, entries) = create.create(authors)
	Model.Upload(Model.Metadata(updated_authors, entries), params.get(MESSAGE).value, "")
	} else validated
	}

	def create(params: Params.Data): Option[Model.T] = {
	upload(params) match {
	case Some(upload: Model.Upload) =>
	mode match {
	case Mode.EDIT =>
	handler.create(Date.now(), upload.meta, upload.message, Bytes.empty, "")
	Some(Model.Created(upload.meta.entries.head.name))
	case Mode.SUBMISSION =>
	val archive = Bytes.decode_base64(params.get(ARCHIVE).get(FILE).value)
	val file_name = params.get(ARCHIVE).value

	if ((archive.is_empty \|\| file_name.isEmpty)) {
	Some(upload.copy(error = "Select a file"))
	} else if (!file_name.endsWith(".zip") && !file_name.endsWith(".tar.gz")) {
	Some(upload.copy(error = "Only .zip and .tar.gz archives allowed"))
	} else {
	val ext = if (file_name.endsWith(".zip")) ".zip" else ".tar.gz"
	val id = handler.create(Date.now(), upload.meta, upload.message, archive, ext)
	Some(Model.Created(id))
	}
	}
	case _ => None
	}
	}

	def empty_submission: Option[Model.T] =
	Some(Model.Create(entries =
	Val.ok(List(Model.Create_Entry(license = licenses.head._2)))))

	def get_submission(props: Properties.T): Option[Model.Submission] =
	Properties.get(props, ID).flatMap(handler.get(_, all_topics, licenses))

	def resubmit(params: Params.Data): Option[Model.T] =
	handler.get(params.get(ACTION).value, all_topics, licenses).map(submission =>
	Model.Upload(submission.meta, submission.message, ""))

	def submit(params: Params.Data): Option[Model.T] =
	handler.get(params.get(ACTION).value, all_topics, licenses).flatMap { submission =>
	if (submission.status.nonEmpty) None
	else {
	handler.submit(submission.id)
	Some(submission.copy(status = Some(Model.Status.Submitted)))
	}
	}

	def abort_build(params: Params.Data): Option[Model.T] =
	handler.get(params.get(ACTION).value, all_topics, licenses).flatMap { submission =>
	if (submission.build != Model.Build.Running) None
	else {
	handler.abort_build(submission.id)
	Some(submission.copy(build = Model.Build.Aborted))
	}
	}

	def set_status(params: Params.Data): Option[Model.T] = {
	for {
	list <- parse_submission_list(params)
	num_entry <- List_Key.num(ENTRY, params.get(ACTION).value)
	changed <- list.submissions.unapply(num_entry)
	} yield {
	if (changed.status == Model.Status.Added && !devel) {
	progress.echo_if(verbose, "Updating server data...")
	val id_before = repo.id()
	repo.pull()
	repo.update()
	val id_after = repo.id()
	if (id_before != id_after) {
	load()
	progress.echo("Updated repo from " + id_before + " to " + id_after)
	}
	}
	handler.set_status(changed.id, changed.status)
	list
	}
	}

	def submission_list: Option[Model.T] = Some(handler.list())

	def download(props: Properties.T): Option[Path] =
	Properties.get(props, ID).flatMap(handler.get_patch)

	def download_archive(props: Properties.T): Option[Path] =
	Properties.get(props, ID).flatMap(handler.get_archive)

	def style_sheet: Option[Path] =
	Some(afp_structure.base_dir + Path.make(List("tools", "main.css")))

	val error = Model.Invalid

	val endpoints = List(
	Get(SUBMIT, "empty submission form", _ => empty_submission),
	Get(SUBMISSION, "get submission", get_submission),
	Get(SUBMISSIONS, "list submissions", _ => submission_list),
	Get_File(API_SUBMISSION_DOWNLOAD, "download patch", download),
	Get_File(API_SUBMISSION_DOWNLOAD_ZIP, "download archive", download_archive),
	Get_File(API_SUBMISSION_DOWNLOAD_TGZ, "download archive", download_archive),
	Get_File(API_CSS, "download css", _ => style_sheet),
	Post(API_RESUBMIT, "get form for resubmit", resubmit),
	Post(API_SUBMIT, "submit to editors", submit),
	Post(API_BUILD_ABORT, "abort the build", abort_build),
	Post(API_SUBMISSION, "get submission form", parse_create),
	Post(API_SUBMISSION_AUTHORS_ADD, "add author", add_new_author),
	Post(API_SUBMISSION_AUTHORS_REMOVE, "remove author", remove_new_author),
	Post(API_SUBMISSION_AFFILIATIONS_ADD, "add affil", add_new_affil),
	Post(API_SUBMISSION_AFFILIATIONS_REMOVE, "remove affil", remove_new_affil),
	Post(API_SUBMISSION_ENTRIES_ADD, "add entry", add_entry),
	Post(API_SUBMISSION_ENTRIES_REMOVE, "remove entry", remove_entry),
	Post(API_SUBMISSION_ENTRY_AUTHORS_ADD, "add entry author", add_author),
	Post(API_SUBMISSION_ENTRY_AUTHORS_REMOVE, "remove entry author", remove_author),
	Post(API_SUBMISSION_ENTRY_NOTIFIES_ADD, "add notify", add_notify),
	Post(API_SUBMISSION_ENTRY_NOTIFIES_REMOVE, "remove notify", remove_notify),
	Post(API_SUBMISSION_ENTRY_TOPICS_ADD, "add topic", add_topic),
	Post(API_SUBMISSION_ENTRY_TOPICS_REMOVE, "remove topic", remove_topic),
	Post(API_SUBMISSION_ENTRY_RELATED_ADD, "add related", add_related),
	Post(API_SUBMISSION_ENTRY_RELATED_REMOVE, "remove related", remove_related),
	Post(API_SUBMISSION_UPLOAD, "upload archive", upload),
	Post(API_SUBMISSION_CREATE, "create submission", create),
	Post(API_SUBMISSION_STATUS, "set submission status", set_status))

	val head =
	List(
	XML.Elem(Markup("script", List(
	"type" -> "text/javascript",
	"id" -> "MathJax-script",
	"async" -> "async",
	"src" -> "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js")), text("\n")),
	script(
	"MathJax={tex:{inlineMath:[['$','$'],['\\\$','\\\$']]},processEscapes:true,svg:{fontCache:'global'}}"),
	style_file(api.api_url(API_CSS)))
	}


	/* Isabelle tool wrapper */

	val isabelle_tool = Isabelle_Tool("afp_submit", "start afp submission server",
	Scala_Project.here,
	{ args =>

	var backend_path = Path.current
	var frontend_url = new URL("http://localhost:8080")
	var devel = false
	var verbose = false
	var port = 8080
	var dir: Option[Path] = None

	val getopts = Getopts("""
	Usage: isabelle afp_submit [OPTIONS]

	Options are:
	-a PATH backend path (if endpoint is not server root)
	-b URL application frontend url. Default: """ + frontend_url + """"
	-d devel mode (serves frontend and skips automatic AFP repository updates)
	-p PORT server port. Default: """ + port + """
	-v verbose
	-D DIR submission directory

	Start afp submission server. Server is in "edit" mode
	unless directory to store submissions in is specified.
	""",
	"a:" -> (arg => backend_path = Path.explode(arg)),
	"b:" -> (arg => frontend_url = new URL(arg)),
	"d" -> (_ => devel = true),
	"p:" -> (arg => port = Value.Int.parse(arg)),
	"v" -> (_ => verbose = true),
	"D:" -> (arg => dir = Some(Path.explode(arg))))

	getopts(args)

	val afp_structure = AFP_Structure()

	val progress = new Console_Progress(verbose = verbose)

	val (handler, mode) = dir match {
	case Some(dir) => (Handler.Adapter(dir, afp_structure), Mode.SUBMISSION)
	case None => (Handler.Edit(afp_structure), Mode.EDIT)
	}

	val api = new API(frontend_url, backend_path, devel = devel)
	val server = new Server(api = api, afp_structure = afp_structure, mode = mode,
	handler = handler, devel = devel, verbose = verbose, progress = progress, port = port)

	server.run()
	})
	}
	diff --git a/web/authors/argyraki/index.html b/web/authors/argyraki/index.html
	--- a/web/authors/argyraki/index.html
	+++ b/web/authors/argyraki/index.html
	@@ -1,187 +1,196 @@
	<!DOCTYPE html>
	<html lang="en">
	<head>
	<meta charset="utf-8" />
	<meta http-equiv="X-UA-Compatible" content="IE=edge">
	<meta name="viewport" content="width=device-width, initial-scale=1"><title>Angeliki Koutsoukou-Argyraki- Archive of Formal Proofs</title><meta name="description" content="A collection of proof libraries, examples, and larger scientific developments, mechanically checked in the theorem prover Isabelle."><link rel="alternate" type="application/rss+xml" href="../../authors/argyraki/index.xml" title="Archive of Formal Proofs" /><meta property="og:title" content="argyraki" />
	<meta property="og:description" content="A collection of proof libraries, examples, and larger scientific developments, mechanically checked in the theorem prover Isabelle." />
	<meta property="og:type" content="website" />
	<meta property="og:url" content="/authors/argyraki/" /><meta property="og:image" content="/images/afp.png"/><meta property="og:site_name" content="Archive of Formal Proofs" />

	<meta name="twitter:card" content="summary_large_image"/>
	<meta name="twitter:image" content="/images/afp.png"/>

	<meta name="twitter:title" content="argyraki"/>
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	<h1 >
	<span class='first'>A</span>ngeliki <span class='first'>K</span>outsoukou-<span class='first'>A</span>rgyraki</h1>
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	-<h2>Entries</h2><h3 class="head">2022</h3><article class="entry">
	+<h2>Entries</h2><h3 class="head">2023</h3><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../authors/leek">Kevin Lee</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJrZmtsMiJdfQ==">📧</a>, <a href="../../authors/yez">Zhengkun Ye</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJ6eTMwNyJdfQ==">📧</a> and <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJhazIxMTAiXX0=">📧</a></div>
	+ <span class="date">
	+ Aug 10
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	+<h3 class="head">2022</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Kneser_Cauchy_Davenport.html">Kneser's Theorem and the Cauchy–Davenport Theorem</a></h5> <br>by <a href="../../authors/baksys">Mantas Bakšys</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJtYjI0MTIiXX0=">📧</a> and <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJhazIxMTAiXX0=">📧</a></div>
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	Nov 21
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Balog_Szemeredi_Gowers.html">The Balog–Szemerédi–Gowers Theorem</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a>, <a href="../../authors/baksys">Mantas Bakšys</a> <a href="https://github.com/MantasBaksys">🌐</a> and <a href="../../authors/edmonds">Chelsea Edmonds</a> <a href="https://www.cst.cam.ac.uk/people/cle47">🌐</a></div>
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	<h5><a class="title" href="../../entries/Khovanskii_Theorem.html">Khovanskii's Theorem</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
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	Sep 02
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	<h5><a class="title" href="../../entries/Pluennecke_Ruzsa_Inequality.html">The Plünnecke-Ruzsa Inequality</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	May 26
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	</article>


	<h3 class="head">2021</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Roth_Arithmetic_Progressions.html">Roth's Theorem on Arithmetic Progressions</a></h5> <br>by <a href="../../authors/edmonds">Chelsea Edmonds</a> <a href="https://www.cst.cam.ac.uk/people/cle47">🌐</a>, <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a> <a href="https://www.cl.cam.ac.uk/~lp15/">🌐</a></div>
	<span class="date">
	Dec 28
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Szemeredi_Regularity.html">Szemerédi's Regularity Lemma</a></h5> <br>by <a href="../../authors/edmonds">Chelsea Edmonds</a> <a href="https://www.cst.cam.ac.uk/people/cle47">🌐</a>, <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cst.cam.ac.uk/people/ak2110">🌐</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a> <a href="https://www.cl.cam.ac.uk/~lp15/">🌐</a></div>
	<span class="date">
	Nov 05
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	<h3 class="head">2020</h3><article class="entry">
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	<h5><a class="title" href="../../entries/Amicable_Numbers.html">Amicable Numbers</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a></div>
	<span class="date">
	Aug 04
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Irrational_Series_Erdos_Straus.html">Irrationality Criteria for Series by Erdős and Straus</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/li">Wenda Li</a> <a href="https://www.cl.cam.ac.uk/~wl302/">🌐</a></div>
	<span class="date">
	May 12
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	<h3 class="head">2019</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Aristotles_Assertoric_Syllogistic.html">Aristotle's Assertoric Syllogistic</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a></div>
	<span class="date">
	Oct 08
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	<h5><a class="title" href="../../entries/Transcendence_Series_Hancl_Rucki.html">The Transcendence of Certain Infinite Series</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/li">Wenda Li</a> <a href="https://www.cl.cam.ac.uk/~wl302/">🌐</a></div>
	<span class="date">
	Mar 27
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	<h3 class="head">2018</h3><article class="entry">
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	<h5><a class="title" href="../../entries/Octonions.html">Octonions</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a></div>
	<span class="date">
	Sep 14
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Irrationality_J_Hancl.html">Irrational Rapidly Convergent Series</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a href="https://www.cl.cam.ac.uk/~ak2110/">🌐</a> and <a href="../../authors/li">Wenda Li</a> <a href="https://www.cl.cam.ac.uk/~wl302/">🌐</a></div>
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	<title>Kneser's Theorem and the Cauchy–Davenport Theorem</title>
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	<title>Irrationality Criteria for Series by Erdős and Straus</title>
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	<title>The Transcendence of Certain Infinite Series</title>
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	<title>Irrational Rapidly Convergent Series</title>
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	<h1 >
	<span class='first'>A</span>uthors</h1>
	<div>


	</div>
	</header><div>


	<table class="entries">
	<tbody>
	<tr>
	<td>
	<ul>

	<li><a href="../authors/abdulaziz/">Mohammad Abdulaziz</a></li>

	<li><a href="../authors/adelsberger/">Stephan Adelsberger</a></li>

	<li><a href="../authors/aehlig/">Klaus Aehlig</a></li>

	<li><a href="../authors/aissat/">Romain Aissat</a></li>

	<li><a href="../authors/amani/">Sidney Amani</a></li>

	<li><a href="../authors/ammer/">Thomas Ammer</a></li>

	<li><a href="../authors/andreka/">Hajnal Andreka</a></li>

	<li><a href="../authors/andronick/">June Andronick</a></li>

	<li><a href="../authors/aransay/">Jesús Aransay</a></li>

	<li><a href="../authors/argyraki/">Angeliki Koutsoukou-Argyraki</a></li>

	<li><a href="../authors/armstrong/">Alasdair Armstrong</a></li>

	<li><a href="../authors/aspinall/">David Aspinall</a></li>

	<li><a href="../authors/ausaf/">Fahad Ausaf</a></li>

	<li><a href="../authors/avigad/">Jeremy Avigad</a></li>

	<li><a href="../authors/back/">Ralph-Johan Back</a></li>

	<li><a href="../authors/baksys/">Mantas Bakšys</a></li>

	<li><a href="../authors/balbach/">Frank J. Balbach</a></li>

	<li><a href="../authors/ballarin/">Clemens Ballarin</a></li>

	<li><a href="../authors/barsotti/">Damián Barsotti</a></li>

	<li><a href="../authors/bauer/">Gertrud Bauer</a></li>

	<li><a href="../authors/bauereiss/">Thomas Bauereiss</a></li>

	<li><a href="../authors/bayer/">Jonas Bayer</a></li>

	<li><a href="../authors/becker/">Heiko Becker</a></li>

	<li><a href="../authors/beeren/">Joel Beeren</a></li>

	<li><a href="../authors/bella/">Giampaolo Bella</a></li>

	<li><a href="../authors/bengtson/">Jesper Bengtson</a></li>

	<li><a href="../authors/bentkamp/">Alexander Bentkamp</a></li>

	<li><a href="../authors/benzmueller/">Christoph Benzmüller</a></li>

	<li><a href="../authors/beresford/">Alastair R. Beresford</a></li>

	<li><a href="../authors/berghofer/">Stefan Berghofer</a></li>

	<li><a href="../authors/beringer/">Lennart Beringer</a></li>

	<li><a href="../authors/bharadwaj/">Abhijith Bharadwaj</a></li>

	<li><a href="../authors/bhatt/">Bhargav Bhatt</a></li>

	<li><a href="../authors/biendarra/">Julian Biendarra</a></li>

	<li><a href="../authors/bisping/">Benjamin Bisping</a></li>

	<li><a href="../authors/blanchette/">Jasmin Christian Blanchette</a></li>

	<li><a href="../authors/blasum/">Holger Blasum</a></li>

	<li><a href="../authors/blumson/">Ben Blumson</a></li>

	<li><a href="../authors/bockenek/">Joshua Bockenek</a></li>

	<li><a href="../authors/boehme/">Sascha Böhme</a></li>

	<li><a href="../authors/bohrer/">Rose Bohrer</a></li>

	<li><a href="../authors/bordg/">Anthony Bordg</a></li>

	<li><a href="../authors/borgstroem/">Johannes Borgström</a></li>

	<li><a href="../authors/bortin/">Maksym Bortin</a></li>

	<li><a href="../authors/bottesch/">Ralph Bottesch</a></li>

	<li><a href="../authors/boulanger/">Frédéric Boulanger</a></li>

	<li><a href="../authors/bourke/">Timothy Bourke</a></li>

	<li><a href="../authors/boutry/">Pierre Boutry</a></li>

	<li><a href="../authors/boyton/">Andrew Boyton</a></li>

	<li><a href="../authors/bracevac/">Oliver Bračevac</a></li>

	<li><a href="../authors/brandt/">Felix Brandt</a></li>

	<li><a href="../authors/breitner/">Joachim Breitner</a></li>

	<li><a href="../authors/brien/">Nicolas Robinson-O'Brien</a></li>

	<li><a href="../authors/brinkop/">Hauke Brinkop</a></li>

	<li><a href="../authors/brodmann/">Paul-David Brodmann</a></li>

	<li><a href="../authors/brucker/">Achim D. Brucker</a></li>

	<li><a href="../authors/bruegger/">Lukas Brügger</a></li>

	<li><a href="../authors/brun/">Matthias Brun</a></li>

	<li><a href="../authors/brunner/">Julian Brunner</a></li>

	<li><a href="../authors/bulwahn/">Lukas Bulwahn</a></li>

	<li><a href="../authors/butler/">David Butler</a></li>

	<li><a href="../authors/buyse/">Maxime Buyse</a></li>

	<li><a href="../authors/caballero/">José Manuel Rodríguez Caballero</a></li>

	+ <li><a href="../authors/calk/">Cameron Calk</a></li>
	+
	<li><a href="../authors/caminati/">Marco B. Caminati</a></li>

	<li><a href="../authors/campo/">Alejandro del Campo</a></li>

	<li><a href="../authors/chapman/">Peter Chapman</a></li>

	<li><a href="../authors/chen/">L. Chen</a></li>

	<li><a href="../authors/chevalier/">Loïc Chevalier</a></li>

	<li><a href="../authors/christfort/">Axel Christfort</a></li>

	<li><a href="../authors/clouston/">Ranald Clouston</a></li>

	<li><a href="../authors/cock/">David Cock</a></li>

	<li><a href="../authors/coghetto/">Roland Coghetto</a></li>

	<li><a href="../authors/coglio/">Alessandro Coglio</a></li>

	<li><a href="../authors/cohen/">Ernie Cohen</a></li>

	<li><a href="../authors/cordwell/">Katherine Kosaian</a></li>

	<li><a href="../authors/cousin/">Marie Cousin</a></li>

	<li><a href="../authors/cremer/">Nils Cremer</a></li>

	<li><a href="../authors/crighton/">Aaron Crighton</a></li>

	<li><a href="../authors/dalvit/">Christian Dalvit</a></li>

	<li><a href="../authors/danilkin/">Anton Danilkin</a></li>

	<li><a href="../authors/dardinier/">Thibault Dardinier</a></li>

	<li><a href="../authors/david/">Marco David</a></li>

	<li><a href="../authors/debois/">Søren Debois</a></li>

	<li><a href="../authors/debrat/">Henri Debrat</a></li>

	<li><a href="../authors/decova/">Sára Decova</a></li>

	<li><a href="../authors/delemazure/">Théo Delemazure</a></li>

	<li><a href="../authors/demeulemeester/">Tom Demeulemeester</a></li>

	<li><a href="../authors/derrick/">John Derrick</a></li>

	<li><a href="../authors/desharnais/">Martin Desharnais</a></li>

	<li><a href="../authors/diaz/">Javier Díaz</a></li>

	<li><a href="../authors/diekmann/">Cornelius Diekmann</a></li>

	<li><a href="../authors/dirix/">Stefan Dirix</a></li>

	<li><a href="../authors/dittmann/">Christoph Dittmann</a></li>

	<li><a href="../authors/divason/">Jose Divasón</a></li>

	<li><a href="../authors/doczkal/">Christian Doczkal</a></li>

	<li><a href="../authors/dongol/">Brijesh Dongol</a></li>

	<li><a href="../authors/doty/">Matthew Doty</a></li>

	<li><a href="../authors/dubut/">Jérémy Dubut</a></li>

	<li><a href="../authors/dunaev/">Georgy Dunaev</a></li>

	<li><a href="../authors/dyckhoff/">Roy Dyckhoff</a></li>

	<li><a href="../authors/eberl/">Manuel Eberl</a></li>

	<li><a href="../authors/echenim/">Mnacho Echenim</a></li>

	<li><a href="../authors/edmonds/">Chelsea Edmonds</a></li>

	<li><a href="../authors/engelhardt/">Kai Engelhardt</a></li>

	<li><a href="../authors/eriksson/">Lars-Henrik Eriksson</a></li>

	<li><a href="../authors/esparza/">Javier Esparza</a></li>

	<li><a href="../authors/essmann/">Robin Eßmann</a></li>

	<li><a href="../authors/felgenhauer/">Bertram Felgenhauer</a></li>

	<li><a href="../authors/feliachi/">Abderrahmane Feliachi</a></li>

	<li><a href="../authors/fell/">Julian Fell</a></li>

	<li><a href="../authors/fernandez/">Matthew Fernandez</a></li>

	<li><a href="../authors/fiedler/">Ben Fiedler</a></li>

	<li><a href="../authors/fleuriot/">Jacques D. Fleuriot</a></li>

	<li><a href="../authors/fleury/">Mathias Fleury</a></li>

	<li><a href="../authors/foster/">Michael Foster</a></li>

	<li><a href="../authors/fosterj/">J. Nathan Foster</a></li>

	<li><a href="../authors/fosters/">Simon Foster</a></li>

	<li><a href="../authors/fouillard/">Valentin Fouillard</a></li>

	<li><a href="../authors/friedrich/">Stefan Friedrich</a></li>

	<li><a href="../authors/from/">Asta Halkjær From</a></li>

	<li><a href="../authors/fuenmayor/">David Fuenmayor</a></li>

	<li><a href="../authors/furusawa/">Hitoshi Furusawa</a></li>

	<li><a href="../authors/gammie/">Peter Gammie</a></li>

	<li><a href="../authors/gao/">Xin Gao</a></li>

	<li><a href="../authors/gaudel/">Marie-Claude Gaudel</a></li>

	<li><a href="../authors/gay/">Richard Gay</a></li>

	<li><a href="../authors/georgescu/">George Georgescu</a></li>

	<li><a href="../authors/gheri/">Lorenzo Gheri</a></li>

	<li><a href="../authors/ghourabi/">Fadoua Ghourabi</a></li>

	<li><a href="../authors/gioiosa/">Gianpaolo Gioiosa</a></li>

	<li><a href="../authors/glabbeek/">Rob van Glabbeek</a></li>

	<li><a href="../authors/gomes/">Victor B. F. Gomes</a></li>

	<li><a href="../authors/gonzalez/">Edgar Gonzàlez</a></li>

	<li><a href="../authors/gore/">Rajeev Gore</a></li>

	<li><a href="../authors/gouezel/">Sebastien Gouezel</a></li>

	<li><a href="../authors/grechuk/">Bogdan Grechuk</a></li>

	<li><a href="../authors/grewe/">Sylvia Grewe</a></li>

	<li><a href="../authors/griebel/">Simon Griebel</a></li>

	<li><a href="../authors/grov/">Gudmund Grov</a></li>

	<li><a href="../authors/guerraoui/">Rachid Guerraoui</a></li>

	<li><a href="../authors/guiol/">Hervé Guiol</a></li>

	<li><a href="../authors/gunther/">Emmanuel Gunther</a></li>

	<li><a href="../authors/gutkovas/">Ramunas Gutkovas</a></li>

	<li><a href="../authors/guttmann/">Walter Guttmann</a></li>

	<li><a href="../authors/guzman/">Laura P. Gamboa Guzman</a></li>

	<li><a href="../authors/haftmann/">Florian Haftmann</a></li>

	<li><a href="../authors/haslbeck/">Max W. Haslbeck</a></li>

	<li><a href="../authors/haslbeckm/">Maximilian P. L. Haslbeck</a></li>

	<li><a href="../authors/havle/">Oto Havle</a></li>

	<li><a href="../authors/hayes/">Ian J. Hayes</a></li>

	<li><a href="../authors/he/">Yijun He</a></li>

	<li><a href="../authors/heimes/">Lukas Heimes</a></li>

	<li><a href="../authors/helke/">Steffen Helke</a></li>

	<li><a href="../authors/hellauer/">Fabian Hellauer</a></li>

	<li><a href="../authors/heller/">Armin Heller</a></li>

	<li><a href="../authors/henrio/">Ludovic Henrio</a></li>

	<li><a href="../authors/herzberg/">Michael Herzberg</a></li>

	<li><a href="../authors/hess/">Andreas V. Hess</a></li>

	<li><a href="../authors/hetzl/">Stefan Hetzl</a></li>

	<li><a href="../authors/hibon/">Quentin Hibon</a></li>

	<li><a href="../authors/higgins/">Edward Higgins</a></li>

	<li><a href="../authors/hirata/">Michikazu Hirata</a></li>

	<li><a href="../authors/hoefner/">Peter Höfner</a></li>

	<li><a href="../authors/hoelzl/">Johannes Hölzl</a></li>

	<li><a href="../authors/hofmann/">Martin Hofmann</a></li>

	<li><a href="../authors/hofmeier/">Paul Hofmeier</a></li>

	<li><a href="../authors/holub/">Štěpán Holub</a></li>

	<li><a href="../authors/hosking/">Tony Hosking</a></li>

	<li><a href="../authors/hou/">Zhe Hou</a></li>

	<li><a href="../authors/hu/">Shuwei Hu</a></li>

	<li><a href="../authors/huffman/">Brian Huffman</a></li>

	<li><a href="../authors/hupel/">Lars Hupel</a></li>

	<li><a href="../authors/ijbema/">Mark Ijbema</a></li>

	<li><a href="../authors/immler/">Fabian Immler</a></li>

	<li><a href="../authors/israel/">Jonas Israel</a></li>

	<li><a href="../authors/ito/">Yosuke Ito</a></li>

	<li><a href="../authors/iwama/">Fumiya Iwama</a></li>

	<li><a href="../authors/jacobsen/">Frederik Krogsdal Jacobsen</a></li>

	<li><a href="../authors/jaskelioff/">Mauro Jaskelioff</a></li>

	<li><a href="../authors/jaskolka/">Jason Jaskolka</a></li>

	<li><a href="../authors/jensen/">Alexander Birch Jensen</a></li>

	<li><a href="../authors/jiang/">Nan Jiang</a></li>

	<li><a href="../authors/jiangd/">Dongchen Jiang</a></li>

	<li><a href="../authors/joosten/">Sebastiaan J. C. Joosten</a></li>

	<li><a href="../authors/jungnickel/">Tim Jungnickel</a></li>

	<li><a href="../authors/kadzioka/">Maya Kądziołka</a></li>

	<li><a href="../authors/kaliszyk/">Cezary Kaliszyk</a></li>

	<li><a href="../authors/kammueller/">Florian Kammüller</a></li>

	<li><a href="../authors/kappelmann/">Kevin Kappelmann</a></li>

	<li><a href="../authors/karayel/">Emin Karayel</a></li>

	<li><a href="../authors/kastermans/">Bart Kastermans</a></li>

	<li><a href="../authors/katovsky/">Alexander Katovsky</a></li>

	<li><a href="../authors/kaufmann/">Daniela Kaufmann</a></li>

	<li><a href="../authors/keefe/">Greg O'Keefe</a></li>

	<li><a href="../authors/keinholz/">Jonas Keinholz</a></li>

	<li><a href="../authors/kerber/">Manfred Kerber</a></li>

	<li><a href="../authors/keskin/">Ata Keskin</a></li>

	<li><a href="../authors/ketland/">Jeffrey Ketland</a></li>

	<li><a href="../authors/kim/">Sunpill Kim</a></li>

	<li><a href="../authors/kirchner/">Daniel Kirchner</a></li>

	<li><a href="../authors/klein/">Gerwin Klein</a></li>

	<li><a href="../authors/klenze/">Tobias Klenze</a></li>

	<li><a href="../authors/kleppmann/">Martin Kleppmann</a></li>

	<li><a href="../authors/kobayashi/">Hidetsune Kobayashi</a></li>

	<li><a href="../authors/koerner/">Stefan Körner</a></li>

	<li><a href="../authors/kolanski/">Rafal Kolanski</a></li>

	<li><a href="../authors/koller/">Lukas Koller</a></li>

	<li><a href="../authors/krauss/">Alexander Krauss</a></li>

	<li><a href="../authors/kreuzer/">Katharina Kreuzer</a></li>

	<li><a href="../authors/kuncak/">Viktor Kuncak</a></li>

	<li><a href="../authors/kuncar/">Ondřej Kunčar</a></li>

	<li><a href="../authors/kurz/">Friedrich Kurz</a></li>

	<li><a href="../authors/lachnitt/">Hanna Lachnitt</a></li>

	<li><a href="../authors/lallemand/">Joseph Lallemand</a></li>

	<li><a href="../authors/lammich/">Peter Lammich</a></li>

	<li><a href="../authors/lange/">Christoph Lange</a></li>

	<li><a href="../authors/langenstein/">Bruno Langenstein</a></li>

	<li><a href="../authors/lattuada/">Andrea Lattuada</a></li>

	<li><a href="../authors/lauermann/">Nils Lauermann</a></li>

	<li><a href="../authors/laursen/">Christian Pardillo-Laursen</a></li>

	<li><a href="../authors/lederer/">Patrick Lederer</a></li>

	<li><a href="../authors/lee/">Holden Lee</a></li>

	+ <li><a href="../authors/leek/">Kevin Lee</a></li>
	+
	<li><a href="../authors/leustean/">Laurentiu Leustean</a></li>

	<li><a href="../authors/lewis/">Corey Lewis</a></li>

	<li><a href="../authors/li/">Wenda Li</a></li>

	<li><a href="../authors/lim/">Japheth Lim</a></li>

	<li><a href="../authors/lindenberg/">Christina Lindenberg</a></li>

	<li><a href="../authors/linker/">Sven Linker</a></li>

	<li><a href="../authors/liu/">Junyi Liu</a></li>

	<li><a href="../authors/liut/">Tao Liu</a></li>

	<li><a href="../authors/liuy/">Yang Liu</a></li>

	<li><a href="../authors/liy/">Yangjia Li</a></li>

	<li><a href="../authors/lochbihler/">Andreas Lochbihler</a></li>

	<li><a href="../authors/lochmann/">Alexander Lochmann</a></li>

	<li><a href="../authors/lohner/">Denis Lohner</a></li>

	<li><a href="../authors/loibl/">Matthias Loibl</a></li>

	<li><a href="../authors/londono/">Alejandro Gómez-Londoño</a></li>

	<li><a href="../authors/losa/">Giuliano Losa</a></li>

	<li><a href="../authors/lutz/">Bianca Lutz</a></li>

	<li><a href="../authors/lux/">Alexander Lux</a></li>

	<li><a href="../authors/madarasz/">Judit Madarasz</a></li>

	<li><a href="../authors/makarios/">T. J. M. Makarios</a></li>

	<li><a href="../authors/maletzky/">Alexander Maletzky</a></li>

	<li><a href="../authors/mansky/">Susannah Mansky</a></li>

	<li><a href="../authors/mantel/">Heiko Mantel</a></li>

	<li><a href="../authors/margetson/">James Margetson</a></li>

	<li><a href="../authors/maric/">Ognjen Marić</a></li>

	<li><a href="../authors/maricf/">Filip Marić</a></li>

	<li><a href="../authors/marmsoler/">Diego Marmsoler</a></li>

	<li><a href="../authors/matache/">Cristina Matache</a></li>

	<li><a href="../authors/mateo/">Adrián Doña Mateo</a></li>

	<li><a href="../authors/matichuk/">Daniel Matichuk</a></li>

	<li><a href="../authors/matiyasevich/">Yuri Matiyasevich</a></li>

	<li><a href="../authors/maximova/">Alexandra Maximova</a></li>

	<li><a href="../authors/meis/">Rene Meis</a></li>

	<li><a href="../authors/merz/">Stephan Merz</a></li>

	<li><a href="../authors/messner/">Florian Messner</a></li>

	<li><a href="../authors/mhalla/">Mehdi Mhalla</a></li>

	<li><a href="../authors/michaelis/">Julius Michaelis</a></li>

	<li><a href="../authors/milehins/">Mihails Milehins</a></li>

	<li><a href="../authors/minamide/">Yasuhiko Minamide</a></li>

	<li><a href="../authors/mitchell/">Neil Mitchell</a></li>

	<li><a href="../authors/mitsch/">Stefan Mitsch</a></li>

	<li><a href="../authors/moedersheim/">Sebastian Mödersheim</a></li>

	<li><a href="../authors/moeller/">Bernhard Möller</a></li>

	<li><a href="../authors/mori/">Coraline Mori</a></li>

	<li><a href="../authors/muendler/">Niels Mündler</a></li>

	<li><a href="../authors/mulligan/">Dominic P. Mulligan</a></li>

	<li><a href="../authors/munive/">Jonathan Julian Huerta y Munive</a></li>

	<li><a href="../authors/murao/">H. Murao</a></li>

	<li><a href="../authors/murray/">Toby Murray</a></li>

	<li><a href="../authors/myreen/">Magnus O. Myreen</a></li>

	<li><a href="../authors/nagashima/">Yutaka Nagashima</a></li>

	<li><a href="../authors/nagele/">Julian Nagele</a></li>

	<li><a href="../authors/naraschewski/">Wolfgang Naraschewski</a></li>

	<li><a href="../authors/nedzelsky/">Michael Nedzelsky</a></li>

	<li><a href="../authors/nemeti/">István Németi</a></li>

	<li><a href="../authors/nemouchi/">Yakoub Nemouchi</a></li>

	<li><a href="../authors/nestmann/">Uwe Nestmann</a></li>

	<li><a href="../authors/neumann/">René Neumann</a></li>

	<li><a href="../authors/nielsen/">Finn Nielsen</a></li>

	<li><a href="../authors/nikiforov/">Denis Nikiforov</a></li>

	<li><a href="../authors/nipkow/">Tobias Nipkow</a></li>

	<li><a href="../authors/nishihara/">Toshiaki Nishihara</a></li>

	<li><a href="../authors/noce/">Pasquale Noce</a></li>

	<li><a href="../authors/nordhoff/">Benedikt Nordhoff</a></li>

	<li><a href="../authors/noschinski/">Lars Noschinski</a></li>

	<li><a href="../authors/obua/">Steven Obua</a></li>

	<li><a href="../authors/ogawa/">Mizuhito Ogawa</a></li>

	<li><a href="../authors/oldenburg/">Lennart Oldenburg</a></li>

	<li><a href="../authors/olm/">Markus Müller-Olm</a></li>

	<li><a href="../authors/oosterhuis/">Roelof Oosterhuis</a></li>

	<li><a href="../authors/oostrom/">Vincent van Oostrom</a></li>

	<li><a href="../authors/ortner/">Veronika Ortner</a></li>

	<li><a href="../authors/overbeek/">Roy Overbeek</a></li>

	<li><a href="../authors/pagano/">Miguel Pagano</a></li>

	<li><a href="../authors/pal/">Abhik Pal</a></li>

	<li><a href="../authors/paleo/">Bruno Woltzenlogel Paleo</a></li>

	<li><a href="../authors/palmer/">Jake Palmer</a></li>

	<li><a href="../authors/park/">Seung Hoon Park</a></li>

	<li><a href="../authors/parkinson/">Matthew Parkinson</a></li>

	<li><a href="../authors/parrow/">Joachim Parrow</a></li>

	<li><a href="../authors/parsert/">Julian Parsert</a></li>

	<li><a href="../authors/paulson/">Lawrence C. Paulson</a></li>

	<li><a href="../authors/peltier/">Nicolas Peltier</a></li>

	<li><a href="../authors/peters/">Kirstin Peters</a></li>

	<li><a href="../authors/petrovic/">Danijela Petrovic</a></li>

	<li><a href="../authors/pierzchalski/">Edward Pierzchalski</a></li>

	<li><a href="../authors/platzer/">André Platzer</a></li>

	<li><a href="../authors/pohjola/">Johannes Åman Pohjola</a></li>

	<li><a href="../authors/pollak/">Florian Pollak</a></li>

	<li><a href="../authors/popescu/">Andrei Popescu</a></li>

	<li><a href="../authors/porter/">Benjamin Porter</a></li>

	<li><a href="../authors/prathamesh/">T.V.H. Prathamesh</a></li>

	<li><a href="../authors/preoteasa/">Viorel Preoteasa</a></li>

	<li><a href="../authors/pusch/">Cornelia Pusch</a></li>

	<li><a href="../authors/qiu/">Qi Qiu</a></li>

	<li><a href="../authors/rabe/">Markus N. Rabe</a></li>

	<li><a href="../authors/raedle/">Jonas Rädle</a></li>

	<li><a href="../authors/raska/">Martin Raška</a></li>

	<li><a href="../authors/raszyk/">Martin Raszyk</a></li>

	<li><a href="../authors/rau/">Martin Rau</a></li>

	<li><a href="../authors/rauch/">Nicole Rauch</a></li>

	<li><a href="../authors/raumer/">Jakob von Raumer</a></li>

	<li><a href="../authors/ravindran/">Binoy Ravindran</a></li>

	<li><a href="../authors/rawson/">Michael Rawson</a></li>

	<li><a href="../authors/raya/">Rodrigo Raya</a></li>

	<li><a href="../authors/regensburger/">Franz Regensburger</a></li>

	<li><a href="../authors/reiche/">Sebastian Reiche</a></li>

	<li><a href="../authors/reiter/">Markus Reiter</a></li>

	<li><a href="../authors/reynaud/">Alban Reynaud</a></li>

	<li><a href="../authors/ribeiro/">Pedro Ribeiro</a></li>

	<li><a href="../authors/richter/">Stefan Richter</a></li>

	<li><a href="../authors/rickmann/">Christina Rickmann</a></li>

	<li><a href="../authors/ridge/">Tom Ridge</a></li>

	<li><a href="../authors/rizaldi/">Albert Rizaldi</a></li>

	<li><a href="../authors/rizkallah/">Christine Rizkallah</a></li>

	<li><a href="../authors/robillard/">Simon Robillard</a></li>

	<li><a href="../authors/roessle/">Ian Roessle</a></li>

	<li><a href="../authors/romanos/">Ralph Romanos</a></li>

	<li><a href="../authors/rosskopf/">Simon Roßkopf</a></li>

	<li><a href="../authors/rowat/">Colin Rowat</a></li>

	<li><a href="../authors/sabouret/">Nicolas Sabouret</a></li>

	<li><a href="../authors/sachtleben/">Robert Sachtleben</a></li>

	<li><a href="../authors/saile/">Christian Saile</a></li>

	<li><a href="../authors/sanan/">David Sanan</a></li>

	<li><a href="../authors/sato/">Tetsuya Sato</a></li>

	<li><a href="../authors/sauer/">Jens Sauer</a></li>

	<li><a href="../authors/schaeffeler/">Maximilian Schäffeler</a></li>

	<li><a href="../authors/scharager/">Matias Scharager</a></li>

	<li><a href="../authors/schimpf/">Alexander Schimpf</a></li>

	<li><a href="../authors/schirmer/">Norbert Schirmer</a></li>

	<li><a href="../authors/schleicher/">Dierk Schleicher</a></li>

	<li><a href="../authors/schlichtkrull/">Anders Schlichtkrull</a></li>

	<li><a href="../authors/schmaltz/">Julien Schmaltz</a></li>

	<li><a href="../authors/schmidinger/">Lukas Schmidinger</a></li>

	<li><a href="../authors/schmoetten/">Richard Schmoetten</a></li>

	<li><a href="../authors/schneider/">Joshua Schneider</a></li>

	<li><a href="../authors/schoepe/">Daniel Schoepe</a></li>

	<li><a href="../authors/schoepf/">Jonas Schöpf</a></li>

	<li><a href="../authors/scott/">Dana Scott</a></li>

	<li><a href="../authors/sefidgar/">S. Reza Sefidgar</a></li>

	<li><a href="../authors/seidl/">Benedikt Seidl</a></li>

	<li><a href="../authors/seidler/">Henning Seidler</a></li>

	<li><a href="../authors/sewell/">Thomas Sewell</a></li>

	<li><a href="../authors/sickert/">Salomon Sickert</a></li>

	<li><a href="../authors/siek/">Jeremy Siek</a></li>

	<li><a href="../authors/simic/">Danijela Simić</a></li>

	<li><a href="../authors/sison/">Robert Sison</a></li>

	<li><a href="../authors/smaus/">Jan-Georg Smaus</a></li>

	<li><a href="../authors/smola/">Filip Smola</a></li>

	<li><a href="../authors/snelting/">Gregor Snelting</a></li>

	<li><a href="../authors/somaini/">Ivano Somaini</a></li>

	<li><a href="../authors/somogyi/">Dániel Somogyi</a></li>

	<li><a href="../authors/spasic/">Mirko Spasić</a></li>

	<li><a href="../authors/spichkova/">Maria Spichkova</a></li>

	<li><a href="../authors/spitz/">Maximilian Spitz</a></li>

	<li><a href="../authors/sprenger/">Christoph Sprenger</a></li>

	<li><a href="../authors/staats/">Charles Staats</a></li>

	<li><a href="../authors/stannett/">Mike Stannett</a></li>

	<li><a href="../authors/stark/">Eugene W. Stark</a></li>

	<li><a href="../authors/starosta/">Štěpán Starosta</a></li>

	<li><a href="../authors/steen/">Alexander Steen</a></li>

	<li><a href="../authors/steinberg/">Matías Steinberg</a></li>

	<li><a href="../authors/stephan/">Werner Stephan</a></li>

	<li><a href="../authors/sternagel/">Christian Sternagel</a></li>

	<li><a href="../authors/sternagelt/">Thomas Sternagel</a></li>

	<li><a href="../authors/stevens/">Lukas Stevens</a></li>

	<li><a href="../authors/stock/">Benedikt Stock</a></li>

	<li><a href="../authors/stoeckl/">Bernhard Stöckl</a></li>

	<li><a href="../authors/stricker/">Christian Stricker</a></li>

	<li><a href="../authors/strnisa/">Rok Strniša</a></li>

	<li><a href="../authors/struth/">Georg Struth</a></li>

	<li><a href="../authors/stueber/">Anke Stüber</a></li>

	<li><a href="../authors/stuewe/">Daniel Stüwe</a></li>

	<li><a href="../authors/sudbrock/">Henning Sudbrock</a></li>

	<li><a href="../authors/sudhof/">Henry Sudhof</a></li>

	<li><a href="../authors/sulejmani/">Ujkan Sulejmani</a></li>

	<li><a href="../authors/sutcliffe/">Geoff Sutcliffe</a></li>

	<li><a href="../authors/sylvestre/">Jeremy Sylvestre</a></li>

	<li><a href="../authors/szekely/">Gergely Szekely</a></li>

	<li><a href="../authors/taha/">Safouan Taha</a></li>

	<li><a href="../authors/tan/">Yong Kiam Tan</a></li>

	<li><a href="../authors/tanaka/">Miki Tanaka</a></li>

	<li><a href="../authors/tasch/">Markus Tasch</a></li>

	<li><a href="../authors/taylor/">Ramsay G. Taylor</a></li>

	<li><a href="../authors/terraf/">Pedro Sánchez Terraf</a></li>

	<li><a href="../authors/thiemann/">René Thiemann</a></li>

	<li><a href="../authors/thommes/">Joseph Thommes</a></li>

	<li><a href="../authors/thomson/">Fox Thomson</a></li>

	<li><a href="../authors/tiu/">Alwen Tiu</a></li>

	<li><a href="../authors/toth/">Balazs Toth</a></li>

	<li><a href="../authors/tourret/">Sophie Tourret</a></li>

	<li><a href="../authors/trachtenherz/">David Trachtenherz</a></li>

	<li><a href="../authors/traut/">Christoph Traut</a></li>

	<li><a href="../authors/traytel/">Dmitriy Traytel</a></li>

	<li><a href="../authors/trelat/">Vincent Trélat</a></li>

	<li><a href="../authors/tuong/">Frédéric Tuong</a></li>

	<li><a href="../authors/tuongj/">Joseph Tuong</a></li>

	<li><a href="../authors/tverdyshev/">Sergey Tverdyshev</a></li>

	<li><a href="../authors/ullrich/">Sebastian Ullrich</a></li>

	<li><a href="../authors/unruh/">Dominique Unruh</a></li>

	<li><a href="../authors/urban/">Christian Urban</a></li>

	<li><a href="../authors/van/">Hai Nguyen Van</a></li>

	<li><a href="../authors/velykis/">Andrius Velykis</a></li>

	<li><a href="../authors/verbeek/">Freek Verbeek</a></li>

	<li><a href="../authors/villadsen/">Jørgen Villadsen</a></li>

	<li><a href="../authors/voisin/">Frederic Voisin</a></li>

	<li><a href="../authors/vytiniotis/">Dimitrios Vytiniotis</a></li>

	<li><a href="../authors/wagner/">Max Wagner</a></li>

	<li><a href="../authors/waldmann/">Uwe Waldmann</a></li>

	<li><a href="../authors/wand/">Daniel Wand</a></li>

	<li><a href="../authors/wang/">Shuling Wang</a></li>

	<li><a href="../authors/wassell/">Mark Wassell</a></li>

	<li><a href="../authors/wasserrab/">Daniel Wasserrab</a></li>

	<li><a href="../authors/watt/">Conrad Watt</a></li>

	<li><a href="../authors/weber/">Tjark Weber</a></li>

	<li><a href="../authors/weerwag/">Timmy Weerwag</a></li>

	<li><a href="../authors/weidner/">Arno Wilhelm-Weidner</a></li>

	<li><a href="../authors/wenninger/">Elias Wenninger</a></li>

	<li><a href="../authors/wenzel/">Makarius Wenzel</a></li>

	<li><a href="../authors/whitley/">A Whitley</a></li>

	<li><a href="../authors/wickerson/">John Wickerson</a></li>

	<li><a href="../authors/willenbrink/">Sebastian Willenbrink</a></li>

	<li><a href="../authors/wimmer/">Simon Wimmer</a></li>

	<li><a href="../authors/wirt/">Kai Wirt</a></li>

	<li><a href="../authors/wolff/">Burkhart Wolff</a></li>

	<li><a href="../authors/wu/">Chunhan Wu</a></li>

	<li><a href="../authors/xu/">Jian Xu</a></li>

	<li><a href="../authors/yamada/">Akihisa Yamada</a></li>

	<li><a href="../authors/ye/">Lina Ye</a></li>

	+ <li><a href="../authors/yez/">Zhengkun Ye</a></li>
	+
	<li><a href="../authors/ying/">Shenggang Ying</a></li>

	<li><a href="../authors/yingm/">Mingsheng Ying</a></li>

	<li><a href="../authors/yu/">Lei Yu</a></li>

	<li><a href="../authors/zankl/">Harald Zankl</a></li>

	<li><a href="../authors/zee/">Karen Zee</a></li>

	<li><a href="../authors/zeller/">Peter Zeller</a></li>

	<li><a href="../authors/zeyda/">Frank Zeyda</a></li>

	<li><a href="../authors/zhan/">Bohua Zhan</a></li>

	<li><a href="../authors/zhang/">Yu Zhang</a></li>

	<li><a href="../authors/zhangx/">Xingyuan Zhang</a></li>

	<li><a href="../authors/zhann/">Naijun Zhan</a></li>

	</ul>
	</td>
	</tr>
	</tbody>
	</table>



	</div>
	</div>
	</body>

	</html>
	\ No newline at end of file
	diff --git a/web/authors/index.json b/web/authors/index.json
	--- a/web/authors/index.json
	+++ b/web/authors/index.json
	@@ -1,2312 +1,2327 @@
	[
	{
	"id": 0,
	"link": "/authors/abdulaziz/",
	"name": "Mohammad Abdulaziz"
	},
	{
	"id": 1,
	"link": "/authors/adelsberger/",
	"name": "Stephan Adelsberger"
	},
	{
	"id": 2,
	"link": "/authors/aehlig/",
	"name": "Klaus Aehlig"
	},
	{
	"id": 3,
	"link": "/authors/aissat/",
	"name": "Romain Aissat"
	},
	{
	"id": 4,
	"link": "/authors/amani/",
	"name": "Sidney Amani"
	},
	{
	"id": 5,
	"link": "/authors/ammer/",
	"name": "Thomas Ammer"
	},
	{
	"id": 6,
	"link": "/authors/andreka/",
	"name": "Hajnal Andreka"
	},
	{
	"id": 7,
	"link": "/authors/andronick/",
	"name": "June Andronick"
	},
	{
	"id": 8,
	"link": "/authors/aransay/",
	"name": "Jesús Aransay"
	},
	{
	"id": 9,
	"link": "/authors/argyraki/",
	"name": "Angeliki Koutsoukou-Argyraki"
	},
	{
	"id": 10,
	"link": "/authors/armstrong/",
	"name": "Alasdair Armstrong"
	},
	{
	"id": 11,
	"link": "/authors/aspinall/",
	"name": "David Aspinall"
	},
	{
	"id": 12,
	"link": "/authors/ausaf/",
	"name": "Fahad Ausaf"
	},
	{
	"id": 13,
	"link": "/authors/avigad/",
	"name": "Jeremy Avigad"
	},
	{
	"id": 14,
	"link": "/authors/back/",
	"name": "Ralph-Johan Back"
	},
	{
	"id": 15,
	"link": "/authors/baksys/",
	"name": "Mantas Bakšys"
	},
	{
	"id": 16,
	"link": "/authors/balbach/",
	"name": "Frank J. Balbach"
	},
	{
	"id": 17,
	"link": "/authors/ballarin/",
	"name": "Clemens Ballarin"
	},
	{
	"id": 18,
	"link": "/authors/barsotti/",
	"name": "Damián Barsotti"
	},
	{
	"id": 19,
	"link": "/authors/bauer/",
	"name": "Gertrud Bauer"
	},
	{
	"id": 20,
	"link": "/authors/bauereiss/",
	"name": "Thomas Bauereiss"
	},
	{
	"id": 21,
	"link": "/authors/bayer/",
	"name": "Jonas Bayer"
	},
	{
	"id": 22,
	"link": "/authors/becker/",
	"name": "Heiko Becker"
	},
	{
	"id": 23,
	"link": "/authors/beeren/",
	"name": "Joel Beeren"
	},
	{
	"id": 24,
	"link": "/authors/bella/",
	"name": "Giampaolo Bella"
	},
	{
	"id": 25,
	"link": "/authors/bengtson/",
	"name": "Jesper Bengtson"
	},
	{
	"id": 26,
	"link": "/authors/bentkamp/",
	"name": "Alexander Bentkamp"
	},
	{
	"id": 27,
	"link": "/authors/benzmueller/",
	"name": "Christoph Benzmüller"
	},
	{
	"id": 28,
	"link": "/authors/beresford/",
	"name": "Alastair R. Beresford"
	},
	{
	"id": 29,
	"link": "/authors/berghofer/",
	"name": "Stefan Berghofer"
	},
	{
	"id": 30,
	"link": "/authors/beringer/",
	"name": "Lennart Beringer"
	},
	{
	"id": 31,
	"link": "/authors/bharadwaj/",
	"name": "Abhijith Bharadwaj"
	},
	{
	"id": 32,
	"link": "/authors/bhatt/",
	"name": "Bhargav Bhatt"
	},
	{
	"id": 33,
	"link": "/authors/biendarra/",
	"name": "Julian Biendarra"
	},
	{
	"id": 34,
	"link": "/authors/bisping/",
	"name": "Benjamin Bisping"
	},
	{
	"id": 35,
	"link": "/authors/blanchette/",
	"name": "Jasmin Christian Blanchette"
	},
	{
	"id": 36,
	"link": "/authors/blasum/",
	"name": "Holger Blasum"
	},
	{
	"id": 37,
	"link": "/authors/blumson/",
	"name": "Ben Blumson"
	},
	{
	"id": 38,
	"link": "/authors/bockenek/",
	"name": "Joshua Bockenek"
	},
	{
	"id": 39,
	"link": "/authors/boehme/",
	"name": "Sascha Böhme"
	},
	{
	"id": 40,
	"link": "/authors/bohrer/",
	"name": "Rose Bohrer"
	},
	{
	"id": 41,
	"link": "/authors/bordg/",
	"name": "Anthony Bordg"
	},
	{
	"id": 42,
	"link": "/authors/borgstroem/",
	"name": "Johannes Borgström"
	},
	{
	"id": 43,
	"link": "/authors/bortin/",
	"name": "Maksym Bortin"
	},
	{
	"id": 44,
	"link": "/authors/bottesch/",
	"name": "Ralph Bottesch"
	},
	{
	"id": 45,
	"link": "/authors/boulanger/",
	"name": "Frédéric Boulanger"
	},
	{
	"id": 46,
	"link": "/authors/bourke/",
	"name": "Timothy Bourke"
	},
	{
	"id": 47,
	"link": "/authors/boutry/",
	"name": "Pierre Boutry"
	},
	{
	"id": 48,
	"link": "/authors/boyton/",
	"name": "Andrew Boyton"
	},
	{
	"id": 49,
	"link": "/authors/bracevac/",
	"name": "Oliver Bračevac"
	},
	{
	"id": 50,
	"link": "/authors/brandt/",
	"name": "Felix Brandt"
	},
	{
	"id": 51,
	"link": "/authors/breitner/",
	"name": "Joachim Breitner"
	},
	{
	"id": 52,
	"link": "/authors/brien/",
	"name": "Nicolas Robinson-O'Brien"
	},
	{
	"id": 53,
	"link": "/authors/brinkop/",
	"name": "Hauke Brinkop"
	},
	{
	"id": 54,
	"link": "/authors/brodmann/",
	"name": "Paul-David Brodmann"
	},
	{
	"id": 55,
	"link": "/authors/brucker/",
	"name": "Achim D. Brucker"
	},
	{
	"id": 56,
	"link": "/authors/bruegger/",
	"name": "Lukas Brügger"
	},
	{
	"id": 57,
	"link": "/authors/brun/",
	"name": "Matthias Brun"
	},
	{
	"id": 58,
	"link": "/authors/brunner/",
	"name": "Julian Brunner"
	},
	{
	"id": 59,
	"link": "/authors/bulwahn/",
	"name": "Lukas Bulwahn"
	},
	{
	"id": 60,
	"link": "/authors/butler/",
	"name": "David Butler"
	},
	{
	"id": 61,
	"link": "/authors/buyse/",
	"name": "Maxime Buyse"
	},
	{
	"id": 62,
	"link": "/authors/caballero/",
	"name": "José Manuel Rodríguez Caballero"
	},
	{
	"id": 63,
	- "link": "/authors/caminati/",
	- "name": "Marco B. Caminati"
	+ "link": "/authors/calk/",
	+ "name": "Cameron Calk"
	},
	{
	"id": 64,
	- "link": "/authors/campo/",
	- "name": "Alejandro del Campo"
	+ "link": "/authors/caminati/",
	+ "name": "Marco B. Caminati"
	},
	{
	"id": 65,
	- "link": "/authors/chapman/",
	- "name": "Peter Chapman"
	+ "link": "/authors/campo/",
	+ "name": "Alejandro del Campo"
	},
	{
	"id": 66,
	- "link": "/authors/chen/",
	- "name": "L. Chen"
	+ "link": "/authors/chapman/",
	+ "name": "Peter Chapman"
	},
	{
	"id": 67,
	- "link": "/authors/chevalier/",
	- "name": "Loïc Chevalier"
	+ "link": "/authors/chen/",
	+ "name": "L. Chen"
	},
	{
	"id": 68,
	- "link": "/authors/christfort/",
	- "name": "Axel Christfort"
	+ "link": "/authors/chevalier/",
	+ "name": "Loïc Chevalier"
	},
	{
	"id": 69,
	- "link": "/authors/clouston/",
	- "name": "Ranald Clouston"
	+ "link": "/authors/christfort/",
	+ "name": "Axel Christfort"
	},
	{
	"id": 70,
	- "link": "/authors/cock/",
	- "name": "David Cock"
	+ "link": "/authors/clouston/",
	+ "name": "Ranald Clouston"
	},
	{
	"id": 71,
	- "link": "/authors/coghetto/",
	- "name": "Roland Coghetto"
	+ "link": "/authors/cock/",
	+ "name": "David Cock"
	},
	{
	"id": 72,
	- "link": "/authors/coglio/",
	- "name": "Alessandro Coglio"
	+ "link": "/authors/coghetto/",
	+ "name": "Roland Coghetto"
	},
	{
	"id": 73,
	- "link": "/authors/cohen/",
	- "name": "Ernie Cohen"
	+ "link": "/authors/coglio/",
	+ "name": "Alessandro Coglio"
	},
	{
	"id": 74,
	- "link": "/authors/cordwell/",
	- "name": "Katherine Kosaian"
	+ "link": "/authors/cohen/",
	+ "name": "Ernie Cohen"
	},
	{
	"id": 75,
	- "link": "/authors/cousin/",
	- "name": "Marie Cousin"
	+ "link": "/authors/cordwell/",
	+ "name": "Katherine Kosaian"
	},
	{
	"id": 76,
	- "link": "/authors/cremer/",
	- "name": "Nils Cremer"
	+ "link": "/authors/cousin/",
	+ "name": "Marie Cousin"
	},
	{
	"id": 77,
	- "link": "/authors/crighton/",
	- "name": "Aaron Crighton"
	+ "link": "/authors/cremer/",
	+ "name": "Nils Cremer"
	},
	{
	"id": 78,
	- "link": "/authors/dalvit/",
	- "name": "Christian Dalvit"
	+ "link": "/authors/crighton/",
	+ "name": "Aaron Crighton"
	},
	{
	"id": 79,
	- "link": "/authors/danilkin/",
	- "name": "Anton Danilkin"
	+ "link": "/authors/dalvit/",
	+ "name": "Christian Dalvit"
	},
	{
	"id": 80,
	- "link": "/authors/dardinier/",
	- "name": "Thibault Dardinier"
	+ "link": "/authors/danilkin/",
	+ "name": "Anton Danilkin"
	},
	{
	"id": 81,
	- "link": "/authors/david/",
	- "name": "Marco David"
	+ "link": "/authors/dardinier/",
	+ "name": "Thibault Dardinier"
	},
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	- "name": "Søren Debois"
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	- "name": "Théo Delemazure"
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	- "link": "/authors/demeulemeester/",
	- "name": "Tom Demeulemeester"
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	- "link": "/authors/desharnais/",
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	- "name": "Javier Díaz"
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	- "link": "/authors/diekmann/",
	- "name": "Cornelius Diekmann"
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	- "name": "Stefan Dirix"
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	- "name": "Jose Divasón"
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	- "name": "Christian Doczkal"
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	- "link": "/authors/dongol/",
	- "name": "Brijesh Dongol"
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	+ "name": "Christian Doczkal"
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	- "link": "/authors/doty/",
	- "name": "Matthew Doty"
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	+ "name": "Brijesh Dongol"
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	- "link": "/authors/dubut/",
	- "name": "Jérémy Dubut"
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	+ "name": "Matthew Doty"
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	- "link": "/authors/dunaev/",
	- "name": "Georgy Dunaev"
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	- "link": "/authors/dyckhoff/",
	- "name": "Roy Dyckhoff"
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	"id": 100,
	- "link": "/authors/eberl/",
	- "name": "Manuel Eberl"
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	+ "name": "Roy Dyckhoff"
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	- "link": "/authors/echenim/",
	- "name": "Mnacho Echenim"
	+ "link": "/authors/eberl/",
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	"id": 102,
	- "link": "/authors/edmonds/",
	- "name": "Chelsea Edmonds"
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	- "link": "/authors/engelhardt/",
	- "name": "Kai Engelhardt"
	+ "link": "/authors/edmonds/",
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	},
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	"id": 104,
	- "link": "/authors/eriksson/",
	- "name": "Lars-Henrik Eriksson"
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	"id": 105,
	- "link": "/authors/esparza/",
	- "name": "Javier Esparza"
	+ "link": "/authors/eriksson/",
	+ "name": "Lars-Henrik Eriksson"
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	"id": 106,
	- "link": "/authors/essmann/",
	- "name": "Robin Eßmann"
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	+ "name": "Javier Esparza"
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	"id": 107,
	- "link": "/authors/felgenhauer/",
	- "name": "Bertram Felgenhauer"
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	+ "name": "Robin Eßmann"
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	- "link": "/authors/feliachi/",
	- "name": "Abderrahmane Feliachi"
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	"id": 109,
	- "link": "/authors/fell/",
	- "name": "Julian Fell"
	+ "link": "/authors/feliachi/",
	+ "name": "Abderrahmane Feliachi"
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	"id": 110,
	- "link": "/authors/fernandez/",
	- "name": "Matthew Fernandez"
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	+ "name": "Julian Fell"
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	- "link": "/authors/fiedler/",
	- "name": "Ben Fiedler"
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	+ "name": "Matthew Fernandez"
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	- "link": "/authors/fleuriot/",
	- "name": "Jacques D. Fleuriot"
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	+ "name": "Ben Fiedler"
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	"id": 113,
	- "link": "/authors/fleury/",
	- "name": "Mathias Fleury"
	+ "link": "/authors/fleuriot/",
	+ "name": "Jacques D. Fleuriot"
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	"id": 114,
	- "link": "/authors/foster/",
	- "name": "Michael Foster"
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	+ "name": "Mathias Fleury"
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	"id": 115,
	- "link": "/authors/fosterj/",
	- "name": "J. Nathan Foster"
	+ "link": "/authors/foster/",
	+ "name": "Michael Foster"
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	"id": 116,
	- "link": "/authors/fosters/",
	- "name": "Simon Foster"
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	+ "name": "J. Nathan Foster"
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	"id": 117,
	- "link": "/authors/fouillard/",
	- "name": "Valentin Fouillard"
	+ "link": "/authors/fosters/",
	+ "name": "Simon Foster"
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	- "link": "/authors/friedrich/",
	- "name": "Stefan Friedrich"
	+ "link": "/authors/fouillard/",
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	"id": 119,
	- "link": "/authors/from/",
	- "name": "Asta Halkjær From"
	+ "link": "/authors/friedrich/",
	+ "name": "Stefan Friedrich"
	},
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	"id": 120,
	- "link": "/authors/fuenmayor/",
	- "name": "David Fuenmayor"
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	+ "name": "Asta Halkjær From"
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	- "link": "/authors/furusawa/",
	- "name": "Hitoshi Furusawa"
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	- "link": "/authors/gammie/",
	- "name": "Peter Gammie"
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	- "link": "/authors/gao/",
	- "name": "Xin Gao"
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	+ "name": "Peter Gammie"
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	"id": 124,
	- "link": "/authors/gaudel/",
	- "name": "Marie-Claude Gaudel"
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	+ "name": "Xin Gao"
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	- "link": "/authors/gay/",
	- "name": "Richard Gay"
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	- "link": "/authors/georgescu/",
	- "name": "George Georgescu"
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	- "link": "/authors/gheri/",
	- "name": "Lorenzo Gheri"
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	- "link": "/authors/ghourabi/",
	- "name": "Fadoua Ghourabi"
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	- "name": "Gianpaolo Gioiosa"
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	- "link": "/authors/glabbeek/",
	- "name": "Rob van Glabbeek"
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	- "link": "/authors/gomes/",
	- "name": "Victor B. F. Gomes"
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	- "link": "/authors/gonzalez/",
	- "name": "Edgar Gonzàlez"
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	- "link": "/authors/gore/",
	- "name": "Rajeev Gore"
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	},
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	- "link": "/authors/gouezel/",
	- "name": "Sebastien Gouezel"
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	- "link": "/authors/grechuk/",
	- "name": "Bogdan Grechuk"
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	"id": 136,
	- "link": "/authors/grewe/",
	- "name": "Sylvia Grewe"
	+ "link": "/authors/grechuk/",
	+ "name": "Bogdan Grechuk"
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	"id": 137,
	- "link": "/authors/griebel/",
	- "name": "Simon Griebel"
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	"id": 138,
	- "link": "/authors/grov/",
	- "name": "Gudmund Grov"
	+ "link": "/authors/griebel/",
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	},
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	"id": 139,
	- "link": "/authors/guerraoui/",
	- "name": "Rachid Guerraoui"
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	+ "name": "Gudmund Grov"
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	"id": 140,
	- "link": "/authors/guiol/",
	- "name": "Hervé Guiol"
	+ "link": "/authors/guerraoui/",
	+ "name": "Rachid Guerraoui"
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	"id": 141,
	- "link": "/authors/gunther/",
	- "name": "Emmanuel Gunther"
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	+ "name": "Hervé Guiol"
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	- "link": "/authors/gutkovas/",
	- "name": "Ramunas Gutkovas"
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	+ "name": "Emmanuel Gunther"
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	"id": 143,
	- "link": "/authors/guttmann/",
	- "name": "Walter Guttmann"
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	},
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	"id": 144,
	- "link": "/authors/guzman/",
	- "name": "Laura P. Gamboa Guzman"
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	+ "name": "Walter Guttmann"
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	"id": 145,
	- "link": "/authors/haftmann/",
	- "name": "Florian Haftmann"
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	+ "name": "Laura P. Gamboa Guzman"
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	"id": 146,
	- "link": "/authors/haslbeck/",
	- "name": "Max W. Haslbeck"
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	+ "name": "Florian Haftmann"
	},
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	"id": 147,
	- "link": "/authors/haslbeckm/",
	- "name": "Maximilian P. L. Haslbeck"
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	+ "name": "Max W. Haslbeck"
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	- "link": "/authors/havle/",
	- "name": "Oto Havle"
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	"id": 149,
	- "link": "/authors/hayes/",
	- "name": "Ian J. Hayes"
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	+ "name": "Oto Havle"
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	"id": 150,
	- "link": "/authors/he/",
	- "name": "Yijun He"
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	+ "name": "Ian J. Hayes"
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	- "link": "/authors/heimes/",
	- "name": "Lukas Heimes"
	+ "link": "/authors/he/",
	+ "name": "Yijun He"
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	- "link": "/authors/helke/",
	- "name": "Steffen Helke"
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	+ "name": "Lukas Heimes"
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	- "link": "/authors/hellauer/",
	- "name": "Fabian Hellauer"
	+ "link": "/authors/helke/",
	+ "name": "Steffen Helke"
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	- "link": "/authors/heller/",
	- "name": "Armin Heller"
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	+ "name": "Fabian Hellauer"
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	- "link": "/authors/henrio/",
	- "name": "Ludovic Henrio"
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	+ "name": "Armin Heller"
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	- "link": "/authors/herzberg/",
	- "name": "Michael Herzberg"
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	+ "name": "Ludovic Henrio"
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	- "link": "/authors/hess/",
	- "name": "Andreas V. Hess"
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	+ "name": "Michael Herzberg"
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	- "link": "/authors/hetzl/",
	- "name": "Stefan Hetzl"
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	+ "name": "Andreas V. Hess"
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	- "link": "/authors/hibon/",
	- "name": "Quentin Hibon"
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	+ "name": "Stefan Hetzl"
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	"id": 160,
	- "link": "/authors/higgins/",
	- "name": "Edward Higgins"
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	+ "name": "Quentin Hibon"
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	"id": 161,
	- "link": "/authors/hirata/",
	- "name": "Michikazu Hirata"
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	+ "name": "Edward Higgins"
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	- "link": "/authors/hoefner/",
	- "name": "Peter Höfner"
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	+ "name": "Michikazu Hirata"
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	- "link": "/authors/hoelzl/",
	- "name": "Johannes Hölzl"
	+ "link": "/authors/hoefner/",
	+ "name": "Peter Höfner"
	},
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	"id": 164,
	- "link": "/authors/hofmann/",
	- "name": "Martin Hofmann"
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	+ "name": "Johannes Hölzl"
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	"id": 165,
	- "link": "/authors/hofmeier/",
	- "name": "Paul Hofmeier"
	+ "link": "/authors/hofmann/",
	+ "name": "Martin Hofmann"
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	"id": 166,
	- "link": "/authors/holub/",
	- "name": "Štěpán Holub"
	+ "link": "/authors/hofmeier/",
	+ "name": "Paul Hofmeier"
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	"id": 167,
	- "link": "/authors/hosking/",
	- "name": "Tony Hosking"
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	+ "name": "Štěpán Holub"
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	"id": 168,
	- "link": "/authors/hou/",
	- "name": "Zhe Hou"
	+ "link": "/authors/hosking/",
	+ "name": "Tony Hosking"
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	"id": 169,
	- "link": "/authors/hu/",
	- "name": "Shuwei Hu"
	+ "link": "/authors/hou/",
	+ "name": "Zhe Hou"
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	"id": 170,
	- "link": "/authors/huffman/",
	- "name": "Brian Huffman"
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	+ "name": "Shuwei Hu"
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	"id": 171,
	- "link": "/authors/hupel/",
	- "name": "Lars Hupel"
	+ "link": "/authors/huffman/",
	+ "name": "Brian Huffman"
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	"id": 172,
	- "link": "/authors/ijbema/",
	- "name": "Mark Ijbema"
	+ "link": "/authors/hupel/",
	+ "name": "Lars Hupel"
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	"id": 173,
	- "link": "/authors/immler/",
	- "name": "Fabian Immler"
	+ "link": "/authors/ijbema/",
	+ "name": "Mark Ijbema"
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	"id": 174,
	- "link": "/authors/israel/",
	- "name": "Jonas Israel"
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	+ "name": "Fabian Immler"
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	{
	"id": 175,
	- "link": "/authors/ito/",
	- "name": "Yosuke Ito"
	+ "link": "/authors/israel/",
	+ "name": "Jonas Israel"
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	{
	"id": 176,
	- "link": "/authors/iwama/",
	- "name": "Fumiya Iwama"
	+ "link": "/authors/ito/",
	+ "name": "Yosuke Ito"
	},
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	"id": 177,
	- "link": "/authors/jacobsen/",
	- "name": "Frederik Krogsdal Jacobsen"
	+ "link": "/authors/iwama/",
	+ "name": "Fumiya Iwama"
	},
	{
	"id": 178,
	- "link": "/authors/jaskelioff/",
	- "name": "Mauro Jaskelioff"
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	+ "name": "Frederik Krogsdal Jacobsen"
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	{
	"id": 179,
	- "link": "/authors/jaskolka/",
	- "name": "Jason Jaskolka"
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	+ "name": "Mauro Jaskelioff"
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	{
	"id": 180,
	- "link": "/authors/jensen/",
	- "name": "Alexander Birch Jensen"
	+ "link": "/authors/jaskolka/",
	+ "name": "Jason Jaskolka"
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	{
	"id": 181,
	- "link": "/authors/jiang/",
	- "name": "Nan Jiang"
	+ "link": "/authors/jensen/",
	+ "name": "Alexander Birch Jensen"
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	{
	"id": 182,
	- "link": "/authors/jiangd/",
	- "name": "Dongchen Jiang"
	+ "link": "/authors/jiang/",
	+ "name": "Nan Jiang"
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	"id": 183,
	- "link": "/authors/joosten/",
	- "name": "Sebastiaan J. C. Joosten"
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	+ "name": "Dongchen Jiang"
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	{
	"id": 184,
	- "link": "/authors/jungnickel/",
	- "name": "Tim Jungnickel"
	+ "link": "/authors/joosten/",
	+ "name": "Sebastiaan J. C. Joosten"
	},
	{
	"id": 185,
	- "link": "/authors/kadzioka/",
	- "name": "Maya Kądziołka"
	+ "link": "/authors/jungnickel/",
	+ "name": "Tim Jungnickel"
	},
	{
	"id": 186,
	- "link": "/authors/kaliszyk/",
	- "name": "Cezary Kaliszyk"
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	+ "name": "Maya Kądziołka"
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	{
	"id": 187,
	- "link": "/authors/kammueller/",
	- "name": "Florian Kammüller"
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	+ "name": "Cezary Kaliszyk"
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	"id": 188,
	- "link": "/authors/kappelmann/",
	- "name": "Kevin Kappelmann"
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	"id": 189,
	- "link": "/authors/karayel/",
	- "name": "Emin Karayel"
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	"id": 190,
	- "link": "/authors/kastermans/",
	- "name": "Bart Kastermans"
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	+ "name": "Emin Karayel"
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	"id": 191,
	- "link": "/authors/katovsky/",
	- "name": "Alexander Katovsky"
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	+ "name": "Bart Kastermans"
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	"id": 192,
	- "link": "/authors/kaufmann/",
	- "name": "Daniela Kaufmann"
	+ "link": "/authors/katovsky/",
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	- "name": "Diego Marmsoler"
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	- "name": "Cristina Matache"
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	- "link": "/authors/mateo/",
	- "name": "Adrián Doña Mateo"
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	- "name": "Daniel Matichuk"
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	- "link": "/authors/maximova/",
	- "name": "Alexandra Maximova"
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	- "name": "Rene Meis"
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	- "name": "Mehdi Mhalla"
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	- "name": "Stefan Mitsch"
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	- "name": "Kirstin Peters"
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	"id": 306,
	- "link": "/authors/petrovic/",
	- "name": "Danijela Petrovic"
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	"id": 307,
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	- "name": "Edward Pierzchalski"
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	"id": 308,
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	"id": 309,
	- "link": "/authors/pohjola/",
	- "name": "Johannes Åman Pohjola"
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	"id": 310,
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	"id": 311,
	- "link": "/authors/popescu/",
	- "name": "Andrei Popescu"
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	+ "name": "Johannes Åman Pohjola"
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	"id": 312,
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	- "name": "Benjamin Porter"
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	"id": 313,
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	- "name": "T.V.H. Prathamesh"
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	"id": 314,
	- "link": "/authors/preoteasa/",
	- "name": "Viorel Preoteasa"
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	- "name": "Cornelia Pusch"
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	"id": 316,
	- "link": "/authors/qiu/",
	- "name": "Qi Qiu"
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	"id": 317,
	- "link": "/authors/rabe/",
	- "name": "Markus N. Rabe"
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	"id": 318,
	- "link": "/authors/raedle/",
	- "name": "Jonas Rädle"
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	+ "name": "Qi Qiu"
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	"id": 319,
	- "link": "/authors/raska/",
	- "name": "Martin Raška"
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	+ "name": "Markus N. Rabe"
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	"id": 320,
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	- "name": "Martin Raszyk"
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	"id": 321,
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	- "name": "Martin Rau"
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	"id": 322,
	- "link": "/authors/rauch/",
	- "name": "Nicole Rauch"
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	+ "name": "Martin Raszyk"
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	"id": 323,
	- "link": "/authors/raumer/",
	- "name": "Jakob von Raumer"
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	+ "name": "Martin Rau"
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	"id": 324,
	- "link": "/authors/ravindran/",
	- "name": "Binoy Ravindran"
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	- "name": "Michael Rawson"
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	"id": 326,
	- "link": "/authors/raya/",
	- "name": "Rodrigo Raya"
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	"id": 327,
	- "link": "/authors/regensburger/",
	- "name": "Franz Regensburger"
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	+ "name": "Michael Rawson"
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	"id": 328,
	- "link": "/authors/reiche/",
	- "name": "Sebastian Reiche"
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	+ "name": "Rodrigo Raya"
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	"id": 329,
	- "link": "/authors/reiter/",
	- "name": "Markus Reiter"
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	- "link": "/authors/reynaud/",
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	+ "name": "Sebastian Reiche"
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	- "link": "/authors/ribeiro/",
	- "name": "Pedro Ribeiro"
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	- "name": "Stefan Richter"
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	+ "name": "Alban Reynaud"
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	"id": 333,
	- "link": "/authors/rickmann/",
	- "name": "Christina Rickmann"
	+ "link": "/authors/ribeiro/",
	+ "name": "Pedro Ribeiro"
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	"id": 334,
	- "link": "/authors/ridge/",
	- "name": "Tom Ridge"
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	+ "name": "Stefan Richter"
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	"id": 335,
	- "link": "/authors/rizaldi/",
	- "name": "Albert Rizaldi"
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	+ "name": "Christina Rickmann"
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	"id": 336,
	- "link": "/authors/rizkallah/",
	- "name": "Christine Rizkallah"
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	- "link": "/authors/robillard/",
	- "name": "Simon Robillard"
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	+ "name": "Albert Rizaldi"
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	"id": 338,
	- "link": "/authors/roessle/",
	- "name": "Ian Roessle"
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	"id": 339,
	- "link": "/authors/romanos/",
	- "name": "Ralph Romanos"
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	- "name": "Simon Roßkopf"
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	"id": 341,
	- "link": "/authors/rowat/",
	- "name": "Colin Rowat"
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	- "link": "/authors/sabouret/",
	- "name": "Nicolas Sabouret"
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	- "link": "/authors/sachtleben/",
	- "name": "Robert Sachtleben"
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	- "name": "Christian Saile"
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	- "name": "David Sanan"
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	- "link": "/authors/sato/",
	- "name": "Tetsuya Sato"
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	"id": 347,
	- "link": "/authors/sauer/",
	- "name": "Jens Sauer"
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	- "name": "Maximilian Schäffeler"
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	- "link": "/authors/scharager/",
	- "name": "Matias Scharager"
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	- "name": "Alexander Schimpf"
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	- "name": "Norbert Schirmer"
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	- "name": "Anders Schlichtkrull"
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	- "name": "Julien Schmaltz"
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	- "name": "Lukas Schmidinger"
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	- "name": "Richard Schmoetten"
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	- "name": "Joshua Schneider"
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	- "name": "Daniel Schoepe"
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	- "name": "Jonas Schöpf"
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	- "name": "Dana Scott"
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	- "name": "S. Reza Sefidgar"
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	- "link": "/authors/seidl/",
	- "name": "Benedikt Seidl"
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	- "link": "/authors/seidler/",
	- "name": "Henning Seidler"
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	- "name": "Thomas Sewell"
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	"id": 365,
	- "link": "/authors/sickert/",
	- "name": "Salomon Sickert"
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	"id": 366,
	- "link": "/authors/siek/",
	- "name": "Jeremy Siek"
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	"id": 367,
	- "link": "/authors/simic/",
	- "name": "Danijela Simić"
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	"id": 368,
	- "link": "/authors/sison/",
	- "name": "Robert Sison"
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	+ "name": "Jeremy Siek"
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	"id": 369,
	- "link": "/authors/smaus/",
	- "name": "Jan-Georg Smaus"
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	- "link": "/authors/smola/",
	- "name": "Filip Smola"
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	- "link": "/authors/snelting/",
	- "name": "Gregor Snelting"
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	"id": 372,
	- "link": "/authors/somaini/",
	- "name": "Ivano Somaini"
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	- "link": "/authors/somogyi/",
	- "name": "Dániel Somogyi"
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	- "link": "/authors/spasic/",
	- "name": "Mirko Spasić"
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	"id": 375,
	- "link": "/authors/spichkova/",
	- "name": "Maria Spichkova"
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	- "link": "/authors/spitz/",
	- "name": "Maximilian Spitz"
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	+ "name": "Mirko Spasić"
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	- "link": "/authors/sprenger/",
	- "name": "Christoph Sprenger"
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	+ "name": "Maria Spichkova"
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	- "name": "Charles Staats"
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	+ "name": "Maximilian Spitz"
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	"id": 379,
	- "link": "/authors/stannett/",
	- "name": "Mike Stannett"
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	"id": 380,
	- "link": "/authors/stark/",
	- "name": "Eugene W. Stark"
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	"id": 381,
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	- "name": "Štěpán Starosta"
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	- "link": "/authors/steen/",
	- "name": "Alexander Steen"
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	- "name": "Matías Steinberg"
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	- "name": "Christian Sternagel"
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	- "name": "Thomas Sternagel"
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	- "name": "Lukas Stevens"
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	"id": 388,
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	- "name": "Benedikt Stock"
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	- "link": "/authors/stoeckl/",
	- "name": "Bernhard Stöckl"
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	"id": 390,
	- "link": "/authors/stricker/",
	- "name": "Christian Stricker"
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	+ "name": "Benedikt Stock"
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	- "name": "Rok Strniša"
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	"id": 392,
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	- "name": "Georg Struth"
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	"id": 393,
	- "link": "/authors/stueber/",
	- "name": "Anke Stüber"
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	- "link": "/authors/stuewe/",
	- "name": "Daniel Stüwe"
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	+ "name": "Georg Struth"
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	- "name": "Henning Sudbrock"
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	- "name": "Henry Sudhof"
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	- "name": "Ujkan Sulejmani"
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	"id": 398,
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	- "name": "Geoff Sutcliffe"
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	- "link": "/authors/sylvestre/",
	- "name": "Jeremy Sylvestre"
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	- "link": "/authors/szekely/",
	- "name": "Gergely Szekely"
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	+ "name": "Geoff Sutcliffe"
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	- "link": "/authors/taha/",
	- "name": "Safouan Taha"
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	- "link": "/authors/tan/",
	- "name": "Yong Kiam Tan"
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	- "name": "Miki Tanaka"
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	- "link": "/authors/taylor/",
	- "name": "Ramsay G. Taylor"
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	- "link": "/authors/terraf/",
	- "name": "Pedro Sánchez Terraf"
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	- "name": "René Thiemann"
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	- "name": "Fox Thomson"
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	- "link": "/authors/tiu/",
	- "name": "Alwen Tiu"
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	- "name": "Balazs Toth"
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	- "name": "Sophie Tourret"
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	- "name": "David Trachtenherz"
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	- "name": "Christoph Traut"
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	- "name": "Dmitriy Traytel"
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	+
	+<h2>E-Mails 📧</h2>
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	+<h2>Entries</h2><h3 class="head">2023</h3><article class="entry">
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	+ <h5><a class="title" href="../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../authors/leek">Kevin Lee</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJrZmtsMiJdfQ==">📧</a>, <a href="../../authors/yez">Zhengkun Ye</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJ6eTMwNyJdfQ==">📧</a> and <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJhazIxMTAiXX0=">📧</a></div>
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	+ <pubDate>Thu, 10 Aug 2023 00:00:00 +0000</pubDate>
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	<h1 >
	<span class='first'>M</span>artin <span class='first'>R</span>au</h1>
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	<h2>E-Mails 📧</h2>
	<ul><li><a class="obfuscated" data="eyJob3N0IjpbInR1bSIsImRlIl0sInVzZXIiOlsibWFydGluIiwicmF1Il19"><span class="rev">ed</span>.<span class="rev">mut</span>@<span class="rev">uar</span>.<span class="rev">nitram</span></a></li></ul>


	-<h2>Entries</h2><h3 class="head">2020</h3><article class="entry">
	+<h2>Entries</h2><h3 class="head">2023</h3><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Earley_Parser.html">Earley Parser</a></h5> <br>by <a href="../../authors/rau">Martin Rau</a> <a class="obfuscated" data="eyJob3N0IjpbInR1bSIsImRlIl0sInVzZXIiOlsibWFydGluIiwicmF1Il19">📧</a></div>
	+ <span class="date">
	+ Jul 16
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	+
	+<h3 class="head">2020</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Closest_Pair_Points.html">Closest Pair of Points Algorithms</a></h5> <br>by <a href="../../authors/rau">Martin Rau</a> <a class="obfuscated" data="eyJob3N0IjpbInR1bSIsImRlIl0sInVzZXIiOlsibWFydGluIiwicmF1Il19">📧</a> and <a href="../../authors/nipkow">Tobias Nipkow</a> <a href="https://www.in.tum.de/~nipkow/">🌐</a></div>
	<span class="date">
	Jan 13
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	</article>


	<h3 class="head">2019</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KD_Tree.html">Multidimensional Binary Search Trees</a></h5> <br>by <a href="../../authors/rau">Martin Rau</a></div>
	<span class="date">
	May 30
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	</article>




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	<title>Multidimensional Binary Search Trees</title>
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	<pubDate>Thu, 30 May 2019 00:00:00 +0000</pubDate>

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	<h2>Entries</h2><h3 class="head">2023</h3><article class="entry">
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	+ <h5><a class="title" href="../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJnIiwic3RydXRoIl19">📧</a> and <a href="../../authors/calk">Cameron Calk</a></div>
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	<h5><a class="title" href="../../entries/Multirelations_Heterogeneous.html">Inner Structure, Determinism and Modal Algebra of Multirelations</a></h5> <br>by <a href="../../authors/guttmann">Walter Guttmann</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbnRlcmJ1cnkiLCJhYyIsIm56Il0sInVzZXIiOlsid2FsdGVyIiwiZ3V0dG1hbm4iXX0=">📧</a> and <a href="../../authors/struth">Georg Struth</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJnIiwic3RydXRoIl19">📧</a></div>
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	<h5><a class="title" href="../../entries/Order_Lattice_Props.html">Properties of Orderings and Lattices</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a></div>
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	<h5><a class="title" href="../../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a> <a class="obfuscated" data="eyJob3N0IjpbImNsIiwiY2FtIiwiYWMiLCJ1ayJdLCJ1c2VyIjpbInZpY3RvciIsImdvbWVzIl19">📧</a> and <a href="../../authors/struth">Georg Struth</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJnIiwic3RydXRoIl19">📧</a></div>
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	Jun 18
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	<h5><a class="title" href="../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a> <a href="http://www.dcs.shef.ac.uk/~victor">🌐</a>, <a href="../../authors/guttmann">Walter Guttmann</a> <a href="https://www.cosc.canterbury.ac.nz/walter.guttmann/">🌐</a>, <a href="../../authors/hoefner">Peter Höfner</a> <a href="http://www.hoefner-online.de/">🌐</a>, <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a> and <a href="../../authors/weber">Tjark Weber</a> <a href="http://user.it.uu.se/~tjawe125/">🌐</a></div>
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	<h3 class="head">2015</h3><article class="entry">
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	<h5><a class="title" href="../../entries/Multirelations.html">Binary Multirelations</a></h5> <br>by <a href="../../authors/furusawa">Hitoshi Furusawa</a> <a href="http://www.sci.kagoshima-u.ac.jp/~furusawa/">🌐</a> and <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a></div>
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	<h5><a class="title" href="../../entries/Residuated_Lattices.html">Residuated Lattices</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJ2Ym9yZ2VzZmVycmVpcmFnb21lczEiXX0=">📧</a> and <a href="../../authors/struth">Georg Struth</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJnIiwic3RydXRoIl19">📧</a></div>
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	<h3 class="head">2014</h3><article class="entry">
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	<h5><a class="title" href="../../entries/Regular_Algebras.html">Regular Algebras</a></h5> <br>by <a href="../../authors/fosters">Simon Foster</a> <a href="https://www-users.cs.york.ac.uk/~simonf/">🌐</a> and <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a></div>
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	<h5><a class="title" href="../../entries/Relation_Algebra.html">Relation Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/fosters">Simon Foster</a> <a class="obfuscated" data="eyJob3N0IjpbInlvcmsiLCJhYyIsInVrIl0sInVzZXIiOlsic2ltb24iLCJmb3N0ZXIiXX0=">📧</a>, <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a> and <a href="../../authors/weber">Tjark Weber</a> <a href="http://user.it.uu.se/~tjawe125/">🌐</a></div>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a> <a href="http://www.dcs.shef.ac.uk/~victor">🌐</a> and <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a></div>
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	<h3 class="head">2013</h3><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a> and <a href="../../authors/weber">Tjark Weber</a> <a href="http://user.it.uu.se/~tjawe125/">🌐</a></div>
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	diff --git a/web/authors/struth/index.xml b/web/authors/struth/index.xml
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	+ <title>Modal quantales, involutive quantales, Dedekind Quantales</title>
	+ <link>/entries/Quantales_Converse.html</link>
	+ <pubDate>Tue, 25 Jul 2023 00:00:00 +0000</pubDate>
	+
	+ <guid>/entries/Quantales_Converse.html</guid>
	+ <description></description>
	+ </item>
	+
	<item>
	<title>Inner Structure, Determinism and Modal Algebra of Multirelations</title>
	<link>/entries/Multirelations_Heterogeneous.html</link>
	<pubDate>Mon, 22 May 2023 00:00:00 +0000</pubDate>

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	<title>Properties of Orderings and Lattices</title>
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	<title>Quantales</title>
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	<title>Transformer Semantics</title>
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	<title>Partial Semigroups and Convolution Algebras</title>
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	<pubDate>Tue, 13 Jun 2017 00:00:00 +0000</pubDate>

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	<title>Program Construction and Verification Components Based on Kleene Algebra</title>
	<link>/entries/Algebraic_VCs.html</link>
	<pubDate>Sat, 18 Jun 2016 00:00:00 +0000</pubDate>

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	<title>Kleene Algebras with Domain</title>
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	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

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	<title>Binary Multirelations</title>
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	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

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	<title>Residuated Lattices</title>
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	<pubDate>Wed, 15 Apr 2015 00:00:00 +0000</pubDate>

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	<title>Regular Algebras</title>
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	<title>Relation Algebra</title>
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	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
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	+<h2>Entries</h2><h3 class="head">2023</h3><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../authors/leek">Kevin Lee</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJrZmtsMiJdfQ==">📧</a>, <a href="../../authors/yez">Zhengkun Ye</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJ6eTMwNyJdfQ==">📧</a> and <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJhazIxMTAiXX0=">📧</a></div>
	+ <span class="date">
	+ Aug 10
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	+
	+
	+
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	</header><div><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../dependencies/three_squares/">Three_Squares</a></h5> <br></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+</article>
	+<article class="entry">
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	+ <h5><a class="title" href="../dependencies/quantales/">Quantales</a></h5> <br></div>
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	+ Jul 25
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../dependencies/kleene_algebra/">Kleene_Algebra</a></h5> <br></div>
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	+ Jul 25
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../dependencies/kad/">KAD</a></h5> <br></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="../dependencies/zeta_function/">Zeta_Function</a></h5> <br></div>
	<span class="date">
	Jun 19
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	<h5><a class="title" href="../dependencies/probabilistic_while/">Probabilistic_While</a></h5> <br></div>
	<span class="date">
	Jun 19
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	<h5><a class="title" href="../dependencies/elliptic_curves_group_law/">Elliptic_Curves_Group_Law</a></h5> <br></div>
	<span class="date">
	Jun 06
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	<h5><a class="title" href="../dependencies/weighted_path_order/">Weighted_Path_Order</a></h5> <br></div>
	<span class="date">
	Jun 01
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	<h5><a class="title" href="../dependencies/complete_non_orders/">Complete_Non_Orders</a></h5> <br></div>
	<span class="date">
	May 24
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	<h5><a class="title" href="../dependencies/undirected_graph_theory/">Undirected_Graph_Theory</a></h5> <br></div>
	<span class="date">
	May 09
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	<h5><a class="title" href="../dependencies/list-index/">List-Index</a></h5> <br></div>
	<span class="date">
	May 05
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	<h5><a class="title" href="../dependencies/hereditarilyfinite/">HereditarilyFinite</a></h5> <br></div>
	<span class="date">
	May 05
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	<h5><a class="title" href="../dependencies/graph_theory/">Graph_Theory</a></h5> <br></div>
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	May 05
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	<h5><a class="title" href="../dependencies/fresh_identifiers/">Fresh_Identifiers</a></h5> <br></div>
	<span class="date">
	May 05
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	<h5><a class="title" href="../dependencies/dirichlet_l/">Dirichlet_L</a></h5> <br></div>
	<span class="date">
	May 03
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	<h5><a class="title" href="../dependencies/skip_lists/">Skip_Lists</a></h5> <br></div>
	<span class="date">
	Apr 27
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	<div class="item-text">
	<h5><a class="title" href="../dependencies/jordan_normal_form/">Jordan_Normal_Form</a></h5> <br></div>
	<span class="date">
	Apr 27
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/factor_algebraic_polynomial/">Factor_Algebraic_Polynomial</a></h5> <br></div>
	<span class="date">
	Apr 27
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	<article class="entry">
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	<h5><a class="title" href="../dependencies/saturation_framework_extensions/">Saturation_Framework_Extensions</a></h5> <br></div>
	<span class="date">
	Apr 20
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	<h5><a class="title" href="../dependencies/saturation_framework/">Saturation_Framework</a></h5> <br></div>
	<span class="date">
	Apr 20
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	<h5><a class="title" href="../dependencies/ordered_resolution_prover/">Ordered_Resolution_Prover</a></h5> <br></div>
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	Apr 20
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	<h5><a class="title" href="../dependencies/functional_ordered_resolution_prover/">Functional_Ordered_Resolution_Prover</a></h5> <br></div>
	<span class="date">
	Apr 20
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	<h5><a class="title" href="../dependencies/first_order_terms/">First_Order_Terms</a></h5> <br></div>
	<span class="date">
	Apr 20
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	<h5><a class="title" href="../dependencies/commuting_hermitian/">Commuting_Hermitian</a></h5> <br></div>
	<span class="date">
	Apr 18
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stirling_formula/">Stirling_Formula</a></h5> <br></div>
	<span class="date">
	Apr 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/expander_graphs/">Expander_Graphs</a></h5> <br></div>
	<span class="date">
	Apr 03
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/discrete_summation/">Discrete_Summation</a></h5> <br></div>
	<span class="date">
	Apr 03
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/weighted_arithmetic_geometric_mean/">Weighted_Arithmetic_Geometric_Mean</a></h5> <br></div>
	<span class="date">
	Mar 03
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/perron_frobenius/">Perron_Frobenius</a></h5> <br></div>
	<span class="date">
	Mar 03
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/frequency_moments/">Frequency_Moments</a></h5> <br></div>
	<span class="date">
	Mar 03
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/suppes_theorem/">Suppes_Theorem</a></h5> <br></div>
	<span class="date">
	Feb 20
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	<div class="item-text">
	<h5><a class="title" href="../dependencies/lll_basis_reduction/">LLL_Basis_Reduction</a></h5> <br></div>
	<span class="date">
	Feb 02
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	<h5><a class="title" href="../dependencies/berlekamp_zassenhaus/">Berlekamp_Zassenhaus</a></h5> <br></div>
	<span class="date">
	Feb 02
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/benor_kozen_reif/">BenOr_Kozen_Reif</a></h5> <br></div>
	<span class="date">
	Feb 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/crypthol/">CryptHOL</a></h5> <br></div>
	<span class="date">
	Jan 27
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/propositional_logic_class/">Propositional_Logic_Class</a></h5> <br></div>
	<span class="date">
	Jan 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/coinductive/">Coinductive</a></h5> <br></div>
	<span class="date">
	Jan 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/combinatorics_words_graph_lemma/">Combinatorics_Words_Graph_Lemma</a></h5> <br></div>
	<span class="date">
	Jan 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/combinatorics_words/">Combinatorics_Words</a></h5> <br></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2022</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/virtual_substitution/">Virtual_Substitution</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/polynomial_interpolation/">Polynomial_Interpolation</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/descartes_sign_rule/">Descartes_Sign_Rule</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/datatype_order_generator/">Datatype_Order_Generator</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/algebraic_numbers/">Algebraic_Numbers</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/word_lib/">Word_Lib</a></h5> <br></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/separation_algebra/">Separation_Algebra</a></h5> <br></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/containers/">Containers</a></h5> <br></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/pluennecke_ruzsa_inequality/">Pluennecke_Ruzsa_Inequality</a></h5> <br></div>
	<span class="date">
	Nov 21
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	<div class="item-text">
	<h5><a class="title" href="../dependencies/jacobson_basic_algebra/">Jacobson_Basic_Algebra</a></h5> <br></div>
	<span class="date">
	Nov 21
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	<div class="item-text">
	<h5><a class="title" href="../dependencies/random_graph_subgraph_threshold/">Random_Graph_Subgraph_Threshold</a></h5> <br></div>
	<span class="date">
	Nov 14
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/girth_chromatic/">Girth_Chromatic</a></h5> <br></div>
	<span class="date">
	Nov 14
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/falling_factorial_sum/">Falling_Factorial_Sum</a></h5> <br></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/card_number_partitions/">Card_Number_Partitions</a></h5> <br></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/randomised_social_choice/">Randomised_Social_Choice</a></h5> <br></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/design_theory/">Design_Theory</a></h5> <br></div>
	<span class="date">
	Sep 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/deriving/">Deriving</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/collections/">Collections</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/epistemic_logic/">Epistemic_Logic</a></h5> <br></div>
	<span class="date">
	Sep 23
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/padic_ints/">Padic_Ints</a></h5> <br></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/localization_ring/">Localization_Ring</a></h5> <br></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/number_theoretic_transform/">Number_Theoretic_Transform</a></h5> <br></div>
	<span class="date">
	Sep 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/bernoulli/">Bernoulli</a></h5> <br></div>
	<span class="date">
	Sep 02
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/native_word/">Native_Word</a></h5> <br></div>
	<span class="date">
	Aug 09
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/projective_measurements/">Projective_Measurements</a></h5> <br></div>
	<span class="date">
	Jul 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/finite-map-extras/">Finite-Map-Extras</a></h5> <br></div>
	<span class="date">
	Jul 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dirichlet_series/">Dirichlet_Series</a></h5> <br></div>
	<span class="date">
	Jun 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lucas_theorem/">Lucas_Theorem</a></h5> <br></div>
	<span class="date">
	Jun 06
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/digit_expansions/">Digit_Expansions</a></h5> <br></div>
	<span class="date">
	Jun 06
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/regular_tree_relations/">Regular_Tree_Relations</a></h5> <br></div>
	<span class="date">
	Jun 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/package_logic/">Package_logic</a></h5> <br></div>
	<span class="date">
	May 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sunflowers/">Sunflowers</a></h5> <br></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/polynomial_factorization/">Polynomial_Factorization</a></h5> <br></div>
	<span class="date">
	Apr 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/groebner_bases/">Groebner_Bases</a></h5> <br></div>
	<span class="date">
	Apr 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/universal_hash_families/">Universal_Hash_Families</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/prefix_free_code_combinators/">Prefix_Free_Code_Combinators</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/median_method/">Median_Method</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lp/">Lp</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/interpolation_polynomials_hol_algebra/">Interpolation_Polynomials_HOL_Algebra</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/equivalence_relation_enumeration/">Equivalence_Relation_Enumeration</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/bertrands_postulate/">Bertrands_Postulate</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/abstract_soundness/">Abstract_Soundness</a></h5> <br></div>
	<span class="date">
	Mar 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/abstract_completeness/">Abstract_Completeness</a></h5> <br></div>
	<span class="date">
	Mar 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/transitive_models/">Transitive_Models</a></h5> <br></div>
	<span class="date">
	Mar 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/delta_system_lemma/">Delta_System_Lemma</a></h5> <br></div>
	<span class="date">
	Mar 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/finite_fields/">Finite_Fields</a></h5> <br></div>
	<span class="date">
	Feb 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/zfc_in_hol/">ZFC_in_HOL</a></h5> <br></div>
	<span class="date">
	Feb 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/card_equiv_relations/">Card_Equiv_Relations</a></h5> <br></div>
	<span class="date">
	Feb 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/linear_inequalities/">Linear_Inequalities</a></h5> <br></div>
	<span class="date">
	Feb 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/fol-fitting/">FOL-Fitting</a></h5> <br></div>
	<span class="date">
	Feb 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/fol_seq_calc1/">FOL_Seq_Calc1</a></h5> <br></div>
	<span class="date">
	Jan 31
	</span>
	</article>


	<h2 class="head">2021</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/szemeredi_regularity/">Szemeredi_Regularity</a></h5> <br></div>
	<span class="date">
	Dec 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ergodic_theory/">Ergodic_Theory</a></h5> <br></div>
	<span class="date">
	Dec 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/mdp-rewards/">MDP-Rewards</a></h5> <br></div>
	<span class="date">
	Dec 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/gauss_jordan/">Gauss_Jordan</a></h5> <br></div>
	<span class="date">
	Dec 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/knuth_bendix_order/">Knuth_Bendix_Order</a></h5> <br></div>
	<span class="date">
	Dec 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sepref_prereq/">Sepref_Prereq</a></h5> <br></div>
	<span class="date">
	Nov 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/robdd/">ROBDD</a></h5> <br></div>
	<span class="date">
	Nov 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/automatic_refinement/">Automatic_Refinement</a></h5> <br></div>
	<span class="date">
	Nov 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/polynomials/">Polynomials</a></h5> <br></div>
	<span class="date">
	Nov 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hermite_lindemann/">Hermite_Lindemann</a></h5> <br></div>
	<span class="date">
	Nov 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/complex_bounded_operators/">Complex_Bounded_Operators</a></h5> <br></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/subset_boolean_algebras/">Subset_Boolean_Algebras</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stone_kleene_relation_algebras/">Stone_Kleene_Relation_Algebras</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/monobooltranalgebra/">MonoBoolTranAlgebra</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/real_impl/">Real_Impl</a></h5> <br></div>
	<span class="date">
	Sep 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/complex_bounded_operators_dependencies/">Complex_Bounded_Operators_Dependencies</a></h5> <br></div>
	<span class="date">
	Sep 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/banach_steinhaus/">Banach_Steinhaus</a></h5> <br></div>
	<span class="date">
	Sep 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/speccheck/">SpecCheck</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/intro_dest_elim/">Intro_Dest_Elim</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/czh_foundations/">CZH_Foundations</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/czh_elementary_categories/">CZH_Elementary_Categories</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/conditional_transfer_rule/">Conditional_Transfer_Rule</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/conditional_simplification/">Conditional_Simplification</a></h5> <br></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/jinja/">Jinja</a></h5> <br></div>
	<span class="date">
	Sep 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/complex_geometry/">Complex_Geometry</a></h5> <br></div>
	<span class="date">
	Sep 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/budan_fourier/">Budan_Fourier</a></h5> <br></div>
	<span class="date">
	Aug 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/bounded_deducibility_security/">Bounded_Deducibility_Security</a></h5> <br></div>
	<span class="date">
	Aug 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/bd_security_compositional/">BD_Security_Compositional</a></h5> <br></div>
	<span class="date">
	Aug 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/nested_multisets_ordinals/">Nested_Multisets_Ordinals</a></h5> <br></div>
	<span class="date">
	Aug 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/card_partitions/">Card_Partitions</a></h5> <br></div>
	<span class="date">
	Aug 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/show/">Show</a></h5> <br></div>
	<span class="date">
	Jun 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/nominal2/">Nominal2</a></h5> <br></div>
	<span class="date">
	Jun 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/szpilrajn/">Szpilrajn</a></h5> <br></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/jinjadci/">JinjaDCI</a></h5> <br></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sturm_tarski/">Sturm_Tarski</a></h5> <br></div>
	<span class="date">
	Apr 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/parity_game/">Parity_Game</a></h5> <br></div>
	<span class="date">
	Apr 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sigma_commit_crypto/">Sigma_Commit_Crypto</a></h5> <br></div>
	<span class="date">
	Mar 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/game_based_crypto/">Game_Based_Crypto</a></h5> <br></div>
	<span class="date">
	Mar 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/constructive_cryptography/">Constructive_Cryptography</a></h5> <br></div>
	<span class="date">
	Mar 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/smith_normal_form/">Smith_Normal_Form</a></h5> <br></div>
	<span class="date">
	Mar 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hermite/">Hermite</a></h5> <br></div>
	<span class="date">
	Mar 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/qhlprover/">QHLProver</a></h5> <br></div>
	<span class="date">
	Mar 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/power_sum_polynomials/">Power_Sum_Polynomials</a></h5> <br></div>
	<span class="date">
	Mar 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/pi_transcendental/">Pi_Transcendental</a></h5> <br></div>
	<span class="date">
	Mar 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/isabelle_marries_dirac/">Isabelle_Marries_Dirac</a></h5> <br></div>
	<span class="date">
	Mar 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/refine_imperative_hol/">Refine_Imperative_HOL</a></h5> <br></div>
	<span class="date">
	Feb 24
	</span>
	</article>


	<h2 class="head">2020</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/relational_disjoint_set_forests/">Relational_Disjoint_Set_Forests</a></h5> <br></div>
	<span class="date">
	Dec 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/vericomp/">VeriComp</a></h5> <br></div>
	<span class="date">
	Dec 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/vectorspace/">VectorSpace</a></h5> <br></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/matrix_tensor/">Matrix_Tensor</a></h5> <br></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hol-csp/">HOL-CSP</a></h5> <br></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/propositional_proof_systems/">Propositional_Proof_Systems</a></h5> <br></div>
	<span class="date">
	Oct 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/certification_monads/">Certification_Monads</a></h5> <br></div>
	<span class="date">
	Oct 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ai_planning_languages_semantics/">AI_Planning_Languages_Semantics</a></h5> <br></div>
	<span class="date">
	Oct 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/shadow_sc_dom/">Shadow_SC_DOM</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/shadow_dom/">Shadow_DOM</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/core_sc_dom/">Core_SC_DOM</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/core_dom/">Core_DOM</a></h5> <br></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/syntax_independent_logic/">Syntax_Independent_Logic</a></h5> <br></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/incompleteness/">Incompleteness</a></h5> <br></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/goedel_incompleteness/">Goedel_Incompleteness</a></h5> <br></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/finfun/">FinFun</a></h5> <br></div>
	<span class="date">
	Sep 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/extended_finite_state_machines/">Extended_Finite_State_Machines</a></h5> <br></div>
	<span class="date">
	Sep 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sepref_iicf/">Sepref_IICF</a></h5> <br></div>
	<span class="date">
	Aug 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/aggregation_algebras/">Aggregation_Algebras</a></h5> <br></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/well_quasi_orders/">Well_Quasi_Orders</a></h5> <br></div>
	<span class="date">
	Aug 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/holcf-prelude/">HOLCF-Prelude</a></h5> <br></div>
	<span class="date">
	Aug 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/pratt_certificate/">Pratt_Certificate</a></h5> <br></div>
	<span class="date">
	Aug 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/nash_williams/">Nash_Williams</a></h5> <br></div>
	<span class="date">
	Aug 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/relation_algebra/">Relation_Algebra</a></h5> <br></div>
	<span class="date">
	Jul 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sturm_sequences/">Sturm_Sequences</a></h5> <br></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/matrix/">Matrix</a></h5> <br></div>
	<span class="date">
	May 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/prime_distribution_elementary/">Prime_Distribution_Elementary</a></h5> <br></div>
	<span class="date">
	May 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ltl_master_theorem/">LTL_Master_Theorem</a></h5> <br></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ltl/">LTL</a></h5> <br></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/symmetric_polynomials/">Symmetric_Polynomials</a></h5> <br></div>
	<span class="date">
	Apr 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hybrid_systems_vcs/">Hybrid_Systems_VCs</a></h5> <br></div>
	<span class="date">
	Apr 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/mfotl_monitor/">MFOTL_Monitor</a></h5> <br></div>
	<span class="date">
	Apr 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lambda_free_rpos/">Lambda_Free_RPOs</a></h5> <br></div>
	<span class="date">
	Apr 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ieee_floating_point/">IEEE_Floating_Point</a></h5> <br></div>
	<span class="date">
	Apr 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/generic_join/">Generic_Join</a></h5> <br></div>
	<span class="date">
	Apr 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stateful_protocol_composition_and_typing/">Stateful_Protocol_Composition_and_Typing</a></h5> <br></div>
	<span class="date">
	Apr 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stone_algebras/">Stone_Algebras</a></h5> <br></div>
	<span class="date">
	Jan 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/probabilistic_prime_tests/">Probabilistic_Prime_Tests</a></h5> <br></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/pell/">Pell</a></h5> <br></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/root_balanced_tree/">Root_Balanced_Tree</a></h5> <br></div>
	<span class="date">
	Jan 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/akra_bazzi/">Akra_Bazzi</a></h5> <br></div>
	<span class="date">
	Jan 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/monad_normalisation/">Monad_Normalisation</a></h5> <br></div>
	<span class="date">
	Jan 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/monoidalcategory/">MonoidalCategory</a></h5> <br></div>
	<span class="date">
	Jan 06
	</span>
	</article>


	<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/prime_number_theorem/">Prime_Number_Theorem</a></h5> <br></div>
	<span class="date">
	Dec 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/e_transcendental/">E_Transcendental</a></h5> <br></div>
	<span class="date">
	Dec 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hol-ode-numerics/">HOL-ODE-Numerics</a></h5> <br></div>
	<span class="date">
	Dec 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/transformer_semantics/">Transformer_Semantics</a></h5> <br></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ordinary_differential_equations/">Ordinary_Differential_Equations</a></h5> <br></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/kat_and_dra/">KAT_and_DRA</a></h5> <br></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	- <h5><a class="title" href="../dependencies/kad/">KAD</a></h5> <br></div>
	- <span class="date">
	- Sep 10
	- </span>
	-</article>
	-<article class="entry">
	- <div class="item-text">
	<h5><a class="title" href="../dependencies/transition_systems_and_automata/">Transition_Systems_and_Automata</a></h5> <br></div>
	<span class="date">
	Aug 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/farkas/">Farkas</a></h5> <br></div>
	<span class="date">
	Aug 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/pairing_heap/">Pairing_Heap</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/huffman/">Huffman</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/higher_order_terms/">Higher_Order_Terms</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dict_construction/">Dict_Construction</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/constructor_funs/">Constructor_Funs</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cakeml/">CakeML</a></h5> <br></div>
	<span class="date">
	Jul 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/priority_search_trees/">Priority_Search_Trees</a></h5> <br></div>
	<span class="date">
	Jun 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/imp2/">IMP2</a></h5> <br></div>
	<span class="date">
	Jun 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/median_of_medians_selection/">Median_Of_Medians_Selection</a></h5> <br></div>
	<span class="date">
	May 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/deep_learning/">Deep_Learning</a></h5> <br></div>
	<span class="date">
	Mar 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/refine_monadic/">Refine_Monadic</a></h5> <br></div>
	<span class="date">
	Feb 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/matroids/">Matroids</a></h5> <br></div>
	<span class="date">
	Feb 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/utp-toolkit/">UTP-Toolkit</a></h5> <br></div>
	<span class="date">
	Feb 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/optics/">Optics</a></h5> <br></div>
	<span class="date">
	Feb 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/simplex/">Simplex</a></h5> <br></div>
	<span class="date">
	Jan 17
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/auto2_hol/">Auto2_HOL</a></h5> <br></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	- <h5><a class="title" href="../dependencies/quantales/">Quantales</a></h5> <br></div>
	- <span class="date">
	- Dec 11
	- </span>
	-</article>
	-<article class="entry">
	- <div class="item-text">
	<h5><a class="title" href="../dependencies/order_lattice_props/">Order_Lattice_Props</a></h5> <br></div>
	<span class="date">
	Dec 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	- <h5><a class="title" href="../dependencies/kleene_algebra/">Kleene_Algebra</a></h5> <br></div>
	- <span class="date">
	- Dec 11
	- </span>
	-</article>
	-<article class="entry">
	- <div class="item-text">
	<h5><a class="title" href="../dependencies/open_induction/">Open_Induction</a></h5> <br></div>
	<span class="date">
	Nov 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/random_bsts/">Random_BSTs</a></h5> <br></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/recursion-theory-i/">Recursion-Theory-I</a></h5> <br></div>
	<span class="date">
	Aug 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/abstract-rewriting/">Abstract-Rewriting</a></h5> <br></div>
	<span class="date">
	Aug 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/first_welfare_theorem/">First_Welfare_Theorem</a></h5> <br></div>
	<span class="date">
	Jul 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stuttering_equivalence/">Stuttering_Equivalence</a></h5> <br></div>
	<span class="date">
	Jun 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/monad_memo_dp/">Monad_Memo_DP</a></h5> <br></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/markov_models/">Markov_Models</a></h5> <br></div>
	<span class="date">
	May 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/timed_automata/">Timed_Automata</a></h5> <br></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lem/">LEM</a></h5> <br></div>
	<span class="date">
	Mar 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dynamicarchitectures/">DynamicArchitectures</a></h5> <br></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/landau_symbols/">Landau_Symbols</a></h5> <br></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/comparison_sort_lower_bound/">Comparison_Sort_Lower_Bound</a></h5> <br></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/affine_arithmetic/">Affine_Arithmetic</a></h5> <br></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2017</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/finitely_generated_abelian_groups/">Finitely_Generated_Abelian_Groups</a></h5> <br></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/crdt/">CRDT</a></h5> <br></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/gabow_scc/">Gabow_SCC</a></h5> <br></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dfs_framework/">DFS_Framework</a></h5> <br></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/winding_number_eval/">Winding_Number_Eval</a></h5> <br></div>
	<span class="date">
	Oct 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/linear_recurrences/">Linear_Recurrences</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/euler_maclaurin/">Euler_MacLaurin</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/count_complex_roots/">Count_Complex_Roots</a></h5> <br></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/amortized_complexity/">Amortized_Complexity</a></h5> <br></div>
	<span class="date">
	Aug 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/triangle/">Triangle</a></h5> <br></div>
	<span class="date">
	Jul 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/stone_relation_algebras/">Stone_Relation_Algebras</a></h5> <br></div>
	<span class="date">
	Jul 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/program-conflict-analysis/">Program-Conflict-Analysis</a></h5> <br></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/flow_networks/">Flow_Networks</a></h5> <br></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cava_automata/">CAVA_Automata</a></h5> <br></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lazy_case/">Lazy_Case</a></h5> <br></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/monomorphic_monad/">Monomorphic_Monad</a></h5> <br></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/mfmc_countable/">MFMC_Countable</a></h5> <br></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/applicative_lifting/">Applicative_Lifting</a></h5> <br></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/category3/">Category3</a></h5> <br></div>
	<span class="date">
	May 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/quick_sort_cost/">Quick_Sort_Cost</a></h5> <br></div>
	<span class="date">
	Apr 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/regular-sets/">Regular-Sets</a></h5> <br></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/formal_ssa/">Formal_SSA</a></h5> <br></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/upf/">UPF</a></h5> <br></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/card_multisets/">Card_Multisets</a></h5> <br></div>
	<span class="date">
	Dec 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/bell_numbers_spivey/">Bell_Numbers_Spivey</a></h5> <br></div>
	<span class="date">
	Dec 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ordinal/">Ordinal</a></h5> <br></div>
	<span class="date">
	Nov 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/iptables_semantics/">Iptables_Semantics</a></h5> <br></div>
	<span class="date">
	Oct 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/subresultants/">Subresultants</a></h5> <br></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/efficient-mergesort/">Efficient-Mergesort</a></h5> <br></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/routing/">Routing</a></h5> <br></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/iptables_semantics_examples/">Iptables_Semantics_Examples</a></h5> <br></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/simple_firewall/">Simple_Firewall</a></h5> <br></div>
	<span class="date">
	Aug 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ip_addresses/">IP_Addresses</a></h5> <br></div>
	<span class="date">
	Aug 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sepref_basic/">Sepref_Basic</a></h5> <br></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/separation_logic_imperative_hol/">Separation_Logic_Imperative_HOL</a></h5> <br></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dijkstra_shortest_path/">Dijkstra_Shortest_Path</a></h5> <br></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/collections_examples/">Collections_Examples</a></h5> <br></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/dependent_sifum_type_systems/">Dependent_SIFUM_Type_Systems</a></h5> <br></div>
	<span class="date">
	Jun 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/noninterference_sequential_composition/">Noninterference_Sequential_Composition</a></h5> <br></div>
	<span class="date">
	Jun 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/rank_nullity_theorem/">Rank_Nullity_Theorem</a></h5> <br></div>
	<span class="date">
	May 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/edmondskarp_maxflow/">EdmondsKarp_Maxflow</a></h5> <br></div>
	<span class="date">
	May 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/noninterference_ipurge_unwinding/">Noninterference_Ipurge_Unwinding</a></h5> <br></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/boolean_expression_checkers/">Boolean_Expression_Checkers</a></h5> <br></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/slicing/">Slicing</a></h5> <br></div>
	<span class="date">
	Feb 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sqrt_babylonian/">Sqrt_Babylonian</a></h5> <br></div>
	<span class="date">
	Jan 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/partial_function_mr/">Partial_Function_MR</a></h5> <br></div>
	<span class="date">
	Jan 29
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/marriage/">Marriage</a></h5> <br></div>
	<span class="date">
	Dec 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/transitive-closure/">Transitive-Closure</a></h5> <br></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/simpl/">Simpl</a></h5> <br></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/case_labeling/">Case_Labeling</a></h5> <br></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/kbps/">KBPs</a></h5> <br></div>
	<span class="date">
	Sep 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/echelon_form/">Echelon_Form</a></h5> <br></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/noninterference_csp/">Noninterference_CSP</a></h5> <br></div>
	<span class="date">
	Jun 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/list_interleaving/">List_Interleaving</a></h5> <br></div>
	<span class="date">
	Jun 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/formula_derivatives/">Formula_Derivatives</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/coinductive_languages/">Coinductive_Languages</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/concurrentimp/">ConcurrentIMP</a></h5> <br></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/heard_of/">Heard_Of</a></h5> <br></div>
	<span class="date">
	Mar 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/launchbury/">Launchbury</a></h5> <br></div>
	<span class="date">
	Feb 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cayley_hamilton/">Cayley_Hamilton</a></h5> <br></div>
	<span class="date">
	Feb 12
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/awn/">AWN</a></h5> <br></div>
	<span class="date">
	Oct 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/secondary_sylow/">Secondary_Sylow</a></h5> <br></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/splay_tree/">Splay_Tree</a></h5> <br></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/skew_heap/">Skew_Heap</a></h5> <br></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sm_base/">SM_Base</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/sm/">SM</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/promela/">Promela</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/partial_order_reduction/">Partial_Order_Reduction</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/ltl_to_gba/">LTL_to_GBA</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cava_setup/">CAVA_Setup</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cava_base/">CAVA_Base</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/strong_security/">Strong_Security</a></h5> <br></div>
	<span class="date">
	Apr 23
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lehmer/">Lehmer</a></h5> <br></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/trie/">Trie</a></h5> <br></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/finger-trees/">Finger-Trees</a></h5> <br></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/cauchy/">Cauchy</a></h5> <br></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lorenz_approximation/">Lorenz_Approximation</a></h5> <br></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/gauss-jordan-elim-fun/">Gauss-Jordan-Elim-Fun</a></h5> <br></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/latticeproperties/">LatticeProperties</a></h5> <br></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/nat-interval-logic/">Nat-Interval-Logic</a></h5> <br></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/list-infinite/">List-Infinite</a></h5> <br></div>
	<span class="date">
	Feb 23
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/datarefinementibp/">DataRefinementIBP</a></h5> <br></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/hrb-slicing/">HRB-Slicing</a></h5> <br></div>
	<span class="date">
	Mar 23
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/binomial-heaps/">Binomial-Heaps</a></h5> <br></div>
	<span class="date">
	Nov 25
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/group-ring-module/">Group-Ring-Module</a></h5> <br></div>
	<span class="date">
	Aug 08
	</span>
	</article>


	<h2 class="head">2006</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/flyspeck-tame/">Flyspeck-Tame</a></h5> <br></div>
	<span class="date">
	May 22
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../dependencies/lazy-lists-ii/">Lazy-Lists-II</a></h5> <br></div>
	<span class="date">
	Apr 26
	</span>
	</article>




	</div>
	</div>
	</body>

	</html>
	\ No newline at end of file
	diff --git a/web/dependencies/index.json b/web/dependencies/index.json
	--- a/web/dependencies/index.json
	+++ b/web/dependencies/index.json
	@@ -1,1522 +1,1527 @@
	[
	{
	"id": 0,
	"link": "/dependencies/abstract-rewriting/",
	"name": "Abstract-Rewriting"
	},
	{
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	"link": "/dependencies/abstract_completeness/",
	"name": "Abstract_Completeness"
	},
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	"id": 2,
	"link": "/dependencies/abstract_soundness/",
	"name": "Abstract_Soundness"
	},
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	"id": 3,
	"link": "/dependencies/affine_arithmetic/",
	"name": "Affine_Arithmetic"
	},
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	"id": 4,
	"link": "/dependencies/aggregation_algebras/",
	"name": "Aggregation_Algebras"
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	"name": "AI_Planning_Languages_Semantics"
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	"link": "/dependencies/akra_bazzi/",
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	"name": "Algebraic_Numbers"
	},
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	"name": "Amortized_Complexity"
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	"id": 9,
	"link": "/dependencies/applicative_lifting/",
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	"name": "Auto2_HOL"
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	"link": "/dependencies/awn/",
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	"link": "/dependencies/banach_steinhaus/",
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	},
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	"id": 14,
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	"id": 15,
	"link": "/dependencies/bell_numbers_spivey/",
	"name": "Bell_Numbers_Spivey"
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	"link": "/dependencies/benor_kozen_reif/",
	"name": "BenOr_Kozen_Reif"
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	"id": 17,
	"link": "/dependencies/berlekamp_zassenhaus/",
	"name": "Berlekamp_Zassenhaus"
	},
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	"id": 18,
	"link": "/dependencies/bernoulli/",
	"name": "Bernoulli"
	},
	{
	"id": 19,
	"link": "/dependencies/bertrands_postulate/",
	"name": "Bertrands_Postulate"
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	"id": 20,
	"link": "/dependencies/binomial-heaps/",
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	},
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	"id": 21,
	"link": "/dependencies/boolean_expression_checkers/",
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	{
	"id": 22,
	"link": "/dependencies/bounded_deducibility_security/",
	"name": "Bounded_Deducibility_Security"
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	{
	"id": 23,
	"link": "/dependencies/budan_fourier/",
	"name": "Budan_Fourier"
	},
	{
	"id": 24,
	"link": "/dependencies/cakeml/",
	"name": "CakeML"
	},
	{
	"id": 25,
	"link": "/dependencies/card_equiv_relations/",
	"name": "Card_Equiv_Relations"
	},
	{
	"id": 26,
	"link": "/dependencies/card_multisets/",
	"name": "Card_Multisets"
	},
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	"id": 27,
	"link": "/dependencies/card_number_partitions/",
	"name": "Card_Number_Partitions"
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	"id": 28,
	"link": "/dependencies/card_partitions/",
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	"link": "/dependencies/case_labeling/",
	"name": "Case_Labeling"
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	"id": 30,
	"link": "/dependencies/category3/",
	"name": "Category3"
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	{
	"id": 31,
	"link": "/dependencies/cauchy/",
	"name": "Cauchy"
	},
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	"id": 32,
	"link": "/dependencies/cava_automata/",
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	},
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	"id": 33,
	"link": "/dependencies/cava_base/",
	"name": "CAVA_Base"
	},
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	"id": 34,
	"link": "/dependencies/cava_setup/",
	"name": "CAVA_Setup"
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	"id": 35,
	"link": "/dependencies/cayley_hamilton/",
	"name": "Cayley_Hamilton"
	},
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	"id": 36,
	"link": "/dependencies/certification_monads/",
	"name": "Certification_Monads"
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	"id": 37,
	"link": "/dependencies/coinductive/",
	"name": "Coinductive"
	},
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	"id": 38,
	"link": "/dependencies/coinductive_languages/",
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	"id": 39,
	"link": "/dependencies/collections/",
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	{
	"id": 40,
	"link": "/dependencies/collections_examples/",
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	"id": 41,
	"link": "/dependencies/combinatorics_words/",
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	"id": 42,
	"link": "/dependencies/combinatorics_words_graph_lemma/",
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	"id": 43,
	"link": "/dependencies/commuting_hermitian/",
	"name": "Commuting_Hermitian"
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	"id": 44,
	"link": "/dependencies/comparison_sort_lower_bound/",
	"name": "Comparison_Sort_Lower_Bound"
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	"id": 45,
	"link": "/dependencies/complete_non_orders/",
	"name": "Complete_Non_Orders"
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	"id": 46,
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	"id": 47,
	"link": "/dependencies/complex_bounded_operators_dependencies/",
	"name": "Complex_Bounded_Operators_Dependencies"
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	"id": 48,
	"link": "/dependencies/complex_geometry/",
	"name": "Complex_Geometry"
	},
	{
	"id": 49,
	"link": "/dependencies/concurrentimp/",
	"name": "ConcurrentIMP"
	},
	{
	"id": 50,
	"link": "/dependencies/conditional_simplification/",
	"name": "Conditional_Simplification"
	},
	{
	"id": 51,
	"link": "/dependencies/conditional_transfer_rule/",
	"name": "Conditional_Transfer_Rule"
	},
	{
	"id": 52,
	"link": "/dependencies/constructive_cryptography/",
	"name": "Constructive_Cryptography"
	},
	{
	"id": 53,
	"link": "/dependencies/constructor_funs/",
	"name": "Constructor_Funs"
	},
	{
	"id": 54,
	"link": "/dependencies/containers/",
	"name": "Containers"
	},
	{
	"id": 55,
	"link": "/dependencies/core_dom/",
	"name": "Core_DOM"
	},
	{
	"id": 56,
	"link": "/dependencies/core_sc_dom/",
	"name": "Core_SC_DOM"
	},
	{
	"id": 57,
	"link": "/dependencies/count_complex_roots/",
	"name": "Count_Complex_Roots"
	},
	{
	"id": 58,
	"link": "/dependencies/crdt/",
	"name": "CRDT"
	},
	{
	"id": 59,
	"link": "/dependencies/crypthol/",
	"name": "CryptHOL"
	},
	{
	"id": 60,
	"link": "/dependencies/czh_elementary_categories/",
	"name": "CZH_Elementary_Categories"
	},
	{
	"id": 61,
	"link": "/dependencies/czh_foundations/",
	"name": "CZH_Foundations"
	},
	{
	"id": 62,
	"link": "/dependencies/datarefinementibp/",
	"name": "DataRefinementIBP"
	},
	{
	"id": 63,
	"link": "/dependencies/datatype_order_generator/",
	"name": "Datatype_Order_Generator"
	},
	{
	"id": 64,
	"link": "/dependencies/deep_learning/",
	"name": "Deep_Learning"
	},
	{
	"id": 65,
	"link": "/dependencies/delta_system_lemma/",
	"name": "Delta_System_Lemma"
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	{
	"id": 66,
	"link": "/dependencies/dependent_sifum_type_systems/",
	"name": "Dependent_SIFUM_Type_Systems"
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	{
	"id": 67,
	"link": "/dependencies/deriving/",
	"name": "Deriving"
	},
	{
	"id": 68,
	"link": "/dependencies/descartes_sign_rule/",
	"name": "Descartes_Sign_Rule"
	},
	{
	"id": 69,
	"link": "/dependencies/design_theory/",
	"name": "Design_Theory"
	},
	{
	"id": 70,
	"link": "/dependencies/dfs_framework/",
	"name": "DFS_Framework"
	},
	{
	"id": 71,
	"link": "/dependencies/dict_construction/",
	"name": "Dict_Construction"
	},
	{
	"id": 72,
	"link": "/dependencies/digit_expansions/",
	"name": "Digit_Expansions"
	},
	{
	"id": 73,
	"link": "/dependencies/dijkstra_shortest_path/",
	"name": "Dijkstra_Shortest_Path"
	},
	{
	"id": 74,
	"link": "/dependencies/dirichlet_l/",
	"name": "Dirichlet_L"
	},
	{
	"id": 75,
	"link": "/dependencies/dirichlet_series/",
	"name": "Dirichlet_Series"
	},
	{
	"id": 76,
	"link": "/dependencies/discrete_summation/",
	"name": "Discrete_Summation"
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	"id": 77,
	"link": "/dependencies/dynamicarchitectures/",
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	{
	"id": 78,
	"link": "/dependencies/e_transcendental/",
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	{
	"id": 79,
	"link": "/dependencies/echelon_form/",
	"name": "Echelon_Form"
	},
	{
	"id": 80,
	"link": "/dependencies/edmondskarp_maxflow/",
	"name": "EdmondsKarp_Maxflow"
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	{
	"id": 81,
	"link": "/dependencies/efficient-mergesort/",
	"name": "Efficient-Mergesort"
	},
	{
	"id": 82,
	"link": "/dependencies/elliptic_curves_group_law/",
	"name": "Elliptic_Curves_Group_Law"
	},
	{
	"id": 83,
	"link": "/dependencies/epistemic_logic/",
	"name": "Epistemic_Logic"
	},
	{
	"id": 84,
	"link": "/dependencies/equivalence_relation_enumeration/",
	"name": "Equivalence_Relation_Enumeration"
	},
	{
	"id": 85,
	"link": "/dependencies/ergodic_theory/",
	"name": "Ergodic_Theory"
	},
	{
	"id": 86,
	"link": "/dependencies/euler_maclaurin/",
	"name": "Euler_MacLaurin"
	},
	{
	"id": 87,
	"link": "/dependencies/expander_graphs/",
	"name": "Expander_Graphs"
	},
	{
	"id": 88,
	"link": "/dependencies/extended_finite_state_machines/",
	"name": "Extended_Finite_State_Machines"
	},
	{
	"id": 89,
	"link": "/dependencies/factor_algebraic_polynomial/",
	"name": "Factor_Algebraic_Polynomial"
	},
	{
	"id": 90,
	"link": "/dependencies/falling_factorial_sum/",
	"name": "Falling_Factorial_Sum"
	},
	{
	"id": 91,
	"link": "/dependencies/farkas/",
	"name": "Farkas"
	},
	{
	"id": 92,
	"link": "/dependencies/finfun/",
	"name": "FinFun"
	},
	{
	"id": 93,
	"link": "/dependencies/finger-trees/",
	"name": "Finger-Trees"
	},
	{
	"id": 94,
	"link": "/dependencies/finite-map-extras/",
	"name": "Finite-Map-Extras"
	},
	{
	"id": 95,
	"link": "/dependencies/finite_fields/",
	"name": "Finite_Fields"
	},
	{
	"id": 96,
	"link": "/dependencies/finitely_generated_abelian_groups/",
	"name": "Finitely_Generated_Abelian_Groups"
	},
	{
	"id": 97,
	"link": "/dependencies/first_order_terms/",
	"name": "First_Order_Terms"
	},
	{
	"id": 98,
	"link": "/dependencies/first_welfare_theorem/",
	"name": "First_Welfare_Theorem"
	},
	{
	"id": 99,
	"link": "/dependencies/flow_networks/",
	"name": "Flow_Networks"
	},
	{
	"id": 100,
	"link": "/dependencies/flyspeck-tame/",
	"name": "Flyspeck-Tame"
	},
	{
	"id": 101,
	"link": "/dependencies/fol-fitting/",
	"name": "FOL-Fitting"
	},
	{
	"id": 102,
	"link": "/dependencies/fol_seq_calc1/",
	"name": "FOL_Seq_Calc1"
	},
	{
	"id": 103,
	"link": "/dependencies/formal_ssa/",
	"name": "Formal_SSA"
	},
	{
	"id": 104,
	"link": "/dependencies/formula_derivatives/",
	"name": "Formula_Derivatives"
	},
	{
	"id": 105,
	"link": "/dependencies/frequency_moments/",
	"name": "Frequency_Moments"
	},
	{
	"id": 106,
	"link": "/dependencies/fresh_identifiers/",
	"name": "Fresh_Identifiers"
	},
	{
	"id": 107,
	"link": "/dependencies/functional_ordered_resolution_prover/",
	"name": "Functional_Ordered_Resolution_Prover"
	},
	{
	"id": 108,
	"link": "/dependencies/gabow_scc/",
	"name": "Gabow_SCC"
	},
	{
	"id": 109,
	"link": "/dependencies/game_based_crypto/",
	"name": "Game_Based_Crypto"
	},
	{
	"id": 110,
	"link": "/dependencies/gauss-jordan-elim-fun/",
	"name": "Gauss-Jordan-Elim-Fun"
	},
	{
	"id": 111,
	"link": "/dependencies/gauss_jordan/",
	"name": "Gauss_Jordan"
	},
	{
	"id": 112,
	"link": "/dependencies/generic_join/",
	"name": "Generic_Join"
	},
	{
	"id": 113,
	"link": "/dependencies/girth_chromatic/",
	"name": "Girth_Chromatic"
	},
	{
	"id": 114,
	"link": "/dependencies/goedel_incompleteness/",
	"name": "Goedel_Incompleteness"
	},
	{
	"id": 115,
	"link": "/dependencies/graph_theory/",
	"name": "Graph_Theory"
	},
	{
	"id": 116,
	"link": "/dependencies/groebner_bases/",
	"name": "Groebner_Bases"
	},
	{
	"id": 117,
	"link": "/dependencies/group-ring-module/",
	"name": "Group-Ring-Module"
	},
	{
	"id": 118,
	"link": "/dependencies/heard_of/",
	"name": "Heard_Of"
	},
	{
	"id": 119,
	"link": "/dependencies/hereditarilyfinite/",
	"name": "HereditarilyFinite"
	},
	{
	"id": 120,
	"link": "/dependencies/hermite/",
	"name": "Hermite"
	},
	{
	"id": 121,
	"link": "/dependencies/hermite_lindemann/",
	"name": "Hermite_Lindemann"
	},
	{
	"id": 122,
	"link": "/dependencies/higher_order_terms/",
	"name": "Higher_Order_Terms"
	},
	{
	"id": 123,
	"link": "/dependencies/hol-csp/",
	"name": "HOL-CSP"
	},
	{
	"id": 124,
	"link": "/dependencies/hol-ode-numerics/",
	"name": "HOL-ODE-Numerics"
	},
	{
	"id": 125,
	"link": "/dependencies/holcf-prelude/",
	"name": "HOLCF-Prelude"
	},
	{
	"id": 126,
	"link": "/dependencies/hrb-slicing/",
	"name": "HRB-Slicing"
	},
	{
	"id": 127,
	"link": "/dependencies/huffman/",
	"name": "Huffman"
	},
	{
	"id": 128,
	"link": "/dependencies/hybrid_systems_vcs/",
	"name": "Hybrid_Systems_VCs"
	},
	{
	"id": 129,
	"link": "/dependencies/ieee_floating_point/",
	"name": "IEEE_Floating_Point"
	},
	{
	"id": 130,
	"link": "/dependencies/imp2/",
	"name": "IMP2"
	},
	{
	"id": 131,
	"link": "/dependencies/incompleteness/",
	"name": "Incompleteness"
	},
	{
	"id": 132,
	"link": "/dependencies/interpolation_polynomials_hol_algebra/",
	"name": "Interpolation_Polynomials_HOL_Algebra"
	},
	{
	"id": 133,
	"link": "/dependencies/intro_dest_elim/",
	"name": "Intro_Dest_Elim"
	},
	{
	"id": 134,
	"link": "/dependencies/ip_addresses/",
	"name": "IP_Addresses"
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	\ No newline at end of file
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	<h1 >
	<span class='first'>K</span><span class='first'>A</span><span class='first'>D</span> Dependents</h1>
	<div>


	</div>
	-</header><div><h2 class="head">2019</h2><article class="entry">
	+</header><div><h2 class="head">2023</h2><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+
	+
	+<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Hybrid_Systems_VCs.html">Verification Components for Hybrid Systems</a></h5> <br>by <a href="../../authors/munive">Jonathan Julian Huerta y Munive</a></div>
	<span class="date">
	Sep 10
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 18
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	\ No newline at end of file
	diff --git a/web/dependencies/kad/index.xml b/web/dependencies/kad/index.xml
	--- a/web/dependencies/kad/index.xml
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	+ <item>
	+ <title>Modal quantales, involutive quantales, Dedekind Quantales</title>
	+ <link>/entries/Quantales_Converse.html</link>
	+ <pubDate>Tue, 25 Jul 2023 00:00:00 +0000</pubDate>
	+
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	+ <description></description>
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	+
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	<title>Verification Components for Hybrid Systems</title>
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	<pubDate>Tue, 10 Sep 2019 00:00:00 +0000</pubDate>

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	<h1 >
	<span class='first'>K</span>leene_<span class='first'>A</span>lgebra Dependents</h1>
	<div>


	</div>
	-</header><div><h2 class="head">2018</h2><article class="entry">
	+</header><div><h2 class="head">2023</h2><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+
	+
	+<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Quantales.html">Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../authors/guttmann">Walter Guttmann</a>, <a href="../../authors/hoefner">Peter Höfner</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Apr 12
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Multirelations.html">Binary Multirelations</a></h5> <br>by <a href="../../authors/furusawa">Hitoshi Furusawa</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 11
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Regular_Algebras.html">Regular Algebras</a></h5> <br>by <a href="../../authors/fosters">Simon Foster</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	May 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Relation_Algebra.html">Relation Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/fosters">Simon Foster</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
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	+ <title>Modal quantales, involutive quantales, Dedekind Quantales</title>
	+ <link>/entries/Quantales_Converse.html</link>
	+ <pubDate>Tue, 25 Jul 2023 00:00:00 +0000</pubDate>
	+
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	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
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	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

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	diff --git a/web/dependencies/quantales/index.html b/web/dependencies/quantales/index.html
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	@@ -1,92 +1,101 @@
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	<h1 >
	<span class='first'>Q</span>uantales Dependents</h1>
	<div>


	</div>
	-</header><div><h2 class="head">2018</h2><article class="entry">
	+</header><div><h2 class="head">2023</h2><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+
	+
	+<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Transformer_Semantics.html">Transformer Semantics</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
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	</article>




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	</body>

	</html>
	\ No newline at end of file
	diff --git a/web/dependencies/quantales/index.xml b/web/dependencies/quantales/index.xml
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	+ <title>Modal quantales, involutive quantales, Dedekind Quantales</title>
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	+
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	+ <h1 >
	+ <span class='first'>T</span>hree_<span class='first'>S</span>quares Dependents</h1>
	+ <div>
	+
	+
	+ </div>
	+</header><div><h2 class="head">2023</h2><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../authors/leek">Kevin Lee</a>, <a href="../../authors/yez">Zhengkun Ye</a> and <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+</article>
	+
	+
	+
	+
	+ </div>
	+ </div>
	+</body>
	+
	+</html>
	\ No newline at end of file
	diff --git a/web/dependencies/three_squares/index.xml b/web/dependencies/three_squares/index.xml
	new file mode 100644
	--- /dev/null
	+++ b/web/dependencies/three_squares/index.xml
	@@ -0,0 +1,20 @@
	+<?xml version="1.0" encoding="utf-8" standalone="yes"?>
	+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
	+ <channel>
	+ <title>Three_Squares on Archive of Formal Proofs</title>
	+ <link>/dependencies/three_squares/</link>
	+ <description>Recent content in Three_Squares on Archive of Formal Proofs</description>
	+ <generator>Hugo -- gohugo.io</generator>
	+ <language>en-gb</language>
	+ <lastBuildDate>Thu, 10 Aug 2023 00:00:00 +0000</lastBuildDate><atom:link href="/dependencies/three_squares/index.xml" rel="self" type="application/rss+xml" />
	+ <item>
	+ <title>Polygonal Number Theorem</title>
	+ <link>/entries/Polygonal_Number_Theorem.html</link>
	+ <pubDate>Thu, 10 Aug 2023 00:00:00 +0000</pubDate>
	+
	+ <guid>/entries/Polygonal_Number_Theorem.html</guid>
	+ <description></description>
	+ </item>
	+
	+ </channel>
	+</rss>
	diff --git a/web/entries/Earley_Parser.html b/web/entries/Earley_Parser.html
	new file mode 100644
	--- /dev/null
	+++ b/web/entries/Earley_Parser.html
	@@ -0,0 +1,171 @@
	+<!DOCTYPE html>
	+<html lang="en">
	+<head>
	+ <meta charset="utf-8" />
	+ <meta http-equiv="X-UA-Compatible" content="IE=edge">
	+ <meta name="viewport" content="width=device-width, initial-scale=1"><title>Earley Parser - Archive of Formal Proofs</title><meta name="description" content="In 1968, Earley introduced his parsing algorithm, capable of parsing all context-free grammars in cubic
	+space and time. This entry contains a formalization..."><meta property="og:title" content="Earley Parser" />
	+<meta property="og:description" content="" />
	+<meta property="og:type" content="article" />
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	+ <a href="../about/"><li >About</li></a>
	+ </ul>
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	+ class='content entries'><header>
	+ <form autocomplete="off" action="../search">
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	+ id="search-button" type="button"><img src="../images/search.svg" alt="Search" /></button>
	+ <datalist id="autocomplete">
	+ </datalist>
	+ </div>
	+ </form>
	+ <h1 >
	+ <span class='first'>E</span>arley <span class='first'>P</span>arser</h1>
	+ <div>
	+
	+ <p><a href="../authors/rau">Martin Rau</a> <a class="obfuscated" data="eyJob3N0IjpbInR1bSIsImRlIl0sInVzZXIiOlsibWFydGluIiwicmF1Il19">📧</a>
	+ </p>
	+
	+
	+ <p class="date">July 16, 2023</p>
	+
	+ </div>
	+</header><div><main><h3>Abstract</h3>
	+
	+ <div class="abstract mathjax_process">In 1968, Earley introduced his parsing algorithm, capable of parsing all context-free grammars in cubic
	+space and time. This entry contains a formalization of an executable Earley parser. We base our development on Jones'
	+extensive paper proof of Earley's recognizer and Obua's formalization of context-free grammars
	+and derivations. We implement and prove correct a functional recognizer modeling Earley's
	+original imperative implementation and extend it with the necessary data structures to enable the construction
	+of parse trees, following the work of Scott. We then develop a functional algorithm that
	+builds a single parse tree, and we prove its correctness. Finally, we generalize this approach to an algorithm
	+for a complete parse forest and prove soundness.</div>
	+
	+ <h3>License</h3>
	+ <div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	+ <ul><li><a href="../topics/computer-science/algorithms">Computer science/Algorithms</a></li></ul>
	+ <h3>Session Earley_Parser</h3>
	+ <ul>
	+ <li><a href="../theories/earley_parser/#Limit">Limit</a></li>
	+ <li><a href="../theories/earley_parser/#CFG">CFG</a></li>
	+ <li><a href="../theories/earley_parser/#Derivations">Derivations</a></li>
	+ <li><a href="../theories/earley_parser/#Earley">Earley</a></li>
	+ <li><a href="../theories/earley_parser/#Earley_Fixpoint">Earley_Fixpoint</a></li>
	+ <li><a href="../theories/earley_parser/#Earley_Recognizer">Earley_Recognizer</a></li>
	+ <li><a href="../theories/earley_parser/#Earley_Parser">Earley_Parser</a></li>
	+ <li><a href="../theories/earley_parser/#Examples">Examples</a></li></ul><div class="flex-wrap">
	+
	+
	+ </div>
	+</main>
	+
	+<nav class='links'>
	+ <a class='popup-button' href="#cite-popup">Cite</a>
	+ <a class='popup-button' href="#download-popup">Download</a>
	+ <h4>PDFs</h4>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Earley_Parser/outline.pdf">Proof
	+ outline</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Earley_Parser/document.pdf">Proof
	+ document</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Earley_Parser/session_graph.pdf">Dependencies</a></nav>
	+
	+<div id="cite-popup" class="overlay">
	+ <a class="cancel" href="#"></a>
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	+ <h2>Cite</h2>
	+ <a class="close" href="#">×</a>
	+ <div>
	+ <p style="display:none;" id="bibtex-filename">Earley_Parser-AFP</p><pre id="copy-text">@article{Earley_Parser-AFP,
	+ author = {Martin Rau},
	+ title = {Earley Parser},
	+ journal = {Archive of Formal Proofs},
	+ month = {July},
	+ year = {2023},
	+ note = {\url{https://isa-afp.org/entries/Earley_Parser.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}</pre>
	+ <button id="copy-bibtex">Copy</button> <a id="download-bibtex">Download</a>
	+ </div>
	+ </div>
	+</div>
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	+
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	+
	+</html>
	\ No newline at end of file
	diff --git a/web/entries/Example-Submission.html b/web/entries/Example-Submission.html
	new file mode 100644
	--- /dev/null
	+++ b/web/entries/Example-Submission.html
	@@ -0,0 +1,242 @@
	+<!DOCTYPE html>
	+<html lang="en">
	+<head>
	+ <meta charset="utf-8" />
	+ <meta http-equiv="X-UA-Compatible" content="IE=edge">
	+ <meta name="viewport" content="width=device-width, initial-scale=1"><title>Example Submission - Archive of Formal Proofs</title><meta name="description" content="This is an example submission to the Archive of Formal Proofs. It shows submission requirements and explains the structure of a simple typical submission...."><meta property="og:title" content="Example Submission" />
	+<meta property="og:description" content="" />
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	+<meta property="article:published_time" content="2004-02-25T00:00:00+00:00" />
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	+ <a href="../about/"><li >About</li></a>
	+ </ul>
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	+
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	+ class='content entries'><header>
	+ <form autocomplete="off" action="../search">
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	+ <datalist id="autocomplete">
	+ </datalist>
	+ </div>
	+ </form>
	+ <h1 >
	+ <span class='first'>E</span>xample <span class='first'>S</span>ubmission</h1>
	+ <div>
	+
	+ <p><a href="../authors/klein">Gerwin Klein</a> <a href="http://www.cse.unsw.edu.au/~kleing/">🌐</a>
	+ </p>
	+
	+
	+ <p class="date">February 25, 2004</p>
	+
	+ </div>
	+</header><div><main><h3>Abstract</h3>
	+
	+ <div class="abstract mathjax_process"><p>This is an example submission to the Archive of Formal Proofs. It shows
	+submission requirements and explains the structure of a simple typical
	+submission.</p>
	+<p>Note that you can use <em>HTML tags</em> and LaTeX formulae like
	+$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ in the abstract. Display formulae like
	+$$ \int_0^1 x^{-x}\,\text{d}x = \sum_{n=1}^\infty n^{-n}$$
	+are also possible. Please read the
	+<a href="../submission">submission guidelines</a> before using this.</p></div>
	+
	+ <h3>License</h3>
	+ <div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	+ <ul><li><a href="../topics/mathematics/analysis">Mathematics/Analysis</a></li><li><a href="../topics/mathematics/number-theory">Mathematics/Number theory</a></li></ul>
	+ <h3>Session Example-Submission</h3>
	+ <ul>
	+ <li><a href="../theories/example-submission/#Submission">Submission</a></li></ul><div class="flex-wrap">
	+
	+
	+ </div>
	+</main>
	+
	+<nav class='links'>
	+ <a class='popup-button' href="#cite-popup">Cite</a>
	+ <a class='popup-button' href="#download-popup">Download</a>
	+ <h4>PDFs</h4>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Example-Submission/outline.pdf">Proof
	+ outline</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Example-Submission/document.pdf">Proof
	+ document</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Example-Submission/session_graph.pdf">Dependencies</a></nav>
	+
	+<div id="cite-popup" class="overlay">
	+ <a class="cancel" href="#"></a>
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	+ <h2>Cite</h2>
	+ <a class="close" href="#">×</a>
	+ <div>
	+ <p style="display:none;" id="bibtex-filename">Example-Submission-AFP</p><pre id="copy-text">@article{Example-Submission-AFP,
	+ author = {Gerwin Klein},
	+ title = {Example Submission},
	+ journal = {Archive of Formal Proofs},
	+ month = {February},
	+ year = {2004},
	+ note = {\url{https://isa-afp.org/entries/Example-Submission.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}</pre>
	+ <button id="copy-bibtex">Copy</button> <a id="download-bibtex">Download</a>
	+ </div>
	+ </div>
	+</div>
	+
	+<div id="download-popup" class="overlay">
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	+ <h2>Download</h2>
	+ <a class="close" href="#">×</a>
	+ <a href="https://www.isa-afp.org/release//afp-Example-Submission-current.tar.gz" download>Download latest</a>
	+ <p>Older releases:</p>
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	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2012-05-24.tar.gz">May 24, 2012</a>: Isabelle2012
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2012-03-15.tar.gz">Mar 15, 2012</a>: Isabelle2011-1
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2011-10-11.tar.gz">Oct 11, 2011</a>: Isabelle2011-1
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2011-02-11.tar.gz">Feb 11, 2011</a>: Isabelle2011
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2010-06-30.tar.gz">Jun 30, 2010</a>: Isabelle2009-2
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2009-12-12.tar.gz">Dec 12, 2009</a>: Isabelle2009-1
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2009-04-29.tar.gz">Apr 29, 2009</a>: Isabelle2009
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2008-06-10.tar.gz">Jun 10, 2008</a>: Isabelle2008
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2007-11-27.tar.gz">Nov 27, 2007</a>: Isabelle2007
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2006-11-13.tar.gz">Nov 13, 2006</a>: Isabelle2005
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2006-11-10.tar.gz">Nov 10, 2006</a>: Isabelle2005
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2005-10-14.tar.gz">Oct 14, 2005</a>: Isabelle2005
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2004-05-21.tar.gz">May 21, 2004</a>: Isabelle2004
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2004-04-20.tar.gz">Apr 20, 2004</a>: Isabelle2004
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2004-03-05.tar.gz">Mar 5, 2004</a>: Isabelle2003
	+ </li><li>
	+ <a href="https://www.isa-afp.org/release/afp-Example-Submission-2004-02-25.tar.gz">Feb 25, 2004</a>: Isabelle2003
	+ </li></ul>
	+ </div>
	+</div>
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	\ No newline at end of file
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	<h1 >
	<span class='first'>K</span>leene <span class='first'>A</span>lgebras <span class='first'>W</span>ith <span class='first'>D</span>omain</h1>
	<div>

	<p><a href="../authors/gomes">Victor B. F. Gomes</a> <a href="http://www.dcs.shef.ac.uk/~victor">🌐</a>, <a href="../authors/guttmann">Walter Guttmann</a> <a href="https://www.cosc.canterbury.ac.nz/walter.guttmann/">🌐</a>, <a href="../authors/hoefner">Peter Höfner</a> <a href="http://www.hoefner-online.de/">🌐</a>, <a href="../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a> and <a href="../authors/weber">Tjark Weber</a> <a href="http://user.it.uu.se/~tjawe125/">🌐</a>
	</p>


	<p class="date">April 12, 2016</p>

	</div>
	</header><div><main><h3>Abstract</h3>

	<div class="abstract mathjax_process">Kleene algebras with domain are Kleene algebras endowed with an
	operation that maps each element of the algebra to its domain of
	definition (or its complement) in abstract fashion. They form a simple
	algebraic basis for Hoare logics, dynamic logics or predicate
	transformer semantics. We formalise a modular hierarchy of algebras
	with domain and antidomain (domain complement) operations in
	Isabelle/HOL that ranges from domain and antidomain semigroups to
	modal Kleene algebras and divergence Kleene algebras. We link these
	algebras with models of binary relations and program traces. We
	include some examples from modal logics, termination and program
	analysis.</div>

	<h3>License</h3>
	<div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	<ul><li><a href="../topics/computer-science/programming-languages/logics">Computer science/Programming languages/Logics</a></li><li><a href="../topics/computer-science/automata-and-formal-languages">Computer science/Automata and formal languages</a></li><li><a href="../topics/mathematics/algebra">Mathematics/Algebra</a></li></ul>
	<h3>Session KAD</h3>
	<ul>
	<li><a href="../theories/kad/#Domain_Semiring">Domain_Semiring</a></li>
	<li><a href="../theories/kad/#Antidomain_Semiring">Antidomain_Semiring</a></li>
	<li><a href="../theories/kad/#Range_Semiring">Range_Semiring</a></li>
	<li><a href="../theories/kad/#Modal_Kleene_Algebra">Modal_Kleene_Algebra</a></li>
	<li><a href="../theories/kad/#Modal_Kleene_Algebra_Models">Modal_Kleene_Algebra_Models</a></li>
	<li><a href="../theories/kad/#Modal_Kleene_Algebra_Applications">Modal_Kleene_Algebra_Applications</a></li></ul><div class="flex-wrap">
	<div>
	<h3>Depends on</h3>
	<ul class="horizontal-list"><li><a href="../entries/Kleene_Algebra.html">Kleene Algebra</a></li></ul>
	</div><div>
	<h3>Used by</h3>
	- <ul class="horizontal-list"><li><a href="../entries/Hybrid_Systems_VCs.html">Verification Components for Hybrid Systems</a></li><li><a href="../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></li></ul>
	+ <ul class="horizontal-list"><li><a href="../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></li><li><a href="../entries/Hybrid_Systems_VCs.html">Verification Components for Hybrid Systems</a></li><li><a href="../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></li></ul>
	</div>

	<div>
	<h3>Auto-related entries</h3>
	<ul class="horizontal-list">

	<li><a href="../entries/Transformer_Semantics.html">Transformer Semantics</a></li>

	<li><a href="../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></li>

	<li><a href="../entries/Multirelations.html">Binary Multirelations</a></li>

	<li><a href="../entries/Relation_Algebra.html">Relation Algebra</a></li>

	<li><a href="../entries/DataRefinementIBP.html">Semantics and Data Refinement of Invariant Based Programs</a></li>

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	<p style="display:none;" id="bibtex-filename">KAD-AFP</p><pre id="copy-text">@article{KAD-AFP,
	author = {Victor B. F. Gomes and Walter Guttmann and Peter Höfner and Georg Struth and Tjark Weber},
	title = {Kleene Algebras with Domain},
	journal = {Archive of Formal Proofs},
	month = {April},
	year = {2016},
	note = {\url{https://isa-afp.org/entries/KAD.html},
	Formal proof development},
	ISSN = {2150-914x},
	}</pre>
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	<p>Older releases:</p>
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	<a href="https://www.isa-afp.org/release/afp-KAD-2022-10-27.tar.gz">Oct 27, 2022</a>: Isabelle2022
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	<a href="https://www.isa-afp.org/release/afp-KAD-2021-12-14.tar.gz">Dec 14, 2021</a>: Isabelle2021-1
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2021-02-23.tar.gz">Feb 23, 2021</a>: Isabelle2021
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2020-04-18.tar.gz">Apr 18, 2020</a>: Isabelle2020
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2019-06-11.tar.gz">Jun 11, 2019</a>: Isabelle2019
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2018-08-16.tar.gz">Aug 16, 2018</a>: Isabelle2018
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2017-10-10.tar.gz">Oct 10, 2017</a>: Isabelle2017
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2016-12-17.tar.gz">Dec 17, 2016</a>: Isabelle2016-1
	</li><li>
	<a href="https://www.isa-afp.org/release/afp-KAD-2016-04-12.tar.gz">Apr 12, 2016</a>: Isabelle2016
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	<h1 >
	<span class='first'>K</span>leene <span class='first'>A</span>lgebra</h1>
	<div>

	<p><a href="../authors/armstrong">Alasdair Armstrong</a>, <a href="../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a> and <a href="../authors/weber">Tjark Weber</a> <a href="http://user.it.uu.se/~tjawe125/">🌐</a>
	</p>


	<p class="date">January 15, 2013</p>

	</div>
	</header><div><main><h3>Abstract</h3>

	<div class="abstract mathjax_process">These files contain a formalisation of variants of Kleene algebras and
	their most important models as axiomatic type classes in Isabelle/HOL.
	Kleene algebras are foundational structures in computing with
	applications ranging from automata and language theory to computational
	modeling, program construction and verification.
	<p>
	We start with formalising dioids, which are additively idempotent
	semirings, and expand them by axiomatisations of the Kleene star for
	finite iteration and an omega operation for infinite iteration. We
	show that powersets over a given monoid, (regular) languages, sets of
	paths in a graph, sets of computation traces, binary relations and
	formal power series form Kleene algebras, and consider further models
	based on lattices, max-plus semirings and min-plus semirings. We also
	demonstrate that dioids are closed under the formation of matrices
	(proofs for Kleene algebras remain to be completed).
	<p>
	On the one hand we have aimed at a reference formalisation of variants
	of Kleene algebras that covers a wide range of variants and the core
	theorems in a structured and modular way and provides readable proofs
	at text book level. On the other hand, we intend to use this algebraic
	hierarchy and its models as a generic algebraic middle-layer from which
	programming applications can quickly be explored, implemented and verified.</div>

	<h3>License</h3>
	<div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	<ul><li><a href="../topics/computer-science/programming-languages/logics">Computer science/Programming languages/Logics</a></li><li><a href="../topics/computer-science/automata-and-formal-languages">Computer science/Automata and formal languages</a></li><li><a href="../topics/mathematics/algebra">Mathematics/Algebra</a></li></ul>
	<h3>Session Kleene_Algebra</h3>
	<ul>
	<li><a href="../theories/kleene_algebra/#Signatures">Signatures</a></li>
	<li><a href="../theories/kleene_algebra/#Dioid">Dioid</a></li>
	<li><a href="../theories/kleene_algebra/#Dioid_Models">Dioid_Models</a></li>
	<li><a href="../theories/kleene_algebra/#Matrix">Matrix</a></li>
	<li><a href="../theories/kleene_algebra/#Conway">Conway</a></li>
	<li><a href="../theories/kleene_algebra/#Kleene_Algebra">Kleene_Algebra</a></li>
	<li><a href="../theories/kleene_algebra/#Kleene_Algebra_Models">Kleene_Algebra_Models</a></li>
	<li><a href="../theories/kleene_algebra/#Omega_Algebra">Omega_Algebra</a></li>
	<li><a href="../theories/kleene_algebra/#Omega_Algebra_Models">Omega_Algebra_Models</a></li>
	<li><a href="../theories/kleene_algebra/#DRA">DRA</a></li>
	<li><a href="../theories/kleene_algebra/#PHL_KA">PHL_KA</a></li>
	<li><a href="../theories/kleene_algebra/#PHL_DRA">PHL_DRA</a></li>
	<li><a href="../theories/kleene_algebra/#Finite_Suprema">Finite_Suprema</a></li>
	<li><a href="../theories/kleene_algebra/#Formal_Power_Series">Formal_Power_Series</a></li>
	<li><a href="../theories/kleene_algebra/#Inf_Matrix">Inf_Matrix</a></li></ul><div class="flex-wrap">
	<div>
	<h3>Used by</h3>
	- <ul class="horizontal-list"><li><a href="../entries/Quantales.html">Quantales</a></li><li><a href="../entries/KAD.html">Kleene Algebras with Domain</a></li><li><a href="../entries/Multirelations.html">Binary Multirelations</a></li><li><a href="../entries/Regular_Algebras.html">Regular Algebras</a></li><li><a href="../entries/Relation_Algebra.html">Relation Algebra</a></li><li><a href="../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></li></ul>
	+ <ul class="horizontal-list"><li><a href="../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></li><li><a href="../entries/Quantales.html">Quantales</a></li><li><a href="../entries/KAD.html">Kleene Algebras with Domain</a></li><li><a href="../entries/Multirelations.html">Binary Multirelations</a></li><li><a href="../entries/Regular_Algebras.html">Regular Algebras</a></li><li><a href="../entries/Relation_Algebra.html">Relation Algebra</a></li><li><a href="../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></li></ul>
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	<p style="display:none;" id="bibtex-filename">Kleene_Algebra-AFP</p><pre id="copy-text">@article{Kleene_Algebra-AFP,
	author = {Alasdair Armstrong and Georg Struth and Tjark Weber},
	title = {Kleene Algebra},
	journal = {Archive of Formal Proofs},
	month = {January},
	year = {2013},
	note = {\url{https://isa-afp.org/entries/Kleene_Algebra.html},
	Formal proof development},
	ISSN = {2150-914x},
	}</pre>
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	+ <h1 >
	+ <span class='first'>P</span>olygonal <span class='first'>N</span>umber <span class='first'>T</span>heorem</h1>
	+ <div>
	+
	+ <p><a href="../authors/leek">Kevin Lee</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJrZmtsMiJdfQ==">📧</a>, <a href="../authors/yez">Zhengkun Ye</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJ6eTMwNyJdfQ==">📧</a> and <a href="../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> <a class="obfuscated" data="eyJob3N0IjpbImNhbSIsImFjIiwidWsiXSwidXNlciI6WyJhazIxMTAiXX0=">📧</a>
	+ </p>
	+
	+
	+ <p class="date">August 10, 2023</p>
	+
	+ </div>
	+</header><div><main><h3>Abstract</h3>
	+
	+ <div class="abstract mathjax_process"><p>We formalize the proofs of Cauchy's and Legendre's Polygonal Number Theorems given in Melvyn B. Nathanson's book "Additive Number Theory: The Classical Bases".</p>
	+
	+<p>For $m \geq 1$, the $k$-th polygonal number of order $m+2$ is defined to be $p_m(k)=\frac{mk(k-1)}{2}+k$. The theorems state that:</p>
	+
	+<p>1. If $m \ge 4$ and $N \geq 108m$, then $N$ can be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four of which are different from $0$ or $1$. If $N \geq 324$, then $N$ can be written as the sum of five pentagonal numbers, at least one of which is $0$ or $1$.</p>
	+
	+<p>2. Let $m \geq 3$ and $N \geq 28m^3$. If $m$ is odd, then $N$ is the sum of four polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.</p>
	+
	+<p>We also formalize the proof of Gauss's theorem which states that every non-negative integer is the sum of three triangular numbers.</p></div>
	+
	+ <h3>License</h3>
	+ <div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	+ <ul><li><a href="../topics/mathematics/number-theory">Mathematics/Number theory</a></li><li><a href="../topics/mathematics/combinatorics">Mathematics/Combinatorics</a></li></ul><h3>Related publications</h3>
	+ <ul><li>
	+ Nathanson, M. B. (1996). Additive Number Theory. Graduate Texts in Mathematics. <a href="https://doi.org/10.1007/978-1-4757-3845-2">https://doi.org/10.1007/978-1-4757-3845-2</a>
	+
	+ </li></ul>
	+ <h3>Session Polygonal_Number_Theorem</h3>
	+ <ul>
	+ <li><a href="../theories/polygonal_number_theorem/#Polygonal_Number_Theorem_Lemmas">Polygonal_Number_Theorem_Lemmas</a></li>
	+ <li><a href="../theories/polygonal_number_theorem/#Polygonal_Number_Theorem_Gauss">Polygonal_Number_Theorem_Gauss</a></li>
	+ <li><a href="../theories/polygonal_number_theorem/#Polygonal_Number_Theorem_Cauchy">Polygonal_Number_Theorem_Cauchy</a></li>
	+ <li><a href="../theories/polygonal_number_theorem/#Polygonal_Number_Theorem_Legendre">Polygonal_Number_Theorem_Legendre</a></li></ul><div class="flex-wrap">
	+ <div>
	+ <h3>Depends on</h3>
	+ <ul class="horizontal-list"><li><a href="../entries/Three_Squares.html">Three Squares Theorem</a></li></ul>
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	+
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	+ <p style="display:none;" id="bibtex-filename">Polygonal_Number_Theorem-AFP</p><pre id="copy-text">@article{Polygonal_Number_Theorem-AFP,
	+ author = {Kevin Lee and Zhengkun Ye and Angeliki Koutsoukou-Argyraki},
	+ title = {Polygonal Number Theorem},
	+ journal = {Archive of Formal Proofs},
	+ month = {August},
	+ year = {2023},
	+ note = {\url{https://isa-afp.org/entries/Polygonal_Number_Theorem.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}</pre>
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	diff --git a/web/entries/Quantales.html b/web/entries/Quantales.html
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	<h1 >
	<span class='first'>Q</span>uantales</h1>
	<div>

	<p><a href="../authors/struth">Georg Struth</a> <a href="http://staffwww.dcs.shef.ac.uk/people/G.Struth/">🌐</a>
	</p>


	<p class="date">December 11, 2018</p>

	</div>
	</header><div><main><h3>Abstract</h3>

	<div class="abstract mathjax_process">These mathematical components formalise basic properties of quantales,
	together with some important models, constructions, and concepts,
	including quantic nuclei and conuclei.</div>

	<h3>License</h3>
	<div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	<ul><li><a href="../topics/mathematics/algebra">Mathematics/Algebra</a></li></ul>
	<h3>Session Quantales</h3>
	<ul>
	<li><a href="../theories/quantales/#Dioid_Models_New">Dioid_Models_New</a></li>
	<li><a href="../theories/quantales/#Quantales">Quantales</a></li>
	<li><a href="../theories/quantales/#Quantale_Star">Quantale_Star</a></li>
	<li><a href="../theories/quantales/#Quantale_Modules">Quantale_Modules</a></li>
	<li><a href="../theories/quantales/#Quantale_Models">Quantale_Models</a></li>
	<li><a href="../theories/quantales/#Quantic_Nuclei_Conuclei">Quantic_Nuclei_Conuclei</a></li>
	<li><a href="../theories/quantales/#Quantale_Left_Sided">Quantale_Left_Sided</a></li></ul><div class="flex-wrap">
	<div>
	<h3>Depends on</h3>
	<ul class="horizontal-list"><li><a href="../entries/Kleene_Algebra.html">Kleene Algebra</a></li><li><a href="../entries/Order_Lattice_Props.html">Properties of Orderings and Lattices</a></li></ul>
	</div><div>
	<h3>Used by</h3>
	- <ul class="horizontal-list"><li><a href="../entries/Transformer_Semantics.html">Transformer Semantics</a></li></ul>
	+ <ul class="horizontal-list"><li><a href="../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></li><li><a href="../entries/Transformer_Semantics.html">Transformer Semantics</a></li></ul>
	</div>

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	outline</a>
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	document</a>
	<a href="https://www.isa-afp.org/browser_info/current/AFP/Quantales/session_graph.pdf">Dependencies</a></nav>

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	<p style="display:none;" id="bibtex-filename">Quantales-AFP</p><pre id="copy-text">@article{Quantales-AFP,
	author = {Georg Struth},
	title = {Quantales},
	journal = {Archive of Formal Proofs},
	month = {December},
	year = {2018},
	note = {\url{https://isa-afp.org/entries/Quantales.html},
	Formal proof development},
	ISSN = {2150-914x},
	}</pre>
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	+ involutive quantales and Dedekind quantales. Modal quantales are
	+ simple extensions..."><meta property="og:title" content="Modal quantales, involutive quantales, Dedekind Quantales" />
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	+ <h1 >
	+ <span class='first'>M</span>odal <span class='first'>Q</span>uantales, <span class='first'>I</span>nvolutive <span class='first'>Q</span>uantales, <span class='first'>D</span>edekind <span class='first'>Q</span>uantales</h1>
	+ <div>
	+
	+ <p><a href="../authors/struth">Georg Struth</a> <a class="obfuscated" data="eyJob3N0IjpbInNoZWZmaWVsZCIsImFjIiwidWsiXSwidXNlciI6WyJnIiwic3RydXRoIl19">📧</a> and <a href="../authors/calk">Cameron Calk</a>
	+ </p>
	+
	+
	+ <p class="date">July 25, 2023</p>
	+
	+ </div>
	+</header><div><main><h3>Abstract</h3>
	+
	+ <div class="abstract mathjax_process">This AFP entry provides mathematical components for modal quantales,
	+ involutive quantales and Dedekind quantales. Modal quantales are
	+ simple extensions of modal Kleene algebras useful for the
	+ verification of recursive programs. Involutive quantales appear in
	+ the study of C*-algebras. Dedekind quantales are relatives of
	+ Tarski's relation algebras, hence relevant to program verification
	+ and beyond that to higher rewriting. We also provide
	+ components for weaker variants such as Kleene algebras with converse
	+ and modal Kleene algebras with converse.</div>
	+
	+ <h3>License</h3>
	+ <div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	+ <ul><li><a href="../topics/computer-science/semantics-and-reasoning">Computer science/Semantics and reasoning</a></li><li><a href="../topics/mathematics/algebra">Mathematics/Algebra</a></li><li><a href="../topics/mathematics/order">Mathematics/Order</a></li></ul><h3>Related publications</h3>
	+ <ul><li>
	+ Fahrenberg, U., Johansen, C., Struth, G., & Ziemiański, K. (2023). Catoids and modal convolution algebras. Algebra Universalis, 84(2). <a href="https://doi.org/10.1007/s00012-023-00805-9">https://doi.org/10.1007/s00012-023-00805-9</a>
	+
	+ </li><li>
	+ Calk, C., Malbos, P., Pous, D., & Struth, G. (2023). <i>Higher Catoids, Higher Quantales and their Correspondences</i> (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2307.09253
	+ </li></ul>
	+ <h3>Session Quantales_Converse</h3>
	+ <ul>
	+ <li><a href="../theories/quantales_converse/#Modal_Kleene_Algebra_Var">Modal_Kleene_Algebra_Var</a></li>
	+ <li><a href="../theories/quantales_converse/#Kleene_Algebra_Converse">Kleene_Algebra_Converse</a></li>
	+ <li><a href="../theories/quantales_converse/#Modal_Kleene_Algebra_Converse">Modal_Kleene_Algebra_Converse</a></li>
	+ <li><a href="../theories/quantales_converse/#Modal_Quantale">Modal_Quantale</a></li>
	+ <li><a href="../theories/quantales_converse/#Quantale_Converse">Quantale_Converse</a></li></ul><div class="flex-wrap">
	+ <div>
	+ <h3>Depends on</h3>
	+ <ul class="horizontal-list"><li><a href="../entries/KAD.html">Kleene Algebras with Domain</a></li><li><a href="../entries/Kleene_Algebra.html">Kleene Algebra</a></li><li><a href="../entries/Quantales.html">Quantales</a></li></ul>
	+ </div>
	+
	+ </div>
	+</main>
	+
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	+ <a class='popup-button' href="#cite-popup">Cite</a>
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	+ <h4>PDFs</h4>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Quantales_Converse/outline.pdf">Proof
	+ outline</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Quantales_Converse/document.pdf">Proof
	+ document</a>
	+ <a href="https://www.isa-afp.org/browser_info/current/AFP/Quantales_Converse/session_graph.pdf">Dependencies</a></nav>
	+
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	+ <div>
	+ <p style="display:none;" id="bibtex-filename">Quantales_Converse-AFP</p><pre id="copy-text">@article{Quantales_Converse-AFP,
	+ author = {Georg Struth and Cameron Calk},
	+ title = {Modal quantales, involutive quantales, Dedekind Quantales},
	+ journal = {Archive of Formal Proofs},
	+ month = {July},
	+ year = {2023},
	+ note = {\url{https://isa-afp.org/entries/Quantales_Converse.html},
	+ Formal proof development},
	+ ISSN = {2150-914x},
	+}</pre>
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	<h1 >
	<span class='first'>T</span>hree <span class='first'>S</span>quares <span class='first'>T</span>heorem</h1>
	<div>

	<p><a href="../authors/danilkin">Anton Danilkin</a> and <a href="../authors/chevalier">Loïc Chevalier</a>
	</p>


	<p class="date">May 3, 2023</p>

	</div>
	</header><div><main><h3>Abstract</h3>

	<div class="abstract mathjax_process">We formalize the Legendre's three squares theorem and its consequences,
	in particular the following results:
	<ol>
	<li>
	A natural number can be represented as the sum of
	three squares of natural numbers if and only if it is not
	of the form $4^a (8 k + 7)$, where $a$ and $k$ are natural numbers.
	</li>
	<li>
	If $n$ is a natural number such that $n \equiv 3 \pmod{8}$,
	then $n$ can be be represented as the sum of three squares
	of odd natural numbers.
	</li>
	</ol>

	Consequences include the following:
	<ol>
	<li>
	An integer $n$ can be written as $n = x^2 + y^2 + z^2 + z$,
	where $x$, $y$, $z$ are integers, if and only if $n \geq 0$.
	</li>
	<li>
	The Legendre's four squares theorem: any natural number
	can be represented as the sum of four squares of natural numbers.
	</li>
	</ol></div>

	<h3>License</h3>
	<div><a href="https://isa-afp.org/LICENSE">BSD License</a></div><h3>Topics</h3>
	<ul><li><a href="../topics/mathematics/number-theory">Mathematics/Number theory</a></li></ul>
	<h3>Session Three_Squares</h3>
	<ul>
	<li><a href="../theories/three_squares/#Residues_Properties">Residues_Properties</a></li>
	<li><a href="../theories/three_squares/#Low_Dimensional_Linear_Algebra">Low_Dimensional_Linear_Algebra</a></li>
	<li><a href="../theories/three_squares/#Quadratic_Forms">Quadratic_Forms</a></li>
	<li><a href="../theories/three_squares/#Three_Squares">Three_Squares</a></li></ul><div class="flex-wrap">
	<div>
	<h3>Depends on</h3>
	<ul class="horizontal-list"><li><a href="../entries/Dirichlet_L.html">Dirichlet L-Functions and Dirichlet’s Theorem</a></li></ul>
	+ </div><div>
	+ <h3>Used by</h3>
	+ <ul class="horizontal-list"><li><a href="../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></li></ul>
	</div>

	<div>
	<h3>Auto-related entries</h3>
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	<li><a href="../entries/DigitsInBase.html">Positional Notation for Natural Numbers in an Arbitrary Base</a></li>

	<li><a href="../entries/Digit_Expansions.html">Digit Expansions</a></li>

	<li><a href="../entries/Dirichlet_Series.html">Dirichlet Series</a></li>

	<li><a href="../entries/SumSquares.html">Sums of Two and Four Squares</a></li>

	<li><a href="../entries/Involutions2Squares.html">From THE BOOK: Two Squares via Involutions</a></li>

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	<p style="display:none;" id="bibtex-filename">Three_Squares-AFP</p><pre id="copy-text">@article{Three_Squares-AFP,
	author = {Anton Danilkin and Loïc Chevalier},
	title = {Three Squares Theorem},
	journal = {Archive of Formal Proofs},
	month = {May},
	year = {2023},
	note = {\url{https://isa-afp.org/entries/Three_Squares.html},
	Formal proof development},
	ISSN = {2150-914x},
	}</pre>
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	</header><div><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../authors/leek">Kevin Lee</a>, <a href="../authors/yez">Zhengkun Ye</a> and <a href="../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../authors/struth">Georg Struth</a> and <a href="../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../entries/Earley_Parser.html">Earley Parser</a></h5> <br>by <a href="../authors/rau">Martin Rau</a></div>
	+ <span class="date">
	+ Jul 16
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="../entries/Gray_Codes.html">Gray Codes for Arbitrary Numeral Systems</a></h5> <br>by <a href="../authors/spitz">Maximilian Spitz</a></div>
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	<h5><a class="title" href="../entries/Efficient_Weighted_Path_Order.html">A Verified Efficient Implementation of the Weighted Path Order</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a> and <a href="../authors/wenninger">Elias Wenninger</a></div>
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	<span class="date">
	Aug 06
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/IMO2019.html">Selected Problems from the International Mathematical Olympiad 2019</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
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	Aug 05
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jul 08
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	<div class="item-text">
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	Apr 26
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LTL_Master_Theorem.html">A Compositional and Unified Translation of LTL into ω-Automata</a></h5> <br>by <a href="../authors/seidl">Benedikt Seidl</a> and <a href="../authors/sickert">Salomon Sickert</a></div>
	<span class="date">
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	<span class="date">
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Mar 09
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	</span>
	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Feb 14
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	Jan 17
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jan 07
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	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Core_DOM.html">A Formal Model of the Document Object Model</a></h5> <br>by <a href="../authors/brucker">Achim D. Brucker</a> and <a href="../authors/herzberg">Michael Herzberg</a></div>
	<span class="date">
	Dec 26
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Concurrent_Revisions.html">Formalization of Concurrent Revisions</a></h5> <br>by <a href="../authors/overbeek">Roy Overbeek</a></div>
	<span class="date">
	Dec 25
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Dec 21
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Transformer_Semantics.html">Transformer Semantics</a></h5> <br>by <a href="../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Dec 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Dec 11
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	<div class="item-text">
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	<span class="date">
	Nov 23
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Nov 20
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Matroids.html">Matroids</a></h5> <br>by <a href="../authors/keinholz">Jonas Keinholz</a></div>
	<span class="date">
	Nov 16
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Nov 06
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	<article class="entry">
	<div class="item-text">
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	<div class="item-text">
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	<span class="date">
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	</article>
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	Oct 22
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	</article>
	<article class="entry">
	<div class="item-text">
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Oct 19
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Oct 12
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Sep 19
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Sep 15
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	</article>
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	<div class="item-text">
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	<span class="date">
	Sep 14
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Quaternions.html">Quaternions</a></h5> <br>by <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Sep 05
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Sep 02
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Simplex.html">An Incremental Simplex Algorithm with Unsatisfiable Core Generation</a></h5> <br>by <a href="../authors/maricf">Filip Marić</a>, <a href="../authors/spasic">Mirko Spasić</a> and <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Aug 24
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Minsky_Machines.html">Minsky Machines</a></h5> <br>by <a href="../authors/felgenhauer">Bertram Felgenhauer</a></div>
	<span class="date">
	Aug 14
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/DiscretePricing.html">Pricing in discrete financial models</a></h5> <br>by <a href="../authors/echenim">Mnacho Echenim</a></div>
	<span class="date">
	Jul 16
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Neumann_Morgenstern_Utility.html">Von-Neumann-Morgenstern Utility Theorem</a></h5> <br>by <a href="../authors/parsert">Julian Parsert</a> and <a href="../authors/kaliszyk">Cezary Kaliszyk</a></div>
	<span class="date">
	Jul 04
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Pell.html">Pell's Equation</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jun 23
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	</article>
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	<span class="date">
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	<span class="date">
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	<article class="entry">
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	<h5><a class="title" href="../entries/Prpu_Maxflow.html">Formalizing Push-Relabel Algorithms</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Flow_Networks.html">Flow Networks and the Min-Cut-Max-Flow Theorem</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Optics.html">Optics</a></h5> <br>by <a href="../authors/fosters">Simon Foster</a>, <a href="../authors/laursen">Christian Pardillo-Laursen</a> and <a href="../authors/zeyda">Frank Zeyda</a></div>
	<span class="date">
	May 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Dict_Construction.html">Dictionary Construction</a></h5> <br>by <a href="../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Security_Protocol_Refinement.html">Developing Security Protocols by Refinement</a></h5> <br>by <a href="../authors/sprenger">Christoph Sprenger</a> and <a href="../authors/somaini">Ivano Somaini</a></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Floyd_Warshall.html">The Floyd-Warshall Algorithm for Shortest Paths</a></h5> <br>by <a href="../authors/wimmer">Simon Wimmer</a> and <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Probabilistic_While.html">Probabilistic while loop</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Monad_Normalisation.html">Monad normalisation</a></h5> <br>by <a href="../authors/schneider">Joshua Schneider</a>, <a href="../authors/eberl">Manuel Eberl</a> and <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Game_Based_Crypto.html">Game-based cryptography in HOL</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a>, <a href="../authors/sefidgar">S. Reza Sefidgar</a> and <a href="../authors/bhatt">Bhargav Bhatt</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Monomorphic_Monad.html">Effect polymorphism in higher-order logic</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CryptHOL.html">CryptHOL</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/MonoidalCategory.html">Monoidal Categories</a></h5> <br>by <a href="../authors/stark">Eugene W. Stark</a></div>
	<span class="date">
	May 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Types_Tableaus_and_Goedels_God.html">Types, Tableaus and Gödel’s God in Isabelle/HOL</a></h5> <br>by <a href="../authors/fuenmayor">David Fuenmayor</a> and <a href="../authors/benzmueller">Christoph Benzmüller</a></div>
	<span class="date">
	May 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LocalLexing.html">Local Lexing</a></h5> <br>by <a href="../authors/obua">Steven Obua</a></div>
	<span class="date">
	Apr 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Constructor_Funs.html">Constructor Functions</a></h5> <br>by <a href="../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Apr 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lazy_Case.html">Lazifying case constants</a></h5> <br>by <a href="../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Apr 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Subresultants.html">Subresultants</a></h5> <br>by <a href="../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../authors/thiemann">René Thiemann</a> and <a href="../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Apr 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Random_BSTs.html">Expected Shape of Random Binary Search Trees</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Apr 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Quick_Sort_Cost.html">The number of comparisons in QuickSort</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Comparison_Sort_Lower_Bound.html">Lower bound on comparison-based sorting algorithms</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Euler_MacLaurin.html">The Euler–MacLaurin Formula</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Elliptic_Curves_Group_Law.html">The Group Law for Elliptic Curves</a></h5> <br>by <a href="../authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Feb 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Menger.html">Menger's Theorem</a></h5> <br>by <a href="../authors/dittmann">Christoph Dittmann</a></div>
	<span class="date">
	Feb 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Differential_Dynamic_Logic.html">Differential Dynamic Logic</a></h5> <br>by <a href="../authors/bohrer">Rose Bohrer</a></div>
	<span class="date">
	Feb 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Abstract_Soundness.html">Abstract Soundness</a></h5> <br>by <a href="../authors/blanchette">Jasmin Christian Blanchette</a>, <a href="../authors/popescu">Andrei Popescu</a> and <a href="../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Feb 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stone_Relation_Algebras.html">Stone Relation Algebras</a></h5> <br>by <a href="../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Key_Agreement_Strong_Adversaries.html">Refining Authenticated Key Agreement with Strong Adversaries</a></h5> <br>by <a href="../authors/lallemand">Joseph Lallemand</a> and <a href="../authors/sprenger">Christoph Sprenger</a></div>
	<span class="date">
	Jan 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Bernoulli.html">Bernoulli Numbers</a></h5> <br>by <a href="../authors/bulwahn">Lukas Bulwahn</a> and <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Minimal_SSA.html">Minimal Static Single Assignment Form</a></h5> <br>by <a href="../authors/wagner">Max Wagner</a> and <a href="../authors/lohner">Denis Lohner</a></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Bertrands_Postulate.html">Bertrand's postulate</a></h5> <br>by <a href="../authors/biendarra">Julian Biendarra</a> and <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/E_Transcendental.html">The Transcendence of e</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/UPF_Firewall.html">Formal Network Models and Their Application to Firewall Policies</a></h5> <br>by <a href="../authors/brucker">Achim D. Brucker</a>, <a href="../authors/bruegger">Lukas Brügger</a> and <a href="../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Password_Authentication_Protocol.html">Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method</a></h5> <br>by <a href="../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FOL_Harrison.html">First-Order Logic According to Harrison</a></h5> <br>by <a href="../authors/jensen">Alexander Birch Jensen</a>, <a href="../authors/schlichtkrull">Anders Schlichtkrull</a> and <a href="../authors/villadsen">Jørgen Villadsen</a></div>
	<span class="date">
	Jan 01
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Concurrent_Ref_Alg.html">Concurrent Refinement Algebra and Rely Quotients</a></h5> <br>by <a href="../authors/fell">Julian Fell</a>, <a href="../authors/hayes">Ian J. Hayes</a> and <a href="../authors/velykis">Andrius Velykis</a></div>
	<span class="date">
	Dec 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Twelvefold_Way.html">The Twelvefold Way</a></h5> <br>by <a href="../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Dec 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Proof_Strategy_Language.html">Proof Strategy Language</a></h5> <br>by <a href="../authors/nagashima">Yutaka Nagashima</a></div>
	<span class="date">
	Dec 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Paraconsistency.html">Paraconsistency</a></h5> <br>by <a href="../authors/schlichtkrull">Anders Schlichtkrull</a> and <a href="../authors/villadsen">Jørgen Villadsen</a></div>
	<span class="date">
	Dec 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Complx.html">COMPLX: A Verification Framework for Concurrent Imperative Programs</a></h5> <br>by <a href="../authors/amani">Sidney Amani</a>, <a href="../authors/andronick">June Andronick</a>, <a href="../authors/bortin">Maksym Bortin</a>, <a href="../authors/lewis">Corey Lewis</a>, <a href="../authors/rizkallah">Christine Rizkallah</a> and <a href="../authors/tuongj">Joseph Tuong</a></div>
	<span class="date">
	Nov 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Abs_Int_ITP2012.html">Abstract Interpretation of Annotated Commands</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Nov 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Separata.html">Separata: Isabelle tactics for Separation Algebra</a></h5> <br>by <a href="../authors/hou">Zhe Hou</a>, <a href="../authors/sanan">David Sanan</a>, <a href="../authors/tiu">Alwen Tiu</a>, <a href="../authors/gore">Rajeev Gore</a> and <a href="../authors/clouston">Ranald Clouston</a></div>
	<span class="date">
	Nov 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Nested_Multisets_Ordinals.html">Formalization of Nested Multisets, Hereditary Multisets, and Syntactic Ordinals</a></h5> <br>by <a href="../authors/blanchette">Jasmin Christian Blanchette</a>, <a href="../authors/fleury">Mathias Fleury</a> and <a href="../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Nov 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lambda_Free_KBOs.html">Formalization of Knuth–Bendix Orders for Lambda-Free Higher-Order Terms</a></h5> <br>by <a href="../authors/becker">Heiko Becker</a>, <a href="../authors/blanchette">Jasmin Christian Blanchette</a>, <a href="../authors/waldmann">Uwe Waldmann</a> and <a href="../authors/wand">Daniel Wand</a></div>
	<span class="date">
	Nov 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Deep_Learning.html">Expressiveness of Deep Learning</a></h5> <br>by <a href="../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Modal_Logics_for_NTS.html">Modal Logics for Nominal Transition Systems</a></h5> <br>by <a href="../authors/weber">Tjark Weber</a>, <a href="../authors/eriksson">Lars-Henrik Eriksson</a>, <a href="../authors/parrow">Joachim Parrow</a>, <a href="../authors/borgstroem">Johannes Borgström</a> and <a href="../authors/gutkovas">Ramunas Gutkovas</a></div>
	<span class="date">
	Oct 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stable_Matching.html">Stable Matching</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Oct 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LOFT.html">LOFT — Verified Migration of Linux Firewalls to SDN</a></h5> <br>by <a href="../authors/michaelis">Julius Michaelis</a> and <a href="../authors/diekmann">Cornelius Diekmann</a></div>
	<span class="date">
	Oct 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Source_Coding_Theorem.html">Source Coding Theorem</a></h5> <br>by <a href="../authors/hibon">Quentin Hibon</a> and <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SPARCv8.html">A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</a></h5> <br>by <a href="../authors/hou">Zhe Hou</a>, <a href="../authors/sanan">David Sanan</a>, <a href="../authors/tiu">Alwen Tiu</a> and <a href="../authors/liuy">Yang Liu</a></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Berlekamp_Zassenhaus.html">The Factorization Algorithm of Berlekamp and Zassenhaus</a></h5> <br>by <a href="../authors/divason">Jose Divasón</a>, <a href="../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../authors/thiemann">René Thiemann</a> and <a href="../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Chord_Segments.html">Intersecting Chords Theorem</a></h5> <br>by <a href="../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Oct 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lp.html">Lp spaces</a></h5> <br>by <a href="../authors/gouezel">Sebastien Gouezel</a></div>
	<span class="date">
	Oct 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Fisher_Yates.html">Fisher–Yates shuffle</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Allen_Calculus.html">Allen's Interval Calculus</a></h5> <br>by <a href="../authors/ghourabi">Fadoua Ghourabi</a></div>
	<span class="date">
	Sep 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lambda_Free_RPOs.html">Formalization of Recursive Path Orders for Lambda-Free Higher-Order Terms</a></h5> <br>by <a href="../authors/blanchette">Jasmin Christian Blanchette</a>, <a href="../authors/waldmann">Uwe Waldmann</a> and <a href="../authors/wand">Daniel Wand</a></div>
	<span class="date">
	Sep 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Iptables_Semantics.html">Iptables Semantics</a></h5> <br>by <a href="../authors/diekmann">Cornelius Diekmann</a> and <a href="../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stone_Algebras.html">Stone Algebras</a></h5> <br>by <a href="../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SuperCalc.html">A Variant of the Superposition Calculus</a></h5> <br>by <a href="../authors/peltier">Nicolas Peltier</a></div>
	<span class="date">
	Sep 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stirling_Formula.html">Stirling's formula</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Routing.html">Routing</a></h5> <br>by <a href="../authors/michaelis">Julius Michaelis</a> and <a href="../authors/diekmann">Cornelius Diekmann</a></div>
	<span class="date">
	Aug 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Simple_Firewall.html">Simple Firewall</a></h5> <br>by <a href="../authors/diekmann">Cornelius Diekmann</a>, <a href="../authors/michaelis">Julius Michaelis</a> and <a href="../authors/haslbeck">Max W. Haslbeck</a></div>
	<span class="date">
	Aug 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/InfPathElimination.html">Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths</a></h5> <br>by <a href="../authors/aissat">Romain Aissat</a>, <a href="../authors/voisin">Frederic Voisin</a> and <a href="../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Aug 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/EdmondsKarp_Maxflow.html">Formalizing the Edmonds-Karp Algorithm</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Refine_Imperative_HOL.html">The Imperative Refinement Framework</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ptolemys_Theorem.html">Ptolemy's Theorem</a></h5> <br>by <a href="../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Aug 07
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	<span class="date">
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	<h2 class="head">2015</h2><article class="entry">
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	<span class="date">
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	<span class="date">
	Jul 14
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	<span class="date">
	Jul 07
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	<article class="entry">
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	<span class="date">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Noninterference_Generic_Unwinding.html">The Generic Unwinding Theorem for CSP Noninterference Security</a></h5> <br>by <a href="../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Jun 11
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	<article class="entry">
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	<span class="date">
	Jun 11
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	<article class="entry">
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	<h5><a class="title" href="../entries/Multirelations.html">Binary Multirelations</a></h5> <br>by <a href="../authors/furusawa">Hitoshi Furusawa</a> and <a href="../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 11
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	<article class="entry">
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	<h5><a class="title" href="../entries/Dynamic_Tables.html">Parameterized Dynamic Tables</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 07
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	<span class="date">
	May 28
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	<span class="date">
	May 27
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	<span class="date">
	Apr 30
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	<article class="entry">
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	<h5><a class="title" href="../entries/Residuated_Lattices.html">Residuated Lattices</a></h5> <br>by <a href="../authors/gomes">Victor B. F. Gomes</a> and <a href="../authors/struth">Georg Struth</a></div>
	<span class="date">
	Apr 15
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	<article class="entry">
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	<span class="date">
	Apr 13
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	<article class="entry">
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	<h5><a class="title" href="../entries/ConcurrentIMP.html">Concurrent IMP</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Apr 13
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Mar 30
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	<article class="entry">
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	<span class="date">
	Mar 18
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	<span class="date">
	Mar 11
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	</article>
	<article class="entry">
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	<span class="date">
	Feb 20
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	<article class="entry">
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	<span class="date">
	Feb 12
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Echelon_Form.html">Echelon Form</a></h5> <br>by <a href="../authors/divason">Jose Divasón</a> and <a href="../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
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	<article class="entry">
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	<h5><a class="title" href="../entries/Finite_Automata_HF.html">Finite Automata in Hereditarily Finite Set Theory</a></h5> <br>by <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Feb 05
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	</article>
	<article class="entry">
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	<h5><a class="title" href="../entries/UpDown_Scheme.html">Verification of the UpDown Scheme</a></h5> <br>by <a href="../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Jan 28
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	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/UPF.html">The Unified Policy Framework (UPF)</a></h5> <br>by <a href="../authors/brucker">Achim D. Brucker</a>, <a href="../authors/bruegger">Lukas Brügger</a> and <a href="../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
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	<article class="entry">
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	<span class="date">
	Oct 23
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Oct 13
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	<article class="entry">
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	<span class="date">
	Oct 10
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	<article class="entry">
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	<span class="date">
	Oct 09
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	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Oct 08
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Certification_Monads.html">Certification Monads</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a> and <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Oct 03
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Sep 25
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Sep 19
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Cayley_Hamilton.html">The Cayley-Hamilton Theorem</a></h5> <br>by <a href="../authors/adelsberger">Stephan Adelsberger</a>, <a href="../authors/hetzl">Stefan Hetzl</a> and <a href="../authors/pollak">Florian Pollak</a></div>
	<span class="date">
	Sep 15
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	<article class="entry">
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	<span class="date">
	Sep 09
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	</article>
	<article class="entry">
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	<span class="date">
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	<div class="item-text">
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	<span class="date">
	Sep 03
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Aug 29
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	</article>
	<article class="entry">
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	<span class="date">
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Aug 13
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Aug 12
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jul 29
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	</article>
	<article class="entry">
	<div class="item-text">
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	Jul 18
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jul 13
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jul 07
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jul 04
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Pop_Refinement.html">Pop-Refinement</a></h5> <br>by <a href="../authors/coglio">Alessandro Coglio</a></div>
	<span class="date">
	Jul 03
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Jun 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Boolean_Expression_Checkers.html">Boolean Expression Checkers</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	May 28
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CAVA_Automata.html">The CAVA Automata Library</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Promela.html">Promela Formalization</a></h5> <br>by <a href="../authors/neumann">René Neumann</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LTL_to_GBA.html">Converting Linear-Time Temporal Logic to Generalized Büchi Automata</a></h5> <br>by <a href="../authors/schimpf">Alexander Schimpf</a> and <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 28
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CAVA_LTL_Modelchecker.html">A Fully Verified Executable LTL Model Checker</a></h5> <br>by <a href="../authors/esparza">Javier Esparza</a>, <a href="../authors/lammich">Peter Lammich</a>, <a href="../authors/neumann">René Neumann</a>, <a href="../authors/nipkow">Tobias Nipkow</a>, <a href="../authors/schimpf">Alexander Schimpf</a> and <a href="../authors/smaus">Jan-Georg Smaus</a></div>
	<span class="date">
	May 28
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Roy_Floyd_Warshall.html">Transitive closure according to Roy-Floyd-Warshall</a></h5> <br>by <a href="../authors/wenzel">Makarius Wenzel</a></div>
	<span class="date">
	May 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Noninterference_CSP.html">Noninterference Security in Communicating Sequential Processes</a></h5> <br>by <a href="../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	May 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Regular_Algebras.html">Regular Algebras</a></h5> <br>by <a href="../authors/fosters">Simon Foster</a> and <a href="../authors/struth">Georg Struth</a></div>
	<span class="date">
	May 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/ComponentDependencies.html">Formalisation and Analysis of Component Dependencies</a></h5> <br>by <a href="../authors/spichkova">Maria Spichkova</a></div>
	<span class="date">
	Apr 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Strong_Security.html">A Formalization of Strong Security</a></h5> <br>by <a href="../authors/grewe">Sylvia Grewe</a>, <a href="../authors/lux">Alexander Lux</a>, <a href="../authors/mantel">Heiko Mantel</a> and <a href="../authors/sauer">Jens Sauer</a></div>
	<span class="date">
	Apr 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/WHATandWHERE_Security.html">A Formalization of Declassification with WHAT-and-WHERE-Security</a></h5> <br>by <a href="../authors/grewe">Sylvia Grewe</a>, <a href="../authors/lux">Alexander Lux</a>, <a href="../authors/mantel">Heiko Mantel</a> and <a href="../authors/sauer">Jens Sauer</a></div>
	<span class="date">
	Apr 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SIFUM_Type_Systems.html">A Formalization of Assumptions and Guarantees for Compositional Noninterference</a></h5> <br>by <a href="../authors/grewe">Sylvia Grewe</a>, <a href="../authors/mantel">Heiko Mantel</a> and <a href="../authors/schoepe">Daniel Schoepe</a></div>
	<span class="date">
	Apr 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Bounded_Deducibility_Security.html">Bounded-Deducibility Security</a></h5> <br>by <a href="../authors/popescu">Andrei Popescu</a>, <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/bauereiss">Thomas Bauereiss</a></div>
	<span class="date">
	Apr 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Apr 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/HyperCTL.html">A shallow embedding of HyperCTL*</a></h5> <br>by <a href="../authors/rabe">Markus N. Rabe</a>, <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/popescu">Andrei Popescu</a></div>
	<span class="date">
	Apr 16
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Discrete_Summation.html">Discrete Summation</a></h5> <br>by <a href="../authors/haftmann">Florian Haftmann</a></div>
	<span class="date">
	Apr 13
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	</article>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Apr 03
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Probabilistic_Noninterference.html">Probabilistic Noninterference</a></h5> <br>by <a href="../authors/popescu">Andrei Popescu</a> and <a href="../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Mar 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/AWN.html">Mechanization of the Algebra for Wireless Networks (AWN)</a></h5> <br>by <a href="../authors/bourke">Timothy Bourke</a></div>
	<span class="date">
	Mar 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Partial_Function_MR.html">Mutually Recursive Partial Functions</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Random_Graph_Subgraph_Threshold.html">Properties of Random Graphs -- Subgraph Containment</a></h5> <br>by <a href="../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Feb 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Selection_Heap_Sort.html">Verification of Selection and Heap Sort Using Locales</a></h5> <br>by <a href="../authors/petrovic">Danijela Petrovic</a></div>
	<span class="date">
	Feb 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Affine_Arithmetic.html">Affine Arithmetic</a></h5> <br>by <a href="../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Real_Impl.html">Implementing field extensions of the form Q[sqrt(b)]</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Regex_Equivalence.html">Unified Decision Procedures for Regular Expression Equivalence</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a> and <a href="../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Secondary_Sylow.html">Secondary Sylow Theorems</a></h5> <br>by <a href="../authors/raumer">Jakob von Raumer</a></div>
	<span class="date">
	Jan 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Relation_Algebra.html">Relation Algebra</a></h5> <br>by <a href="../authors/armstrong">Alasdair Armstrong</a>, <a href="../authors/fosters">Simon Foster</a>, <a href="../authors/struth">Georg Struth</a> and <a href="../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../authors/armstrong">Alasdair Armstrong</a>, <a href="../authors/gomes">Victor B. F. Gomes</a> and <a href="../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Featherweight_OCL.html">Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5</a></h5> <br>by <a href="../authors/brucker">Achim D. Brucker</a>, <a href="../authors/tuong">Frédéric Tuong</a> and <a href="../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Sturm_Sequences.html">Sturm's Theorem</a></h5> <br>by <a href="../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CryptoBasedCompositionalProperties.html">Compositional Properties of Crypto-Based Components</a></h5> <br>by <a href="../authors/spichkova">Maria Spichkova</a></div>
	<span class="date">
	Jan 11
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Tail_Recursive_Functions.html">A General Method for the Proof of Theorems on Tail-recursive Functions</a></h5> <br>by <a href="../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Dec 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/HereditarilyFinite.html">The Hereditarily Finite Sets</a></h5> <br>by <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Incompleteness.html">Gödel's Incompleteness Theorems</a></h5> <br>by <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Coinductive_Languages.html">A Codatatype of Formal Languages</a></h5> <br>by <a href="../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Nov 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FocusStreamsCaseStudies.html">Stream Processing Components: Isabelle/HOL Formalisation and Case Studies</a></h5> <br>by <a href="../authors/spichkova">Maria Spichkova</a></div>
	<span class="date">
	Nov 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/GoedelGod.html">Gödel's God in Isabelle/HOL</a></h5> <br>by <a href="../authors/benzmueller">Christoph Benzmüller</a> and <a href="../authors/paleo">Bruno Woltzenlogel Paleo</a></div>
	<span class="date">
	Nov 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Decreasing-Diagrams.html">Decreasing Diagrams</a></h5> <br>by <a href="../authors/zankl">Harald Zankl</a></div>
	<span class="date">
	Nov 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Automatic_Refinement.html">Automatic Data Refinement</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Native_Word.html">Native Word</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Sep 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/IEEE_Floating_Point.html">A Formal Model of IEEE Floating Point Arithmetic</a></h5> <br>by <a href="../authors/yu">Lei Yu</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Pratt_Certificate.html">Pratt's Primality Certificates</a></h5> <br>by <a href="../authors/wimmer">Simon Wimmer</a> and <a href="../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lehmer.html">Lehmer's Theorem</a></h5> <br>by <a href="../authors/wimmer">Simon Wimmer</a> and <a href="../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Koenigsberg_Friendship.html">The Königsberg Bridge Problem and the Friendship Theorem</a></h5> <br>by <a href="../authors/li">Wenda Li</a></div>
	<span class="date">
	Jul 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Sort_Encodings.html">Sound and Complete Sort Encodings for First-Order Logic</a></h5> <br>by <a href="../authors/blanchette">Jasmin Christian Blanchette</a> and <a href="../authors/popescu">Andrei Popescu</a></div>
	<span class="date">
	Jun 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/ShortestPath.html">An Axiomatic Characterization of the Single-Source Shortest Path Problem</a></h5> <br>by <a href="../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Graph_Theory.html">Graph Theory</a></h5> <br>by <a href="../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Apr 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Containers.html">Light-weight Containers</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Nominal2.html">Nominal 2</a></h5> <br>by <a href="../authors/urban">Christian Urban</a>, <a href="../authors/berghofer">Stefan Berghofer</a> and <a href="../authors/kaliszyk">Cezary Kaliszyk</a></div>
	<span class="date">
	Feb 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Launchbury.html">The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</a></h5> <br>by <a href="../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jan 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ribbon_Proofs.html">Ribbon Proofs</a></h5> <br>by <a href="../authors/wickerson">John Wickerson</a></div>
	<span class="date">
	Jan 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Rank_Nullity_Theorem.html">Rank-Nullity Theorem in Linear Algebra</a></h5> <br>by <a href="../authors/divason">Jose Divasón</a> and <a href="../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../authors/armstrong">Alasdair Armstrong</a>, <a href="../authors/struth">Georg Struth</a> and <a href="../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Sqrt_Babylonian.html">Computing N-th Roots using the Babylonian Method</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Separation_Logic_Imperative_HOL.html">A Separation Logic Framework for Imperative HOL</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/meis">Rene Meis</a></div>
	<span class="date">
	Nov 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Open_Induction.html">Open Induction</a></h5> <br>by <a href="../authors/ogawa">Mizuhito Ogawa</a> and <a href="../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Tarskis_Geometry.html">The independence of Tarski's Euclidean axiom</a></h5> <br>by <a href="../authors/makarios">T. J. M. Makarios</a></div>
	<span class="date">
	Oct 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Bondy.html">Bondy's Theorem</a></h5> <br>by <a href="../authors/avigad">Jeremy Avigad</a> and <a href="../authors/hetzl">Stefan Hetzl</a></div>
	<span class="date">
	Oct 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Possibilistic_Noninterference.html">Possibilistic Noninterference</a></h5> <br>by <a href="../authors/popescu">Andrei Popescu</a> and <a href="../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Datatype_Order_Generator.html">Generating linear orders for datatypes</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Aug 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Impossible_Geometry.html">Proving the Impossibility of Trisecting an Angle and Doubling the Cube</a></h5> <br>by <a href="../authors/romanos">Ralph Romanos</a> and <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Heard_Of.html">Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</a></h5> <br>by <a href="../authors/debrat">Henri Debrat</a> and <a href="../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/PCF.html">Logical Relations for PCF</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Jul 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Tycon.html">Type Constructor Classes and Monad Transformers</a></h5> <br>by <a href="../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Jun 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Pi_Calculus.html">The pi-calculus in nominal logic</a></h5> <br>by <a href="../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Psi_Calculi.html">Psi-calculi in Isabelle</a></h5> <br>by <a href="../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CCS.html">CCS in nominal logic</a></h5> <br>by <a href="../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Circus.html">Isabelle/Circus</a></h5> <br>by <a href="../authors/feliachi">Abderrahmane Feliachi</a>, <a href="../authors/wolff">Burkhart Wolff</a> and <a href="../authors/gaudel">Marie-Claude Gaudel</a></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Separation_Algebra.html">Separation Algebra</a></h5> <br>by <a href="../authors/klein">Gerwin Klein</a>, <a href="../authors/kolanski">Rafal Kolanski</a> and <a href="../authors/boyton">Andrew Boyton</a></div>
	<span class="date">
	May 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stuttering_Equivalence.html">Stuttering Equivalence</a></h5> <br>by <a href="../authors/merz">Stephan Merz</a></div>
	<span class="date">
	May 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Inductive_Confidentiality.html">Inductive Study of Confidentiality</a></h5> <br>by <a href="../authors/bella">Giampaolo Bella</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ordinary_Differential_Equations.html">Ordinary Differential Equations</a></h5> <br>by <a href="../authors/immler">Fabian Immler</a> and <a href="../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Well_Quasi_Orders.html">Well-Quasi-Orders</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Abortable_Linearizable_Modules.html">Abortable Linearizable Modules</a></h5> <br>by <a href="../authors/guerraoui">Rachid Guerraoui</a>, <a href="../authors/kuncak">Viktor Kuncak</a> and <a href="../authors/losa">Giuliano Losa</a></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Transitive-Closure-II.html">Executable Transitive Closures</a></h5> <br>by <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Girth_Chromatic.html">A Probabilistic Proof of the Girth-Chromatic Number Theorem</a></h5> <br>by <a href="../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Refine_Monadic.html">Refinement for Monadic Programs</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Dijkstra_Shortest_Path.html">Dijkstra's Shortest Path Algorithm</a></h5> <br>by <a href="../authors/nordhoff">Benedikt Nordhoff</a> and <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Markov_Models.html">Markov Models</a></h5> <br>by <a href="../authors/hoelzl">Johannes Hölzl</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/TLA.html">A Definitional Encoding of TLA* in Isabelle/HOL</a></h5> <br>by <a href="../authors/grov">Gudmund Grov</a> and <a href="../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Efficient-Mergesort.html">Efficient Mergesort</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/PseudoHoops.html">Pseudo Hoops</a></h5> <br>by <a href="../authors/georgescu">George Georgescu</a>, <a href="../authors/leustean">Laurentiu Leustean</a> and <a href="../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LatticeProperties.html">Lattice Properties</a></h5> <br>by <a href="../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/MonoBoolTranAlgebra.html">Algebra of Monotonic Boolean Transformers</a></h5> <br>by <a href="../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Myhill-Nerode.html">The Myhill-Nerode Theorem Based on Regular Expressions</a></h5> <br>by <a href="../authors/wu">Chunhan Wu</a>, <a href="../authors/zhangx">Xingyuan Zhang</a> and <a href="../authors/urban">Christian Urban</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Max-Card-Matching.html">Maximum Cardinality Matching</a></h5> <br>by <a href="../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jul 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/KBPs.html">Knowledge-based programs</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	May 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/General-Triangle.html">The General Triangle Is Unique</a></h5> <br>by <a href="../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Apr 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Transitive-Closure.html">Executable Transitive Closures of Finite Relations</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a> and <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Nat-Interval-Logic.html">Interval Temporal Logic on Natural Numbers</a></h5> <br>by <a href="../authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/List-Infinite.html">Infinite Lists</a></h5> <br>by <a href="../authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/AutoFocus-Stream.html">AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</a></h5> <br>by <a href="../authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LightweightJava.html">Lightweight Java</a></h5> <br>by <a href="../authors/strnisa">Rok Strniša</a> and <a href="../authors/parkinson">Matthew Parkinson</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/RIPEMD-160-SPARK.html">RIPEMD-160</a></h5> <br>by <a href="../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jan 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lower_Semicontinuous.html">Lower Semicontinuous Functions</a></h5> <br>by <a href="../authors/grechuk">Bogdan Grechuk</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Marriage.html">Hall's Marriage Theorem</a></h5> <br>by <a href="../authors/jiangd">Dongchen Jiang</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Dec 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Shivers-CFA.html">Shivers' Control Flow Analysis</a></h5> <br>by <a href="../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Nov 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Binomial-Queues.html">Functional Binomial Queues</a></h5> <br>by <a href="../authors/neumann">René Neumann</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Finger-Trees.html">Finger Trees</a></h5> <br>by <a href="../authors/nordhoff">Benedikt Nordhoff</a>, <a href="../authors/koerner">Stefan Körner</a> and <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Binomial-Heaps.html">Binomial Heaps and Skew Binomial Heaps</a></h5> <br>by <a href="../authors/meis">Rene Meis</a>, <a href="../authors/nielsen">Finn Nielsen</a> and <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lam-ml-Normalization.html">Strong Normalization of Moggis's Computational Metalanguage</a></h5> <br>by <a href="../authors/doczkal">Christian Doczkal</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a>, <a href="../authors/thiemann">René Thiemann</a>, <a href="../authors/maletzky">Alexander Maletzky</a>, <a href="../authors/immler">Fabian Immler</a>, <a href="../authors/haftmann">Florian Haftmann</a>, <a href="../authors/lochbihler">Andreas Lochbihler</a> and <a href="../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Statecharts.html">Formalizing Statecharts using Hierarchical Automata</a></h5> <br>by <a href="../authors/helke">Steffen Helke</a> and <a href="../authors/kammueller">Florian Kammüller</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Free-Groups.html">Free Groups</a></h5> <br>by <a href="../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Category2.html">Category Theory</a></h5> <br>by <a href="../authors/katovsky">Alexander Katovsky</a></div>
	<span class="date">
	Jun 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Matrix.html">Executable Matrix Operations on Matrices of Arbitrary Dimensions</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a> and <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Abstract-Rewriting.html">Abstract Rewriting</a></h5> <br>by <a href="../authors/sternagel">Christian Sternagel</a> and <a href="../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/GraphMarkingIBP.html">Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</a></h5> <br>by <a href="../authors/preoteasa">Viorel Preoteasa</a> and <a href="../authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/DataRefinementIBP.html">Semantics and Data Refinement of Invariant Based Programs</a></h5> <br>by <a href="../authors/preoteasa">Viorel Preoteasa</a> and <a href="../authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Robbins-Conjecture.html">A Complete Proof of the Robbins Conjecture</a></h5> <br>by <a href="../authors/doty">Matthew Doty</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Regular-Sets.html">Regular Sets and Expressions</a></h5> <br>by <a href="../authors/krauss">Alexander Krauss</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Locally-Nameless-Sigma.html">Locally Nameless Sigma Calculus</a></h5> <br>by <a href="../authors/henrio">Ludovic Henrio</a>, <a href="../authors/kammueller">Florian Kammüller</a>, <a href="../authors/lutz">Bianca Lutz</a> and <a href="../authors/sudhof">Henry Sudhof</a></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Free-Boolean-Algebra.html">Free Boolean Algebra</a></h5> <br>by <a href="../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Mar 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/InformationFlowSlicing_Inter.html">Inter-Procedural Information Flow Noninterference via Slicing</a></h5> <br>by <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/InformationFlowSlicing.html">Information Flow Noninterference via Slicing</a></h5> <br>by <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/List-Index.html">List Index</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Coinductive.html">Coinductive</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/DPT-SAT-Solver.html">A Fast SAT Solver for Isabelle in Standard ML</a></h5> <br>by <a href="../authors/heller">Armin Heller</a></div>
	<span class="date">
	Dec 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Presburger-Automata.html">Formalizing the Logic-Automaton Connection</a></h5> <br>by <a href="../authors/berghofer">Stefan Berghofer</a> and <a href="../authors/reiter">Markus Reiter</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Tree-Automata.html">Tree Automata</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Collections.html">Collections Framework</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Perfect-Number-Thm.html">Perfect Number Theorem</a></h5> <br>by <a href="../authors/ijbema">Mark Ijbema</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/HRB-Slicing.html">Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</a></h5> <br>by <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Nov 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/WorkerWrapper.html">The Worker/Wrapper Transformation</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Oct 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ordinals_and_Cardinals.html">Ordinals and Cardinals</a></h5> <br>by <a href="../authors/popescu">Andrei Popescu</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SequentInvertibility.html">Invertibility in Sequent Calculi</a></h5> <br>by <a href="../authors/chapman">Peter Chapman</a></div>
	<span class="date">
	Aug 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CofGroups.html">An Example of a Cofinitary Group in Isabelle/HOL</a></h5> <br>by <a href="../authors/kastermans">Bart Kastermans</a></div>
	<span class="date">
	Aug 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FinFun.html">Code Generation for Functions as Data</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Stream-Fusion.html">Stream Fusion</a></h5> <br>by <a href="../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Apr 29
	</span>
	</article>


	<h2 class="head">2008</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/BytecodeLogicJmlTypes.html">A Bytecode Logic for JML and Types</a></h5> <br>by <a href="../authors/beringer">Lennart Beringer</a> and <a href="../authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Dec 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SIFPL.html">Secure information flow and program logics</a></h5> <br>by <a href="../authors/beringer">Lennart Beringer</a> and <a href="../authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SenSocialChoice.html">Some classical results in Social Choice Theory</a></h5> <br>by <a href="../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FunWithTilings.html">Fun With Tilings</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a> and <a href="../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Huffman.html">The Textbook Proof of Huffman's Algorithm</a></h5> <br>by <a href="../authors/blanchette">Jasmin Christian Blanchette</a></div>
	<span class="date">
	Oct 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Slicing.html">Towards Certified Slicing</a></h5> <br>by <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/VolpanoSmith.html">A Correctness Proof for the Volpano/Smith Security Typing System</a></h5> <br>by <a href="../authors/snelting">Gregor Snelting</a> and <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/ArrowImpossibilityGS.html">Arrow and Gibbard-Satterthwaite</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FunWithFunctions.html">Fun With Functions</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SATSolverVerification.html">Formal Verification of Modern SAT Solvers</a></h5> <br>by <a href="../authors/maricf">Filip Marić</a></div>
	<span class="date">
	Jul 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Recursion-Theory-I.html">Recursion Theory I</a></h5> <br>by <a href="../authors/nedzelsky">Michael Nedzelsky</a></div>
	<span class="date">
	Apr 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/BDD.html">BDD Normalisation</a></h5> <br>by <a href="../authors/ortner">Veronika Ortner</a> and <a href="../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Simpl.html">A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</a></h5> <br>by <a href="../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/NormByEval.html">Normalization by Evaluation</a></h5> <br>by <a href="../authors/aehlig">Klaus Aehlig</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/LinearQuantifierElim.html">Quantifier Elimination for Linear Arithmetic</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 11
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Program-Conflict-Analysis.html">Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors</a></h5> <br>by <a href="../authors/lammich">Peter Lammich</a> and <a href="../authors/olm">Markus Müller-Olm</a></div>
	<span class="date">
	Dec 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/JinjaThreads.html">Jinja with Threads</a></h5> <br>by <a href="../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/MuchAdoAboutTwo.html">Much Ado About Two</a></h5> <br>by <a href="../authors/boehme">Sascha Böhme</a></div>
	<span class="date">
	Nov 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/SumSquares.html">Sums of Two and Four Squares</a></h5> <br>by <a href="../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Fermat3_4.html">Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</a></h5> <br>by <a href="../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Valuation.html">Fundamental Properties of Valuation Theory and Hensel's Lemma</a></h5> <br>by <a href="../authors/kobayashi">Hidetsune Kobayashi</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/POPLmark-deBruijn.html">POPLmark Challenge Via de Bruijn Indices</a></h5> <br>by <a href="../authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Aug 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FOL-Fitting.html">First-Order Logic According to Fitting</a></h5> <br>by <a href="../authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Aug 02
	</span>
	</article>


	<h2 class="head">2006</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/HotelKeyCards.html">Hotel Key Card System</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Abstract-Hoare-Logics.html">Abstract Hoare Logics</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Flyspeck-Tame.html">Flyspeck I: Tame Graphs</a></h5> <br>by <a href="../authors/bauer">Gertrud Bauer</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/CoreC++.html">CoreC++</a></h5> <br>by <a href="../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	May 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FeatherweightJava.html">A Theory of Featherweight Java in Isabelle/HOL</a></h5> <br>by <a href="../authors/fosterj">J. Nathan Foster</a> and <a href="../authors/vytiniotis">Dimitrios Vytiniotis</a></div>
	<span class="date">
	Mar 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/ClockSynchInst.html">Instances of Schneider's generalized protocol of clock synchronization</a></h5> <br>by <a href="../authors/barsotti">Damián Barsotti</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Cauchy.html">Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality</a></h5> <br>by <a href="../authors/porter">Benjamin Porter</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>


	<h2 class="head">2005</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ordinal.html">Countable Ordinals</a></h5> <br>by <a href="../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FFT.html">Fast Fourier Transform</a></h5> <br>by <a href="../authors/ballarin">Clemens Ballarin</a></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/GenClock.html">Formalization of a Generalized Protocol for Clock Synchronization</a></h5> <br>by <a href="../authors/tiu">Alwen Tiu</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/DiskPaxos.html">Proving the Correctness of Disk Paxos</a></h5> <br>by <a href="../authors/jaskelioff">Mauro Jaskelioff</a> and <a href="../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jun 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/JiveDataStoreModel.html">Jive Data and Store Model</a></h5> <br>by <a href="../authors/rauch">Nicole Rauch</a> and <a href="../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Jun 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Jinja.html">Jinja is not Java</a></h5> <br>by <a href="../authors/klein">Gerwin Klein</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/RSAPSS.html">SHA1, RSA, PSS and more</a></h5> <br>by <a href="../authors/lindenberg">Christina Lindenberg</a> and <a href="../authors/wirt">Kai Wirt</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Category.html">Category Theory to Yoneda's Lemma</a></h5> <br>by <a href="../authors/keefe">Greg O’Keefe</a></div>
	<span class="date">
	Apr 21
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/FileRefinement.html">File Refinement</a></h5> <br>by <a href="../authors/zee">Karen Zee</a> and <a href="../authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Dec 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Integration.html">Integration theory and random variables</a></h5> <br>by <a href="../authors/richter">Stefan Richter</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Verified-Prover.html">A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic</a></h5> <br>by <a href="../authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Ramsey-Infinite.html">Ramsey's theorem, infinitary version</a></h5> <br>by <a href="../authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Completeness.html">Completeness theorem</a></h5> <br>by <a href="../authors/margetson">James Margetson</a> and <a href="../authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Compiling-Exceptions-Correctly.html">Compiling Exceptions Correctly</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jul 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Depth-First-Search.html">Depth First Search</a></h5> <br>by <a href="../authors/nishihara">Toshiaki Nishihara</a> and <a href="../authors/minamide">Yasuhiko Minamide</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Group-Ring-Module.html">Groups, Rings and Modules</a></h5> <br>by <a href="../authors/kobayashi">Hidetsune Kobayashi</a>, <a href="../authors/chen">L. Chen</a> and <a href="../authors/murao">H. Murao</a></div>
	<span class="date">
	May 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Topology.html">Topology</a></h5> <br>by <a href="../authors/friedrich">Stefan Friedrich</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Lazy-Lists-II.html">Lazy Lists II</a></h5> <br>by <a href="../authors/friedrich">Stefan Friedrich</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/BinarySearchTree.html">Binary Search Trees</a></h5> <br>by <a href="../authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Apr 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/Functional-Automata.html">Functional Automata</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Mar 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/MiniML.html">Mini ML</a></h5> <br>by <a href="../authors/naraschewski">Wolfgang Naraschewski</a> and <a href="../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Mar 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../entries/AVL-Trees.html">AVL Trees</a></h5> <br>by <a href="../authors/nipkow">Tobias Nipkow</a> and <a href="../authors/pusch">Cornelia Pusch</a></div>
	<span class="date">
	Mar 19
	</span>
	</article>




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	<title>A Verified Efficient Implementation of the Weighted Path Order</title>
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	<title>Formalizing Results on Directed Sets</title>
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	<title>Inner Structure, Determinism and Modal Algebra of Multirelations</title>
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	<title>Tree Enumeration</title>
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	<title>MLSS Decision Procedure</title>
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	<title>The Halting Problem is Soluble in Malament-Hogarth Spacetimes</title>
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	<title>The Schwartz-Zippel Lemma</title>
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	<title>A Formalization of the SCL(FOL) Calculus: Simple Clause Learning for First-Order Logic</title>
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	<title>The CHSH inequality: Tsirelson's upper-bound and other results</title>
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	<title>Formalization of Hyper Hoare Logic: A Logic to (Dis-)Prove Program Hyperproperties</title>
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	<title>Formalization of CommCSL: A Relational Concurrent Separation Logic for Proving Information Flow Security in Concurrent Programs</title>
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	<title>No Faster-Than-Light Observers (GenRel)</title>
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	<title>Hardness of Lattice Problems</title>
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	<title>ABY3 Multiplication and Array Shuffling</title>
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	<title>Suppes' Theorem For Probability Logic</title>
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	<title>Strict Omega Categories</title>
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	<title>The Cook-Levin theorem</title>
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	<title>Binary codes that do not preserve primitivity</title>
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	<title>Intersection of two monoids generated by two element codes</title>
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	<title>A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL</title>
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	<title>A Formal CHERI-C Memory Model</title>
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	<item>
	<title>Sauer-Shelah Lemma</title>
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	<item>
	<title>Kneser's Theorem and the Cauchy–Davenport Theorem</title>
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	<title>Turán's Graph Theorem</title>
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	<item>
	<title>Combinatorial Enumeration Algorithms</title>
	<link>/entries/Combinatorial_Enumeration_Algorithms.html</link>
	<pubDate>Fri, 11 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorial_Enumeration_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Balog–Szemerédi–Gowers Theorem</title>
	<link>/entries/Balog_Szemeredi_Gowers.html</link>
	<pubDate>Fri, 11 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Balog_Szemeredi_Gowers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of Strategy-Proofness and Representation in Party-Approval Multi-Winner Elections</title>
	<link>/entries/PAPP_Impossibility.html</link>
	<pubDate>Thu, 10 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/PAPP_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of Query Optimization Algorithms</title>
	<link>/entries/Query_Optimization.html</link>
	<pubDate>Tue, 04 Oct 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Query_Optimization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum Segment Sum</title>
	<link>/entries/Maximum_Segment_Sum.html</link>
	<pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Maximum_Segment_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Undirected Graph Theory</title>
	<link>/entries/Undirected_Graph_Theory.html</link>
	<pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Undirected_Graph_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Making Arbitrary Relational Calculus Queries Safe-Range</title>
	<link>/entries/Safe_Range_RC.html</link>
	<pubDate>Wed, 28 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Range_RC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stalnaker's Epistemic Logic</title>
	<link>/entries/Stalnaker_Logic.html</link>
	<pubDate>Fri, 23 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Stalnaker_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>p-adic Fields and p-adic Semialgebraic Sets</title>
	<link>/entries/Padic_Field.html</link>
	<pubDate>Thu, 22 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Padic_Field.html</guid>
	<description></description>
	</item>

	<item>
	<title>Risk-Free Lending</title>
	<link>/entries/Risk_Free_Lending.html</link>
	<pubDate>Sun, 18 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Risk_Free_Lending.html</guid>
	<description></description>
	</item>

	<item>
	<title>Soundness and Completeness of Implicational Logic</title>
	<link>/entries/Implicational_Logic.html</link>
	<pubDate>Tue, 13 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Implicational_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>CRYSTALS-Kyber</title>
	<link>/entries/CRYSTALS-Kyber.html</link>
	<pubDate>Thu, 08 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/CRYSTALS-Kyber.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unbounded Separation Logic</title>
	<link>/entries/Separation_Logic_Unbounded.html</link>
	<pubDate>Mon, 05 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Unbounded.html</guid>
	<description></description>
	</item>

	<item>
	<title>Khovanskii&#x27;s Theorem</title>
	<link>/entries/Khovanskii_Theorem.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Khovanskii_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hales–Jewett Theorem</title>
	<link>/entries/Hales_Jewett.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Hales_Jewett.html</guid>
	<description></description>
	</item>

	<item>
	<title>Number Theoretic Transform</title>
	<link>/entries/Number_Theoretic_Transform.html</link>
	<pubDate>Thu, 18 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Number_Theoretic_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Correctness of a Set-based Algorithm for Computing Strongly Connected Components of a Graph</title>
	<link>/entries/SCC_Bloemen_Sequential.html</link>
	<pubDate>Wed, 17 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/SCC_Bloemen_Sequential.html</guid>
	<description></description>
	</item>

	<item>
	<title>From THE BOOK: Two Squares via Involutions</title>
	<link>/entries/Involutions2Squares.html</link>
	<pubDate>Mon, 15 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Involutions2Squares.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Complete Test Strategies for Finite State Machines</title>
	<link>/entries/FSM_Tests.html</link>
	<pubDate>Tue, 09 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FSM_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Nano JSON: Working with JSON formatted data in Isabelle/HOL and Isabelle/ML</title>
	<link>/entries/Nano_JSON.html</link>
	<pubDate>Fri, 29 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Nano_JSON.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL</title>
	<link>/entries/Solidity.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Solidity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simultaneous diagonalization of pairwise commuting Hermitian matrices</title>
	<link>/entries/Commuting_Hermitian.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Commuting_Hermitian.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pólya’s Proof of the Weighted Arithmetic–Geometric Mean Inequality</title>
	<link>/entries/Weighted_Arithmetic_Geometric_Mean.html</link>
	<pubDate>Mon, 11 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Weighted_Arithmetic_Geometric_Mean.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reuse-Based Multi-Stage Compiler Verification for Language IMP</title>
	<link>/entries/IMP_Compiler_Reuse.html</link>
	<pubDate>Sun, 10 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler_Reuse.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Time Double-Ended Queue</title>
	<link>/entries/Real_Time_Deque.html</link>
	<pubDate>Thu, 23 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Time_Deque.html</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos's Curious Inference in Isabelle/HOL</title>
	<link>/entries/Boolos_Curious_Inference.html</link>
	<pubDate>Mon, 20 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Boolos_Curious_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Fields</title>
	<link>/entries/Finite_Fields.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Fields.html</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet: Formalization of a Verification Framework for Secure Data Plane Protocols</title>
	<link>/entries/IsaNet.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IsaNet.html</guid>
	<description></description>
	</item>

	<item>
	<title>Diophantine Equations and the DPRM Theorem</title>
	<link>/entries/DPRM_Theorem.html</link>
	<pubDate>Mon, 06 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/DPRM_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Reducing Rewrite Properties to Properties on Ground Terms</title>
	<link>/entries/Rewrite_Properties_Reduction.html</link>
	<pubDate>Thu, 02 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Rewrite_Properties_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Restricted Definition of the Magic Wand to Soundly Combine Fractions of a Wand</title>
	<link>/entries/Combinable_Wands.html</link>
	<pubDate>Mon, 30 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Combinable_Wands.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Plünnecke-Ruzsa Inequality</title>
	<link>/entries/Pluennecke_Ruzsa_Inequality.html</link>
	<pubDate>Thu, 26 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Pluennecke_Ruzsa_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Framework for the Sound Automation of Magic Wands</title>
	<link>/entries/Package_logic.html</link>
	<pubDate>Wed, 18 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Package_logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clique is not solvable by monotone circuits of polynomial size</title>
	<link>/entries/Clique_and_Monotone_Circuits.html</link>
	<pubDate>Sun, 08 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Clique_and_Monotone_Circuits.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</title>
	<link>/entries/Fishers_Inequality.html</link>
	<pubDate>Thu, 21 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Fishers_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Digit Expansions</title>
	<link>/entries/Digit_Expansions.html</link>
	<pubDate>Wed, 20 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Digit_Expansions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generalized Multiset Ordering is NP-Complete</title>
	<link>/entries/Multiset_Ordering_NPC.html</link>
	<pubDate>Wed, 20 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Multiset_Ordering_NPC.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sophomore's Dream</title>
	<link>/entries/Sophomores_Dream.html</link>
	<pubDate>Sun, 10 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Sophomores_Dream.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Combinator Library for Prefix-Free Codes</title>
	<link>/entries/Prefix_Free_Code_Combinators.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Prefix_Free_Code_Combinators.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Randomized Approximation Algorithms for Frequency Moments</title>
	<link>/entries/Frequency_Moments.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Frequency_Moments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructing the Reals as Dedekind Cuts of Rationals</title>
	<link>/entries/Dedekind_Real.html</link>
	<pubDate>Thu, 24 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Dedekind_Real.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ackermann's Function Is Not Primitive Recursive</title>
	<link>/entries/Ackermanns_not_PR.html</link>
	<pubDate>Wed, 23 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Ackermanns_not_PR.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Naive Prover for First-Order Logic</title>
	<link>/entries/FOL_Seq_Calc3.html</link>
	<pubDate>Tue, 22 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc3.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent</title>
	<link>/entries/Cotangent_PFD_Formula.html</link>
	<pubDate>Tue, 15 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Cotangent_PFD_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Independence of the Continuum Hypothesis in Isabelle/ZF</title>
	<link>/entries/Independence_CH.html</link>
	<pubDate>Sun, 06 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Independence_CH.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive Models of Fragments of ZFC</title>
	<link>/entries/Transitive_Models.html</link>
	<pubDate>Thu, 03 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Transition Systems</title>
	<link>/entries/ResiduatedTransitionSystem.html</link>
	<pubDate>Mon, 28 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/ResiduatedTransitionSystem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Hash Families</title>
	<link>/entries/Universal_Hash_Families.html</link>
	<pubDate>Sun, 20 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Hash_Families.html</guid>
	<description></description>
	</item>

	<item>
	<title>Wetzel's Problem and the Continuum Hypothesis</title>
	<link>/entries/Wetzels_Problem.html</link>
	<pubDate>Fri, 18 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Wetzels_Problem.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Query Evaluation</title>
	<link>/entries/Eval_FO.html</link>
	<pubDate>Tue, 15 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Eval_FO.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Head Monitoring of Metric Dynamic Logic</title>
	<link>/entries/VYDRA_MDL.html</link>
	<pubDate>Sun, 13 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/VYDRA_MDL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Enumeration of Equivalence Relations</title>
	<link>/entries/Equivalence_Relation_Enumeration.html</link>
	<pubDate>Fri, 04 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Equivalence_Relation_Enumeration.html</guid>
	<description></description>
	</item>

	<item>
	<title>Duality of Linear Programming</title>
	<link>/entries/LP_Duality.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/LP_Duality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi-Borel Spaces</title>
	<link>/entries/Quasi_Borel_Spaces.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Quasi_Borel_Spaces.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Theory of Rewriting</title>
	<link>/entries/FO_Theory_Rewriting.html</link>
	<pubDate>Wed, 02 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FO_Theory_Rewriting.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sequent Calculus Prover for First-Order Logic with Functions</title>
	<link>/entries/FOL_Seq_Calc2.html</link>
	<pubDate>Mon, 31 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc2.html</guid>
	<description></description>
	</item>

	<item>
	<title>Young's Inequality for Increasing Functions</title>
	<link>/entries/Youngs_Inequality.html</link>
	<pubDate>Mon, 31 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Youngs_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interpolation Polynomials (in HOL-Algebra)</title>
	<link>/entries/Interpolation_Polynomials_HOL_Algebra.html</link>
	<pubDate>Sat, 29 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Interpolation_Polynomials_HOL_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Median Method</title>
	<link>/entries/Median_Method.html</link>
	<pubDate>Tue, 25 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Actuarial Mathematics</title>
	<link>/entries/Actuarial_Mathematics.html</link>
	<pubDate>Sun, 23 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Actuarial_Mathematics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational numbers from THE BOOK</title>
	<link>/entries/Irrationals_From_THEBOOK.html</link>
	<pubDate>Sat, 08 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationals_From_THEBOOK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knight's Tour Revisited Revisited</title>
	<link>/entries/Knights_Tour.html</link>
	<pubDate>Tue, 04 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Knights_Tour.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual Numbers and Forward Differentiation</title>
	<link>/entries/Hyperdual.html</link>
	<pubDate>Fri, 31 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hyperdual.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Shapley Algorithm</title>
	<link>/entries/Gale_Shapley.html</link>
	<pubDate>Wed, 29 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Gale_Shapley.html</guid>
	<description></description>
	</item>

	<item>
	<title>Roth's Theorem on Arithmetic Progressions</title>
	<link>/entries/Roth_Arithmetic_Progressions.html</link>
	<pubDate>Tue, 28 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Roth_Arithmetic_Progressions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Decision Processes with Rewards</title>
	<link>/entries/MDP-Rewards.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Rewards.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Algorithms for Solving Markov Decision Processes</title>
	<link>/entries/MDP-Algorithms.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Tree Relations</title>
	<link>/entries/Regular_Tree_Relations.html</link>
	<pubDate>Wed, 15 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Tree_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simplicial Complexes and Boolean functions</title>
	<link>/entries/Simplicial_complexes_and_boolean_functions.html</link>
	<pubDate>Mon, 29 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Simplicial_complexes_and_boolean_functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>van Emde Boas Trees</title>
	<link>/entries/Van_Emde_Boas_Trees.html</link>
	<pubDate>Tue, 23 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_Emde_Boas_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Foundation of geometry in planes, and some complements: Excluding the parallel axioms</title>
	<link>/entries/Foundation_of_geometry.html</link>
	<pubDate>Mon, 22 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Foundation_of_geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hahn and Jordan Decomposition Theorems</title>
	<link>/entries/Hahn_Jordan_Decomposition.html</link>
	<pubDate>Fri, 19 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hahn_Jordan_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automating Public Announcement Logic and the Wise Men Puzzle in Isabelle/HOL</title>
	<link>/entries/PAL.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/PAL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Exploring Simplified Variants of Gödel’s Ontological Argument in Isabelle/HOL</title>
	<link>/entries/SimplifiedOntologicalArgument.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/SimplifiedOntologicalArgument.html</guid>
	<description></description>
	</item>

	<item>
	<title>Factorization of Polynomials with Algebraic Coefficients</title>
	<link>/entries/Factor_Algebraic_Polynomial.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Factor_Algebraic_Polynomial.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real Exponents as the Limits of Sequences of Rational Exponents</title>
	<link>/entries/Real_Power.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Power.html</guid>
	<description></description>
	</item>

	<item>
	<title>Szemerédi's Regularity Lemma</title>
	<link>/entries/Szemeredi_Regularity.html</link>
	<pubDate>Fri, 05 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Szemeredi_Regularity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum and Classical Registers</title>
	<link>/entries/Registers.html</link>
	<pubDate>Thu, 28 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Registers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Belief Revision Theory</title>
	<link>/entries/Belief_Revision.html</link>
	<pubDate>Tue, 19 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Belief_Revision.html</guid>
	<description></description>
	</item>

	<item>
	<title>X86 instruction semantics and basic block symbolic execution</title>
	<link>/entries/X86_Semantics.html</link>
	<pubDate>Wed, 13 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/X86_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebras for Iteration, Infinite Executions and Correctness of Sequential Computations</title>
	<link>/entries/Correctness_Algebras.html</link>
	<pubDate>Tue, 12 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Correctness_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Quadratic Virtual Substitution for Real Arithmetic</title>
	<link>/entries/Virtual_Substitution.html</link>
	<pubDate>Sat, 02 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Virtual_Substitution.html</guid>
	<description></description>
	</item>

	<item>
	<title>Soundness and Completeness of an Axiomatic System for First-Order Logic</title>
	<link>/entries/FOL_Axiomatic.html</link>
	<pubDate>Fri, 24 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Axiomatic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Bounded Operators</title>
	<link>/entries/Complex_Bounded_Operators.html</link>
	<pubDate>Sat, 18 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Bounded_Operators.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Weighted Path Orders and Recursive Path Orders</title>
	<link>/entries/Weighted_Path_Order.html</link>
	<pubDate>Thu, 16 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Weighted_Path_Order.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL I: Foundations: Design Patterns, Set Theory, Digraphs, Semicategories</title>
	<link>/entries/CZH_Foundations.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Foundations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL II: Elementary Theory of 1-Categories</title>
	<link>/entries/CZH_Elementary_Categories.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Elementary_Categories.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL III: Universal Constructions</title>
	<link>/entries/CZH_Universal_Constructions.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Universal_Constructions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional Simplification</title>
	<link>/entries/Conditional_Simplification.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Conditional_Simplification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional Transfer Rule</title>
	<link>/entries/Conditional_Transfer_Rule.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Conditional_Transfer_Rule.html</guid>
	<description></description>
	</item>

	<item>
	<title>Extension of Types-To-Sets</title>
	<link>/entries/Types_To_Sets_Extension.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Types_To_Sets_Extension.html</guid>
	<description></description>
	</item>

	<item>
	<title>IDE: Introduction, Destruction, Elimination</title>
	<link>/entries/Intro_Dest_Elim.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Intro_Dest_Elim.html</guid>
	<description></description>
	</item>

	<item>
	<title>A data flow analysis algorithm for computing dominators</title>
	<link>/entries/Dominance_CHK.html</link>
	<pubDate>Sun, 05 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Dominance_CHK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Solving Cubic and Quartic Equations</title>
	<link>/entries/Cubic_Quartic_Equations.html</link>
	<pubDate>Fri, 03 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Cubic_Quartic_Equations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logging-independent Message Anonymity in the Relational Method</title>
	<link>/entries/Logging_Independent_Anonymity.html</link>
	<pubDate>Thu, 26 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Logging_Independent_Anonymity.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Theorem of Three Circles</title>
	<link>/entries/Three_Circles.html</link>
	<pubDate>Sat, 21 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Three_Circles.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoCon: A Confidentiality-Verified Conference Management System</title>
	<link>/entries/CoCon.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoCon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional BD Security</title>
	<link>/entries/BD_Security_Compositional.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BD_Security_Compositional.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMed: A confidentiality-verified social media platform</title>
	<link>/entries/CoSMed.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMed.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMeDis: A confidentiality-verified distributed social media platform</title>
	<link>/entries/CoSMeDis.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMeDis.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fresh identifiers</title>
	<link>/entries/Fresh_Identifiers.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Fresh_Identifiers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial Design Theory</title>
	<link>/entries/Design_Theory.html</link>
	<pubDate>Fri, 13 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Design_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Forests</title>
	<link>/entries/Relational_Forests.html</link>
	<pubDate>Tue, 03 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz' Independent Axioms for Minkowski Spacetime</title>
	<link>/entries/Schutz_Spacetime.html</link>
	<pubDate>Tue, 27 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Schutz_Spacetime.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finitely Generated Abelian Groups</title>
	<link>/entries/Finitely_Generated_Abelian_Groups.html</link>
	<pubDate>Wed, 07 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Finitely_Generated_Abelian_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>SpecCheck - Specification-Based Testing for Isabelle/ML</title>
	<link>/entries/SpecCheck.html</link>
	<pubDate>Thu, 01 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/SpecCheck.html</guid>
	<description></description>
	</item>

	<item>
	<title>Van der Waerden's Theorem</title>
	<link>/entries/Van_der_Waerden.html</link>
	<pubDate>Tue, 22 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_der_Waerden.html</guid>
	<description></description>
	</item>

	<item>
	<title>MiniSail - A kernel language for the ISA specification language SAIL</title>
	<link>/entries/MiniSail.html</link>
	<pubDate>Fri, 18 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MiniSail.html</guid>
	<description></description>
	</item>

	<item>
	<title>Public Announcement Logic</title>
	<link>/entries/Public_Announcement_Logic.html</link>
	<pubDate>Thu, 17 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Public_Announcement_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Shorter Compiler Correctness Proof for Language IMP</title>
	<link>/entries/IMP_Compiler.html</link>
	<pubDate>Fri, 04 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics on Words Basics</title>
	<link>/entries/Combinatorics_Words.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Lemma</title>
	<link>/entries/Combinatorics_Words_Graph_Lemma.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Graph_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lyndon words</title>
	<link>/entries/Combinatorics_Words_Lyndon.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Lyndon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regression Test Selection</title>
	<link>/entries/Regression_Test_Selection.html</link>
	<pubDate>Fri, 30 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regression_Test_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle's Metalogic: Formalization and Proof Checker</title>
	<link>/entries/Metalogic_ProofChecker.html</link>
	<pubDate>Tue, 27 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Metalogic_ProofChecker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting the Exponent</title>
	<link>/entries/Lifting_the_Exponent.html</link>
	<pubDate>Tue, 27 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_the_Exponent.html</guid>
	<description></description>
	</item>

	<item>
	<title>The BKR Decision Procedure for Univariate Real Arithmetic</title>
	<link>/entries/BenOr_Kozen_Reif.html</link>
	<pubDate>Sat, 24 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BenOr_Kozen_Reif.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Stewart Games</title>
	<link>/entries/GaleStewart_Games.html</link>
	<pubDate>Fri, 23 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/GaleStewart_Games.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Timely Dataflow's Progress Tracking Protocol</title>
	<link>/entries/Progress_Tracking.html</link>
	<pubDate>Tue, 13 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Progress_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Control via Dependency Tracking</title>
	<link>/entries/IFC_Tracking.html</link>
	<pubDate>Thu, 01 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IFC_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck's Schemes in Algebraic Geometry</title>
	<link>/entries/Grothendieck_Schemes.html</link>
	<pubDate>Mon, 29 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Grothendieck_Schemes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hensel's Lemma for the p-adic Integers</title>
	<link>/entries/Padic_Ints.html</link>
	<pubDate>Tue, 23 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Padic_Ints.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL: the Communication Modeling Aspect</title>
	<link>/entries/Constructive_Cryptography_CM.html</link>
	<pubDate>Wed, 17 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography_CM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation</title>
	<link>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</link>
	<pubDate>Fri, 12 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum projective measurements and the CHSH inequality</title>
	<link>/entries/Projective_Measurements.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Measurements.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hermite–Lindemann–Weierstraß Transcendence Theorem</title>
	<link>/entries/Hermite_Lindemann.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite_Lindemann.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mereology</title>
	<link>/entries/Mereology.html</link>
	<pubDate>Mon, 01 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Mereology.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sunflower Lemma of Erdős and Rado</title>
	<link>/entries/Sunflowers.html</link>
	<pubDate>Thu, 25 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Sunflowers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Imperative Implementation of B-Trees</title>
	<link>/entries/BTree.html</link>
	<pubDate>Wed, 24 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BTree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Puiseux Series</title>
	<link>/entries/Formal_Puiseux_Series.html</link>
	<pubDate>Wed, 17 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_Puiseux_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Laws of Large Numbers</title>
	<link>/entries/Laws_of_Large_Numbers.html</link>
	<pubDate>Wed, 10 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Laws_of_Large_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid</title>
	<link>/entries/IsaGeoCoq.html</link>
	<pubDate>Sun, 31 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IsaGeoCoq.html</guid>
	<description></description>
	</item>

	<item>
	<title>Solution to the xkcd Blue Eyes puzzle</title>
	<link>/entries/Blue_Eyes.html</link>
	<pubDate>Sat, 30 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Blue_Eyes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hood-Melville Queue</title>
	<link>/entries/Hood_Melville_Queue.html</link>
	<pubDate>Mon, 18 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hood_Melville_Queue.html</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaDCI: a Java semantics with dynamic class initialization</title>
	<link>/entries/JinjaDCI.html</link>
	<pubDate>Mon, 11 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/JinjaDCI.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cofinality and the Delta System Lemma</title>
	<link>/entries/Delta_System_Lemma.html</link>
	<pubDate>Sun, 27 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Delta_System_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Topological semantics for paraconsistent and paracomplete logics</title>
	<link>/entries/Topological_Semantics.html</link>
	<pubDate>Thu, 17 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Topological_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Minimum Spanning Tree Algorithms</title>
	<link>/entries/Relational_Minimum_Spanning_Trees.html</link>
	<pubDate>Tue, 08 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Minimum_Spanning_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inline Caching and Unboxing Optimization for Interpreters</title>
	<link>/entries/Interpreter_Optimizations.html</link>
	<pubDate>Mon, 07 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Interpreter_Optimizations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Relational Method with Message Anonymity for the Verification of Cryptographic Protocols</title>
	<link>/entries/Relational_Method.html</link>
	<pubDate>Sat, 05 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information</title>
	<link>/entries/Isabelle_Marries_Dirac.html</link>
	<pubDate>Sun, 22 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Marries_Dirac.html</guid>
	<description></description>
	</item>

	<item>
	<title>The HOL-CSP Refinement Toolkit</title>
	<link>/entries/CSP_RefTK.html</link>
	<pubDate>Thu, 19 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/CSP_RefTK.html</guid>
	<description></description>
	</item>

	<item>
	<title>AI Planning Languages Semantics</title>
	<link>/entries/AI_Planning_Languages_Semantics.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/AI_Planning_Languages_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified SAT-Based AI Planning</title>
	<link>/entries/Verified_SAT_Based_AI_Planning.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Verified_SAT_Based_AI_Planning.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sound Type System for Physical Quantities, Units, and Measurements</title>
	<link>/entries/Physical_Quantities.html</link>
	<pubDate>Tue, 20 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Physical_Quantities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Map Extras</title>
	<link>/entries/Finite-Map-Extras.html</link>
	<pubDate>Mon, 12 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Finite-Map-Extras.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Safely Composable Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Safely Composable Web Components</title>
	<link>/entries/SC_DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/SC_DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Web Components</title>
	<link>/entries/DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safely Composable DOM</title>
	<link>/entries/Core_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Core_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Abstract Formalization of G&ouml;del's Incompleteness Theorems</title>
	<link>/entries/Goedel_Incompleteness.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_Incompleteness.html</guid>
	<description></description>
	</item>

	<item>
	<title>From Abstract to Concrete G&ouml;del's Incompleteness Theorems&mdash;Part I</title>
	<link>/entries/Goedel_HFSet_Semantic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_HFSet_Semantic.html</guid>
	<description></description>
	</item>

	<item>
	<title>From Abstract to Concrete G&ouml;del's Incompleteness Theorems&mdash;Part II</title>
	<link>/entries/Goedel_HFSet_Semanticless.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_HFSet_Semanticless.html</guid>
	<description></description>
	</item>

	<item>
	<title>Robinson Arithmetic</title>
	<link>/entries/Robinson_Arithmetic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Robinson_Arithmetic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax-Independent Logic Infrastructure</title>
	<link>/entries/Syntax_Independent_Logic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Syntax_Independent_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machines.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machines.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inference of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machine_Inference.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machine_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Practical Algebraic Calculus Checker</title>
	<link>/entries/PAC_Checker.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/PAC_Checker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Some classical results in inductive inference of recursive functions</title>
	<link>/entries/Inductive_Inference.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Disjoint-Set Forests</title>
	<link>/entries/Relational_Disjoint_Set_Forests.html</link>
	<pubDate>Wed, 26 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Disjoint_Set_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Extensions to the Comprehensive Framework for Saturation Theorem Proving</title>
	<link>/entries/Saturation_Framework_Extensions.html</link>
	<pubDate>Tue, 25 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Saturation_Framework_Extensions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Putting the `K' into Bird's derivation of Knuth-Morris-Pratt string matching</title>
	<link>/entries/BirdKMP.html</link>
	<pubDate>Tue, 25 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/BirdKMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amicable Numbers</title>
	<link>/entries/Amicable_Numbers.html</link>
	<pubDate>Tue, 04 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Amicable_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal Partitions</title>
	<link>/entries/Ordinal_Partitions.html</link>
	<pubDate>Mon, 03 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinal_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of The Chandy--Lamport Distributed Snapshot Algorithm</title>
	<link>/entries/Chandy_Lamport.html</link>
	<pubDate>Tue, 21 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Chandy_Lamport.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Characterisations of Paths</title>
	<link>/entries/Relational_Paths.html</link>
	<pubDate>Mon, 13 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Paths.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles</title>
	<link>/entries/Safe_Distance.html</link>
	<pubDate>Mon, 01 Jun 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Distance.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified algorithm for computing the Smith normal form of a matrix</title>
	<link>/entries/Smith_Normal_Form.html</link>
	<pubDate>Sat, 23 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Smith_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Nash-Williams Partition Theorem</title>
	<link>/entries/Nash_Williams.html</link>
	<pubDate>Sat, 16 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Nash_Williams.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Knuth–Bendix Orders</title>
	<link>/entries/Knuth_Bendix_Order.html</link>
	<pubDate>Wed, 13 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Bendix_Order.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality Criteria for Series by Erdős and Straus</title>
	<link>/entries/Irrational_Series_Erdos_Straus.html</link>
	<pubDate>Tue, 12 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Irrational_Series_Erdos_Straus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion Theorem in ZF</title>
	<link>/entries/Recursion-Addition.html</link>
	<pubDate>Mon, 11 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Recursion-Addition.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Normalisation Procedure for Linear Temporal Logic: Isabelle/HOL Formalisation</title>
	<link>/entries/LTL_Normal_Form.html</link>
	<pubDate>Fri, 08 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Forcing in Isabelle/ZF</title>
	<link>/entries/Forcing.html</link>
	<pubDate>Wed, 06 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Forcing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Banach-Steinhaus Theorem</title>
	<link>/entries/Banach_Steinhaus.html</link>
	<pubDate>Sat, 02 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Banach_Steinhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Attack Trees in Isabelle for GDPR compliance of IoT healthcare systems</title>
	<link>/entries/Attack_Trees.html</link>
	<pubDate>Mon, 27 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Attack_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gaussian Integers</title>
	<link>/entries/Gaussian_Integers.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Gaussian_Integers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Power Sum Polynomials</title>
	<link>/entries/Power_Sum_Polynomials.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Power_Sum_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Lambert W Function on the Reals</title>
	<link>/entries/Lambert_W.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lambert_W.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices for ODEs</title>
	<link>/entries/Matrices_for_ODEs.html</link>
	<pubDate>Sun, 19 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Matrices_for_ODEs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Authenticated Data Structures As Functors</title>
	<link>/entries/ADS_Functor.html</link>
	<pubDate>Thu, 16 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/ADS_Functor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Algorithm for Greedily Computing Associative Aggregations on Sliding Windows</title>
	<link>/entries/Sliding_Window_Algorithm.html</link>
	<pubDate>Fri, 10 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Sliding_Window_Algorithm.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Comprehensive Framework for Saturation Theorem Proving</title>
	<link>/entries/Saturation_Framework.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Saturation_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations</title>
	<link>/entries/MFODL_Monitor_Optimized.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/MFODL_Monitor_Optimized.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automated Stateful Protocol Verification</title>
	<link>/entries/Automated_Stateful_Protocol_Verification.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Automated_Stateful_Protocol_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stateful Protocol Composition and Typing</title>
	<link>/entries/Stateful_Protocol_Composition_and_Typing.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Stateful_Protocol_Composition_and_Typing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lucas's Theorem</title>
	<link>/entries/Lucas_Theorem.html</link>
	<pubDate>Tue, 07 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lucas_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Eventual Consistency of the Collaborative Editing Framework WOOT</title>
	<link>/entries/WOOT_Strong_Eventual_Consistency.html</link>
	<pubDate>Wed, 25 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/WOOT_Strong_Eventual_Consistency.html</guid>
	<description></description>
	</item>

	<item>
	<title>Furstenberg's topology and his proof of the infinitude of primes</title>
	<link>/entries/Furstenberg_Topology.html</link>
	<pubDate>Sun, 22 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Furstenberg_Topology.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Under-Approximate Relational Logic</title>
	<link>/entries/Relational-Incorrectness-Logic.html</link>
	<pubDate>Thu, 12 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational-Incorrectness-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hello World</title>
	<link>/entries/Hello_World.html</link>
	<pubDate>Sat, 07 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Hello_World.html</guid>
	<description></description>
	</item>

	<item>
	<title>Implementing the Goodstein Function in &lambda;-Calculus</title>
	<link>/entries/Goodstein_Lambda.html</link>
	<pubDate>Fri, 21 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goodstein_Lambda.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Generic Framework for Verified Compilers</title>
	<link>/entries/VeriComp.html</link>
	<pubDate>Mon, 10 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/VeriComp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Arithmetic progressions and relative primes</title>
	<link>/entries/Arith_Prog_Rel_Primes.html</link>
	<pubDate>Sat, 01 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Arith_Prog_Rel_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Hierarchy of Algebras for Boolean Subsets</title>
	<link>/entries/Subset_Boolean_Algebras.html</link>
	<pubDate>Fri, 31 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Subset_Boolean_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mersenne primes and the Lucas–Lehmer test</title>
	<link>/entries/Mersenne_Primes.html</link>
	<pubDate>Fri, 17 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Mersenne_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Approximation Algorithms</title>
	<link>/entries/Approximation_Algorithms.html</link>
	<pubDate>Thu, 16 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Approximation_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Closest Pair of Points Algorithms</title>
	<link>/entries/Closest_Pair_Points.html</link>
	<pubDate>Mon, 13 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Closest_Pair_Points.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skip Lists</title>
	<link>/entries/Skip_Lists.html</link>
	<pubDate>Thu, 09 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Skip_Lists.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bicategories</title>
	<link>/entries/Bicategory.html</link>
	<pubDate>Mon, 06 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Bicategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Irrationality of ζ(3)</title>
	<link>/entries/Zeta_3_Irrational.html</link>
	<pubDate>Fri, 27 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_3_Irrational.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing a Seligman-Style Tableau System for Hybrid Logic</title>
	<link>/entries/Hybrid_Logic.html</link>
	<pubDate>Fri, 20 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Poincaré-Bendixson Theorem</title>
	<link>/entries/Poincare_Bendixson.html</link>
	<pubDate>Wed, 18 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Bendixson.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Geometry</title>
	<link>/entries/Complex_Geometry.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Poincaré Disc Model</title>
	<link>/entries/Poincare_Disc.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Disc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss Sums and the Pólya–Vinogradov Inequality</title>
	<link>/entries/Gauss_Sums.html</link>
	<pubDate>Tue, 10 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Sums.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</title>
	<link>/entries/Generalized_Counting_Sort.html</link>
	<pubDate>Wed, 04 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generalized_Counting_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interval Arithmetic on 32-bit Words</title>
	<link>/entries/Interval_Arithmetic_Word32.html</link>
	<pubDate>Wed, 27 Nov 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Interval_Arithmetic_Word32.html</guid>
	<description></description>
	</item>

	<item>
	<title>Zermelo Fraenkel Set Theory in Higher-Order Logic</title>
	<link>/entries/ZFC_in_HOL.html</link>
	<pubDate>Thu, 24 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/ZFC_in_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/C</title>
	<link>/entries/Isabelle_C.html</link>
	<pubDate>Tue, 22 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_C.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2019 -- Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2019.html</link>
	<pubDate>Wed, 16 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Aristotle's Assertoric Syllogistic</title>
	<link>/entries/Aristotles_Assertoric_Syllogistic.html</link>
	<pubDate>Tue, 08 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Aristotles_Assertoric_Syllogistic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sigma Protocols and Commitment Schemes</title>
	<link>/entries/Sigma_Commit_Crypto.html</link>
	<pubDate>Mon, 07 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Sigma_Commit_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clean - An Abstract Imperative Programming Language and its Theory</title>
	<link>/entries/Clean.html</link>
	<pubDate>Fri, 04 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Clean.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Multiway-Join Algorithms</title>
	<link>/entries/Generic_Join.html</link>
	<pubDate>Mon, 16 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Join.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification Components for Hybrid Systems</title>
	<link>/entries/Hybrid_Systems_VCs.html</link>
	<pubDate>Tue, 10 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Systems_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fourier Series</title>
	<link>/entries/Fourier.html</link>
	<pubDate>Fri, 06 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Case Study in Basic Algebra</title>
	<link>/entries/Jacobson_Basic_Algebra.html</link>
	<pubDate>Fri, 30 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Jacobson_Basic_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation of an Adaptive State Counting Algorithm</title>
	<link>/entries/Adaptive_State_Counting.html</link>
	<pubDate>Fri, 16 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Adaptive_State_Counting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Laplace Transform</title>
	<link>/entries/Laplace_Transform.html</link>
	<pubDate>Wed, 14 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Laplace_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Communicating Concurrent Kleene Algebra for Distributed Systems Specification</title>
	<link>/entries/C2KA_DistributedSystems.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/C2KA_DistributedSystems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Programming</title>
	<link>/entries/Linear_Programming.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Programming.html</guid>
	<description></description>
	</item>

	<item>
	<title>Selected Problems from the International Mathematical Olympiad 2019</title>
	<link>/entries/IMO2019.html</link>
	<pubDate>Mon, 05 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMO2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar Quorum Systems</title>
	<link>/entries/Stellar_Quorums.html</link>
	<pubDate>Thu, 01 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Stellar_Quorums.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Development of a Polychronous Polytimed Coordination Language</title>
	<link>/entries/TESL_Language.html</link>
	<pubDate>Tue, 30 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/TESL_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>Order Extension and Szpilrajn's Extension Theorem</title>
	<link>/entries/Szpilrajn.html</link>
	<pubDate>Sat, 27 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Szpilrajn.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sequent Calculus for First-Order Logic</title>
	<link>/entries/FOL_Seq_Calc1.html</link>
	<pubDate>Thu, 18 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc1.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Code Generator from Isabelle/HOL to CakeML</title>
	<link>/entries/CakeML_Codegen.html</link>
	<pubDate>Mon, 08 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML_Codegen.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</title>
	<link>/entries/MFOTL_Monitor.html</link>
	<pubDate>Thu, 04 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/MFOTL_Monitor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complete Non-Orders and Fixed Points</title>
	<link>/entries/Complete_Non_Orders.html</link>
	<pubDate>Thu, 27 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complete_Non_Orders.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Search Trees</title>
	<link>/entries/Priority_Search_Trees.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Search_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra</title>
	<link>/entries/Prim_Dijkstra_Simple.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prim_Dijkstra_Simple.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Inequalities</title>
	<link>/entries/Linear_Inequalities.html</link>
	<pubDate>Fri, 21 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Inequalities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hilbert's Nullstellensatz</title>
	<link>/entries/Nullstellensatz.html</link>
	<pubDate>Sun, 16 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Nullstellensatz.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds</title>
	<link>/entries/Groebner_Macaulay.html</link>
	<pubDate>Sat, 15 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Macaulay.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Heaps for IMP2</title>
	<link>/entries/IMP2_Binary_Heap.html</link>
	<pubDate>Thu, 13 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2_Binary_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Game Logic</title>
	<link>/entries/Differential_Game_Logic.html</link>
	<pubDate>Mon, 03 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Game_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multidimensional Binary Search Trees</title>
	<link>/entries/KD_Tree.html</link>
	<pubDate>Thu, 30 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/KD_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Generic Authenticated Data Structures</title>
	<link>/entries/LambdaAuth.html</link>
	<pubDate>Tue, 14 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaAuth.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Party Computation</title>
	<link>/entries/Multi_Party_Computation.html</link>
	<pubDate>Thu, 09 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Multi_Party_Computation.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP Version 2.0</title>
	<link>/entries/HOL-CSP.html</link>
	<pubDate>Fri, 26 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/HOL-CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Compositional and Unified Translation of LTL into ω-Automata</title>
	<link>/entries/LTL_Master_Theorem.html</link>
	<pubDate>Tue, 16 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Master_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Theory of Syntax with Bindings</title>
	<link>/entries/Binding_Syntax_Theory.html</link>
	<pubDate>Sat, 06 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Binding_Syntax_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of Certain Infinite Series</title>
	<link>/entries/Transcendence_Series_Hancl_Rucki.html</link>
	<pubDate>Wed, 27 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Transcendence_Series_Hancl_Rucki.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum Hoare Logic</title>
	<link>/entries/QHLProver.html</link>
	<pubDate>Sun, 24 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/QHLProver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Safe OCL</title>
	<link>/entries/Safe_OCL.html</link>
	<pubDate>Sat, 09 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Elementary Facts About the Distribution of Primes</title>
	<link>/entries/Prime_Distribution_Elementary.html</link>
	<pubDate>Thu, 21 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Distribution_Elementary.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal's Algorithm for Minimum Spanning Forest</title>
	<link>/entries/Kruskal.html</link>
	<pubDate>Thu, 14 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Kruskal.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Primality Testing</title>
	<link>/entries/Probabilistic_Prime_Tests.html</link>
	<pubDate>Mon, 11 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Prime_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Turing Machine</title>
	<link>/entries/Universal_Turing_Machine.html</link>
	<pubDate>Fri, 08 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Turing_Machine.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/UTP: Mechanised Theory Engineering for Unifying Theories of Programming</title>
	<link>/entries/UTP.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/UTP.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inversions of a List</title>
	<link>/entries/List_Inversions.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/List_Inversions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Farkas' Lemma and Motzkin's Transposition Theorem</title>
	<link>/entries/Farkas.html</link>
	<pubDate>Thu, 17 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Farkas.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Algebra for Higher-Order Terms</title>
	<link>/entries/Higher_Order_Terms.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Higher_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2 – Simple Program Verification in Isabelle/HOL</title>
	<link>/entries/IMP2.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reduction Theorem for Store Buffers</title>
	<link>/entries/Store_Buffer_Reduction.html</link>
	<pubDate>Mon, 07 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Store_Buffer_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model</title>
	<link>/entries/Core_DOM.html</link>
	<pubDate>Wed, 26 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Core_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Concurrent Revisions</title>
	<link>/entries/Concurrent_Revisions.html</link>
	<pubDate>Tue, 25 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Revisions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Imperative Programs using Auto2</title>
	<link>/entries/Auto2_Imperative_HOL.html</link>
	<pubDate>Fri, 21 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL</title>
	<link>/entries/Constructive_Cryptography.html</link>
	<pubDate>Mon, 17 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Orderings and Lattices</title>
	<link>/entries/Order_Lattice_Props.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Order_Lattice_Props.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantales</title>
	<link>/entries/Quantales.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quantales.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer Semantics</title>
	<link>/entries/Transformer_Semantics.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Transformer_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Functional Implementation of Bachmair and Ganzinger's Ordered Resolution Prover</title>
	<link>/entries/Functional_Ordered_Resolution_Prover.html</link>
	<pubDate>Fri, 23 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Functional_Ordered_Resolution_Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Saturation</title>
	<link>/entries/Graph_Saturation.html</link>
	<pubDate>Fri, 23 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Graph_Saturation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2 Prover</title>
	<link>/entries/Auto2_HOL.html</link>
	<pubDate>Tue, 20 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matroids</title>
	<link>/entries/Matroids.html</link>
	<pubDate>Fri, 16 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Matroids.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving generic class instances for datatypes</title>
	<link>/entries/Generic_Deriving.html</link>
	<pubDate>Tue, 06 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation and Evaluation of Alan Gewirth's Proof for the Principle of Generic Consistency in Isabelle/HOL</title>
	<link>/entries/GewirthPGCProof.html</link>
	<pubDate>Tue, 30 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/GewirthPGCProof.html</guid>
	<description></description>
	</item>

	<item>
	<title>Epistemic Logic: Completeness of Modal Logics</title>
	<link>/entries/Epistemic_Logic.html</link>
	<pubDate>Mon, 29 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Epistemic_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth Manifolds</title>
	<link>/entries/Smooth_Manifolds.html</link>
	<pubDate>Mon, 22 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Smooth_Manifolds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_EPO.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_EPO.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Binary Search Trees</title>
	<link>/entries/Randomised_BSTs.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Upper Bounding Diameters of State Spaces of Factored Transition Systems</title>
	<link>/entries/Factored_Transition_System_Bounding.html</link>
	<pubDate>Fri, 12 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Factored_Transition_System_Bounding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of π</title>
	<link>/entries/Pi_Transcendental.html</link>
	<pubDate>Fri, 28 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>Symmetric Polynomials</title>
	<link>/entries/Symmetric_Polynomials.html</link>
	<pubDate>Tue, 25 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Symmetric_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
	<link>/entries/Signature_Groebner.html</link>
	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Signature_Groebner.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Prime Number Theorem</title>
	<link>/entries/Prime_Number_Theorem.html</link>
	<pubDate>Wed, 19 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Number_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Aggregation Algebras</title>
	<link>/entries/Aggregation_Algebras.html</link>
	<pubDate>Sat, 15 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Aggregation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/entries/Octonions.html</link>
	<pubDate>Fri, 14 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Octonions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/entries/Quaternions.html</link>
	<pubDate>Wed, 05 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quaternions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Budan–Fourier Theorem and Counting Real Roots with Multiplicity</title>
	<link>/entries/Budan_Fourier.html</link>
	<pubDate>Sun, 02 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Budan_Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Incremental Simplex Algorithm with Unsatisfiable Core Generation</title>
	<link>/entries/Simplex.html</link>
	<pubDate>Fri, 24 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Simplex.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minsky Machines</title>
	<link>/entries/Minsky_Machines.html</link>
	<pubDate>Tue, 14 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Minsky_Machines.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pricing in discrete financial models</title>
	<link>/entries/DiscretePricing.html</link>
	<pubDate>Mon, 16 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/DiscretePricing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Von-Neumann-Morgenstern Utility Theorem</title>
	<link>/entries/Neumann_Morgenstern_Utility.html</link>
	<pubDate>Wed, 04 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Neumann_Morgenstern_Utility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pell's Equation</title>
	<link>/entries/Pell.html</link>
	<pubDate>Sat, 23 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pell.html</guid>
	<description></description>
	</item>

	<item>
	<title>Projective Geometry</title>
	<link>/entries/Projective_Geometry.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Localization of a Commutative Ring</title>
	<link>/entries/Localization_Ring.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Localization_Ring.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Order Reduction</title>
	<link>/entries/Partial_Order_Reduction.html</link>
	<pubDate>Tue, 05 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Order_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal Binary Search Trees</title>
	<link>/entries/Optimal_BST.html</link>
	<pubDate>Sun, 27 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Optimal_BST.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden Markov Models</title>
	<link>/entries/Hidden_Markov_Models.html</link>
	<pubDate>Fri, 25 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hidden_Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Timed Automata</title>
	<link>/entries/Probabilistic_Timed_Automata.html</link>
	<pubDate>Thu, 24 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Axiom Systems for Category Theory in Free Logic</title>
	<link>/entries/AxiomaticCategoryTheory.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/AxiomaticCategoryTheory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational Rapidly Convergent Series</title>
	<link>/entries/Irrationality_J_Hancl.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationality_J_Hancl.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monadification, Memoization and Dynamic Programming</title>
	<link>/entries/Monad_Memo_DP.html</link>
	<pubDate>Tue, 22 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Memo_DP.html</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets: Sequential Specifications for Replicated Datatypes</title>
	<link>/entries/OpSets.html</link>
	<pubDate>Thu, 10 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/OpSets.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL Formalization of the Modular Assembly Kit for Security Properties</title>
	<link>/entries/Modular_Assembly_Kit_Security.html</link>
	<pubDate>Mon, 07 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_Assembly_Kit_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>WebAssembly</title>
	<link>/entries/WebAssembly.html</link>
	<pubDate>Sun, 29 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/WebAssembly.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2018 - Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2018.html</link>
	<pubDate>Fri, 27 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2018.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded Natural Functors with Covariance and Contravariance</title>
	<link>/entries/BNF_CC.html</link>
	<pubDate>Tue, 24 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/BNF_CC.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of Fishburn-Strategyproofness and Pareto-Efficiency</title>
	<link>/entries/Fishburn_Impossibility.html</link>
	<pubDate>Thu, 22 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Fishburn_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Weight-Balanced Trees</title>
	<link>/entries/Weight_Balanced_Trees.html</link>
	<pubDate>Tue, 13 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Weight_Balanced_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML</title>
	<link>/entries/CakeML.html</link>
	<pubDate>Mon, 12 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Theory of Architectural Design Patterns</title>
	<link>/entries/Architectural_Design_Patterns.html</link>
	<pubDate>Thu, 01 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Architectural_Design_Patterns.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hoare Logics for Time Bounds</title>
	<link>/entries/Hoare_Time.html</link>
	<pubDate>Mon, 26 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hoare_Time.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified factorization algorithm for integer polynomials with polynomial complexity</title>
	<link>/entries/LLL_Factorization.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Terms</title>
	<link>/entries/First_Order_Terms.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/First_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Error Function</title>
	<link>/entries/Error_Function.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Error_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>Treaps</title>
	<link>/entries/Treaps.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Treaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified LLL algorithm</title>
	<link>/entries/LLL_Basis_Reduction.html</link>
	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Bachmair and Ganzinger's Ordered Resolution Prover</title>
	<link>/entries/Ordered_Resolution_Prover.html</link>
	<pubDate>Thu, 18 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Ordered_Resolution_Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gromov Hyperbolicity</title>
	<link>/entries/Gromov_Hyperbolicity.html</link>
	<pubDate>Tue, 16 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Gromov_Hyperbolicity.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL formalisation of Green's Theorem</title>
	<link>/entries/Green.html</link>
	<pubDate>Thu, 11 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Green.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Falling Factorial of a Sum</title>
	<link>/entries/Falling_Factorial_Sum.html</link>
	<pubDate>Fri, 22 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Falling_Factorial_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet L-Functions and Dirichlet's Theorem</title>
	<link>/entries/Dirichlet_L.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_L.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Mason–Stothers Theorem</title>
	<link>/entries/Mason_Stothers.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Mason_Stothers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Median-of-Medians Selection Algorithm</title>
	<link>/entries/Median_Of_Medians_Selection.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Of_Medians_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>Operations on Bounded Natural Functors</title>
	<link>/entries/BNF_Operations.html</link>
	<pubDate>Tue, 19 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/BNF_Operations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The string search algorithm by Knuth, Morris and Pratt</title>
	<link>/entries/Knuth_Morris_Pratt.html</link>
	<pubDate>Mon, 18 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Morris_Pratt.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic Matrices and the Perron-Frobenius Theorem</title>
	<link>/entries/Stochastic_Matrices.html</link>
	<pubDate>Wed, 22 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stochastic_Matrices.html</guid>
	<description></description>
	</item>

	<item>
	<title>The IMAP CmRDT</title>
	<link>/entries/IMAP-CRDT.html</link>
	<pubDate>Thu, 09 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/IMAP-CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid Multi-Lane Spatial Logic</title>
	<link>/entries/Hybrid_Multi_Lane_Spatial_Logic.html</link>
	<pubDate>Mon, 06 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Multi_Lane_Spatial_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Kuratowski Closure-Complement Theorem</title>
	<link>/entries/Kuratowski_Closure_Complement.html</link>
	<pubDate>Thu, 26 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Kuratowski_Closure_Complement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Büchi Complementation</title>
	<link>/entries/Buchi_Complementation.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buchi_Complementation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transition Systems and Automata</title>
	<link>/entries/Transition_Systems_and_Automata.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Transition_Systems_and_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Count the Number of Complex Roots</title>
	<link>/entries/Count_Complex_Roots.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Count_Complex_Roots.html</guid>
	<description></description>
	</item>

	<item>
	<title>Evaluate Winding Numbers through Cauchy Indices</title>
	<link>/entries/Winding_Number_Eval.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Winding_Number_Eval.html</guid>
	<description></description>
	</item>

	<item>
	<title>Homogeneous Linear Diophantine Equations</title>
	<link>/entries/Diophantine_Eqns_Lin_Hom.html</link>
	<pubDate>Sat, 14 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Diophantine_Eqns_Lin_Hom.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet Series</title>
	<link>/entries/Dirichlet_Series.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Recurrences</title>
	<link>/entries/Linear_Recurrences.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Recurrences.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hurwitz and Riemann ζ Functions</title>
	<link>/entries/Zeta_Function.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>Computer-assisted Reconstruction and Assessment of E. J. Lowe's Modal Ontological Argument</title>
	<link>/entries/Lowe_Ontological_Argument.html</link>
	<pubDate>Thu, 21 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Lowe_Ontological_Argument.html</guid>
	<description></description>
	</item>

	<item>
	<title>Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL</title>
	<link>/entries/PLM.html</link>
	<pubDate>Sun, 17 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PLM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Anselm's God in Isabelle/HOL</title>
	<link>/entries/AnselmGod.html</link>
	<pubDate>Wed, 06 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/AnselmGod.html</guid>
	<description></description>
	</item>

	<item>
	<title>Microeconomics and the First Welfare Theorem</title>
	<link>/entries/First_Welfare_Theorem.html</link>
	<pubDate>Fri, 01 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/First_Welfare_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Orbit-Stabiliser Theorem with Application to Rotational Symmetries</title>
	<link>/entries/Orbit_Stabiliser.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Orbit_Stabiliser.html</guid>
	<description></description>
	</item>

	<item>
	<title>Root-Balanced Tree</title>
	<link>/entries/Root_Balanced_Tree.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Root_Balanced_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>The LambdaMu-calculus</title>
	<link>/entries/LambdaMu.html</link>
	<pubDate>Wed, 16 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaMu.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stewart's Theorem and Apollonius' Theorem</title>
	<link>/entries/Stewart_Apollonius.html</link>
	<pubDate>Mon, 31 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stewart_Apollonius.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dynamic Architectures</title>
	<link>/entries/DynamicArchitectures.html</link>
	<pubDate>Fri, 28 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/DynamicArchitectures.html</guid>
	<description></description>
	</item>

	<item>
	<title>Declarative Semantics for Functional Languages</title>
	<link>/entries/Decl_Sem_Fun_PL.html</link>
	<pubDate>Fri, 21 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Decl_Sem_Fun_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOLCF-Prelude</title>
	<link>/entries/HOLCF-Prelude.html</link>
	<pubDate>Sat, 15 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/HOLCF-Prelude.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowski's Theorem</title>
	<link>/entries/Minkowskis_Theorem.html</link>
	<pubDate>Thu, 13 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minkowskis_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Metatheory and Type Inference for a Name-Carrying Simply-Typed Lambda Calculus</title>
	<link>/entries/Name_Carrying_Type_Inference.html</link>
	<pubDate>Sun, 09 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Name_Carrying_Type_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</title>
	<link>/entries/CRDT.html</link>
	<pubDate>Fri, 07 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone-Kleene Relation Algebras</title>
	<link>/entries/Stone_Kleene_Relation_Algebras.html</link>
	<pubDate>Thu, 06 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Kleene_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional Proof Systems</title>
	<link>/entries/Propositional_Proof_Systems.html</link>
	<pubDate>Wed, 21 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Propositional_Proof_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Semigroups and Convolution Algebras</title>
	<link>/entries/PSemigroupsConvolution.html</link>
	<pubDate>Tue, 13 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PSemigroupsConvolution.html</guid>
	<description></description>
	</item>

	<item>
	<title>Buffon's Needle Problem</title>
	<link>/entries/Buffons_Needle.html</link>
	<pubDate>Tue, 06 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buffons_Needle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Flow Networks and the Min-Cut-Max-Flow Theorem</title>
	<link>/entries/Flow_Networks.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Flow_Networks.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Push-Relabel Algorithms</title>
	<link>/entries/Prpu_Maxflow.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Prpu_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/entries/Optics.html</link>
	<pubDate>Thu, 25 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Optics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Developing Security Protocols by Refinement</title>
	<link>/entries/Security_Protocol_Refinement.html</link>
	<pubDate>Wed, 24 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Security_Protocol_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dictionary Construction</title>
	<link>/entries/Dict_Construction.html</link>
	<pubDate>Wed, 24 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dict_Construction.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Floyd-Warshall Algorithm for Shortest Paths</title>
	<link>/entries/Floyd_Warshall.html</link>
	<pubDate>Mon, 08 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/entries/CryptHOL.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CryptHOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Effect polymorphism in higher-order logic</title>
	<link>/entries/Monomorphic_Monad.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monomorphic_Monad.html</guid>
	<description></description>
	</item>

	<item>
	<title>Game-based cryptography in HOL</title>
	<link>/entries/Game_Based_Crypto.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Game_Based_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monad normalisation</title>
	<link>/entries/Monad_Normalisation.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Normalisation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
	<link>/entries/Probabilistic_While.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_While.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monoidal Categories</title>
	<link>/entries/MonoidalCategory.html</link>
	<pubDate>Thu, 04 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/MonoidalCategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Types, Tableaus and Gödel’s God in Isabelle/HOL</title>
	<link>/entries/Types_Tableaus_and_Goedels_God.html</link>
	<pubDate>Mon, 01 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Types_Tableaus_and_Goedels_God.html</guid>
	<description></description>
	</item>

	<item>
	<title>Local Lexing</title>
	<link>/entries/LocalLexing.html</link>
	<pubDate>Fri, 28 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LocalLexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructor Functions</title>
	<link>/entries/Constructor_Funs.html</link>
	<pubDate>Wed, 19 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Constructor_Funs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lazifying case constants</title>
	<link>/entries/Lazy_Case.html</link>
	<pubDate>Tue, 18 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Lazy_Case.html</guid>
	<description></description>
	</item>

	<item>
	<title>Subresultants</title>
	<link>/entries/Subresultants.html</link>
	<pubDate>Thu, 06 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Subresultants.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expected Shape of Random Binary Search Trees</title>
	<link>/entries/Random_BSTs.html</link>
	<pubDate>Tue, 04 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Random_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower bound on comparison-based sorting algorithms</title>
	<link>/entries/Comparison_Sort_Lower_Bound.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Comparison_Sort_Lower_Bound.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
	<link>/entries/Quick_Sort_Cost.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Euler–MacLaurin Formula</title>
	<link>/entries/Euler_MacLaurin.html</link>
	<pubDate>Fri, 10 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_MacLaurin.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Group Law for Elliptic Curves</title>
	<link>/entries/Elliptic_Curves_Group_Law.html</link>
	<pubDate>Tue, 28 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Elliptic_Curves_Group_Law.html</guid>
	<description></description>
	</item>

	<item>
	<title>Menger's Theorem</title>
	<link>/entries/Menger.html</link>
	<pubDate>Sun, 26 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Menger.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Dynamic Logic</title>
	<link>/entries/Differential_Dynamic_Logic.html</link>
	<pubDate>Mon, 13 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Dynamic_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Soundness</title>
	<link>/entries/Abstract_Soundness.html</link>
	<pubDate>Fri, 10 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract_Soundness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Relation Algebras</title>
	<link>/entries/Stone_Relation_Algebras.html</link>
	<pubDate>Tue, 07 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refining Authenticated Key Agreement with Strong Adversaries</title>
	<link>/entries/Key_Agreement_Strong_Adversaries.html</link>
	<pubDate>Tue, 31 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Key_Agreement_Strong_Adversaries.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli Numbers</title>
	<link>/entries/Bernoulli.html</link>
	<pubDate>Tue, 24 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bernoulli.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bertrand's postulate</title>
	<link>/entries/Bertrands_Postulate.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bertrands_Postulate.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minimal Static Single Assignment Form</title>
	<link>/entries/Minimal_SSA.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minimal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of e</title>
	<link>/entries/E_Transcendental.html</link>
	<pubDate>Thu, 12 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/E_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Network Models and Their Application to Firewall Policies</title>
	<link>/entries/UPF_Firewall.html</link>
	<pubDate>Sun, 08 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/UPF_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method</title>
	<link>/entries/Password_Authentication_Protocol.html</link>
	<pubDate>Tue, 03 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Password_Authentication_Protocol.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Logic According to Harrison</title>
	<link>/entries/FOL_Harrison.html</link>
	<pubDate>Sun, 01 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Harrison.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent Refinement Algebra and Rely Quotients</title>
	<link>/entries/Concurrent_Ref_Alg.html</link>
	<pubDate>Fri, 30 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Ref_Alg.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Twelvefold Way</title>
	<link>/entries/Twelvefold_Way.html</link>
	<pubDate>Thu, 29 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Twelvefold_Way.html</guid>
	<description></description>
	</item>

	<item>
	<title>Proof Strategy Language</title>
	<link>/entries/Proof_Strategy_Language.html</link>
	<pubDate>Tue, 20 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Proof_Strategy_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>Paraconsistency</title>
	<link>/entries/Paraconsistency.html</link>
	<pubDate>Wed, 07 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Paraconsistency.html</guid>
	<description></description>
	</item>

	<item>
	<title>COMPLX: A Verification Framework for Concurrent Imperative Programs</title>
	<link>/entries/Complx.html</link>
	<pubDate>Tue, 29 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Complx.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Interpretation of Annotated Commands</title>
	<link>/entries/Abs_Int_ITP2012.html</link>
	<pubDate>Wed, 23 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Abs_Int_ITP2012.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separata: Isabelle tactics for Separation Algebra</title>
	<link>/entries/Separata.html</link>
	<pubDate>Wed, 16 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Separata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Knuth–Bendix Orders for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_KBOs.html</link>
	<pubDate>Sat, 12 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_KBOs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Nested Multisets, Hereditary Multisets, and Syntactic Ordinals</title>
	<link>/entries/Nested_Multisets_Ordinals.html</link>
	<pubDate>Sat, 12 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Nested_Multisets_Ordinals.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expressiveness of Deep Learning</title>
	<link>/entries/Deep_Learning.html</link>
	<pubDate>Thu, 10 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Deep_Learning.html</guid>
	<description></description>
	</item>

	<item>
	<title>Modal Logics for Nominal Transition Systems</title>
	<link>/entries/Modal_Logics_for_NTS.html</link>
	<pubDate>Tue, 25 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Modal_Logics_for_NTS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stable Matching</title>
	<link>/entries/Stable_Matching.html</link>
	<pubDate>Mon, 24 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stable_Matching.html</guid>
	<description></description>
	</item>

	<item>
	<title>LOFT — Verified Migration of Linux Firewalls to SDN</title>
	<link>/entries/LOFT.html</link>
	<pubDate>Fri, 21 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LOFT.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</title>
	<link>/entries/SPARCv8.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SPARCv8.html</guid>
	<description></description>
	</item>

	<item>
	<title>Source Coding Theorem</title>
	<link>/entries/Source_Coding_Theorem.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Source_Coding_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Factorization Algorithm of Berlekamp and Zassenhaus</title>
	<link>/entries/Berlekamp_Zassenhaus.html</link>
	<pubDate>Fri, 14 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Berlekamp_Zassenhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Intersecting Chords Theorem</title>
	<link>/entries/Chord_Segments.html</link>
	<pubDate>Tue, 11 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Chord_Segments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lp spaces</title>
	<link>/entries/Lp.html</link>
	<pubDate>Wed, 05 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher–Yates shuffle</title>
	<link>/entries/Fisher_Yates.html</link>
	<pubDate>Fri, 30 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Fisher_Yates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Allen's Interval Calculus</title>
	<link>/entries/Allen_Calculus.html</link>
	<pubDate>Thu, 29 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Allen_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Recursive Path Orders for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_RPOs.html</link>
	<pubDate>Fri, 23 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_RPOs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables Semantics</title>
	<link>/entries/Iptables_Semantics.html</link>
	<pubDate>Fri, 09 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Iptables_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Variant of the Superposition Calculus</title>
	<link>/entries/SuperCalc.html</link>
	<pubDate>Tue, 06 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SuperCalc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Algebras</title>
	<link>/entries/Stone_Algebras.html</link>
	<pubDate>Tue, 06 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stirling's formula</title>
	<link>/entries/Stirling_Formula.html</link>
	<pubDate>Thu, 01 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stirling_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>Routing</title>
	<link>/entries/Routing.html</link>
	<pubDate>Wed, 31 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Routing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simple Firewall</title>
	<link>/entries/Simple_Firewall.html</link>
	<pubDate>Wed, 24 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Simple_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths</title>
	<link>/entries/InfPathElimination.html</link>
	<pubDate>Thu, 18 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/InfPathElimination.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Edmonds-Karp Algorithm</title>
	<link>/entries/EdmondsKarp_Maxflow.html</link>
	<pubDate>Fri, 12 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/EdmondsKarp_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Imperative Refinement Framework</title>
	<link>/entries/Refine_Imperative_HOL.html</link>
	<pubDate>Mon, 08 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ptolemy's Theorem</title>
	<link>/entries/Ptolemys_Theorem.html</link>
	<pubDate>Sun, 07 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Ptolemys_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Surprise Paradox</title>
	<link>/entries/Surprise_Paradox.html</link>
	<pubDate>Sun, 17 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Surprise_Paradox.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pairing Heap</title>
	<link>/entries/Pairing_Heap.html</link>
	<pubDate>Thu, 14 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Pairing_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Framework for Verifying Depth-First Search Algorithms</title>
	<link>/entries/DFS_Framework.html</link>
	<pubDate>Tue, 05 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/DFS_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Chamber Complexes, Coxeter Systems, and Buildings</title>
	<link>/entries/Buildings.html</link>
	<pubDate>Fri, 01 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Buildings.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Resolution Calculus for First-Order Logic</title>
	<link>/entries/Resolution_FOL.html</link>
	<pubDate>Thu, 30 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Resolution_FOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Z Property</title>
	<link>/entries/Rewriting_Z.html</link>
	<pubDate>Thu, 30 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Rewriting_Z.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Security-Preserving Refinement for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Refinement.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>IP Addresses</title>
	<link>/entries/IP_Addresses.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/IP_Addresses.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Multisets</title>
	<link>/entries/Card_Multisets.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Multisets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory with Adjunctions and Limits</title>
	<link>/entries/Category3.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Category3.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Dependent Security Type System for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Type_Systems.html</link>
	<pubDate>Sat, 25 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Catalan Numbers</title>
	<link>/entries/Catalan_Numbers.html</link>
	<pubDate>Tue, 21 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Catalan_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Program Construction and Verification Components Based on Kleene Algebra</title>
	<link>/entries/Algebraic_VCs.html</link>
	<pubDate>Sat, 18 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Concurrent Composition</title>
	<link>/entries/Noninterference_Concurrent_Composition.html</link>
	<pubDate>Mon, 13 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Concurrent_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Machine Word Library</title>
	<link>/entries/Word_Lib.html</link>
	<pubDate>Thu, 09 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Word_Lib.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Decomposition</title>
	<link>/entries/Tree_Decomposition.html</link>
	<pubDate>Tue, 31 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Tree_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Equivalence Relations</title>
	<link>/entries/Card_Equiv_Relations.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Equiv_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>POSIX Lexing with Derivatives of Regular Expressions</title>
	<link>/entries/Posix-Lexing.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Posix-Lexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Perron-Frobenius Theorem for Spectral Radius Analysis</title>
	<link>/entries/Perron_Frobenius.html</link>
	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Perron_Frobenius.html</guid>
	<description></description>
	</item>

	<item>
	<title>The meta theory of the Incredible Proof Machine</title>
	<link>/entries/Incredible_Proof_Machine.html</link>
	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Incredible_Proof_Machine.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Constructive Proof for FLP</title>
	<link>/entries/FLP.html</link>
	<pubDate>Wed, 18 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/FLP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks</title>
	<link>/entries/MFMC_Countable.html</link>
	<pubDate>Mon, 09 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/MFMC_Countable.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Social Choice Theory</title>
	<link>/entries/Randomised_Social_Choice.html</link>
	<pubDate>Thu, 05 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_Social_Choice.html</guid>
	<description></description>
	</item>

	<item>
	<title>Spivey's Generalized Recurrence for Bell Numbers</title>
	<link>/entries/Bell_Numbers_Spivey.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Bell_Numbers_Spivey.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of SD-Efficiency and SD-Strategy-Proofness</title>
	<link>/entries/SDS_Impossibility.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SDS_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases Theory</title>
	<link>/entries/Groebner_Bases.html</link>
	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Bases.html</guid>
	<description></description>
	</item>

	<item>
	<title>No Faster-Than-Light Observers</title>
	<link>/entries/No_FTL_observers.html</link>
	<pubDate>Thu, 28 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/No_FTL_observers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formalisation of the Cocke-Younger-Kasami algorithm</title>
	<link>/entries/CYK.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/CYK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algorithms for Reduced Ordered Binary Decision Diagrams</title>
	<link>/entries/ROBDD.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/ROBDD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Sequential Composition</title>
	<link>/entries/Noninterference_Sequential_Composition.html</link>
	<pubDate>Tue, 26 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Sequential_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebras with Domain</title>
	<link>/entries/KAD.html</link>
	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional Resolution and Prime Implicates Generation</title>
	<link>/entries/PropResPI.html</link>
	<pubDate>Fri, 11 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/PropResPI.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cartan Fixed Point Theorems</title>
	<link>/entries/Cartan_FP.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Cartan_FP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Timed Automata</title>
	<link>/entries/Timed_Automata.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Temporal Logic</title>
	<link>/entries/LTL.html</link>
	<pubDate>Tue, 01 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
	<link>/entries/List_Update.html</link>
	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/List_Update.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Construction of Static Single Assignment Form</title>
	<link>/entries/Formal_SSA.html</link>
	<pubDate>Fri, 05 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Factorization</title>
	<link>/entries/Polynomial_Factorization.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Interpolation</title>
	<link>/entries/Polynomial_Interpolation.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Interpolation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knot Theory</title>
	<link>/entries/Knot_Theory.html</link>
	<pubDate>Wed, 20 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Knot_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tensor Product of Matrices</title>
	<link>/entries/Matrix_Tensor.html</link>
	<pubDate>Mon, 18 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix_Tensor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Number Partitions</title>
	<link>/entries/Card_Number_Partitions.html</link>
	<pubDate>Thu, 14 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Number_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Basic Geometric Properties of Triangles</title>
	<link>/entries/Triangle.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Triangle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Descartes' Rule of Signs</title>
	<link>/entries/Descartes_Sign_Rule.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Descartes_Sign_Rule.html</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville numbers</title>
	<link>/entries/Liouville_Numbers.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Liouville_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Divergence of the Prime Harmonic Series</title>
	<link>/entries/Prime_Harmonic_Series.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Harmonic_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic Numbers in Isabelle/HOL</title>
	<link>/entries/Algebraic_Numbers.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Applicative Lifting</title>
	<link>/entries/Applicative_Lifting.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Applicative_Lifting.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Stern-Brocot Tree</title>
	<link>/entries/Stern_Brocot.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Stern_Brocot.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Set Partitions</title>
	<link>/entries/Card_Partitions.html</link>
	<pubDate>Sat, 12 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Latin Square</title>
	<link>/entries/Latin_Square.html</link>
	<pubDate>Wed, 02 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Latin_Square.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ergodic Theory</title>
	<link>/entries/Ergodic_Theory.html</link>
	<pubDate>Tue, 01 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Ergodic_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Euler's Partition Theorem</title>
	<link>/entries/Euler_Partition.html</link>
	<pubDate>Thu, 19 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_Partition.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Tortoise and Hare Algorithm</title>
	<link>/entries/TortoiseHare.html</link>
	<pubDate>Wed, 18 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/TortoiseHare.html</guid>
	<description></description>
	</item>

	<item>
	<title>Planarity Certificates</title>
	<link>/entries/Planarity_Certificates.html</link>
	<pubDate>Wed, 11 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Planarity_Certificates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Positional Determinacy of Parity Games</title>
	<link>/entries/Parity_Game.html</link>
	<pubDate>Mon, 02 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Parity_Game.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Meta-Model for the Isabelle API</title>
	<link>/entries/Isabelle_Meta_Model.html</link>
	<pubDate>Wed, 16 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Meta_Model.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear Temporal Logic to Deterministic (Generalized) Rabin Automata</title>
	<link>/entries/LTL_to_DRA.html</link>
	<pubDate>Fri, 04 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices, Jordan Normal Forms, and Spectral Radius Theory</title>
	<link>/entries/Jordan_Normal_Form.html</link>
	<pubDate>Fri, 21 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing Diagrams II</title>
	<link>/entries/Decreasing-Diagrams-II.html</link>
	<pubDate>Thu, 20 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Decreasing-Diagrams-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inductive Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Inductive_Unwinding.html</link>
	<pubDate>Tue, 18 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Inductive_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Representations of Finite Groups</title>
	<link>/entries/Rep_Fin_Groups.html</link>
	<pubDate>Wed, 12 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Rep_Fin_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysing and Comparing Encodability Criteria for Process Calculi</title>
	<link>/entries/Encodability_Process_Calculi.html</link>
	<pubDate>Mon, 10 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Encodability_Process_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating Cases from Labeled Subgoals</title>
	<link>/entries/Case_Labeling.html</link>
	<pubDate>Tue, 21 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Case_Labeling.html</guid>
	<description></description>
	</item>

	<item>
	<title>Landau Symbols</title>
	<link>/entries/Landau_Symbols.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Landau_Symbols.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Akra-Bazzi theorem and the Master theorem</title>
	<link>/entries/Akra_Bazzi.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Akra_Bazzi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite Normal Form</title>
	<link>/entries/Hermite.html</link>
	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derangements Formula</title>
	<link>/entries/Derangements.html</link>
	<pubDate>Sat, 27 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Derangements.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Multirelations</title>
	<link>/entries/Multirelations.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Multirelations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Reasoning about Lists via List Interleaving</title>
	<link>/entries/List_Interleaving.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/List_Interleaving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generic Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Generic_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Generic_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Ipurge Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Ipurge_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Ipurge_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Parameterized Dynamic Tables</title>
	<link>/entries/Dynamic_Tables.html</link>
	<pubDate>Sun, 07 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Dynamic_Tables.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derivatives of Logical Formulas</title>
	<link>/entries/Formula_Derivatives.html</link>
	<pubDate>Thu, 28 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Formula_Derivatives.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Zoo of Probabilistic Systems</title>
	<link>/entries/Probabilistic_System_Zoo.html</link>
	<pubDate>Wed, 27 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_System_Zoo.html</guid>
	<description></description>
	</item>

	<item>
	<title>VCG - Combinatorial Vickrey-Clarke-Groves Auctions</title>
	<link>/entries/Vickrey_Clarke_Groves.html</link>
	<pubDate>Thu, 30 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Vickrey_Clarke_Groves.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Lattices</title>
	<link>/entries/Residuated_Lattices.html</link>
	<pubDate>Wed, 15 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Residuated_Lattices.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent IMP</title>
	<link>/entries/ConcurrentIMP.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentIMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO</title>
	<link>/entries/ConcurrentGC.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentGC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Trie</title>
	<link>/entries/Trie.html</link>
	<pubDate>Mon, 30 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Trie.html</guid>
	<description></description>
	</item>

	<item>
	<title>Consensus Refined</title>
	<link>/entries/Consensus_Refined.html</link>
	<pubDate>Wed, 18 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Consensus_Refined.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving class instances for datatypes</title>
	<link>/entries/Deriving.html</link>
	<pubDate>Wed, 11 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safety of Call Arity</title>
	<link>/entries/Call_Arity.html</link>
	<pubDate>Fri, 20 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Call_Arity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Echelon Form</title>
	<link>/entries/Echelon_Form.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Echelon_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>QR Decomposition</title>
	<link>/entries/QR_Decomposition.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/QR_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Automata in Hereditarily Finite Set Theory</title>
	<link>/entries/Finite_Automata_HF.html</link>
	<pubDate>Thu, 05 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Automata_HF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the UpDown Scheme</title>
	<link>/entries/UpDown_Scheme.html</link>
	<pubDate>Wed, 28 Jan 2015 00:00:00 +0000</pubDate>

	<guid>/entries/UpDown_Scheme.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Unified Policy Framework (UPF)</title>
	<link>/entries/UPF.html</link>
	<pubDate>Fri, 28 Nov 2014 00:00:00 +0000</pubDate>

	<guid>/entries/UPF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Loop freedom of the (untimed) AODV routing protocol</title>
	<link>/entries/AODV.html</link>
	<pubDate>Thu, 23 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AODV.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting Definition Option</title>
	<link>/entries/Lifting_Definition_Option.html</link>
	<pubDate>Mon, 13 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_Definition_Option.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion in HOL with Code Generation</title>
	<link>/entries/Stream_Fusion_Code.html</link>
	<pubDate>Fri, 10 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Stream_Fusion_Code.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Compiler for Probability Density Functions</title>
	<link>/entries/Density_Compiler.html</link>
	<pubDate>Thu, 09 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Density_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Refinement Calculus for Reactive Systems</title>
	<link>/entries/RefinementReactive.html</link>
	<pubDate>Wed, 08 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/RefinementReactive.html</guid>
	<description></description>
	</item>

	<item>
	<title>Certification Monads</title>
	<link>/entries/Certification_Monads.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Certification_Monads.html</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/entries/XML.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/XML.html</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative Insertion Sort</title>
	<link>/entries/Imperative_Insertion_Sort.html</link>
	<pubDate>Thu, 25 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Imperative_Insertion_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sturm–Tarski Theorem</title>
	<link>/entries/Sturm_Tarski.html</link>
	<pubDate>Fri, 19 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Sturm_Tarski.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cayley-Hamilton Theorem</title>
	<link>/entries/Cayley_Hamilton.html</link>
	<pubDate>Mon, 15 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Cayley_Hamilton.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Jordan-Hölder Theorem</title>
	<link>/entries/Jordan_Hoelder.html</link>
	<pubDate>Tue, 09 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Hoelder.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Queues Based on Braun Trees</title>
	<link>/entries/Priority_Queue_Braun.html</link>
	<pubDate>Thu, 04 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Queue_Braun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Algorithm and Its Applications</title>
	<link>/entries/Gauss_Jordan.html</link>
	<pubDate>Wed, 03 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Jordan.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Valued Special Functions: Upper and Lower Bounds</title>
	<link>/entries/Special_Function_Bounds.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Special_Function_Bounds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Vector Spaces</title>
	<link>/entries/VectorSpace.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/VectorSpace.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skew Heap</title>
	<link>/entries/Skew_Heap.html</link>
	<pubDate>Wed, 13 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Skew_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Splay Tree</title>
	<link>/entries/Splay_Tree.html</link>
	<pubDate>Tue, 12 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Splay_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Haskell's Show Class in Isabelle/HOL</title>
	<link>/entries/Show.html</link>
	<pubDate>Tue, 29 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Show.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Specification of a Generic Separation Kernel</title>
	<link>/entries/CISC-Kernel.html</link>
	<pubDate>Fri, 18 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CISC-Kernel.html</guid>
	<description></description>
	</item>

	<item>
	<title>pGCL for Isabelle</title>
	<link>/entries/pGCL.html</link>
	<pubDate>Sun, 13 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/pGCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amortized Complexity Verified</title>
	<link>/entries/Amortized_Complexity.html</link>
	<pubDate>Mon, 07 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Amortized_Complexity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Network Security Policy Verification</title>
	<link>/entries/Network_Security_Policy_Verification.html</link>
	<pubDate>Fri, 04 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Network_Security_Policy_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pop-Refinement</title>
	<link>/entries/Pop_Refinement.html</link>
	<pubDate>Thu, 03 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Pop_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decision Procedures for MSO on Words Based on Derivatives of Regular Expressions</title>
	<link>/entries/MSO_Regex_Equivalence.html</link>
	<pubDate>Thu, 12 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/MSO_Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Boolean Expression Checkers</title>
	<link>/entries/Boolean_Expression_Checkers.html</link>
	<pubDate>Sun, 08 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Boolean_Expression_Checkers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Fully Verified Executable LTL Model Checker</title>
	<link>/entries/CAVA_LTL_Modelchecker.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_LTL_Modelchecker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear-Time Temporal Logic to Generalized Büchi Automata</title>
	<link>/entries/LTL_to_GBA.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_GBA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Promela Formalization</title>
	<link>/entries/Promela.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Promela.html</guid>
	<description></description>
	</item>

	<item>
	<title>The CAVA Automata Library</title>
	<link>/entries/CAVA_Automata.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</title>
	<link>/entries/Gabow_SCC.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gabow_SCC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference Security in Communicating Sequential Processes</title>
	<link>/entries/Noninterference_CSP.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive closure according to Roy-Floyd-Warshall</title>
	<link>/entries/Roy_Floyd_Warshall.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Roy_Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Algebras</title>
	<link>/entries/Regular_Algebras.html</link>
	<pubDate>Wed, 21 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation and Analysis of Component Dependencies</title>
	<link>/entries/ComponentDependencies.html</link>
	<pubDate>Mon, 28 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/ComponentDependencies.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Assumptions and Guarantees for Compositional Noninterference</title>
	<link>/entries/SIFUM_Type_Systems.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Declassification with WHAT-and-WHERE-Security</title>
	<link>/entries/WHATandWHERE_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/WHATandWHERE_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Strong Security</title>
	<link>/entries/Strong_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Strong_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded-Deducibility Security</title>
	<link>/entries/Bounded_Deducibility_Security.html</link>
	<pubDate>Tue, 22 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Bounded_Deducibility_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A shallow embedding of HyperCTL*</title>
	<link>/entries/HyperCTL.html</link>
	<pubDate>Wed, 16 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/HyperCTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Completeness</title>
	<link>/entries/Abstract_Completeness.html</link>
	<pubDate>Wed, 16 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract_Completeness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Discrete Summation</title>
	<link>/entries/Discrete_Summation.html</link>
	<pubDate>Sun, 13 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Discrete_Summation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax and semantics of a GPU kernel programming language</title>
	<link>/entries/GPU_Kernel_PL.html</link>
	<pubDate>Thu, 03 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/GPU_Kernel_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Noninterference</title>
	<link>/entries/Probabilistic_Noninterference.html</link>
	<pubDate>Tue, 11 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mechanization of the Algebra for Wireless Networks (AWN)</title>
	<link>/entries/AWN.html</link>
	<pubDate>Sat, 08 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AWN.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mutually Recursive Partial Functions</title>
	<link>/entries/Partial_Function_MR.html</link>
	<pubDate>Tue, 18 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Function_MR.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Random Graphs -- Subgraph Containment</title>
	<link>/entries/Random_Graph_Subgraph_Threshold.html</link>
	<pubDate>Thu, 13 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Random_Graph_Subgraph_Threshold.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of Selection and Heap Sort Using Locales</title>
	<link>/entries/Selection_Heap_Sort.html</link>
	<pubDate>Tue, 11 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Selection_Heap_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Affine Arithmetic</title>
	<link>/entries/Affine_Arithmetic.html</link>
	<pubDate>Fri, 07 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Affine_Arithmetic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Implementing field extensions of the form Q[sqrt(b)]</title>
	<link>/entries/Real_Impl.html</link>
	<pubDate>Thu, 06 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Impl.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unified Decision Procedures for Regular Expression Equivalence</title>
	<link>/entries/Regex_Equivalence.html</link>
	<pubDate>Thu, 30 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secondary Sylow Theorems</title>
	<link>/entries/Secondary_Sylow.html</link>
	<pubDate>Tue, 28 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Secondary_Sylow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relation Algebra</title>
	<link>/entries/Relation_Algebra.html</link>
	<pubDate>Sat, 25 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Relation_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
	<link>/entries/KAT_and_DRA.html</link>
	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/KAT_and_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5</title>
	<link>/entries/Featherweight_OCL.html</link>
	<pubDate>Thu, 16 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Featherweight_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Properties of Crypto-Based Components</title>
	<link>/entries/CryptoBasedCompositionalProperties.html</link>
	<pubDate>Sat, 11 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CryptoBasedCompositionalProperties.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm's Theorem</title>
	<link>/entries/Sturm_Sequences.html</link>
	<pubDate>Sat, 11 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Sturm_Sequences.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Method for the Proof of Theorems on Tail-recursive Functions</title>
	<link>/entries/Tail_Recursive_Functions.html</link>
	<pubDate>Sun, 01 Dec 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Tail_Recursive_Functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gödel's Incompleteness Theorems</title>
	<link>/entries/Incompleteness.html</link>
	<pubDate>Sun, 17 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Incompleteness.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hereditarily Finite Sets</title>
	<link>/entries/HereditarilyFinite.html</link>
	<pubDate>Sun, 17 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/HereditarilyFinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Codatatype of Formal Languages</title>
	<link>/entries/Coinductive_Languages.html</link>
	<pubDate>Fri, 15 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive_Languages.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Processing Components: Isabelle/HOL Formalisation and Case Studies</title>
	<link>/entries/FocusStreamsCaseStudies.html</link>
	<pubDate>Thu, 14 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/FocusStreamsCaseStudies.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gödel's God in Isabelle/HOL</title>
	<link>/entries/GoedelGod.html</link>
	<pubDate>Tue, 12 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/GoedelGod.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing Diagrams</title>
	<link>/entries/Decreasing-Diagrams.html</link>
	<pubDate>Fri, 01 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Decreasing-Diagrams.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automatic Data Refinement</title>
	<link>/entries/Automatic_Refinement.html</link>
	<pubDate>Wed, 02 Oct 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Automatic_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Native Word</title>
	<link>/entries/Native_Word.html</link>
	<pubDate>Tue, 17 Sep 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Native_Word.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of IEEE Floating Point Arithmetic</title>
	<link>/entries/IEEE_Floating_Point.html</link>
	<pubDate>Sat, 27 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/IEEE_Floating_Point.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lehmer's Theorem</title>
	<link>/entries/Lehmer.html</link>
	<pubDate>Mon, 22 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Lehmer.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pratt's Primality Certificates</title>
	<link>/entries/Pratt_Certificate.html</link>
	<pubDate>Mon, 22 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Pratt_Certificate.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Königsberg Bridge Problem and the Friendship Theorem</title>
	<link>/entries/Koenigsberg_Friendship.html</link>
	<pubDate>Fri, 19 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Koenigsberg_Friendship.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sound and Complete Sort Encodings for First-Order Logic</title>
	<link>/entries/Sort_Encodings.html</link>
	<pubDate>Thu, 27 Jun 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Sort_Encodings.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Axiomatic Characterization of the Single-Source Shortest Path Problem</title>
	<link>/entries/ShortestPath.html</link>
	<pubDate>Wed, 22 May 2013 00:00:00 +0000</pubDate>

	<guid>/entries/ShortestPath.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Theory</title>
	<link>/entries/Graph_Theory.html</link>
	<pubDate>Sun, 28 Apr 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Graph_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Light-weight Containers</title>
	<link>/entries/Containers.html</link>
	<pubDate>Mon, 15 Apr 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Containers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Nominal 2</title>
	<link>/entries/Nominal2.html</link>
	<pubDate>Thu, 21 Feb 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Nominal2.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</title>
	<link>/entries/Launchbury.html</link>
	<pubDate>Thu, 31 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Launchbury.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ribbon Proofs</title>
	<link>/entries/Ribbon_Proofs.html</link>
	<pubDate>Sat, 19 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Ribbon_Proofs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Rank-Nullity Theorem in Linear Algebra</title>
	<link>/entries/Rank_Nullity_Theorem.html</link>
	<pubDate>Wed, 16 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Rank_Nullity_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra</title>
	<link>/entries/Kleene_Algebra.html</link>
	<pubDate>Tue, 15 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Kleene_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Computing N-th Roots using the Babylonian Method</title>
	<link>/entries/Sqrt_Babylonian.html</link>
	<pubDate>Thu, 03 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Sqrt_Babylonian.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Separation Logic Framework for Imperative HOL</title>
	<link>/entries/Separation_Logic_Imperative_HOL.html</link>
	<pubDate>Wed, 14 Nov 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Open Induction</title>
	<link>/entries/Open_Induction.html</link>
	<pubDate>Fri, 02 Nov 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Open_Induction.html</guid>
	<description></description>
	</item>

	<item>
	<title>The independence of Tarski's Euclidean axiom</title>
	<link>/entries/Tarskis_Geometry.html</link>
	<pubDate>Tue, 30 Oct 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Tarskis_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bondy's Theorem</title>
	<link>/entries/Bondy.html</link>
	<pubDate>Sat, 27 Oct 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Bondy.html</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic Noninterference</title>
	<link>/entries/Possibilistic_Noninterference.html</link>
	<pubDate>Mon, 10 Sep 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Possibilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating linear orders for datatypes</title>
	<link>/entries/Datatype_Order_Generator.html</link>
	<pubDate>Tue, 07 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Datatype_Order_Generator.html</guid>
	<description></description>
	</item>

	<item>
	<title>Proving the Impossibility of Trisecting an Angle and Doubling the Cube</title>
	<link>/entries/Impossible_Geometry.html</link>
	<pubDate>Sun, 05 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Impossible_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</title>
	<link>/entries/Heard_Of.html</link>
	<pubDate>Fri, 27 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Heard_Of.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logical Relations for PCF</title>
	<link>/entries/PCF.html</link>
	<pubDate>Sun, 01 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/PCF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Type Constructor Classes and Monad Transformers</title>
	<link>/entries/Tycon.html</link>
	<pubDate>Tue, 26 Jun 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Tycon.html</guid>
	<description></description>
	</item>

	<item>
	<title>CCS in nominal logic</title>
	<link>/entries/CCS.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/CCS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Psi-calculi in Isabelle</title>
	<link>/entries/Psi_Calculi.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Psi_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>The pi-calculus in nominal logic</title>
	<link>/entries/Pi_Calculus.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Circus</title>
	<link>/entries/Circus.html</link>
	<pubDate>Sun, 27 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Circus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separation Algebra</title>
	<link>/entries/Separation_Algebra.html</link>
	<pubDate>Fri, 11 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stuttering Equivalence</title>
	<link>/entries/Stuttering_Equivalence.html</link>
	<pubDate>Mon, 07 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Stuttering_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive Study of Confidentiality</title>
	<link>/entries/Inductive_Confidentiality.html</link>
	<pubDate>Wed, 02 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Confidentiality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinary Differential Equations</title>
	<link>/entries/Ordinary_Differential_Equations.html</link>
	<pubDate>Thu, 26 Apr 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinary_Differential_Equations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Well-Quasi-Orders</title>
	<link>/entries/Well_Quasi_Orders.html</link>
	<pubDate>Fri, 13 Apr 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Well_Quasi_Orders.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abortable Linearizable Modules</title>
	<link>/entries/Abortable_Linearizable_Modules.html</link>
	<pubDate>Thu, 01 Mar 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Abortable_Linearizable_Modules.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures</title>
	<link>/entries/Transitive-Closure-II.html</link>
	<pubDate>Wed, 29 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Probabilistic Proof of the Girth-Chromatic Number Theorem</title>
	<link>/entries/Girth_Chromatic.html</link>
	<pubDate>Mon, 06 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Girth_Chromatic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dijkstra's Shortest Path Algorithm</title>
	<link>/entries/Dijkstra_Shortest_Path.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Dijkstra_Shortest_Path.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refinement for Monadic Programs</title>
	<link>/entries/Refine_Monadic.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Monadic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Models</title>
	<link>/entries/Markov_Models.html</link>
	<pubDate>Tue, 03 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Definitional Encoding of TLA* in Isabelle/HOL</title>
	<link>/entries/TLA.html</link>
	<pubDate>Sat, 19 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/TLA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient Mergesort</title>
	<link>/entries/Efficient-Mergesort.html</link>
	<pubDate>Wed, 09 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Efficient-Mergesort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebra of Monotonic Boolean Transformers</title>
	<link>/entries/MonoBoolTranAlgebra.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/MonoBoolTranAlgebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lattice Properties</title>
	<link>/entries/LatticeProperties.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/LatticeProperties.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pseudo Hoops</title>
	<link>/entries/PseudoHoops.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/PseudoHoops.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Myhill-Nerode Theorem Based on Regular Expressions</title>
	<link>/entries/Myhill-Nerode.html</link>
	<pubDate>Fri, 26 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Myhill-Nerode.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
	<link>/entries/Gauss-Jordan-Elim-Fun.html</link>
	<pubDate>Fri, 19 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss-Jordan-Elim-Fun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum Cardinality Matching</title>
	<link>/entries/Max-Card-Matching.html</link>
	<pubDate>Thu, 21 Jul 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Max-Card-Matching.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knowledge-based programs</title>
	<link>/entries/KBPs.html</link>
	<pubDate>Tue, 17 May 2011 00:00:00 +0000</pubDate>

	<guid>/entries/KBPs.html</guid>
	<description></description>
	</item>

	<item>
	<title>The General Triangle Is Unique</title>
	<link>/entries/General-Triangle.html</link>
	<pubDate>Fri, 01 Apr 2011 00:00:00 +0000</pubDate>

	<guid>/entries/General-Triangle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures of Finite Relations</title>
	<link>/entries/Transitive-Closure.html</link>
	<pubDate>Mon, 14 Mar 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure.html</guid>
	<description></description>
	</item>

	<item>
	<title>AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</title>
	<link>/entries/AutoFocus-Stream.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/AutoFocus-Stream.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infinite Lists</title>
	<link>/entries/List-Infinite.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/List-Infinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interval Temporal Logic on Natural Numbers</title>
	<link>/entries/Nat-Interval-Logic.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Nat-Interval-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lightweight Java</title>
	<link>/entries/LightweightJava.html</link>
	<pubDate>Mon, 07 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/LightweightJava.html</guid>
	<description></description>
	</item>

	<item>
	<title>RIPEMD-160</title>
	<link>/entries/RIPEMD-160-SPARK.html</link>
	<pubDate>Mon, 10 Jan 2011 00:00:00 +0000</pubDate>

	<guid>/entries/RIPEMD-160-SPARK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower Semicontinuous Functions</title>
	<link>/entries/Lower_Semicontinuous.html</link>
	<pubDate>Sat, 08 Jan 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Lower_Semicontinuous.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hall's Marriage Theorem</title>
	<link>/entries/Marriage.html</link>
	<pubDate>Fri, 17 Dec 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Marriage.html</guid>
	<description></description>
	</item>

	<item>
	<title>Shivers' Control Flow Analysis</title>
	<link>/entries/Shivers-CFA.html</link>
	<pubDate>Tue, 16 Nov 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Shivers-CFA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial Heaps and Skew Binomial Heaps</title>
	<link>/entries/Binomial-Heaps.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Heaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finger Trees</title>
	<link>/entries/Finger-Trees.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Finger-Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Functional Binomial Queues</title>
	<link>/entries/Binomial-Queues.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Queues.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Normalization of Moggis's Computational Metalanguage</title>
	<link>/entries/Lam-ml-Normalization.html</link>
	<pubDate>Sun, 29 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Lam-ml-Normalization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Multivariate Polynomials</title>
	<link>/entries/Polynomials.html</link>
	<pubDate>Tue, 10 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Statecharts using Hierarchical Automata</title>
	<link>/entries/Statecharts.html</link>
	<pubDate>Sun, 08 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Statecharts.html</guid>
	<description></description>
	</item>

	<item>
	<title>Free Groups</title>
	<link>/entries/Free-Groups.html</link>
	<pubDate>Thu, 24 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Free-Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory</title>
	<link>/entries/Category2.html</link>
	<pubDate>Sun, 20 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Category2.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Matrix Operations on Matrices of Arbitrary Dimensions</title>
	<link>/entries/Matrix.html</link>
	<pubDate>Thu, 17 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Rewriting</title>
	<link>/entries/Abstract-Rewriting.html</link>
	<pubDate>Mon, 14 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract-Rewriting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Semantics and Data Refinement of Invariant Based Programs</title>
	<link>/entries/DataRefinementIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/DataRefinementIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</title>
	<link>/entries/GraphMarkingIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/GraphMarkingIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Complete Proof of the Robbins Conjecture</title>
	<link>/entries/Robbins-Conjecture.html</link>
	<pubDate>Sat, 22 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Robbins-Conjecture.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Sets and Expressions</title>
	<link>/entries/Regular-Sets.html</link>
	<pubDate>Wed, 12 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Regular-Sets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Locally Nameless Sigma Calculus</title>
	<link>/entries/Locally-Nameless-Sigma.html</link>
	<pubDate>Fri, 30 Apr 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Locally-Nameless-Sigma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Free Boolean Algebra</title>
	<link>/entries/Free-Boolean-Algebra.html</link>
	<pubDate>Mon, 29 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Free-Boolean-Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inter-Procedural Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing_Inter.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing_Inter.html</guid>
	<description></description>
	</item>

	<item>
	<title>List Index</title>
	<link>/entries/List-Index.html</link>
	<pubDate>Sat, 20 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/List-Index.html</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive</title>
	<link>/entries/Coinductive.html</link>
	<pubDate>Fri, 12 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Fast SAT Solver for Isabelle in Standard ML</title>
	<link>/entries/DPT-SAT-Solver.html</link>
	<pubDate>Wed, 09 Dec 2009 00:00:00 +0000</pubDate>

	<guid>/entries/DPT-SAT-Solver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Logic-Automaton Connection</title>
	<link>/entries/Presburger-Automata.html</link>
	<pubDate>Thu, 03 Dec 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Presburger-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Collections Framework</title>
	<link>/entries/Collections.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Collections.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Automata</title>
	<link>/entries/Tree-Automata.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Tree-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Perfect Number Theorem</title>
	<link>/entries/Perfect-Number-Thm.html</link>
	<pubDate>Sun, 22 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Perfect-Number-Thm.html</guid>
	<description></description>
	</item>

	<item>
	<title>Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</title>
	<link>/entries/HRB-Slicing.html</link>
	<pubDate>Fri, 13 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/HRB-Slicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Worker/Wrapper Transformation</title>
	<link>/entries/WorkerWrapper.html</link>
	<pubDate>Fri, 30 Oct 2009 00:00:00 +0000</pubDate>

	<guid>/entries/WorkerWrapper.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinals and Cardinals</title>
	<link>/entries/Ordinals_and_Cardinals.html</link>
	<pubDate>Tue, 01 Sep 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinals_and_Cardinals.html</guid>
	<description></description>
	</item>

	<item>
	<title>Invertibility in Sequent Calculi</title>
	<link>/entries/SequentInvertibility.html</link>
	<pubDate>Fri, 28 Aug 2009 00:00:00 +0000</pubDate>

	<guid>/entries/SequentInvertibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Example of a Cofinitary Group in Isabelle/HOL</title>
	<link>/entries/CofGroups.html</link>
	<pubDate>Tue, 04 Aug 2009 00:00:00 +0000</pubDate>

	<guid>/entries/CofGroups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Code Generation for Functions as Data</title>
	<link>/entries/FinFun.html</link>
	<pubDate>Wed, 06 May 2009 00:00:00 +0000</pubDate>

	<guid>/entries/FinFun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion</title>
	<link>/entries/Stream-Fusion.html</link>
	<pubDate>Wed, 29 Apr 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Stream-Fusion.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Bytecode Logic for JML and Types</title>
	<link>/entries/BytecodeLogicJmlTypes.html</link>
	<pubDate>Fri, 12 Dec 2008 00:00:00 +0000</pubDate>

	<guid>/entries/BytecodeLogicJmlTypes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secure information flow and program logics</title>
	<link>/entries/SIFPL.html</link>
	<pubDate>Mon, 10 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SIFPL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Some classical results in Social Choice Theory</title>
	<link>/entries/SenSocialChoice.html</link>
	<pubDate>Sun, 09 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SenSocialChoice.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fun With Tilings</title>
	<link>/entries/FunWithTilings.html</link>
	<pubDate>Fri, 07 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/FunWithTilings.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Textbook Proof of Huffman's Algorithm</title>
	<link>/entries/Huffman.html</link>
	<pubDate>Wed, 15 Oct 2008 00:00:00 +0000</pubDate>

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	<title>A Correctness Proof for the Volpano/Smith Security Typing System</title>
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	<title>Formal Verification of Modern SAT Solvers</title>
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	<title>Recursion Theory I</title>
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	<title>A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</title>
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	<pubDate>Fri, 29 Feb 2008 00:00:00 +0000</pubDate>

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	<title>Quantifier Elimination for Linear Arithmetic</title>
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	<title>Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors</title>
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	<pubDate>Fri, 14 Dec 2007 00:00:00 +0000</pubDate>

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	<title>Jinja with Threads</title>
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	<title>Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</title>
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	<pubDate>Sun, 12 Aug 2007 00:00:00 +0000</pubDate>

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	<title>Sums of Two and Four Squares</title>
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	<pubDate>Sun, 12 Aug 2007 00:00:00 +0000</pubDate>

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	<item>
	<title>Fundamental Properties of Valuation Theory and Hensel's Lemma</title>
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	<pubDate>Wed, 08 Aug 2007 00:00:00 +0000</pubDate>

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	<title>First-Order Logic According to Fitting</title>
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	<pubDate>Thu, 02 Aug 2007 00:00:00 +0000</pubDate>

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	<pubDate>Sat, 09 Sep 2006 00:00:00 +0000</pubDate>

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	<title>Abstract Hoare Logics</title>
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	<pubDate>Tue, 08 Aug 2006 00:00:00 +0000</pubDate>

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	<title>Flyspeck I: Tame Graphs</title>
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	<pubDate>Mon, 22 May 2006 00:00:00 +0000</pubDate>

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	<pubDate>Mon, 15 May 2006 00:00:00 +0000</pubDate>

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	<pubDate>Fri, 31 Mar 2006 00:00:00 +0000</pubDate>

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	<title>Instances of Schneider's generalized protocol of clock synchronization</title>
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	<pubDate>Wed, 15 Mar 2006 00:00:00 +0000</pubDate>

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	<pubDate>Tue, 14 Mar 2006 00:00:00 +0000</pubDate>

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	<pubDate>Fri, 11 Nov 2005 00:00:00 +0000</pubDate>

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	<pubDate>Wed, 12 Oct 2005 00:00:00 +0000</pubDate>

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	<title>Formalization of a Generalized Protocol for Clock Synchronization</title>
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	<pubDate>Fri, 24 Jun 2005 00:00:00 +0000</pubDate>

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	<title>Proving the Correctness of Disk Paxos</title>
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	<pubDate>Wed, 22 Jun 2005 00:00:00 +0000</pubDate>

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	<item>
	<title>Jive Data and Store Model</title>
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	<pubDate>Mon, 20 Jun 2005 00:00:00 +0000</pubDate>

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	<title>Jinja is not Java</title>
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	<pubDate>Wed, 01 Jun 2005 00:00:00 +0000</pubDate>

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	<item>
	<title>SHA1, RSA, PSS and more</title>
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	<pubDate>Mon, 02 May 2005 00:00:00 +0000</pubDate>

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	<item>
	<title>Category Theory to Yoneda's Lemma</title>
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	<pubDate>Thu, 21 Apr 2005 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>File Refinement</title>
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	<pubDate>Thu, 09 Dec 2004 00:00:00 +0000</pubDate>

	<guid>/entries/FileRefinement.html</guid>
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	</item>

	<item>
	<title>Integration theory and random variables</title>
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	<pubDate>Fri, 19 Nov 2004 00:00:00 +0000</pubDate>

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	<item>
	<title>A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic</title>
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	<pubDate>Tue, 28 Sep 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Verified-Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Completeness theorem</title>
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	<pubDate>Mon, 20 Sep 2004 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Ramsey's theorem, infinitary version</title>
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	<pubDate>Mon, 20 Sep 2004 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Compiling Exceptions Correctly</title>
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	<pubDate>Fri, 09 Jul 2004 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Depth First Search</title>
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	<pubDate>Thu, 24 Jun 2004 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Groups, Rings and Modules</title>
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	<pubDate>Tue, 18 May 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Group-Ring-Module.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy Lists II</title>
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	<pubDate>Mon, 26 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Lazy-Lists-II.html</guid>
	<description></description>
	</item>

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	<title>Topology</title>
	<link>/entries/Topology.html</link>
	<pubDate>Mon, 26 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Topology.html</guid>
	<description></description>
	</item>

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	<title>Binary Search Trees</title>
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	<pubDate>Mon, 05 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/BinarySearchTree.html</guid>
	<description></description>
	</item>

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	<title>Functional Automata</title>
	<link>/entries/Functional-Automata.html</link>
	<pubDate>Tue, 30 Mar 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Functional-Automata.html</guid>
	<description></description>
	</item>

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	<title>AVL Trees</title>
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	<pubDate>Fri, 19 Mar 2004 00:00:00 +0000</pubDate>

	<guid>/entries/AVL-Trees.html</guid>
	<description></description>
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	<item>
	<title>Mini ML</title>
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	<pubDate>Fri, 19 Mar 2004 00:00:00 +0000</pubDate>

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	The Archive of Formal Proofs is a collection of proof libraries, examples, and larger scientific developments,
	mechanically checked in the theorem prover <a href="https://isabelle.in.tum.de/">Isabelle</a>.
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	<h2 class="year">2023</h2>
	<article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="./entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>
	+ by <a href="./authors/leek">Kevin Lee</a>, <a href="./authors/yez">Zhengkun Ye</a> and <a href="./authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+ </article>
	+ <article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="./entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>
	+ by <a href="./authors/struth">Georg Struth</a> and <a href="./authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+ </article>
	+ <article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="./entries/Earley_Parser.html">Earley Parser</a></h5> <br>
	+ by <a href="./authors/rau">Martin Rau</a></div>
	+ <span class="date">
	+ Jul 16
	+ </span>
	+ </article>
	+ <article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="./entries/Gray_Codes.html">Gray Codes for Arbitrary Numeral Systems</a></h5> <br>
	by <a href="./authors/spitz">Maximilian Spitz</a></div>
	<span class="date">
	Jul 11
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	<h5><a class="title" href="./entries/Executable_Randomized_Algorithms.html">Executable Randomized Algorithms</a></h5> <br>
	by <a href="./authors/karayel">Emin Karayel</a> and <a href="./authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jun 19
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	<h5><a class="title" href="./entries/DCR-ExecutionEquivalence.html">DCR Syntax and Execution Equivalent Markings</a></h5> <br>
	by <a href="./authors/christfort">Axel Christfort</a> and <a href="./authors/debois">Søren Debois</a></div>
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	Jun 16
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	<h5><a class="title" href="./entries/Zeckendorf.html">Zeckendorf’s Theorem</a></h5> <br>
	by <a href="./authors/dalvit">Christian Dalvit</a></div>
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	Jun 12
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	<h5><a class="title" href="./entries/Crypto_Standards.html">Cryptographic Standards</a></h5> <br>
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	<span class="date">
	Jun 06
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	<h5><a class="title" href="./entries/Efficient_Weighted_Path_Order.html">A Verified Efficient Implementation of the Weighted Path Order</a></h5> <br>
	by <a href="./authors/thiemann">René Thiemann</a> and <a href="./authors/wenninger">Elias Wenninger</a></div>
	<span class="date">
	Jun 01
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	<h5><a class="title" href="./entries/Directed_Sets.html">Formalizing Results on Directed Sets</a></h5> <br>
	by <a href="./authors/yamada">Akihisa Yamada</a> and <a href="./authors/dubut">Jérémy Dubut</a></div>
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	May 24
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	<h5><a class="title" href="./entries/Multirelations_Heterogeneous.html">Inner Structure, Determinism and Modal Algebra of Multirelations</a></h5> <br>
	by <a href="./authors/guttmann">Walter Guttmann</a> and <a href="./authors/struth">Georg Struth</a></div>
	<span class="date">
	May 22
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	<h5><a class="title" href="./entries/Tree_Enumeration.html">Tree Enumeration</a></h5> <br>
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	<span class="date">
	May 09
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	<h5><a class="title" href="./entries/MLSS_Decision_Proc.html">MLSS Decision Procedure</a></h5> <br>
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	May 05
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	<h5><a class="title" href="./entries/Three_Squares.html">Three Squares Theorem</a></h5> <br>
	by <a href="./authors/danilkin">Anton Danilkin</a> and <a href="./authors/chevalier">Loïc Chevalier</a></div>
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	May 03
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	<h5><a class="title" href="./entries/MHComputation.html">The Halting Problem is Soluble in Malament-Hogarth Spacetimes</a></h5> <br>
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	Apr 29
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	Apr 27
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	<h5><a class="title" href="./entries/Simple_Clause_Learning.html">A Formalization of the SCL(FOL) Calculus: Simple Clause Learning for First-Order Logic</a></h5> <br>
	by <a href="./authors/desharnais">Martin Desharnais</a></div>
	<span class="date">
	Apr 20
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	<div class="item-text">
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	Jun 08
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	Jan 23
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	Jan 16
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	<span class="date">
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	Oct 02
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	<span class="date">
	Jul 22
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	<span class="date">
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	<span class="date">
	May 22
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	<span class="date">
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	<h5><a class="title" href="./entries/Nominal2.html">Nominal 2</a></h5> <br>
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	<span class="date">
	Feb 21
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	<span class="date">
	Jan 31
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	<span class="date">
	Jan 19
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	<span class="date">
	Jan 16
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	<div class="item-text">
	<h5><a class="title" href="./entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>
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	<span class="date">
	Jan 15
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	<span class="date">
	Jan 03
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	<h2 class="year">2012</h2>
	<article class="entry">
	<div class="item-text">
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	<span class="date">
	Nov 14
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	<span class="date">
	Nov 02
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	<h5><a class="title" href="./entries/Tarskis_Geometry.html">The independence of Tarski's Euclidean axiom</a></h5> <br>
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	<span class="date">
	Oct 30
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	<div class="item-text">
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	<span class="date">
	Oct 27
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	<span class="date">
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	<span class="date">
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	<div class="item-text">
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	<span class="date">
	Aug 05
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	<h5><a class="title" href="./entries/Heard_Of.html">Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</a></h5> <br>
	by <a href="./authors/debrat">Henri Debrat</a> and <a href="./authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/PCF.html">Logical Relations for PCF</a></h5> <br>
	by <a href="./authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Jul 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Tycon.html">Type Constructor Classes and Monad Transformers</a></h5> <br>
	by <a href="./authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Jun 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Pi_Calculus.html">The pi-calculus in nominal logic</a></h5> <br>
	by <a href="./authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Psi_Calculi.html">Psi-calculi in Isabelle</a></h5> <br>
	by <a href="./authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/CCS.html">CCS in nominal logic</a></h5> <br>
	by <a href="./authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Circus.html">Isabelle/Circus</a></h5> <br>
	by <a href="./authors/feliachi">Abderrahmane Feliachi</a>, <a href="./authors/wolff">Burkhart Wolff</a> and <a href="./authors/gaudel">Marie-Claude Gaudel</a></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Separation_Algebra.html">Separation Algebra</a></h5> <br>
	by <a href="./authors/klein">Gerwin Klein</a>, <a href="./authors/kolanski">Rafal Kolanski</a> and <a href="./authors/boyton">Andrew Boyton</a></div>
	<span class="date">
	May 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Stuttering_Equivalence.html">Stuttering Equivalence</a></h5> <br>
	by <a href="./authors/merz">Stephan Merz</a></div>
	<span class="date">
	May 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Inductive_Confidentiality.html">Inductive Study of Confidentiality</a></h5> <br>
	by <a href="./authors/bella">Giampaolo Bella</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Ordinary_Differential_Equations.html">Ordinary Differential Equations</a></h5> <br>
	by <a href="./authors/immler">Fabian Immler</a> and <a href="./authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Well_Quasi_Orders.html">Well-Quasi-Orders</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Abortable_Linearizable_Modules.html">Abortable Linearizable Modules</a></h5> <br>
	by <a href="./authors/guerraoui">Rachid Guerraoui</a>, <a href="./authors/kuncak">Viktor Kuncak</a> and <a href="./authors/losa">Giuliano Losa</a></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Transitive-Closure-II.html">Executable Transitive Closures</a></h5> <br>
	by <a href="./authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Girth_Chromatic.html">A Probabilistic Proof of the Girth-Chromatic Number Theorem</a></h5> <br>
	by <a href="./authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Refine_Monadic.html">Refinement for Monadic Programs</a></h5> <br>
	by <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Dijkstra_Shortest_Path.html">Dijkstra's Shortest Path Algorithm</a></h5> <br>
	by <a href="./authors/nordhoff">Benedikt Nordhoff</a> and <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Markov_Models.html">Markov Models</a></h5> <br>
	by <a href="./authors/hoelzl">Johannes Hölzl</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 03
	</span>
	</article></div><div>
	<h2 class="year">2011</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/TLA.html">A Definitional Encoding of TLA* in Isabelle/HOL</a></h5> <br>
	by <a href="./authors/grov">Gudmund Grov</a> and <a href="./authors/merz">Stephan Merz</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Efficient-Mergesort.html">Efficient Mergesort</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/PseudoHoops.html">Pseudo Hoops</a></h5> <br>
	by <a href="./authors/georgescu">George Georgescu</a>, <a href="./authors/leustean">Laurentiu Leustean</a> and <a href="./authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/LatticeProperties.html">Lattice Properties</a></h5> <br>
	by <a href="./authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/MonoBoolTranAlgebra.html">Algebra of Monotonic Boolean Transformers</a></h5> <br>
	by <a href="./authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Myhill-Nerode.html">The Myhill-Nerode Theorem Based on Regular Expressions</a></h5> <br>
	by <a href="./authors/wu">Chunhan Wu</a>, <a href="./authors/zhangx">Xingyuan Zhang</a> and <a href="./authors/urban">Christian Urban</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Max-Card-Matching.html">Maximum Cardinality Matching</a></h5> <br>
	by <a href="./authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jul 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/KBPs.html">Knowledge-based programs</a></h5> <br>
	by <a href="./authors/gammie">Peter Gammie</a></div>
	<span class="date">
	May 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/General-Triangle.html">The General Triangle Is Unique</a></h5> <br>
	by <a href="./authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Apr 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Transitive-Closure.html">Executable Transitive Closures of Finite Relations</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a> and <a href="./authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Nat-Interval-Logic.html">Interval Temporal Logic on Natural Numbers</a></h5> <br>
	by <a href="./authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/List-Infinite.html">Infinite Lists</a></h5> <br>
	by <a href="./authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/AutoFocus-Stream.html">AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</a></h5> <br>
	by <a href="./authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/LightweightJava.html">Lightweight Java</a></h5> <br>
	by <a href="./authors/strnisa">Rok Strniša</a> and <a href="./authors/parkinson">Matthew Parkinson</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/RIPEMD-160-SPARK.html">RIPEMD-160</a></h5> <br>
	by <a href="./authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jan 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Lower_Semicontinuous.html">Lower Semicontinuous Functions</a></h5> <br>
	by <a href="./authors/grechuk">Bogdan Grechuk</a></div>
	<span class="date">
	Jan 08
	</span>
	</article></div><div>
	<h2 class="year">2010</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Marriage.html">Hall's Marriage Theorem</a></h5> <br>
	by <a href="./authors/jiangd">Dongchen Jiang</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Dec 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Shivers-CFA.html">Shivers' Control Flow Analysis</a></h5> <br>
	by <a href="./authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Nov 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Binomial-Queues.html">Functional Binomial Queues</a></h5> <br>
	by <a href="./authors/neumann">René Neumann</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Finger-Trees.html">Finger Trees</a></h5> <br>
	by <a href="./authors/nordhoff">Benedikt Nordhoff</a>, <a href="./authors/koerner">Stefan Körner</a> and <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Binomial-Heaps.html">Binomial Heaps and Skew Binomial Heaps</a></h5> <br>
	by <a href="./authors/meis">Rene Meis</a>, <a href="./authors/nielsen">Finn Nielsen</a> and <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Lam-ml-Normalization.html">Strong Normalization of Moggis's Computational Metalanguage</a></h5> <br>
	by <a href="./authors/doczkal">Christian Doczkal</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a>, <a href="./authors/thiemann">René Thiemann</a>, <a href="./authors/maletzky">Alexander Maletzky</a>, <a href="./authors/immler">Fabian Immler</a>, <a href="./authors/haftmann">Florian Haftmann</a>, <a href="./authors/lochbihler">Andreas Lochbihler</a> and <a href="./authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Statecharts.html">Formalizing Statecharts using Hierarchical Automata</a></h5> <br>
	by <a href="./authors/helke">Steffen Helke</a> and <a href="./authors/kammueller">Florian Kammüller</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Free-Groups.html">Free Groups</a></h5> <br>
	by <a href="./authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Category2.html">Category Theory</a></h5> <br>
	by <a href="./authors/katovsky">Alexander Katovsky</a></div>
	<span class="date">
	Jun 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Matrix.html">Executable Matrix Operations on Matrices of Arbitrary Dimensions</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a> and <a href="./authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Abstract-Rewriting.html">Abstract Rewriting</a></h5> <br>
	by <a href="./authors/sternagel">Christian Sternagel</a> and <a href="./authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/GraphMarkingIBP.html">Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</a></h5> <br>
	by <a href="./authors/preoteasa">Viorel Preoteasa</a> and <a href="./authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/DataRefinementIBP.html">Semantics and Data Refinement of Invariant Based Programs</a></h5> <br>
	by <a href="./authors/preoteasa">Viorel Preoteasa</a> and <a href="./authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Robbins-Conjecture.html">A Complete Proof of the Robbins Conjecture</a></h5> <br>
	by <a href="./authors/doty">Matthew Doty</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Regular-Sets.html">Regular Sets and Expressions</a></h5> <br>
	by <a href="./authors/krauss">Alexander Krauss</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Locally-Nameless-Sigma.html">Locally Nameless Sigma Calculus</a></h5> <br>
	by <a href="./authors/henrio">Ludovic Henrio</a>, <a href="./authors/kammueller">Florian Kammüller</a>, <a href="./authors/lutz">Bianca Lutz</a> and <a href="./authors/sudhof">Henry Sudhof</a></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Free-Boolean-Algebra.html">Free Boolean Algebra</a></h5> <br>
	by <a href="./authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Mar 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/InformationFlowSlicing_Inter.html">Inter-Procedural Information Flow Noninterference via Slicing</a></h5> <br>
	by <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/InformationFlowSlicing.html">Information Flow Noninterference via Slicing</a></h5> <br>
	by <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/List-Index.html">List Index</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Coinductive.html">Coinductive</a></h5> <br>
	by <a href="./authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Feb 12
	</span>
	</article></div><div>
	<h2 class="year">2009</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/DPT-SAT-Solver.html">A Fast SAT Solver for Isabelle in Standard ML</a></h5> <br>
	by <a href="./authors/heller">Armin Heller</a></div>
	<span class="date">
	Dec 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Presburger-Automata.html">Formalizing the Logic-Automaton Connection</a></h5> <br>
	by <a href="./authors/berghofer">Stefan Berghofer</a> and <a href="./authors/reiter">Markus Reiter</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Tree-Automata.html">Tree Automata</a></h5> <br>
	by <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Collections.html">Collections Framework</a></h5> <br>
	by <a href="./authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Perfect-Number-Thm.html">Perfect Number Theorem</a></h5> <br>
	by <a href="./authors/ijbema">Mark Ijbema</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/HRB-Slicing.html">Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</a></h5> <br>
	by <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Nov 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/WorkerWrapper.html">The Worker/Wrapper Transformation</a></h5> <br>
	by <a href="./authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Oct 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Ordinals_and_Cardinals.html">Ordinals and Cardinals</a></h5> <br>
	by <a href="./authors/popescu">Andrei Popescu</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/SequentInvertibility.html">Invertibility in Sequent Calculi</a></h5> <br>
	by <a href="./authors/chapman">Peter Chapman</a></div>
	<span class="date">
	Aug 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/CofGroups.html">An Example of a Cofinitary Group in Isabelle/HOL</a></h5> <br>
	by <a href="./authors/kastermans">Bart Kastermans</a></div>
	<span class="date">
	Aug 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FinFun.html">Code Generation for Functions as Data</a></h5> <br>
	by <a href="./authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Stream-Fusion.html">Stream Fusion</a></h5> <br>
	by <a href="./authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Apr 29
	</span>
	</article></div><div>
	<h2 class="year">2008</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/BytecodeLogicJmlTypes.html">A Bytecode Logic for JML and Types</a></h5> <br>
	by <a href="./authors/beringer">Lennart Beringer</a> and <a href="./authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Dec 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/SIFPL.html">Secure information flow and program logics</a></h5> <br>
	by <a href="./authors/beringer">Lennart Beringer</a> and <a href="./authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/SenSocialChoice.html">Some classical results in Social Choice Theory</a></h5> <br>
	by <a href="./authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FunWithTilings.html">Fun With Tilings</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a> and <a href="./authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Huffman.html">The Textbook Proof of Huffman's Algorithm</a></h5> <br>
	by <a href="./authors/blanchette">Jasmin Christian Blanchette</a></div>
	<span class="date">
	Oct 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Slicing.html">Towards Certified Slicing</a></h5> <br>
	by <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/VolpanoSmith.html">A Correctness Proof for the Volpano/Smith Security Typing System</a></h5> <br>
	by <a href="./authors/snelting">Gregor Snelting</a> and <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/ArrowImpossibilityGS.html">Arrow and Gibbard-Satterthwaite</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FunWithFunctions.html">Fun With Functions</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/SATSolverVerification.html">Formal Verification of Modern SAT Solvers</a></h5> <br>
	by <a href="./authors/maricf">Filip Marić</a></div>
	<span class="date">
	Jul 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Recursion-Theory-I.html">Recursion Theory I</a></h5> <br>
	by <a href="./authors/nedzelsky">Michael Nedzelsky</a></div>
	<span class="date">
	Apr 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/BDD.html">BDD Normalisation</a></h5> <br>
	by <a href="./authors/ortner">Veronika Ortner</a> and <a href="./authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Simpl.html">A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</a></h5> <br>
	by <a href="./authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/NormByEval.html">Normalization by Evaluation</a></h5> <br>
	by <a href="./authors/aehlig">Klaus Aehlig</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/LinearQuantifierElim.html">Quantifier Elimination for Linear Arithmetic</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 11
	</span>
	</article></div><div>
	<h2 class="year">2007</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Program-Conflict-Analysis.html">Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors</a></h5> <br>
	by <a href="./authors/lammich">Peter Lammich</a> and <a href="./authors/olm">Markus Müller-Olm</a></div>
	<span class="date">
	Dec 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/JinjaThreads.html">Jinja with Threads</a></h5> <br>
	by <a href="./authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/MuchAdoAboutTwo.html">Much Ado About Two</a></h5> <br>
	by <a href="./authors/boehme">Sascha Böhme</a></div>
	<span class="date">
	Nov 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/SumSquares.html">Sums of Two and Four Squares</a></h5> <br>
	by <a href="./authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Fermat3_4.html">Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</a></h5> <br>
	by <a href="./authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Valuation.html">Fundamental Properties of Valuation Theory and Hensel's Lemma</a></h5> <br>
	by <a href="./authors/kobayashi">Hidetsune Kobayashi</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/POPLmark-deBruijn.html">POPLmark Challenge Via de Bruijn Indices</a></h5> <br>
	by <a href="./authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Aug 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FOL-Fitting.html">First-Order Logic According to Fitting</a></h5> <br>
	by <a href="./authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Aug 02
	</span>
	</article></div><div>
	<h2 class="year">2006</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/HotelKeyCards.html">Hotel Key Card System</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Abstract-Hoare-Logics.html">Abstract Hoare Logics</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Flyspeck-Tame.html">Flyspeck I: Tame Graphs</a></h5> <br>
	by <a href="./authors/bauer">Gertrud Bauer</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/CoreC++.html">CoreC++</a></h5> <br>
	by <a href="./authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	May 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FeatherweightJava.html">A Theory of Featherweight Java in Isabelle/HOL</a></h5> <br>
	by <a href="./authors/fosterj">J. Nathan Foster</a> and <a href="./authors/vytiniotis">Dimitrios Vytiniotis</a></div>
	<span class="date">
	Mar 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/ClockSynchInst.html">Instances of Schneider's generalized protocol of clock synchronization</a></h5> <br>
	by <a href="./authors/barsotti">Damián Barsotti</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Cauchy.html">Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality</a></h5> <br>
	by <a href="./authors/porter">Benjamin Porter</a></div>
	<span class="date">
	Mar 14
	</span>
	</article></div><div>
	<h2 class="year">2005</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Ordinal.html">Countable Ordinals</a></h5> <br>
	by <a href="./authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FFT.html">Fast Fourier Transform</a></h5> <br>
	by <a href="./authors/ballarin">Clemens Ballarin</a></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/GenClock.html">Formalization of a Generalized Protocol for Clock Synchronization</a></h5> <br>
	by <a href="./authors/tiu">Alwen Tiu</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/DiskPaxos.html">Proving the Correctness of Disk Paxos</a></h5> <br>
	by <a href="./authors/jaskelioff">Mauro Jaskelioff</a> and <a href="./authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jun 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/JiveDataStoreModel.html">Jive Data and Store Model</a></h5> <br>
	by <a href="./authors/rauch">Nicole Rauch</a> and <a href="./authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Jun 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Jinja.html">Jinja is not Java</a></h5> <br>
	by <a href="./authors/klein">Gerwin Klein</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/RSAPSS.html">SHA1, RSA, PSS and more</a></h5> <br>
	by <a href="./authors/lindenberg">Christina Lindenberg</a> and <a href="./authors/wirt">Kai Wirt</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Category.html">Category Theory to Yoneda's Lemma</a></h5> <br>
	by <a href="./authors/keefe">Greg O’Keefe</a></div>
	<span class="date">
	Apr 21
	</span>
	</article></div><div>
	<h2 class="year">2004</h2>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/FileRefinement.html">File Refinement</a></h5> <br>
	by <a href="./authors/zee">Karen Zee</a> and <a href="./authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Dec 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Integration.html">Integration theory and random variables</a></h5> <br>
	by <a href="./authors/richter">Stefan Richter</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Verified-Prover.html">A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic</a></h5> <br>
	by <a href="./authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Ramsey-Infinite.html">Ramsey's theorem, infinitary version</a></h5> <br>
	by <a href="./authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Completeness.html">Completeness theorem</a></h5> <br>
	by <a href="./authors/margetson">James Margetson</a> and <a href="./authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Compiling-Exceptions-Correctly.html">Compiling Exceptions Correctly</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jul 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Depth-First-Search.html">Depth First Search</a></h5> <br>
	by <a href="./authors/nishihara">Toshiaki Nishihara</a> and <a href="./authors/minamide">Yasuhiko Minamide</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Group-Ring-Module.html">Groups, Rings and Modules</a></h5> <br>
	by <a href="./authors/kobayashi">Hidetsune Kobayashi</a>, <a href="./authors/chen">L. Chen</a> and <a href="./authors/murao">H. Murao</a></div>
	<span class="date">
	May 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Topology.html">Topology</a></h5> <br>
	by <a href="./authors/friedrich">Stefan Friedrich</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Lazy-Lists-II.html">Lazy Lists II</a></h5> <br>
	by <a href="./authors/friedrich">Stefan Friedrich</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/BinarySearchTree.html">Binary Search Trees</a></h5> <br>
	by <a href="./authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Apr 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/Functional-Automata.html">Functional Automata</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Mar 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/MiniML.html">Mini ML</a></h5> <br>
	by <a href="./authors/naraschewski">Wolfgang Naraschewski</a> and <a href="./authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Mar 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="./entries/AVL-Trees.html">AVL Trees</a></h5> <br>
	by <a href="./authors/nipkow">Tobias Nipkow</a> and <a href="./authors/pusch">Cornelia Pusch</a></div>
	<span class="date">
	Mar 19
	</span>
	</article></div>
	</div>
	</div>
	</body>

	</html>
	\ No newline at end of file
	diff --git a/web/index.json b/web/index.json
	--- a/web/index.json
	+++ b/web/index.json
	@@ -1,15351 +1,15417 @@
	[
	{
	+ "abstract": "\u003cp\u003eWe formalize the proofs of Cauchy's and Legendre's Polygonal Number Theorems given in Melvyn B. Nathanson's book \"Additive Number Theory: The Classical Bases\".\u003c/p\u003e \u003cp\u003eFor $m \\geq 1$, the $k$-th polygonal number of order $m+2$ is defined to be $p_m(k)=\\frac{mk(k-1)}{2}+k$. The theorems state that:\u003c/p\u003e \u003cp\u003e1. If $m \\ge 4$ and $N \\geq 108m$, then $N$ can be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four of which are different from $0$ or $1$. If $N \\geq 324$, then $N$ can be written as the sum of five pentagonal numbers, at least one of which is $0$ or $1$.\u003c/p\u003e \u003cp\u003e2. Let $m \\geq 3$ and $N \\geq 28m^3$. If $m$ is odd, then $N$ is the sum of four polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.\u003c/p\u003e \u003cp\u003eWe also formalize the proof of Gauss's theorem which states that every non-negative integer is the sum of three triangular numbers.\u003c/p\u003e",
	+ "authors": [
	+ "Kevin Lee",
	+ "Zhengkun Ye",
	+ "Angeliki Koutsoukou-Argyraki"
	+ ],
	+ "date": "2023-08-10",
	+ "id": 0,
	+ "link": "/entries/Polygonal_Number_Theorem.html",
	+ "permalink": "/entries/Polygonal_Number_Theorem.html",
	+ "shortname": "Polygonal_Number_Theorem",
	+ "title": "Polygonal Number Theorem",
	+ "topic_links": [
	+ "mathematics/number-theory",
	+ "mathematics/combinatorics"
	+ ],
	+ "topics": [
	+ "Mathematics/Number theory",
	+ "Mathematics/Combinatorics"
	+ ],
	+ "used_by": 0
	+ },
	+ {
	+ "abstract": "This AFP entry provides mathematical components for modal quantales, involutive quantales and Dedekind quantales. Modal quantales are simple extensions of modal Kleene algebras useful for the verification of recursive programs. Involutive quantales appear in the study of C*-algebras. Dedekind quantales are relatives of Tarski's relation algebras, hence relevant to program verification and beyond that to higher rewriting. We also provide components for weaker variants such as Kleene algebras with converse and modal Kleene algebras with converse.",
	+ "authors": [
	+ "Georg Struth",
	+ "Cameron Calk"
	+ ],
	+ "date": "2023-07-25",
	+ "id": 1,
	+ "link": "/entries/Quantales_Converse.html",
	+ "permalink": "/entries/Quantales_Converse.html",
	+ "shortname": "Quantales_Converse",
	+ "title": "Modal quantales, involutive quantales, Dedekind Quantales",
	+ "topic_links": [
	+ "computer-science/semantics-and-reasoning",
	+ "mathematics/algebra",
	+ "mathematics/order"
	+ ],
	+ "topics": [
	+ "Computer science/Semantics and reasoning",
	+ "Mathematics/Algebra",
	+ "Mathematics/Order"
	+ ],
	+ "used_by": 0
	+ },
	+ {
	+ "abstract": "In 1968, Earley introduced his parsing algorithm, capable of parsing all context-free grammars in cubic space and time. This entry contains a formalization of an executable Earley parser. We base our development on Jones' extensive paper proof of Earley's recognizer and Obua's formalization of context-free grammars and derivations. We implement and prove correct a functional recognizer modeling Earley's original imperative implementation and extend it with the necessary data structures to enable the construction of parse trees, following the work of Scott. We then develop a functional algorithm that builds a single parse tree, and we prove its correctness. Finally, we generalize this approach to an algorithm for a complete parse forest and prove soundness.",
	+ "authors": [
	+ "Martin Rau"
	+ ],
	+ "date": "2023-07-16",
	+ "id": 2,
	+ "link": "/entries/Earley_Parser.html",
	+ "permalink": "/entries/Earley_Parser.html",
	+ "shortname": "Earley_Parser",
	+ "title": "Earley Parser",
	+ "topic_links": [
	+ "computer-science/algorithms"
	+ ],
	+ "topics": [
	+ "Computer science/Algorithms"
	+ ],
	+ "used_by": 0
	+ },
	+ {
	"abstract": "The original Gray code after Frank Gray, also known as reflected binary code (RBC), is an ordering of the binary numeral system such that two successive values differ only in one bit. We provide a theory for a non-Boolean Gray code, which is a generalisation of the idea for an arbitrary base. Contained is the necessary theoretical environment to express and reason about the respective properties.",
	"authors": [
	"Maximilian Spitz"
	],
	"date": "2023-07-11",
	- "id": 0,
	+ "id": 3,
	"link": "/entries/Gray_Codes.html",
	"permalink": "/entries/Gray_Codes.html",
	"shortname": "Gray_Codes",
	"title": "Gray Codes for Arbitrary Numeral Systems",
	"topic_links": [
	"mathematics/combinatorics",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "In Isabelle, randomized algorithms are usually represented using probability mass functions (PMFs), with which it is possible to verify their correctness, particularly properties about the distribution of their result. However, that approach does not provide a way to generate executable code for such algorithms. In this entry, we introduce a new monad for randomized algorithms, for which it is possible to generate code and simultaneously reason about the correctness of randomized algorithms. The latter works by a Scott-continuous monad morphism between the newly introduced random monad and PMFs. On the other hand, when supplied with an external source of random coin flips, the randomized algorithms can be executed.",
	"authors": [
	"Emin Karayel",
	"Manuel Eberl"
	],
	"date": "2023-06-19",
	- "id": 1,
	+ "id": 4,
	"link": "/entries/Executable_Randomized_Algorithms.html",
	"permalink": "/entries/Executable_Randomized_Algorithms.html",
	"shortname": "Executable_Randomized_Algorithms",
	"title": "Executable Randomized Algorithms",
	"topic_links": [
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 0
	},
	{
	"abstract": "We present an Isabelle formalization of the basics of Dynamic Condition Response (DCR) graphs before defining Execution Equivalent markings. We then prove that execution equivalent markings are perfectly interchangeable during process execution, yielding significant state-space reduction for execution-based model-checking of DCR graphs.",
	"authors": [
	"Axel Christfort",
	"Søren Debois"
	],
	"date": "2023-06-16",
	- "id": 2,
	+ "id": 5,
	"link": "/entries/DCR-ExecutionEquivalence.html",
	"permalink": "/entries/DCR-ExecutionEquivalence.html",
	"shortname": "DCR-ExecutionEquivalence",
	"title": "DCR Syntax and Execution Equivalent Markings",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This work formalizes Zeckendorf's theorem. The theorem states that every positive integer can be uniquely represented as a sum of one or more non-consecutive Fibonacci numbers. More precisely, if $N$ is a positive integer, there exist unique positive integers $c_i \\ge 2$ with $c_{i+1} \u003e c_i + 1$, such that \\[ N = \\sum_{i=0}^k F_{c_i} \\] where $F_n$ is the $n$-th Fibonacci number.",
	"authors": [
	"Christian Dalvit"
	],
	"date": "2023-06-12",
	- "id": 3,
	+ "id": 6,
	"link": "/entries/Zeckendorf.html",
	"permalink": "/entries/Zeckendorf.html",
	"shortname": "Zeckendorf",
	"title": "Zeckendorf’s Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003chtml\u003e \u003chead\u003e \u003cmeta http-equiv=Content-Type content=\"text/html; charset=utf-8\"\u003e \u003cstyle\u003e \u003c!-- /* Font Definitions / @font-face \t{font-family:\"Cambria Math\"; \tpanose-1:2 4 5 3 5 4 6 3 2 4;} @font-face \t{font-family:Calibri; \tpanose-1:2 15 5 2 2 2 4 3 2 4;} @font-face \t{font-family:Cambria; \tpanose-1:2 4 5 3 5 4 6 3 2 4;} @font-face \t{font-family:Menlo; \tpanose-1:2 11 6 9 3 8 4 2 2 4;} / Style Definitions / p.MsoNormal, li.MsoNormal, div.MsoNormal \t{margin:0in; \tfont-size:12.0pt; \tfont-family:\"Calibri\",sans-serif;} @page WordSection1 \t{size:8.5in 11.0in; \tmargin:1.0in 1.0in 1.0in 1.0in;} div.WordSection1 \t{page:WordSection1;} --\u003e \u003c/style\u003e \u003c/head\u003e \u003cbody lang=EN-US style='word-wrap:break-word'\u003e \u003cdiv class=WordSection1\u003e \u003cp class=MsoNormal\u003eIn this set of theories, we express well-known crytographic standards in the language of Isabelle. The standards we have translated so far are:\u003c/p\u003e \u003cp class=MsoNormal\u003e\u0026nbsp;\u003c/p\u003e \u003ctable class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 style='border-collapse:collapse;border:none'\u003e \u003ctr\u003e \u003ctd width=143 valign=top style='width:107.5pt;border:solid windowtext 1.0pt; padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003e FIPS 180-4\u003c/p\u003e \u003c/td\u003e \u003ctd width=618 valign=top style='width:463.5pt;border:solid windowtext 1.0pt; border-left:none;padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003eNIST's Secure Hash Standard, rev 4.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd width=143 valign=top style='width:107.5pt;border:solid windowtext 1.0pt; border-top:none;padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003e FIPS 186-4\u003c/p\u003e \u003c/td\u003e \u003ctd width=618 valign=top style='width:463.5pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003eOnly the elliptic curves over prime fields, i.e. NIST's \u0026quot;P-\u0026quot; curves\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd width=143 valign=top style='width:107.5pt;border:solid windowtext 1.0pt; border-top:none;padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003e FIPS 198-1\u003c/p\u003e \u003c/td\u003e \u003ctd width=618 valign=top style='width:463.5pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003eNIST's The Keyed-Hash Message Authentication Code (HMAC Standard)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd width=143 valign=top style='width:107.5pt;border:solid windowtext 1.0pt; border-top:none;padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003e PKCS #1 v2.2\u003c/p\u003e \u003c/td\u003e \u003ctd width=618 valign=top style='width:463.5pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003eRSA Laboratories' RSA Cryptography Standard, version 2.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd width=143 valign=top style='width:107.5pt;border:solid windowtext 1.0pt; border-top:none;padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003e SEC1 v2.0\u003c/p\u003e \u003c/td\u003e \u003ctd width=618 valign=top style='width:463.5pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; padding:2.9pt 5.75pt 2.9pt 5.75pt'\u003e \u003cp class=MsoNormal\u003eSEC's Elliptic Curve Cryptography, version 2.0 \u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/table\u003e \u003cp class=MsoNormal\u003e\u0026nbsp;\u003c/p\u003e \u003cp class=MsoNormal\u003eThe intention is that these translations will be used to prove that any particular implementation matches the relevant standard. With that in mind, the overriding principle is to adhere as closely as possible, given the syntax of HOL, to the written standard. It should be obvious to any reader, regardless of their past experience with Isabelle, that these translations exactly match the standards. Thus we use the same function and variable names as in the written standards whenever possible and explain in the comments the few times when that is not possible. \u003c/p\u003e \u003cp class=MsoNormal\u003e\u0026nbsp;\u003c/p\u003e \u003cp class=MsoNormal\u003eWe want the users of these theories to have faith that errors were not made in the translations. We do two things to achieve this. First, in addition to translating a standard, we provide a robust supporting theory that proves anything that one might wish to know about the primitives that the standard defines. For example, we prove that encryption and decryption are inverse operations. We prove when RSA keys are equivalent. We prove that if a message is signed, then that signature and message will pass the verification operation. Any fact that you may need in using these standards, we hope and believe we have already proved for you. \u003c/p\u003e \u003cp class=MsoNormal\u003e\u0026nbsp;\u003c/p\u003e \u003cp class=MsoNormal\u003eSecond, we prove (by evaluation) that the test vectors provided by NIST, et al, check as intended (whether a passing or failing test case.) The test vectors may be found in theories named _Test_Vectors.thy. These files can be large and take time for Isabelle to process. Thus, they are not imported by this main Crypto_Standards theory. Users may open those separately. As an aside, Isabelle must be told how to compute certain operations efficiently. For example, modular exponentiation or scalar multiplication of a point on an elliptic curve. The Efficient*.thy files are necessary to check the test vectors. \u003c/p\u003e \u003cp class=MsoNormal\u003e\u0026nbsp;\u003c/p\u003e \u003cp class=MsoNormal\u003eWe attempt to be as agnostic to implementation details as possible. For example, we do not assume any particular data type has been used as input or output. Many standards operate on octets, or 8-bit values. For these theories, an octet string is modeled as a list of natural numbers. Then a nat x is a \u0026quot;valid octet\u0026quot; exactly when x \u0026lt; 256. Words.thy contains all the operations needed to convert natural number to a string of n-bit values and back to a natural number. There you will also find definitions for handling padding and bit manipulations that are found in the above standards. Again, we believe that we have proved anything you may need to apply these theories. We have erred on the side of including lemmas that may be of practical use as opposed to proving a minimal set of lemmas required. \u003c/p\u003e \u003c/div\u003e \u003c/body\u003e \u003c/html\u003e",
	"authors": [
	"A Whitley"
	],
	"date": "2023-06-06",
	- "id": 4,
	+ "id": 7,
	"link": "/entries/Crypto_Standards.html",
	"permalink": "/entries/Crypto_Standards.html",
	"shortname": "Crypto_Standards",
	"title": "Cryptographic Standards",
	"topic_links": [
	"computer-science/security/cryptography"
	],
	"topics": [
	"Computer science/Security/Cryptography"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe Weighted Path Order (WPO) of Yamada is a powerful technique for proving termination. In a previous AFP entry, the WPO was defined and properties of WPO have been formally verified. However, the implementation of WPO was naive, leading to an exponential runtime in the worst case. \u003c/p\u003e\u003cp\u003e Therefore, in this AFP entry we provide a poly-time implementation of WPO. The implementation is based on memoization. Since WPO generalizes the recursive path order (RPO), we also easily derive an efficient implementation of RPO. \u003c/p\u003e",
	"authors": [
	"René Thiemann",
	"Elias Wenninger"
	],
	"date": "2023-06-01",
	- "id": 5,
	+ "id": 8,
	"link": "/entries/Efficient_Weighted_Path_Order.html",
	"permalink": "/entries/Efficient_Weighted_Path_Order.html",
	"shortname": "Efficient_Weighted_Path_Order",
	"title": "A Verified Efficient Implementation of the Weighted Path Order",
	"topic_links": [
	"logic/rewriting",
	"computer-science/algorithms"
	],
	"topics": [
	"Logic/Rewriting",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eDirected sets are of fundamental interest in domain theory and topology. In this paper, we formalize some results on directed sets in Isabelle/HOL, most notably: under the axiom of choice, a poset has a supremum for every directed set if and only if it does so for every chain; and a function between such posets preserves suprema of directed sets if and only if it preserves suprema of chains. The known pen-and-paper proofs of these results crucially use uncountable transfinite sequences, which are not directly implementable in Isabelle/HOL. We show how to emulate such proofs by utilizing Isabelle/HOL's ordinal and cardinal library. Thanks to the formalization, we relax some conditions for the above results.\u003c/p\u003e",
	"authors": [
	"Akihisa Yamada",
	"Jérémy Dubut"
	],
	"date": "2023-05-24",
	- "id": 6,
	+ "id": 9,
	"link": "/entries/Directed_Sets.html",
	"permalink": "/entries/Directed_Sets.html",
	"shortname": "Directed_Sets",
	"title": "Formalizing Results on Directed Sets",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 0
	},
	{
	"abstract": "Binary multirelations form a model of alternating nondeterminism useful for analysing games, interactions of computing systems with their environments or abstract interpretations of probabilistic programs. We investigate this alternating structure in a relational language based on power allegories extended with specific operations on multirelations. We develop algebras of modal operators over multirelations, related to concurrent dynamic logics, in this language.",
	"authors": [
	"Walter Guttmann",
	"Georg Struth"
	],
	"date": "2023-05-22",
	- "id": 7,
	+ "id": 10,
	"link": "/entries/Multirelations_Heterogeneous.html",
	"permalink": "/entries/Multirelations_Heterogeneous.html",
	"shortname": "Multirelations_Heterogeneous",
	"title": "Inner Structure, Determinism and Modal Algebra of Multirelations",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This thesis presents the verification of enumeration algorithms for trees. The first algorithm is based on the well known Prüfer-correspondence and allows the enumeration of all possible labeled trees over a fixed finite set of vertices. The second algorithm enumerates rooted, unlabeled trees of a specified size up to graph isomorphism. It allows for the efficient enumeration without the use of an intermediate encoding of the trees with level sequences, unlike the \u003ca href=\"https://doi.org/10.1137/0209055\"\u003ealgorithm\u003c/a\u003e by Beyer and Hedetniemi it is based on. Both algorithms are formalized and verified in Isabelle/HOL. The formalization of trees and other graph theoretic results is also presented.",
	"authors": [
	"Nils Cremer"
	],
	"date": "2023-05-09",
	- "id": 8,
	+ "id": 11,
	"link": "/entries/Tree_Enumeration.html",
	"permalink": "/entries/Tree_Enumeration.html",
	"shortname": "Tree_Enumeration",
	"title": "Tree Enumeration",
	"topic_links": [
	"computer-science/algorithms/graph",
	"mathematics/graph-theory",
	"mathematics/combinatorics"
	],
	"topics": [
	"Computer science/Algorithms/Graph",
	"Mathematics/Graph theory",
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalization verifies a decision procedure due to Cantone and Zarba for a quantifier-free fragment of set theory. The fragment is called multi-level syllogistic with singleton, or MLSS for short. Its syntax syntax includes the usual set operations union, intersection, difference, membership, equality as well as the construction of a set containing a single element. We specify the semantics of MLSS in terms of hereditarily finite sets and provide a sound and complete tableau calculus for it. We also provide an executable specification of a decision procedure that applies the rules of the calculus exhaustively and prove its termination. Furthermore, we extend the calculus with a light-weight type system that paves the way for an integration of the procedure into Isabelle/HOL.",
	"authors": [
	"Lukas Stevens"
	],
	"date": "2023-05-05",
	- "id": 9,
	+ "id": 12,
	"link": "/entries/MLSS_Decision_Proc.html",
	"permalink": "/entries/MLSS_Decision_Proc.html",
	"shortname": "MLSS_Decision_Proc",
	"title": "MLSS Decision Procedure",
	"topic_links": [
	"logic/general-logic/classical-propositional-logic",
	"logic/set-theory",
	"logic/general-logic/decidability-of-theories"
	],
	"topics": [
	"Logic/General logic/Classical propositional logic",
	"Logic/Set theory",
	"Logic/General logic/Decidability of theories"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the Legendre's three squares theorem and its consequences, in particular the following results: \u003col\u003e \u003cli\u003e A natural number can be represented as the sum of three squares of natural numbers if and only if it is not of the form $4^a (8 k + 7)$, where $a$ and $k$ are natural numbers. \u003c/li\u003e \u003cli\u003e If $n$ is a natural number such that $n \\equiv 3 \\pmod{8}$, then $n$ can be be represented as the sum of three squares of odd natural numbers. \u003c/li\u003e \u003c/ol\u003e Consequences include the following: \u003col\u003e \u003cli\u003e An integer $n$ can be written as $n = x^2 + y^2 + z^2 + z$, where $x$, $y$, $z$ are integers, if and only if $n \\geq 0$. \u003c/li\u003e \u003cli\u003e The Legendre's four squares theorem: any natural number can be represented as the sum of four squares of natural numbers. \u003c/li\u003e \u003c/ol\u003e",
	"authors": [
	"Anton Danilkin",
	"Loïc Chevalier"
	],
	"date": "2023-05-03",
	- "id": 10,
	+ "id": 13,
	"link": "/entries/Three_Squares.html",
	"permalink": "/entries/Three_Squares.html",
	"shortname": "Three_Squares",
	"title": "Three Squares Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	- "used_by": 0
	+ "used_by": 1
	},
	{
	"abstract": "We provide an Isabelle verification that the Halting Problem can be solved in Malament-Hogarth (MH) spacetimes. Our proof is quite general -- rather than assume the full machinery of general relativity, we only assume the existence of a reachability relation defined on an abstract space of locations. An MH spacetime can then be described as a space in which there exists an unboundedly long path together with a location which is reachable from all points on that path. Likewise, we use a very general notion of computation - the current state of a computation is assumed to be representable as a machine configuration containing all the information required to determine how the system changes with the execution of each ensuing instruction. The program is deemed to halt if the system enters a stable configuration. Since this situation is generally detectable by an operating system, we can use its occurrence to trigger events that exploit the nature of MH spacetimes, thereby enabling us to detect whether or not halting will eventually have occurred. Our verification follows existing arguments in the literature, albeit translated into this more general setting.",
	"authors": [
	"Mike Stannett"
	],
	"date": "2023-04-29",
	- "id": 11,
	+ "id": 14,
	"link": "/entries/MHComputation.html",
	"permalink": "/entries/MHComputation.html",
	"shortname": "MHComputation",
	"title": "The Halting Problem is Soluble in Malament-Hogarth Spacetimes",
	"topic_links": [
	"computer-science",
	"logic/computability",
	"mathematics/physics"
	],
	"topics": [
	"Computer science",
	"Logic/Computability",
	"Mathematics/Physics"
	],
	"used_by": 0
	},
	{
	"abstract": "This short entry formalizes a version of the Schwartz-Zippel lemma for probabilistic (multivariate) polynomial identity testing. The entry includes a textbook example using the lemma to test for perfect matchings in a bipartite graph. The lemma is attributed to several independent authors, including Schwartz, Zippel, and DeMillo and Lipton; a historical perspective is given by Lipton.",
	"authors": [
	"Sunpill Kim",
	"Yong Kiam Tan"
	],
	"date": "2023-04-27",
	- "id": 12,
	+ "id": 15,
	"link": "/entries/Schwartz_Zippel.html",
	"permalink": "/entries/Schwartz_Zippel.html",
	"shortname": "Schwartz_Zippel",
	"title": "The Schwartz-Zippel Lemma",
	"topic_links": [
	"computer-science/algorithms/randomized",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Randomized",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization covers the unexecutable specification of Simple Clause Learning for first-order logic without equality: SCL(FOL). The main results are formal proofs of soundness, non-redundancy of learned clauses, termination, and refutational completeness. Compared to the unformalized version, the formalized calculus is simpler, a number of results were generalized, and the non-redundancy statement was strengthened. We found and corrected one bug in a previously published version of the SCL Backtrack rule. Compared to related formalizations, we introduce a new technique for showing termination based on non-redundant clause learning.",
	"authors": [
	"Martin Desharnais"
	],
	"date": "2023-04-20",
	- "id": 13,
	+ "id": 16,
	"link": "/entries/Simple_Clause_Learning.html",
	"permalink": "/entries/Simple_Clause_Learning.html",
	"shortname": "Simple_Clause_Learning",
	"title": "A Formalization of the SCL(FOL) Calculus: Simple Clause Learning for First-Order Logic",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "The CHSH inequality, named after Clauser, Horne, Shimony and Holt, was used by Alain Aspect to prove experimentally that Einstein's hypothesis stating that quantum mechanics could be defined using local hidden variables was incorrect. The CHSH inequality is based on a setting in which an experiment consisting of two separate parties performing joint measurements is run several times, and a score is derived from these runs. If the local hidden variable hypothesis had been correct, this score would have been bounded by $2$, but a suitable choice of observables in a quantum setting permits to violate this inequality when measuring the Bell state; this is the result that Aspect obtained experimentally. Tsirelson answered the question of how large this violation could be by proving that in the quantum setting, the highest score that can be obtained when running this experiment is $2\\sqrt{2}$. Along with elementary results on density matrices which represent quantum states in the finite dimensional setting, we formalize Tsirelson's result and summarize the main results on the CHSH score: \u003col\u003e \u003cli\u003eUnder the local hidden variable hypothesis, this score admits 2 as an upper-bound.\u003c/li\u003e \u003cli\u003eWhen the density matrix under consideration is separable, the upper-bound cannot be violated.\u003c/li\u003e \u003cli\u003eWhen one of the parties in the experiment performs measures using commuting observables, this upper-bound remains valid.\u003c/li\u003e \u003cli\u003eOtherwise, the upper-bound of this score is $2\\sqrt{2}$, regardless of the observables that are used and the quantum state that is measured, and\u003c/li\u003e \u003cli\u003eThis upper-bound is reached for a suitable choice of observables when measuring the Bell state.\u003c/li\u003e \u003c/ol\u003e",
	"authors": [
	"Mnacho Echenim",
	"Mehdi Mhalla",
	"Coraline Mori"
	],
	"date": "2023-04-18",
	- "id": 14,
	+ "id": 17,
	"link": "/entries/TsirelsonBound.html",
	"permalink": "/entries/TsirelsonBound.html",
	"shortname": "TsirelsonBound",
	"title": "The CHSH inequality: Tsirelson's upper-bound and other results",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/physics/quantum-information"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Physics/Quantum information"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry formalizes a randomized cardinality estimation data structure with asymptotically optimal space usage. It is inspired by the streaming algorithm presented by \u003ca href=\"https://doi.org/10.1145/3309193\"\u003eB\u0026lstrok;asiok\u003c/a\u003e in 2018. His work closed the gap between the best-known lower bound and upper bound after a long line of research started by \u003ca href=\"https://doi.org/10.1016/0022-0000(85)90041-8\"\u003eFlajolet and Martin\u003c/a\u003e in 1984 and was to first to apply expander graphs (in addition to hash families) to the problem. The formalized algorithm has two improvements compared to the algorithm by B\u0026lstrok;asiok. It supports operation in parallel mode, and it relies on a simpler pseudo-random construction avoiding the use of code based extractors.",
	"authors": [
	"Emin Karayel"
	],
	"date": "2023-04-03",
	- "id": 15,
	+ "id": 18,
	"link": "/entries/Distributed_Distinct_Elements.html",
	"permalink": "/entries/Distributed_Distinct_Elements.html",
	"shortname": "Distributed_Distinct_Elements",
	"title": "Distributed Distinct Elements",
	"topic_links": [
	"computer-science/algorithms/distributed",
	"computer-science/algorithms/approximation",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Distributed",
	"Computer science/Algorithms/Approximation",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 0
	},
	{
	"abstract": "Hoare logics are proof systems that allow one to formally establish properties of computer programs. Traditional Hoare logics prove properties of individual program executions (so-called trace properties, such as functional correctness). On the one hand, Hoare logic has been generalized to prove properties of multiple executions of a program (so-called hyperproperties, such as determinism or non-interference). These program logics prove the absence of (bad combinations of) executions. On the other hand, program logics similar to Hoare logic have been proposed to disprove program properties (e.g., Incorrectness Logic [8]), by proving the existence of (bad combinations of) executions. All of these logics have in common that they specify program properties using assertions over a fixed number of states, for instance, a single pre- and post-state for functional properties or pairs of pre- and post-states for non-interference. In this entry, we formalize Hyper Hoare Logic, a generalization of Hoare logic that lifts assertions to properties of arbitrary sets of states. The resulting logic is simple yet expressive: its judgments can express arbitrary trace- and hyperproperties over the terminating executions of a program. By allowing assertions to reason about sets of states, Hyper Hoare Logic can reason about both the absence and the existence of (combinations of) executions, and, thereby, supports both proving and disproving program (hyper-)properties within the same logic. In fact, we prove that Hyper Hoare Logic subsumes the properties handled by numerous existing correctness and incorrectness logics, and can express hyperproperties that no existing Hoare logic can. We also prove that Hyper Hoare Logic is sound and complete.",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2023-04-03",
	- "id": 16,
	+ "id": 19,
	"link": "/entries/HyperHoareLogic.html",
	"permalink": "/entries/HyperHoareLogic.html",
	"shortname": "HyperHoareLogic",
	"title": "Formalization of Hyper Hoare Logic: A Logic to (Dis-)Prove Program Hyperproperties",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We demonstrate the existence and uniqueness of the base-$n$ representation of a natural number, where $n$ is any natural number greater than 1. This comes up when trying to translate mathematical contest problems and solutions into Isabelle/HOL.",
	"authors": [
	"Charles Staats"
	],
	"date": "2023-04-03",
	- "id": 17,
	+ "id": 20,
	"link": "/entries/DigitsInBase.html",
	"permalink": "/entries/DigitsInBase.html",
	"shortname": "DigitsInBase",
	"title": "Positional Notation for Natural Numbers in an Arbitrary Base",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eInformation flow security ensures that the secret data manipulated by a program does not influence its observable output. Proving information flow security is especially challenging for concurrent programs, where operations on secret data may influence the execution time of a thread and, thereby, the interleaving between threads. Such internal timing channels may affect the observable outcome of a program even if an attacker does not observe execution times. Existing verification techniques for information flow security in concurrent programs attempt to prove that secret data does not influence the relative timing of threads. However, these techniques are often restrictive (for instance because they disallow branching on secret data) and make strong assumptions about the execution platform (ignoring caching, processor instructions with data-dependent execution time, and other common features that affect execution time). \u003c/p\u003e\u003cp\u003e In this entry, we formalize and prove the soundness of CommCSL, a novel relational concurrent separation logic for proving secure information flow in concurrent programs that lifts these restrictions and does not make any assumptions about timing behavior. The key idea is to prove that all mutating operations performed on shared data commute, such that different thread interleavings do not influence its final value. Crucially, commutativity is required only for an abstraction of the shared data that contains the information that will be leaked to a public output. Abstract commutativity is satisfied by many more operations than standard commutativity, which makes our technique widely applicable.\u003c/p\u003e",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2023-03-15",
	- "id": 18,
	+ "id": 21,
	"link": "/entries/CommCSL.html",
	"permalink": "/entries/CommCSL.html",
	"shortname": "CommCSL",
	"title": "Formalization of CommCSL: A Relational Concurrent Separation Logic for Proving Information Flow Security in Concurrent Programs",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/concurrency",
	"computer-science/security"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Concurrency",
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "We have \u003ca href=\"https://www.isa-afp.org/entries/No_FTL_observers.html\"\u003epreviously verified\u003c/a\u003e, in the first order theory SpecRel of Special Relativity, that inertial observers cannot travel faster than light. We now prove the corresponding result for GenRel, the first-order theory of General Relativity. Specifically, we prove that whenever an observer \u003ci\u003em\u003c/i\u003e encounters another observer \u003ci\u003ek\u003c/i\u003e (so that \u003ci\u003em\u003c/i\u003e and \u003ci\u003ek\u003c/i\u003e are both present at some spacetime location \u003ci\u003ex\u003c/i\u003e), \u003ci\u003ek\u003c/i\u003e will necessarily be observed by \u003ci\u003em\u003c/i\u003e to be traveling at less than light speed.",
	"authors": [
	"Mike Stannett",
	"Edward Higgins",
	"Hajnal Andreka",
	"Judit Madarasz",
	"István Németi",
	"Gergely Szekely"
	],
	"date": "2023-03-05",
	- "id": 19,
	+ "id": 22,
	"link": "/entries/No_FTL_observers_Gen_Rel.html",
	"permalink": "/entries/No_FTL_observers_Gen_Rel.html",
	"shortname": "No_FTL_observers_Gen_Rel",
	"title": "No Faster-Than-Light Observers (GenRel)",
	"topic_links": [
	"mathematics/physics"
	],
	"topics": [
	"Mathematics/Physics"
	],
	"used_by": 0
	},
	{
	"abstract": "Expander Graphs are low-degree graphs that are highly connected. They have diverse applications, for example in derandomization and pseudo-randomness, error-correcting codes, as well as pure mathematical subjects such as metric embeddings. This entry formalizes the concept and derives main theorems about them such as Cheeger's inequality or tail bounds on distribution of random walks on them. It includes a strongly explicit construction for every size and spectral gap. The latter is based on the Margulis-Gabber-Galil graphs and several graph operations that preserve spectral properties. The proofs are based on the survey papers/monographs by \u003ca href=\"http://dx.doi.org/10.1090/S0273-0979-06-01126-8\"\u003eHoory et al.\u003c/a\u003e and \u003ca href=\"http://dx.doi.org/10.1561/0400000010\"\u003eVadhan\u003c/a\u003e, as well as results from \u003ca href=\"https://doi.org/10.1007/978-3-642-15369-3_46\"\u003eImpagliazzo and Kabanets\u003c/a\u003e and \u003ca href=\"http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.42\"\u003eMurtagh et al.\u003c/a\u003e",
	"authors": [
	"Emin Karayel"
	],
	"date": "2023-03-03",
	- "id": 20,
	+ "id": 23,
	"link": "/entries/Expander_Graphs.html",
	"permalink": "/entries/Expander_Graphs.html",
	"shortname": "Expander_Graphs",
	"title": "Expander Graphs",
	"topic_links": [
	"computer-science/algorithms/graph",
	"computer-science/algorithms/randomized",
	"mathematics/graph-theory"
	],
	"topics": [
	"Computer science/Algorithms/Graph",
	"Computer science/Algorithms/Randomized",
	"Mathematics/Graph theory"
	],
	"used_by": 1
	},
	{
	"abstract": "I formalize the notion of \u003ci\u003erenaming-enriched sets\u003c/i\u003e (\u003ci\u003erensets\u003c/i\u003e for short) and renaming-based recursion introduced in my \u003ca href=\"https://link.springer.com/book/10.1007/978-3-031-10769-6\"\u003eIJCAR 2022\u003c/a\u003e paper \u003ca href=\"https://link.springer.com/chapter/10.1007/978-3-031-10769-6_36\"\u003e“Rensets and Renaming-Based Recursion for Syntax with Bindings”\u003c/a\u003e. Rensets are an algebraic axiomatization of renaming (variable-for-variable substitution). The formalization includes a connection with nominal sets, showing that any renset naturally gives rise to a nominal set. It also includes examples of deploying the renaming-based recursor: semantic interpretation, counting functions for free and bound occurrences, unary and parallel substitution, etc. Finally, it includes a variation of rensets that axiomatize term-for-variable substitution, called \u003ci\u003esubstitutive sets\u003c/i\u003e, which yields a corresponding recursion principle.",
	"authors": [
	"Andrei Popescu"
	],
	"date": "2023-02-28",
	- "id": 21,
	+ "id": 24,
	"link": "/entries/Rensets.html",
	"permalink": "/entries/Rensets.html",
	"shortname": "Rensets",
	"title": "Renaming-Enriched Sets (Rensets) and Renaming-Based Recursion",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We give a sound an complete multiple-conclusion calculus \$\\$\\vdash\$ for finitely additive probability inequalities. In particular, we show $$\\mathbf{\\sim} \\Gamma \\$\\vdash \\mathbf{\\sim} \\Phi \\equiv \\forall \\mathcal{P} \\in probabilities. \\sum \\phi \\leftarrow \\Phi.\\ \\mathcal{P} \\phi \\leq \\sum \\gamma \\leftarrow \\Gamma.\\ \\mathcal{P} \\gamma $$ ... where $\\sim \\Gamma$ is the negation of all of the formulae in $\\Gamma$ (and similarly for $\\sim\\Phi$). We prove this by using an abstract form of \u003cem\u003eMaxSAT\u003c/em\u003e. We also show $$MaxSAT (\\mathbf{\\sim} \\Gamma\\ @\\ \\Phi) + c\\leq length\\ \\Gamma \\equiv \\forall \\mathcal{P} \\in probabilities. \\left(\\sum \\phi \\leftarrow \\Phi.\\ \\mathcal{P} \\phi\\right) + c \\leq \\sum \\gamma \\leftarrow \\Gamma.\\ \\mathcal{P} \\gamma $$ Finally, we establish a \u003cem\u003ecollapse theorem\u003c/em\u003e, which asserts that $\\left(\\sum \\phi \\leftarrow \\Phi.\\ \\mathcal{P} \\phi\\right) + c \\leq \\sum \\gamma \\leftarrow \\Gamma.\\ \\mathcal{P} \\gamma$ holds for all probabilities $\\mathcal{P}$ if and only if $\\left(\\sum \\phi \\leftarrow \\Phi.\\ \\delta \\phi\\right) + c \\leq \\sum \\gamma \\leftarrow \\Gamma.\\ \\delta \\gamma$ holds for all binary-valued probabilities $\\delta$.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2023-02-20",
	- "id": 22,
	+ "id": 25,
	"link": "/entries/Probability_Inequality_Completeness.html",
	"permalink": "/entries/Probability_Inequality_Completeness.html",
	"shortname": "Probability_Inequality_Completeness",
	"title": "A Sound and Complete Calculus for Probability Inequalities",
	"topic_links": [
	"mathematics/probability-theory",
	"logic/general-logic/classical-propositional-logic",
	"logic/general-logic"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Logic/General logic/Classical propositional logic",
	"Logic/General logic"
	],
	"used_by": 0
	},
	{
	"abstract": "This article gives an elementary computational proof of the group law for Edwards elliptic curves. The associative law is expressed as a polynomial identity over the integers that is directly checked by polynomial division. Unlike other proofs, no preliminaries such as intersection numbers, B́ezout’s theorem, projective geometry, divisors, or Riemann Roch are required.",
	"authors": [
	"Rodrigo Raya"
	],
	"date": "2023-02-16",
	- "id": 23,
	+ "id": 26,
	"link": "/entries/Edwards_Elliptic_Curves_Group.html",
	"permalink": "/entries/Edwards_Elliptic_Curves_Group.html",
	"shortname": "Edwards_Elliptic_Curves_Group",
	"title": "Group Law of Edwards Elliptic Curves",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes the NP-hardness proofs of the Closest Vector Problem (CVP) and the Shortest Vector Problem (SVP) in maximum norm as well as the CVP in any p-norm for p\u003e=1. CVP and SVP are two fundamental problems in lattice theory. Lattices are a discrete, additive subgroup of R^n and are used for lattice-based cryptography. The CVP asks to find the nearest lattice vector to a target. The SVP asks to find the shortest non-zero lattice vector. This entry formalizes the basic properties of lattices, the reduction from CVP to Subset Sum in both maximum and p-norm for a finite p with 1\u003c= p and the reduction of SVP to Partition using the Bounded Homogeneous Linear Equations problem (BHLE) as an intermediate step. The formalization uncovered a number of problems with the existing proofs in the literature.",
	"authors": [
	"Katharina Kreuzer"
	],
	"date": "2023-02-02",
	- "id": 24,
	+ "id": 27,
	"link": "/entries/CVP_Hardness.html",
	"permalink": "/entries/CVP_Hardness.html",
	"shortname": "CVP_Hardness",
	"title": "Hardness of Lattice Problems",
	"topic_links": [
	"computer-science/security/cryptography",
	"mathematics/misc"
	],
	"topics": [
	"Computer science/Security/Cryptography",
	"Mathematics/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalizes two protocols from a privacy-preserving machine-learning framework based on ABY3, a particular three-party computation framework where inputs are systematically ‘reshared’ without being considered as privacy leakage. In particular, we consider the multiplication protocol and the array shuffling protocol, both based on ABY3's additive secret sharing scheme. We proved their security in the semi-honest setting under the ideal/real simulation paradigm. These two proof-of-concept opens the door to further verification of more protocols within the framework.",
	"authors": [
	"Shuwei Hu"
	],
	"date": "2023-01-27",
	- "id": 25,
	+ "id": 28,
	"link": "/entries/ABY3_Protocols.html",
	"permalink": "/entries/ABY3_Protocols.html",
	"shortname": "ABY3_Protocols",
	"title": "ABY3 Multiplication and Array Shuffling",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization extends the Saturation_Framework and Saturation_Framework_Extensions entries of the Archive of Formal Proofs with the specification and verification of four semiabstract given clause procedures, or \"loops\": the DISCOUNT, Otter, iProver, and Zipperposition loops. For each loop, (dynamic) refutational completeness is proved under the assumption that the underlying calculus is (statically) refutationally complete and that the used queue data structures are fair. The formalization is inspired by the proof sketches found in the article \"A comprehensive framework for saturation theorem proving\" by Uwe Waldmann, Sophie Tourret, Simon Robillard, and Jasmin Blanchette (Journal of Automated Reasoning 66(4): 499-539, 2022). A paper titled \"Verified given clause procedures\" about the present formalization is in the works.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Qi Qiu",
	"Sophie Tourret"
	],
	"date": "2023-01-25",
	- "id": 26,
	+ "id": 29,
	"link": "/entries/Given_Clause_Loops.html",
	"permalink": "/entries/Given_Clause_Loops.html",
	"shortname": "Given_Clause_Loops",
	"title": "Given Clause Loops",
	"topic_links": [
	"logic/general-logic/classical-first-order-logic"
	],
	"topics": [
	"Logic/General logic/Classical first-order logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We develop finitely additive probability logic and prove a theorem of Patrick Suppes that asserts that $\\Psi \\vdash \\phi$ in classical propositional logic if and only if $(\\sum \\psi \\leftarrow \\Psi.\\; 1 - \\mathcal{P} \\psi) \\geq 1 - \\mathcal{P} \\phi$ holds for all probabilities $\\mathcal{P}$. We also provide a novel \u003cem\u003edual\u003c/em\u003e form of Suppes' Theorem, which holds that $(\\sum \\phi \\leftarrow \\Phi.\\; \\mathcal{P} \\phi) \\leq \\mathcal{P} \\psi$ for all probabilities $\\mathcal{P}$ if and only $\\left(\\bigvee \\Phi\\right) \\vdash \\psi$ and all of the formulae in $\\Phi$ are logically exclusive from one another. Our proofs use \u003cem\u003eMaximally Consistent Sets\u003c/em\u003e, and as a consequence, we obtain two \u003cem\u003ecollapse\u003c/em\u003e theorems. In particular, we show $(\\sum \\phi \\leftarrow \\Phi.\\; \\mathcal{P} \\phi) \\geq \\mathcal{P} \\psi$ holds for all probabilities $\\mathcal{P}$ if and only if $(\\sum \\phi \\leftarrow \\Phi.\\; \\delta\\; \\phi) \\geq \\delta\\; \\psi$ holds for all binary-valued probabilities $\\delta$, along with the dual assertion that $(\\sum \\phi \\leftarrow \\Phi. \\;\\mathcal{P} \\phi) \\leq \\mathcal{P} \\psi$ holds for all probabilities $\\mathcal{P}$ if and only if $(\\sum \\phi \\leftarrow \\Phi.\\; \\delta\\; \\phi) \\leq \\delta\\; \\psi$ holds for all binary-valued probabilities $\\delta$.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2023-01-22",
	- "id": 27,
	+ "id": 30,
	"link": "/entries/Suppes_Theorem.html",
	"permalink": "/entries/Suppes_Theorem.html",
	"shortname": "Suppes_Theorem",
	"title": "Suppes' Theorem For Probability Logic",
	"topic_links": [
	"logic/general-logic/classical-propositional-logic",
	"mathematics/probability-theory"
	],
	"topics": [
	"Logic/General logic/Classical propositional logic",
	"Mathematics/Probability theory"
	],
	"used_by": 1
	},
	{
	"abstract": "This submission contains: \u003col\u003e \u003cli\u003e a formalisation of a small While language with support for output;\u003c/li\u003e \u003cli\u003e a standard total-correctness Hoare logic that has been proved sound and complete; and\u003c/li\u003e \u003cli\u003e a new Hoare logic for proofs about programs that diverge: this new logic has also been proved sound and complete.\u003c/li\u003e \u003c/ol\u003e",
	"authors": [
	"Johannes Åman Pohjola",
	"Magnus O. Myreen",
	"Miki Tanaka"
	],
	"date": "2023-01-20",
	- "id": 28,
	+ "id": 31,
	"link": "/entries/HoareForDivergence.html",
	"permalink": "/entries/HoareForDivergence.html",
	"shortname": "HoareForDivergence",
	"title": "A Hoare Logic for Diverging Programs",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory formalises a definition of strict $\\omega$-categories and the strict $\\omega$-category of pasting diagrams. It is the first step towards a formalisation of weak infinity categories à la Batanin\u0026ndash;Leinster.",
	"authors": [
	"Anthony Bordg",
	"Adrián Doña Mateo"
	],
	"date": "2023-01-14",
	- "id": 29,
	+ "id": 32,
	"link": "/entries/StrictOmegaCategories.html",
	"permalink": "/entries/StrictOmegaCategories.html",
	"shortname": "StrictOmegaCategories",
	"title": "Strict Omega Categories",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eIn this work, I provide an abstract framework for proving the completeness of a logical calculus using the synthetic method. The synthetic method is based on maximal consistent saturated sets (MCSs). A set of formulas is consistent (with respect to the calculus) when we cannot derive a contradiction from it. It is maximally consistent when it contains every formula that is consistent with it. For logics where it is relevant, it is saturated when it contains a witness for every existential formula. To prove completeness using these maximal consistent saturated sets, we prove a truth lemma: every formula in an MCS has a satisfying model. Here, Hintikka sets provide a useful stepping stone. These can be seen as characterizations of the MCSs based on simple subformula conditions rather than via the calculus. We then prove that every Hintikka set gives rise to a satisfying model and that MCSs are Hintikka sets. Now, assume a valid formula cannot be derived. Then its negation must be consistent and therefore satisfiable. This contradicts validity and the original formula must be derivable.\u003c/p\u003e \u003cp\u003eTo start, I build maximal consistent saturated sets for any logic that satisfies a small set of assumptions. I do this using a transfinite version of Lindenbaum's lemma, which allows me to support languages of any cardinality. I then prove useful abstract results about derivations and refutations as they relate to MCSs. Finally, I show how Hintikka sets can be derived from the logic's semantics, outlining one way to prove the required truth lemma.\u003c/p\u003e \u003cp\u003eTo demonstrate the versatility of the framework, I instantiate it with five different examples. The formalization contains soundness and completeness results for: a propositional tableau calculus, a propositional sequent calculus, an axiomatic system for modal logic, a labelled natural deduction system for hybrid logic and a natural deduction system for first-order logic. The tableau example uses custom Hintikka sets based on the calculus, but the other four examples derive them from the semantics in the style of the framework. The hybrid and first-order logic examples rely on saturated MCSs. This places requirements on the cardinalities of their languages to ensure that there are enough witnesses available. In both cases, the type of witnesses must be infinite and have cardinality at least that of the type of propositional/predicate symbols.\u003c/p\u003e",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2023-01-09",
	- "id": 30,
	+ "id": 33,
	"link": "/entries/Synthetic_Completeness.html",
	"permalink": "/entries/Synthetic_Completeness.html",
	"shortname": "Synthetic_Completeness",
	"title": "Synthetic Completeness",
	"topic_links": [
	"logic/general-logic",
	"logic/proof-theory"
	],
	"topics": [
	"Logic/General logic",
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The Cook-Levin theorem states that deciding the satisfiability of Boolean formulas in conjunctive normal form is $\\mathcal{NP}$-complete. This entry formalizes a proof of this theorem based on the textbook \u003cem\u003eComputational Complexity: A Modern Approach\u003c/em\u003e by Arora and Barak. It contains definitions of deterministic multi-tape Turing machines, the complexity classes $\\mathcal{P}$ and $\\mathcal{NP}$, polynomial-time many-one reduction, and the decision problem $\\mathtt{SAT}$. For the $\\mathcal{NP}$-hardness of $\\mathtt{SAT}$, the proof first shows that every polynomial-time computation can be performed by a two-tape oblivious Turing machine. An $\\mathcal{NP}$ problem can then be reduced to $\\mathtt{SAT}$ by a polynomial-time Turing machine that encodes computations of the problem's oblivious two-tape verifier Turing machine as formulas in conjunctive normal form.",
	"authors": [
	"Frank J. Balbach"
	],
	"date": "2023-01-08",
	- "id": 31,
	+ "id": 34,
	"link": "/entries/Cook_Levin.html",
	"permalink": "/entries/Cook_Levin.html",
	"shortname": "Cook_Levin",
	"title": "The Cook-Levin theorem",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/computability"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/Computability"
	],
	"used_by": 0
	},
	{
	"abstract": "A code $X$ is not primitivity preserving if there is a primitive list $\\mathtt{ws} \\in \\mathtt{lists} \\ X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\\{x,y\\}$-interpretations of the square $xx$ if $\\mathtt{length}\\ y \\leq \\mathtt{length} \\ x$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\\ell$. The core of the theory is an investigation of imprimitive words which are concatenations of copies of two noncommuting words (such a pair of words is called a binary code). We follow the article \u003ca href=\"https://dx.doi.org/10.1016/0304-3975(85)90060-X\"\u003e[Barbin-Le Rest, Le Rest, 85]\u003c/a\u003e (mainly Théorème 2.1 and Lemme 3.1), while substantially optimizing the proof. See also [J.-C. Spehner. Quelques problèmes d’extension, de conjugaison et de présentation des sous-monoïdes d’un monoïde libre. PhD thesis, Université Paris VII, 1976] for an earlier result on this question, and \u003ca href=\"https://dx.doi.org/10.46298/dmtcs.279\"\u003e[Maňuch, 01]\u003c/a\u003e for another proof.",
	"authors": [
	"Štěpán Holub",
	"Martin Raška"
	],
	"date": "2023-01-03",
	- "id": 32,
	+ "id": 35,
	"link": "/entries/Binary_Code_Imprimitive.html",
	"permalink": "/entries/Binary_Code_Imprimitive.html",
	"shortname": "Binary_Code_Imprimitive",
	"title": "Binary codes that do not preserve primitivity",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This article provides a formalization of the classification of intersection \$ \\{x,y\\}^* \\cap \\{u,v\\}^\$ of two monoids generated by two element codes. Namely, the intersection has one of the following forms \$ \\{\\beta,\\gamma\\}^ \\quad \\text{ or } \\quad \\left(\\beta_0 + \\beta(\\gamma(1+\\delta+ \\cdots + \\delta^t))^\\epsilon\\right)^.\$ Note that it can be infinitely generated. The result is due to \u003ca href=\"https://doi.org/10.1007/BFb0036924\"\u003e[Karhumäki, 84]\u003c/a\u003e. Our proof uses the terminology of morphisms which allows us to formulate the result in a shorter and more transparent way.",
	"authors": [
	"Štěpán Holub",
	"Štěpán Starosta"
	],
	"date": "2023-01-03",
	- "id": 33,
	+ "id": 36,
	"link": "/entries/Two_Generated_Word_Monoids_Intersection.html",
	"permalink": "/entries/Two_Generated_Word_Monoids_Intersection.html",
	"shortname": "Two_Generated_Word_Monoids_Intersection",
	"title": "Intersection of two monoids generated by two element codes",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize a multivariate quantifier elimination (QE) algorithm in the theorem prover Isabelle/HOL. Our algorithm is complete, in that it is able to reduce \u003ci\u003eany\u003c/i\u003e quantified formula in the first-order logic of real arithmetic to a logically equivalent quantifier-free formula. The algorithm we formalize is a hybrid mixture of Tarski's original QE algorithm and the Ben-Or, Kozen, and Reif algorithm, and it is the first complete multivariate QE algorithm formalized in Isabelle/HOL.",
	"authors": [
	"Katherine Kosaian",
	"Yong Kiam Tan",
	"André Platzer"
	],
	"date": "2022-12-15",
	- "id": 34,
	+ "id": 37,
	"link": "/entries/Quantifier_Elimination_Hybrid.html",
	"permalink": "/entries/Quantifier_Elimination_Hybrid.html",
	"shortname": "Quantifier_Elimination_Hybrid",
	"title": "A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "We formulate classical propositional logic as an axiom class. Our class represents a Hilbert-style proof system with the axioms \$\\vdash \\varphi \\to \\psi \\to \\varphi\$, \$\\vdash (\\varphi \\to \\psi \\to \\chi) \\to (\\varphi \\to \\psi) \\to \\varphi \\to \\chi\$, and \$\\vdash ((\\varphi \\to \\bot) \\to \\bot) \\to \\varphi\$ along with the rule \u003cem\u003emodus ponens\u003c/em\u003e \$\\vdash \\varphi \\to \\psi \\Longrightarrow \\; \\vdash \\varphi \\Longrightarrow \\; \\vdash \\psi\$. In this axiom class we provide lemmas to obtain \u003cem\u003eMaximally Consistent Sets\u003c/em\u003e via Zorn's lemma. We define the concrete classical propositional calculus inductively and show it instantiates our axiom class. We formulate the usual semantics for the propositional calculus and show strong soundness and completeness. We provide conventional definitions of the other logical connectives and prove various common identities. Finally, we show that the propositional calculus \u003cem\u003eembeds\u003c/em\u003e into any logic in our axiom class.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2022-12-15",
	- "id": 35,
	+ "id": 38,
	"link": "/entries/Propositional_Logic_Class.html",
	"permalink": "/entries/Propositional_Logic_Class.html",
	"shortname": "Propositional_Logic_Class",
	"title": "Class-based Classical Propositional Logic",
	"topic_links": [
	"logic/general-logic/classical-propositional-logic",
	"logic/proof-theory"
	],
	"topics": [
	"Logic/General logic/Classical propositional logic",
	"Logic/Proof theory"
	],
	"used_by": 1
	},
	{
	"abstract": "This theory proves a theorem of Birkhoff that asserts that any finite distributive lattice is isomorphic to the set of \u003cem\u003edown-sets\u003c/em\u003e of that lattice's join-irreducible elements. The isomorphism preserves order, meets and joins as well as complementation in the case the lattice is a Boolean algebra. A consequence of this representation theorem is that every finite Boolean algebra is isomorphic to a powerset algebra.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2022-12-06",
	- "id": 36,
	+ "id": 39,
	"link": "/entries/Birkhoff_Finite_Distributive_Lattices.html",
	"permalink": "/entries/Birkhoff_Finite_Distributive_Lattices.html",
	"shortname": "Birkhoff_Finite_Distributive_Lattices",
	"title": "Birkhoff's Representation Theorem For Finite Distributive Lattices",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "Boolos’ Curious Inference is automated in Isabelle/HOL after interactive speculation of a suitable shorthand notation (one or two definitions).",
	"authors": [
	"Christoph Benzmüller",
	"David Fuenmayor",
	"Alexander Steen",
	"Geoff Sutcliffe"
	],
	"date": "2022-12-05",
	- "id": 37,
	+ "id": 40,
	"link": "/entries/Boolos_Curious_Inference_Automated.html",
	"permalink": "/entries/Boolos_Curious_Inference_Automated.html",
	"shortname": "Boolos_Curious_Inference_Automated",
	"title": "Automation of Boolos' Curious Inference in Isabelle/HOL",
	"topic_links": [
	"logic",
	"logic/philosophical-aspects",
	"logic/computability",
	"logic/proof-theory",
	"logic/general-logic",
	"computer-science/artificial-intelligence",
	"tools"
	],
	"topics": [
	"Logic",
	"Logic/Philosophical aspects",
	"Logic/Computability",
	"Logic/Proof theory",
	"Logic/General logic",
	"Computer science/Artificial intelligence",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "We define single- and multitape Turing machines (TMs) and verify a translation from multitape TMs to singletape TMs. In particular, the following results have been formalized: the accepted languages coincide, and whenever the multitape TM runs in $\\mathcal{O}(f(n))$ time, then the singletape TM has a worst-time complexity of $\\mathcal{O}(f(n)^2 + n)$. The translation is applicable both on deterministic and non-deterministic TMs.",
	"authors": [
	"Christian Dalvit",
	"René Thiemann"
	],
	"date": "2022-11-30",
	- "id": 38,
	+ "id": 41,
	"link": "/entries/Multitape_To_Singletape_TM.html",
	"permalink": "/entries/Multitape_To_Singletape_TM.html",
	"shortname": "Multitape_To_Singletape_TM",
	"title": "A Verified Translation of Multitape Turing Machines into Singletape Turing Machines",
	"topic_links": [
	"logic/computability",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Logic/Computability",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We utilize and extend the method of shallow semantic embeddings (SSEs) in classical higher-order logic (HOL) to construct a custom theorem proving environment for abstract objects theory (AOT) on the basis of Isabelle/HOL. SSEs are a means for universal logical reasoning by translating a target logic to HOL using a representation of its semantics. AOT is a foundational metaphysical theory, developed by Edward Zalta, that explains the objects presupposed by the sciences as abstract objects that reify property patterns. In particular, AOT aspires to provide a philosphically grounded basis for the construction and analysis of the objects of mathematics. We can support this claim by verifying Uri Nodelman's and Edward Zalta's reconstruction of Frege's theorem: we can confirm that the Dedekind-Peano postulates for natural numbers are consistently derivable in AOT using Frege's method. Furthermore, we can suggest and discuss generalizations and variants of the construction and can thereby provide theoretical insights into, and contribute to the philosophical justification of, the construction. In the process, we can demonstrate that our method allows for a nearly transparent exchange of results between traditional pen-and-paper-based reasoning and the computerized implementation, which in turn can retain the automation mechanisms available for Isabelle/HOL. During our work, we could significantly contribute to the evolution of our target theory itself, while simultaneously solving the technical challenge of using an SSE to implement a theory that is based on logical foundations that significantly differ from the meta-logic HOL. In general, our results demonstrate the fruitfulness of the practice of Computational Metaphysics, i.e. the application of computational methods to metaphysical questions and theories.",
	"authors": [
	"Daniel Kirchner"
	],
	"date": "2022-11-28",
	- "id": 39,
	+ "id": 42,
	"link": "/entries/AOT.html",
	"permalink": "/entries/AOT.html",
	"shortname": "AOT",
	"title": "Abstract Object Theory",
	"topic_links": [
	"logic/general-logic",
	"mathematics"
	],
	"topics": [
	"Logic/General logic",
	"Mathematics"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work, we present a formal memory model that provides a memory semantics for CHERI-C programs with uncompressed capabilities in a 'purecap' environment. We present a CHERI-C memory model theory with properties suitable for verification and potentially other types of analyses. Our theory generates an OCaml executable instance of the memory model, which is then used to instantiate the parametric Gillian program analysis framework, enabling concrete execution of CHERI-C programs. The tool can run a CHERI-C test suite, demonstrating the correctness of our tool, and catch a good class of safety violations that the CHERI hardware might miss.",
	"authors": [
	"Seung Hoon Park"
	],
	"date": "2022-11-25",
	- "id": 40,
	+ "id": 43,
	"link": "/entries/CHERI-C_Memory_Model.html",
	"permalink": "/entries/CHERI-C_Memory_Model.html",
	"shortname": "CHERI-C_Memory_Model",
	"title": "A Formal CHERI-C Memory Model",
	"topic_links": [
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "The Sauer-Shelah Lemma is a fundamental result in extremal set theory and combinatorics. It guarantees the existence of a set $T$ of size $k$ that is shattered by a family of sets $\\mathcal{F}$ if the cardinality of the family is greater than some bound dependent on $k$. A set $T$ is said to be \u003cem\u003eshattered\u003c/em\u003e by a family $\\mathcal{F}$ if every subset of $T$ can be obtained as an intersection of $T$ with some set $S \\in \\mathcal{F}$. The Sauer-Shelah Lemma has found use in diverse fields such as computational geometry, approximation algorithms and machine learning. In this entry we formalize the notion of shattering and prove the generalized and standard versions of the Sauer-Shelah Lemma.",
	"authors": [
	"Ata Keskin"
	],
	"date": "2022-11-24",
	- "id": 41,
	+ "id": 44,
	"link": "/entries/Sauer_Shelah_Lemma.html",
	"permalink": "/entries/Sauer_Shelah_Lemma.html",
	"shortname": "Sauer_Shelah_Lemma",
	"title": "Sauer-Shelah Lemma",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise Kneser's Theorem in combinatorics following the proof from the 2014 paper \u003ca href=\"https://link.springer.com/chapter/10.1007/978-1-4939-1601-6_3\"\u003eA short proof of Kneser’s addition theorem for abelian groups\u003c/a\u003e by Matt DeVos. We also show a strict version of Kneser's Theorem as well as the Cauchy–Davenport Theorem as a corollary of Kneser's Theorem.",
	"authors": [
	"Mantas Bakšys",
	"Angeliki Koutsoukou-Argyraki"
	],
	"date": "2022-11-21",
	- "id": 42,
	+ "id": 45,
	"link": "/entries/Kneser_Cauchy_Davenport.html",
	"permalink": "/entries/Kneser_Cauchy_Davenport.html",
	"shortname": "Kneser_Cauchy_Davenport",
	"title": "Kneser's Theorem and the Cauchy–Davenport Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003ca href=\"https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem\"\u003eTurán's Graph Theorem\u003c/a\u003e states that any undirected, simple graph with $n$ vertices that does not contain a $p$-clique, contains at most $\\left( 1 - \\frac{1}{p-1} \\right) \\frac{n^2}{2}$ edges. The theorem is an important result in graph theory and the foundation of the field of extremal graph theory. The formalisation follows Aigner and Ziegler's presentation in \u003ci\u003eProofs from THE BOOK\u003c/i\u003e of Turán's initial proof. Besides a direct adaptation of the textbook proof, a simplified, second proof is presented which decreases the size of the formalised proof significantly.",
	"authors": [
	"Nils Lauermann"
	],
	"date": "2022-11-14",
	- "id": 43,
	+ "id": 46,
	"link": "/entries/Turans_Graph_Theorem.html",
	"permalink": "/entries/Turans_Graph_Theorem.html",
	"shortname": "Turans_Graph_Theorem",
	"title": "Turán's Graph Theorem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Combinatorial objects have configurations which can be enumerated by algorithms, but especially for imperative programs, it is difficult to find out if they produce the correct output and don’t generate duplicates. Therefore, for some of the most common combinatorial objects, namely n_Sequences, n_Permutations, n_Subsets, Powerset, Integer_Compositions, Integer_Partitions, Weak_Integer_Compositions, Derangements and Trees, this entry formalizes efficient functional programs and verifies their correctness. In addition, it provides cardinality proofs for those combinatorial objects. Some cardinalities are verified using the enumeration functions and others are shown using existing libraries including other AFP entries.",
	"authors": [
	"Paul Hofmeier",
	"Emin Karayel"
	],
	"date": "2022-11-11",
	- "id": 44,
	+ "id": 47,
	"link": "/entries/Combinatorial_Enumeration_Algorithms.html",
	"permalink": "/entries/Combinatorial_Enumeration_Algorithms.html",
	"shortname": "Combinatorial_Enumeration_Algorithms",
	"title": "Combinatorial Enumeration Algorithms",
	"topic_links": [
	"mathematics/combinatorics",
	"computer-science/algorithms"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise the Balog–Szemerédi–Gowers Theorem, a profound result in additive combinatorics which played a central role in \u003ca href=\"https://link.springer.com/article/10.1007/s00039-001-0332-9\"\u003eGowers's proof deriving the first effective bounds for Szemerédi's Theorem\u003c/a\u003e. The proof is of great mathematical interest given that it involves an interplay between different mathematical areas, namely applications of graph theory and probability theory to additive combinatorics involving algebraic objects. This interplay is what made the process of the formalisation, for which we had to develop formalisations of new background material in the aforementioned areas, more rich and technically challenging. We demonstrate how locales, Isabelle’s module system, can be employed to handle such interplays. To treat the graph-theoretic aspects of the proof, we make use of a new, more general \u003ca href=\"https://www.isa-afp.org/entries/Undirected_Graph_Theory.html\"\u003eundirected graph theory library\u003c/a\u003e developed recently by Chelsea Edmonds, which is both flexible and extensible. For the formalisation we followed a proof presented in the 2022 lecture notes by Timothy Gowers \u003ca href=\"https://drive.google.com/file/d/1ut0mUqSyPMweoxoDTfhXverEONyFgcuO/view\"\u003eIntroduction to Additive Combinatorics \u003c/a\u003e for Part III of the Mathematical Tripos taught at the University of Cambridge. In addition to the main theorem, which, following our source, is formulated for difference sets, we also give an alternative version for sumsets which required a formalisation of an auxiliary triangle inequality following a proof by Yufei Zhao from his book \u003ca href=\"https://yufeizhao.com/gtacbook/\"\u003eGraph Theory and Additive Combinatorics \u003c/a\u003e. We moreover formalise a few additional results in additive combinatorics that are not used in the proof of the main theorem. This is the first formalisation of the Balog–Szemerédi–Gowers Theorem in any proof assistant to our knowledge.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Mantas Bakšys",
	"Chelsea Edmonds"
	],
	"date": "2022-11-11",
	- "id": 45,
	+ "id": 48,
	"link": "/entries/Balog_Szemeredi_Gowers.html",
	"permalink": "/entries/Balog_Szemeredi_Gowers.html",
	"shortname": "Balog_Szemeredi_Gowers",
	"title": "The Balog–Szemerédi–Gowers Theorem",
	"topic_links": [
	"mathematics/combinatorics",
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e In party-approval multi-winner elections, the goal is to allocate the seats of a fixed-size committee to parties based on approval ballots of the voters over the parties. In particular, each can approve multiple parties and each party can be assigned multiple seats. \u003c/p\u003e \u003cp\u003eThree central requirements in this settings are:\u003c/p\u003e \u003cul\u003e \u003cli\u003eAnonymity: The result is invariant under renaming the voters.\u003c/li\u003e \u003cli\u003eRepresentation:Every sufficiently large group of voters with similar preferences is represented by some committee members.\u003c/li\u003e \u003cli\u003eStrategy-proofness: No voter can benefit by misreporting her true preferences.\u003c/li\u003e \u003c/ul\u003e \u003cp\u003e We show that these three basic axioms are incompatible for party-approval multi-winner voting rules, thus proving a far-reaching impossibility theorem.\u003c/p\u003e \u003cp\u003eThe proof of this result is obtained by formulating the problem in propositional logic and then letting a SAT solver show that the formula is unsatisfiable. The DRUP proof output by the SAT solver is then converted into \u003ca href=\"https://www21.in.tum.de/~lammich/grat/\"\u003eLammich's GRAT format\u003c/a\u003e and imported into Isabelle/HOL with some custom-written ML code.\u003c/p\u003e \u003cp\u003eThis transformation is proof-producing, so the final Isabelle/HOL theorem does not rely on any oracles or other trusted external trusted components.\u003c/p\u003e",
	"authors": [
	"Théo Delemazure",
	"Tom Demeulemeester",
	"Manuel Eberl",
	"Jonas Israel",
	"Patrick Lederer"
	],
	"date": "2022-11-10",
	- "id": 46,
	+ "id": 49,
	"link": "/entries/PAPP_Impossibility.html",
	"permalink": "/entries/PAPP_Impossibility.html",
	"shortname": "PAPP_Impossibility",
	"title": "The Incompatibility of Strategy-Proofness and Representation in Party-Approval Multi-Winner Elections",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalization includes a general framework for query optimization consisting of the definitions of selectivities, query graphs, join trees, and cost functions. Furthermore, it implements the join ordering algorithm IKKBZ using these definitions. It verifies the correctness of these definitions and proves that IKKBZ produces an optimal solution within a restricted solution space.",
	"authors": [
	"Lukas Stevens",
	"Bernhard Stöckl"
	],
	"date": "2022-10-04",
	- "id": 47,
	+ "id": 50,
	"link": "/entries/Query_Optimization.html",
	"permalink": "/entries/Query_Optimization.html",
	"shortname": "Query_Optimization",
	"title": "Verification of Query Optimization Algorithms",
	"topic_links": [
	"computer-science/data-management-systems"
	],
	"topics": [
	"Computer science/Data management systems"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work we consider the \u003ca href=\"https://en.wikipedia.org/wiki/Maximum_subarray_problem\"\u003emaximum segment sum\u003c/a\u003e problem, that is to compute, given a list of numbers, the largest of the sums of the contiguous segments of that list. We assume that the elements of the list are not necessarily numbers but just elements of some linearly ordered group. Both a naive algorithm ($\\mathcal{O}(n^2)$) and \u003ca href=\"https://en.wikipedia.org/wiki/Maximum_subarray_problem#Kadane's_algorithm\"\u003eKadane's algorithm\u003c/a\u003e ($\\mathcal{O}(n)$) are given and their correctness is proved.",
	"authors": [
	"Nils Cremer"
	],
	"date": "2022-09-29",
	- "id": 48,
	+ "id": 51,
	"link": "/entries/Maximum_Segment_Sum.html",
	"permalink": "/entries/Maximum_Segment_Sum.html",
	"shortname": "Maximum_Segment_Sum",
	"title": "Maximum Segment Sum",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry presents a general library for undirected graph theory - enabling reasoning on simple graphs and undirected graphs with loops. It primarily is inspired by Noschinski's basic ugraph definition in the \u003ca href=\"/entries/Girth_Chromatic.html\"\u003eGirth Chromatic Entry\u003c/a\u003e, however generalises it in a number of ways and significantly expands on the range of basic graph theory definitions formalised. Notably, this library removes the constraint of vertices being a type synonym with the natural numbers which causes issues in more complex mathematical reasoning using graphs, such as the Balog Szemeredi Gowers theorem which this library is used for. Secondly this library also presents a locale-centric approach, enabling more concise, flexible, and reusable modelling of different types of graphs. Using this approach enables easy links to be made with more expansive formalisations of other combinatorial structures, such as incidence systems, as well as various types of formal representations of graphs. Further inspiration is also taken from Noschinski's Directed Graph library for some proofs and definitions on walks, paths and cycles, however these are much simplified using the set based representation of graphs, and also extended on in this formalisation.",
	"authors": [
	"Chelsea Edmonds"
	],
	"date": "2022-09-29",
	- "id": 49,
	+ "id": 52,
	"link": "/entries/Undirected_Graph_Theory.html",
	"permalink": "/entries/Undirected_Graph_Theory.html",
	"shortname": "Undirected_Graph_Theory",
	"title": "Undirected Graph Theory",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 2
	},
	{
	"abstract": "The relational calculus (RC), i.e., first-order logic with equality but without function symbols, is a concise, declarative database query language. In contrast to relational algebra or SQL, which are the traditional query languages of choice in the database community, RC queries can evaluate to an infinite relation. Moreover, even in cases where the evaluation result of an RC query would be finite it is not clear how to efficiently compute it. Safe-range RC is an interesting syntactic subclass of RC, because all safe-range queries evaluate to a finite result and it is \u003ca href=\"http://webdam.inria.fr/Alice/pdfs/Chapter-5.pdf\"\u003ewell-known\u003c/a\u003e how to evaluate such queries by translating them to relational algebra. We formalize and prove correct \u003ca href=\"https://doi.org/10.4230/LIPIcs.ICDT.2022.11\"\u003eour recent translation\u003c/a\u003e of an arbitrary RC query into a pair of safe-range queries. Assuming an infinite domain, the two queries have the following meaning: The first is closed and characterizes the original query's relative safety, i.e., whether given a fixed database (interpretation of atomic predicates with finite relations), the original query evaluates to a finite relation. The second safe-range query is equivalent to the original query, if the latter is relatively safe. The formalization uses the Refinement Framework to go from the non-deterministic algorithm described in the paper to a deterministic, executable query translation. Our executable query translation is a first step towards a verified tool that efficiently evaluates arbitrary RC queries. This very problem is also solved by the AFP entry \u003ca href=\"https://isa-afp.org/entries/Eval_FO.html\"\u003eEval_FO\u003c/a\u003e with a theoretically incomparable but practically worse time complexity. (The latter is demonstrated by \u003ca href=\"https://doi.org/10.4230/LIPIcs.ICDT.2022.11\"\u003eour empirical evaluation\u003c/a\u003e.)",
	"authors": [
	"Martin Raszyk",
	"Dmitriy Traytel"
	],
	"date": "2022-09-28",
	- "id": 50,
	+ "id": 53,
	"link": "/entries/Safe_Range_RC.html",
	"permalink": "/entries/Safe_Range_RC.html",
	"shortname": "Safe_Range_RC",
	"title": "Making Arbitrary Relational Calculus Queries Safe-Range",
	"topic_links": [
	"computer-science/data-management-systems",
	"logic/general-logic/classical-first-order-logic"
	],
	"topics": [
	"Computer science/Data management systems",
	"Logic/General logic/Classical first-order logic"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is a formalization of Stalnaker's epistemic logic and its soundness and completeness theorems, as well as the equivalence between the axiomatization of S4 available in the Epistemic Logic theory and the topological one. It builds on the Epistemic Logic theory.",
	"authors": [
	"Laura P. Gamboa Guzman"
	],
	"date": "2022-09-23",
	- "id": 51,
	+ "id": 54,
	"link": "/entries/Stalnaker_Logic.html",
	"permalink": "/entries/Stalnaker_Logic.html",
	"shortname": "Stalnaker_Logic",
	"title": "Stalnaker's Epistemic Logic",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 0
	},
	{
	"abstract": "The field of p-adic numbers for a prime integer p is constructed. Basic facts about p-adic topology including Hensel's Lemma are proved, building on a prior submission by the author. The theory of semialgebraic sets and semialgebraic functions on cartesian powers of p-adic fields is also developed, following a formalization of these concepts due to Denef. This is done towards a formalization of Denef's proof of Macintyre's quantifier elimination theorem for p-adic fields. Theories developing general multivariable polynomial rings over a commutative ring are developed, as well as some general theory of cartesian powers of an arbitrary ring.",
	"authors": [
	"Aaron Crighton"
	],
	"date": "2022-09-22",
	- "id": 52,
	+ "id": 55,
	"link": "/entries/Padic_Field.html",
	"permalink": "/entries/Padic_Field.html",
	"shortname": "Padic_Field",
	"title": "p-adic Fields and p-adic Semialgebraic Sets",
	"topic_links": [
	"mathematics/number-theory",
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Number theory",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "We construct an abstract ledger supporting the \u003cem\u003erisk-free lending\u003c/em\u003e protocol. The risk-free lending protocol is a system for issuing and exchanging novel financial products we call \u003cem\u003erisk-free loans\u003c/em\u003e. The system allows one party to lend money at 0\u0026percnt; APY to another party in exchange for a good or service. On every update of the ledger, accounts have interest distributed to them. Holders of lent assets keep interest accrued by those assets. After distributing interest, the system returns a fixed fraction of each loan. These fixed fractions determine \u003cem\u003eloan periods\u003c/em\u003e. Loans for longer periods have a smaller fixed fraction returned. Loans may be re-lent or used as collateral for other loans. We give a sufficient criterion to enforce that all accounts will forever be solvent. We give a protocol for maintaining this invariant when transferring or lending funds. We also show that this invariant holds after an update. Even though the system does not track counter-party obligations, we show that all credited and debited loans cancel, and the monetary supply grows at a specified interest rate.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2022-09-18",
	- "id": 53,
	+ "id": 56,
	"link": "/entries/Risk_Free_Lending.html",
	"permalink": "/entries/Risk_Free_Lending.html",
	"shortname": "Risk_Free_Lending",
	"title": "Risk-Free Lending",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is a formalization of soundness and completeness of the Bernays-Tarski axiom system for classical implicational logic. The completeness proof is constructive following the approach by László Kalmár, Elliott Mendelson and others. The result can be extended to full classical propositional logic by uncommenting a few lines for falsehood. ",
	"authors": [
	"Asta Halkjær From",
	"Jørgen Villadsen"
	],
	"date": "2022-09-13",
	- "id": 54,
	+ "id": 57,
	"link": "/entries/Implicational_Logic.html",
	"permalink": "/entries/Implicational_Logic.html",
	"shortname": "Implicational_Logic",
	"title": "Soundness and Completeness of Implicational Logic",
	"topic_links": [
	"logic/general-logic/classical-propositional-logic",
	"logic/proof-theory"
	],
	"topics": [
	"Logic/General logic/Classical propositional logic",
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes the specification and the algorithm of the cryptographic scheme CRYSTALS-KYBER with multiplication using the Number Theoretic Transform and verifies its (1-δ)-correctness proof. CRYSTALS-KYBER is a key encapsulation mechanism in lattice-based post-quantum cryptography. This entry formalizes the key generation, encryption and decryption algorithms and shows that the algorithm decodes correctly under a highly probable assumption ((1-δ)-correctness). Moreover, the Number Theoretic Transform (NTT) in the case of Kyber and the convolution theorem thereon is formalized.",
	"authors": [
	"Katharina Kreuzer"
	],
	"date": "2022-09-08",
	- "id": 55,
	+ "id": 58,
	"link": "/entries/CRYSTALS-Kyber.html",
	"permalink": "/entries/CRYSTALS-Kyber.html",
	"shortname": "CRYSTALS-Kyber",
	"title": "CRYSTALS-Kyber",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Many separation logics support fractional permissions to distinguish between read and write access to a heap location, for instance, to allow concurrent reads while enforcing exclusive writes. Fractional permissions extend to composite assertions such as (co)inductive predicates and magic wands by allowing those to be multiplied by a fraction. Typical separation logic proofs require that this multiplication has three key properties: it needs to distribute over assertions, it should permit fractions to be factored out from assertions, and two fractions of the same assertion should be combinable into one larger fraction. Existing formal semantics incorporating fractional assertions into a separation logic define multiplication semantically (via models), resulting in a semantics in which distributivity and combinability do not hold for key resource assertions such as magic wands, and fractions cannot be factored out from a separating conjunction. By contrast, existing automatic separation logic verifiers define multiplication syntactically, resulting in a different semantics for which it is unknown whether distributivity and combinability hold for all assertions. In this entry (which accompanies an \u003ca href=\"https://dardinier.me/papers/multiplication.pdf\"\u003eOOPSLA'22 paper\u003c/a\u003e), we present and formalize an unbounded version of separation logic, a novel semantics for separation logic assertions that allows states to hold more than a full permission to a heap location during the evaluation of an assertion. By reimposing upper bounds on the permissions held per location at statement boundaries, we retain key properties of separation logic, in particular, we prove that the frame rule still holds. We also prove that our assertion semantics unifies semantic and syntactic multiplication and thereby reconciles the discrepancy between separation logic theory and tools and enjoys distributivity, factorisability, and combinability.",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2022-09-05",
	- "id": 56,
	+ "id": 59,
	"link": "/entries/Separation_Logic_Unbounded.html",
	"permalink": "/entries/Separation_Logic_Unbounded.html",
	"shortname": "Separation_Logic_Unbounded",
	"title": "Unbounded Separation Logic",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise the proof of an important theorem in additive combinatorics due to Khovanskii, attesting that the cardinality of the set of all sums of $n$ many elements of $A$, where $A$ is a finite subset of an abelian group, is a polynomial in $n$ for all sufficiently large $n$. We follow a proof due to Nathanson and Ruzsa as presented in the notes “Introduction to Additive Combinatorics” by Timothy Gowers for the University of Cambridge.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Lawrence C. Paulson"
	],
	"date": "2022-09-02",
	- "id": 57,
	+ "id": 60,
	"link": "/entries/Khovanskii_Theorem.html",
	"permalink": "/entries/Khovanskii_Theorem.html",
	"shortname": "Khovanskii_Theorem",
	"title": "Khovanskii\u0026#x27;s Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article is a formalisation of a proof of the Hales–Jewett theorem presented in the textbook \u003cem\u003eRamsey Theory\u003c/em\u003e by Graham et al.\u003c/p\u003e \u003cp\u003eThe Hales–Jewett theorem is a result in Ramsey Theory which states that, for any non-negative integers $r$ and $t$, there exists a minimal dimension $N$, such that any $r$-coloured $N'$-dimensional cube over $t$ elements (with $N' \\geq N$) contains a monochromatic line. This theorem generalises Van der Waerden's Theorem, which has already been formalised in another \u003ca href=\"https://www.isa-afp.org/entries/Van_der_Waerden.html\"\u003eAFP entry\u003c/a\u003e.\u003c/p\u003e",
	"authors": [
	"Ujkan Sulejmani",
	"Manuel Eberl",
	"Katharina Kreuzer"
	],
	"date": "2022-09-02",
	- "id": 58,
	+ "id": 61,
	"link": "/entries/Hales_Jewett.html",
	"permalink": "/entries/Hales_Jewett.html",
	"shortname": "Hales_Jewett",
	"title": "The Hales–Jewett Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry contains an Isabelle formalization of the \u003cem\u003eNumber Theoretic Transform (NTT)\u003c/em\u003e which is the analogue to a \u003cem\u003eDiscrete Fourier Transform (DFT)\u003c/em\u003e over a finite field. Roots of unity in the complex numbers are replaced by those in a finite field. \u003c/p\u003e\u003cp\u003eFirst, we define both \u003cem\u003eNTT\u003c/em\u003e and the inverse transform \u003cem\u003eINTT\u003c/em\u003e in Isabelle and prove them to be mutually inverse. \u003c/p\u003e\u003cp\u003e\u003cem\u003eDFT\u003c/em\u003e can be efficiently computed by the recursive \u003cem\u003eFast Fourier Transform (FFT)\u003c/em\u003e. In our formalization, this algorithm is adapted to the setting of the \u003cem\u003eNTT\u003c/em\u003e: We implement a \u003cem\u003eFast Number Theoretic Transform (FNTT)\u003c/em\u003e based on the Butterfly scheme by Cooley and Tukey. Additionally, we provide an inverse transform \u003cem\u003eIFNTT\u003c/em\u003e and prove it mutually inverse to \u003cem\u003eFNTT\u003c/em\u003e. \u003c/p\u003e\u003cp\u003e Afterwards, a recursive formalization of the \u003cem\u003eFNTT\u003c/em\u003e running time is examined and the famous $O(n \\log n)$ bounds are proven.\u003c/p\u003e",
	"authors": [
	"Thomas Ammer",
	"Katharina Kreuzer"
	],
	"date": "2022-08-18",
	- "id": 59,
	+ "id": 62,
	"link": "/entries/Number_Theoretic_Transform.html",
	"permalink": "/entries/Number_Theoretic_Transform.html",
	"shortname": "Number_Theoretic_Transform",
	"title": "Number Theoretic Transform",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 1
	},
	{
	"abstract": "We prove the correctness of a sequential algorithm for computing maximal strongly connected components (SCCs) of a graph due to Vincent Bloemen.",
	"authors": [
	"Stephan Merz",
	"Vincent Trélat"
	],
	"date": "2022-08-17",
	- "id": 60,
	+ "id": 63,
	"link": "/entries/SCC_Bloemen_Sequential.html",
	"permalink": "/entries/SCC_Bloemen_Sequential.html",
	"shortname": "SCC_Bloemen_Sequential",
	"title": "Correctness of a Set-based Algorithm for Computing Strongly Connected Components of a Graph",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory contains the involution-based proof of the two squares theorem from \u003ca href=\"https://dx.doi.org/10.1007/978-3-662-57265-8\"\u003eTHE BOOK\u003c/a\u003e.",
	"authors": [
	"Maksym Bortin"
	],
	"date": "2022-08-15",
	- "id": 61,
	+ "id": 64,
	"link": "/entries/Involutions2Squares.html",
	"permalink": "/entries/Involutions2Squares.html",
	"shortname": "Involutions2Squares",
	"title": "From THE BOOK: Two Squares via Involutions",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides executable formalisations of complete test generation algorithms for finite state machines. It covers testing for the language-equivalence and reduction conformance relations, supporting the former via the W, Wp, HSI, H, SPY and SPYH-methods, and the latter via adaptive state counting. The test strategies are implemented using generic frameworks, allowing for reuse of shared components between related strategies. This work is described in the author\u0026#x27;s \u003ca href=\"https://doi.org/10.26092/elib/1665\"\u003edoctoral thesis\u003c/a\u003e.",
	"authors": [
	"Robert Sachtleben"
	],
	"date": "2022-08-09",
	- "id": 62,
	+ "id": 65,
	"link": "/entries/FSM_Tests.html",
	"permalink": "/entries/FSM_Tests.html",
	"shortname": "FSM_Tests",
	"title": "Verified Complete Test Strategies for Finite State Machines",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "JSON (JavaScript Object Notation) is a common format for exchanging data, based on a collection of key/value-pairs (the JSON objects) and lists. Its syntax is inspired by JavaScript with the aim of being easy to read and write for humans and easy to parse and generate for machines. Despite its origin in the JavaScript world, JSON is language-independent and many programming languages support working with JSON-encoded data. This makes JSON an interesting format for exchanging data with Isabelle/HOL. This AFP entry provides a JSON-like import-expert format for both Isabelle/ML and Isabelle/HOL. On the one hand, this AFP entry provides means for Isabelle/HOL users to work with JSON encoded data without the need using Isabelle/ML. On the other and, the provided Isabelle/ML interfaces allow additional extensions or integration into Isabelle extensions written in Isabelle/ML. While format is not fully JSON compliant (e.g., due to limitations in the range of supported Unicode characters), it works in most situations: the provided implementation in Isabelle/ML and its representation in Isabelle/HOL have been used successfully in several projects for exchanging data sets of several hundredths of megabyte between Isabelle and external tools.",
	"authors": [
	"Achim D. Brucker"
	],
	"date": "2022-07-29",
	- "id": 63,
	+ "id": 66,
	"link": "/entries/Nano_JSON.html",
	"permalink": "/entries/Nano_JSON.html",
	"shortname": "Nano_JSON",
	"title": "Nano JSON: Working with JSON formatted data in Isabelle/HOL and Isabelle/ML",
	"topic_links": [
	"tools",
	"computer-science/data-structures"
	],
	"topics": [
	"Tools",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Smart contracts are automatically executed programs, usually representing legal agreements such as financial transactions. Thus, bugs in smart contracts can lead to large financial losses. For example, an incorrectly initialized contract was the root cause of the Parity Wallet bug that saw $280M worth of Ether destroyed. Ether is the cryptocurrency of the Ethereum blockchain that uses Solidity for expressing smart contracts. We address this problem by formalizing an executable denotational semantics for Solidity in the interactive theorem prover Isabelle/HOL. This formal semantics builds the foundation of an interactive program verification environment for Solidity programs and allows for inspecting them by (symbolic) execution. We combine the latter with grammar based fuzzing to ensure that our formal semantics complies to the Solidity implementation on the Ethereum Blockchain. Finally, we demonstrate the formal verification of Solidity programs by two examples: constant folding and a simple verified token.",
	"authors": [
	"Diego Marmsoler",
	"Achim D. Brucker"
	],
	"date": "2022-07-18",
	- "id": 64,
	+ "id": 67,
	"link": "/entries/Solidity.html",
	"permalink": "/entries/Solidity.html",
	"shortname": "Solidity",
	"title": "Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL",
	"topic_links": [
	"computer-science/programming-languages",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Programming languages",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "A Hermitian matrix is a square complex matrix that is equal to its conjugate transpose. The (finite-dimensional) spectral theorem states that any such matrix can be decomposed into a product of a unitary matrix and a diagonal matrix containing only real elements. We formalize the generalization of this result, which states that any finite set of Hermitian and pairwise commuting matrices can be decomposed as previously, using the same unitary matrix; in other words, they are simultaneously diagonalizable. Sets of pairwise commuting Hermitian matrices are called \u003cem\u003eComplete Sets of Commuting Observables\u003c/em\u003e in Quantum Mechanics, where they represent physical quantities that can be simultaneously measured to uniquely distinguish quantum states.",
	"authors": [
	"Mnacho Echenim"
	],
	"date": "2022-07-18",
	- "id": 65,
	+ "id": 68,
	"link": "/entries/Commuting_Hermitian.html",
	"permalink": "/entries/Commuting_Hermitian.html",
	"shortname": "Commuting_Hermitian",
	"title": "Simultaneous diagonalization of pairwise commuting Hermitian matrices",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of the Weighted Arithmetic–Geometric Mean Inequality: given non-negative reals $a_1, \\ldots, a_n$ and non-negative weights $w_1, \\ldots, w_n$ such that $w_1 + \\ldots + w_n = 1$, we have \\[\\prod\\limits_{i=1}^n a_i^{w_i} \\leq \\sum\\limits_{i=1}^n w_i a_i\\ .\\] If the weights are additionally all non-zero, equality holds if and only if $a_1 = \\ldots = a_n$.\u003c/p\u003e \u003cp\u003eAs a corollary with $w_1 = \\ldots = w_n = 1/n$, the regular arithmetic–geometric mean inequality follows, namely that \\[\\sqrt[n]{a_1\\,\\cdots\\, a_n} \\leq \\tfrac{1}{n}(a_1 + \\ldots + a_n)\\ .\\]\u003c/p\u003e \u003cp\u003eI follow Pólya's elegant proof, which uses the inequality $1 + x \\leq e^x$ as a starting point. Pólya claims that this proof came to him in a dream, and that it was “the best mathematics he had ever dreamt.”\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2022-07-11",
	- "id": 66,
	+ "id": 69,
	"link": "/entries/Weighted_Arithmetic_Geometric_Mean.html",
	"permalink": "/entries/Weighted_Arithmetic_Geometric_Mean.html",
	"shortname": "Weighted_Arithmetic_Geometric_Mean",
	"title": "Pólya’s Proof of the Weighted Arithmetic–Geometric Mean Inequality",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "After introducing the didactic imperative programming language IMP, Nipkow and Klein's book on formal programming language semantics (version of March 2021) specifies compilation of IMP commands into a lower-level language based on a stack machine, and expounds a formal verification of that compiler. Exercise 8.4 asks the reader to adjust such proof for a new compilation target, consisting of a machine language that (i) accesses memory locations through their addresses instead of variable names, and (ii) maintains a stack in memory via a stack pointer rather than relying upon a built-in stack. A natural strategy to maximize reuse of the original proof is keeping the original language as an assembly one and splitting compilation into multiple steps, namely a source-to-assembly step matching the original compilation process followed by an assembly-to-machine step. In this way, proving assembly code-machine code equivalence is the only extant task. A previous paper by the present author introduces a reasoning toolbox that allows for a compiler correctness proof shorter than the book's one, as such promising to constitute a further enhanced reference for the formal verification of real-world compilers. This paper in turn shows that such toolbox can be reused to accomplish the aforesaid task as well, which demonstrates that the proposed approach also promotes proof reuse in multi-stage compiler verifications.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2022-07-10",
	- "id": 67,
	+ "id": 70,
	"link": "/entries/IMP_Compiler_Reuse.html",
	"permalink": "/entries/IMP_Compiler_Reuse.html",
	"shortname": "IMP_Compiler_Reuse",
	"title": "A Reuse-Based Multi-Stage Compiler Verification for Language IMP",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "A double-ended queue (\u003cem\u003edeque\u003c/em\u003e) is a queue where one can enqueue and dequeue at both ends. We define and verify the \u003ca href=\"https://doi.org/10.1145/165180.165225\"\u003edeque implementation by Chuang and Goldberg\u003c/a\u003e. It is purely functional and all operations run in constant time.",
	"authors": [
	"Balazs Toth",
	"Tobias Nipkow"
	],
	"date": "2022-06-23",
	- "id": 68,
	+ "id": 71,
	"link": "/entries/Real_Time_Deque.html",
	"permalink": "/entries/Real_Time_Deque.html",
	"shortname": "Real_Time_Deque",
	"title": "Real-Time Double-Ended Queue",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "In 1987, George Boolos gave an interesting and vivid concrete example of the considerable speed-up afforded by higher-order logic over first-order logic. (A phenomenon first noted by Kurt Gödel in 1936.) Boolos's example concerned an inference $I$ with five premises, and a conclusion, such that the shortest derivation of the conclusion from the premises in a standard system for first-order logic is astronomically huge; while there exists a second-order derivation whose length is of the order of a page or two. Boolos gave a short sketch of that second-order derivation, which relies on the comprehension principle of second-order logic. Here, Boolos's inference is formalized into fourteen lemmas, each quickly verified by the automated-theorem-proving assistant Isabelle/HOL.",
	"authors": [
	"Jeffrey Ketland"
	],
	"date": "2022-06-20",
	- "id": 69,
	+ "id": 72,
	"link": "/entries/Boolos_Curious_Inference.html",
	"permalink": "/entries/Boolos_Curious_Inference.html",
	"shortname": "Boolos_Curious_Inference",
	"title": "Boolos's Curious Inference in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry formalizes the classification of the finite fields (also called Galois fields): For each prime power $p^n$ there exists exactly one (up to isomorphisms) finite field of that size and there are no other finite fields. The derivation includes a formalization of the characteristic of rings, the Frobenius endomorphism, formal differentiation for polynomials in HOL-Algebra and Gauss' formula for the number of monic irreducible polynomials over finite fields: \\[ \\frac{1}{n} \\sum_{d \| n} \\mu(d) p^{n/d} \\textrm{.} \\] The proofs are based on the books from \u003ca href=\"https://doi.org/10.1007/978-1-4757-2103-4\"\u003eIreland and Rosen\u003c/a\u003e, as well as, \u003ca href=\"https://doi.org/10.1017/CBO9781139172769\"\u003eLidl and Niederreiter\u003c/a\u003e.",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-06-08",
	- "id": 70,
	+ "id": 73,
	"link": "/entries/Finite_Fields.html",
	"permalink": "/entries/Finite_Fields.html",
	"shortname": "Finite_Fields",
	"title": "Finite Fields",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "Today's Internet is built on decades-old networking protocols that lack scalability, reliability and security. In response, the networking community has developed \u003cem\u003epath-aware\u003c/em\u003e Internet architectures that solve these issues while simultaneously empowering end hosts. In these architectures, autonomous systems authorize forwarding paths in accordance with their routing policies, and protect paths using cryptographic authenticators. For each packet, the sending end host selects an authorized path and embeds it and its authenticators in the packet header. This allows routers to efficiently determine how to forward the packet. The central security property of the data plane, i.e., of forwarding, is that packets can only travel along authorized paths. This property, which we call \u003cem\u003epath authorization\u003c/em\u003e, protects the routing policies of autonomous systems from malicious senders. The fundamental role of packet forwarding in the Internet's ecosystem and the complexity of the authentication mechanisms employed call for a formal analysis. We develop IsaNet, a parameterized verification framework for data plane protocols in Isabelle/HOL. We first formulate an abstract model without an attacker for which we prove path authorization. We then refine this model by introducing a Dolev--Yao attacker and by protecting authorized paths using (generic) cryptographic validation fields. This model is parametrized by the path authorization mechanism and assumes five simple verification conditions. We propose novel attacker models and different sets of assumptions on the underlying routing protocol. We validate our framework by instantiating it with nine concrete protocols variants and prove that they each satisfy the verification conditions (and hence path authorization). The invariants needed for the security proof are proven in the parametrized model instead of the instance models. Our framework thus supports low-effort security proofs for data plane protocols. In contrast to what could be achieved with state-of-the-art automated protocol verifiers, our results hold for arbitrary network topologies and sets of authorized paths.",
	"authors": [
	"Tobias Klenze",
	"Christoph Sprenger"
	],
	"date": "2022-06-08",
	- "id": 71,
	+ "id": 74,
	"link": "/entries/IsaNet.html",
	"permalink": "/entries/IsaNet.html",
	"shortname": "IsaNet",
	"title": "IsaNet: Formalization of a Verification Framework for Secure Data Plane Protocols",
	"topic_links": [
	"computer-science/security",
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Networks"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of Matiyasevich's proof of the DPRM theorem, which states that every recursively enumerable set of natural numbers is Diophantine. This result from 1970 yields a negative solution to Hilbert's 10th problem over the integers. To represent recursively enumerable sets in equations, we implement and arithmetize register machines. We formalize a general theory of Diophantine sets and relations to reason about them abstractly. Using several number-theoretic lemmas, we prove that exponentiation has a Diophantine representation.",
	"authors": [
	"Jonas Bayer",
	"Marco David",
	"Benedikt Stock",
	"Abhik Pal",
	"Yuri Matiyasevich",
	"Dierk Schleicher"
	],
	"date": "2022-06-06",
	- "id": 72,
	+ "id": 75,
	"link": "/entries/DPRM_Theorem.html",
	"permalink": "/entries/DPRM_Theorem.html",
	"shortname": "DPRM_Theorem",
	"title": "Diophantine Equations and the DPRM Theorem",
	"topic_links": [
	"logic/computability",
	"mathematics/number-theory"
	],
	"topics": [
	"Logic/Computability",
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This AFP entry relates important rewriting properties between the set of terms and the set of ground terms induced by a given signature. The properties considered are confluence, strong/local confluence, the normal form property, unique normal forms with respect to reduction and conversion, commutation, conversion equivalence, and normalization equivalence.",
	"authors": [
	"Alexander Lochmann"
	],
	"date": "2022-06-02",
	- "id": 73,
	+ "id": 76,
	"link": "/entries/Rewrite_Properties_Reduction.html",
	"permalink": "/entries/Rewrite_Properties_Reduction.html",
	"shortname": "Rewrite_Properties_Reduction",
	"title": "Reducing Rewrite Properties to Properties on Ground Terms",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "Many separation logics support \u003cem\u003efractional permissions\u003c/em\u003e to distinguish between read and write access to a heap location, for instance, to allow concurrent reads while enforcing exclusive writes. The concept has been generalized to fractional assertions. $A^p$ (where $A$ is a separation logic assertion and $p$ a fraction between $0$ and $1$) represents a fraction $p$ of $A$. $A^p$ holds in a state $\\sigma$ iff there exists a state $\\sigma_A$ in which $A$ holds and $\\sigma$ is obtained from $\\sigma_A$ by multiplying all permission amounts held by $p$. While $A^{p + q}$ can always be split into $A^p * A^q$, recombining $A^p * A^q$ into $A^{p+q}$ is not always sound. We say that $A$ is \u003cem\u003ecombinable\u003c/em\u003e iff the entailment $A^p * A^q \\models A^{p+q}$ holds for any two positive fractions $p$ and $q$ such that $p + q \\le 1$. Combinable assertions are particularly useful to reason about concurrent programs, for instance, to combine the postconditions of parallel branches when they terminate. Unfortunately, the magic wand assertion $A \\mathbin{-\\!\\!*} B$, commonly used to specify properties of partial data structures, is typically \u003cem\u003enot\u003c/em\u003e combinable. In this entry, we formalize a novel, restricted definition of the magic wand, described in \u003ca href=\"https://arxiv.org/abs/2205.11325\"\u003ea paper at CAV 22\u003c/a\u003e, which we call the \u003cem\u003ecombinable wand\u003c/em\u003e. We prove some key properties of the combinable wand; in particular, a combinable wand is combinable if its right-hand side is combinable.",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2022-05-30",
	- "id": 74,
	+ "id": 77,
	"link": "/entries/Combinable_Wands.html",
	"permalink": "/entries/Combinable_Wands.html",
	"shortname": "Combinable_Wands",
	"title": "A Restricted Definition of the Magic Wand to Soundly Combine Fractions of a Wand",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise Plünnecke's inequality and the Plünnecke-Ruzsa inequality, following the notes by Timothy Gowers: \"Introduction to Additive Combinatorics\" (2022) for the University of Cambridge. To this end, we first introduce basic definitions and prove elementary facts on sumsets and difference sets. Then, we show two versions of the Ruzsa triangle inequality. We follow with a proof due to Petridis.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Lawrence C. Paulson"
	],
	"date": "2022-05-26",
	- "id": 75,
	+ "id": 78,
	"link": "/entries/Pluennecke_Ruzsa_Inequality.html",
	"permalink": "/entries/Pluennecke_Ruzsa_Inequality.html",
	"shortname": "Pluennecke_Ruzsa_Inequality",
	"title": "The Plünnecke-Ruzsa Inequality",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 3
	},
	{
	"abstract": "The magic wand $\\mathbin{-\\!\\!}$ (also called separating implication) is a separation logic connective commonly used to specify properties of partial data structures, for instance during iterative traversals. A \u003cem\u003efootprint\u003c/em\u003e of a magic wand formula $$A \\mathbin{-\\!\\!} B$$ is a state that, combined with any state in which $A$ holds, yields a state in which $B$ holds. The key challenge of proving a magic wand (also called \u003cem\u003epackaging\u003c/em\u003e a wand) is to find such a footprint. Existing package algorithms either have a high annotation overhead or are unsound. In this entry, we formally define a framework for the sound automation of magic wands, described in an \u003ca href=\"https://www.cs.ubc.ca/~alexsumm/papers/DardinierParthasarathyWeeksMuellerSummers22.pdf\"\u003eupcoming paper at CAV 2022\u003c/a\u003e, and prove that it is sound and complete. This framework, called the \u003cem\u003epackage logic\u003c/em\u003e, precisely characterises a wide design space of possible package algorithms applicable to a large class of separation logics.",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2022-05-18",
	- "id": 76,
	+ "id": 79,
	"link": "/entries/Package_logic.html",
	"permalink": "/entries/Package_logic.html",
	"shortname": "Package_logic",
	"title": "Formalization of a Framework for the Sound Automation of Magic Wands",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e Given a graph $G$ with $n$ vertices and a number $s$, the decision problem Clique asks whether $G$ contains a fully connected subgraph with $s$ vertices. For this NP-complete problem there exists a non-trivial lower bound: no monotone circuit of a size that is polynomial in $n$ can solve Clique. \u003c/p\u003e\u003cp\u003e This entry provides an Isabelle/HOL formalization of a concrete lower bound (the bound is $\\sqrt[7]{n}^{\\sqrt[8]{n}}$ for the fixed choice of $s = \\sqrt[4]{n}$), following a proof by Gordeev. \u003c/p\u003e",
	"authors": [
	"René Thiemann"
	],
	"date": "2022-05-08",
	- "id": 77,
	+ "id": 80,
	"link": "/entries/Clique_and_Monotone_Circuits.html",
	"permalink": "/entries/Clique_and_Monotone_Circuits.html",
	"shortname": "Clique_and_Monotone_Circuits",
	"title": "Clique is not solvable by monotone circuits of polynomial size",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "Linear algebraic techniques are powerful, yet often underrated tools in combinatorial proofs. This formalisation provides a library including matrix representations of incidence set systems, general formal proof techniques for the rank argument and linear bound argument, and finally a formalisation of a number of variations of the well-known Fisher's inequality. We build on our prior work formalising combinatorial design theory using a locale-centric approach, including extensions such as constant intersect designs and dual incidence systems. In addition to Fisher's inequality, we also formalise proofs on other incidence system properties using the incidence matrix representation, such as design existence, dual system relationships and incidence system isomorphisms. This formalisation is presented in the paper \"Formalising Fisher's Inequality: Formal Linear Algebraic Techniques in Combinatorics\", accepted to ITP 2022.",
	"authors": [
	"Chelsea Edmonds",
	"Lawrence C. Paulson"
	],
	"date": "2022-04-21",
	- "id": 78,
	+ "id": 81,
	"link": "/entries/Fishers_Inequality.html",
	"permalink": "/entries/Fishers_Inequality.html",
	"shortname": "Fishers_Inequality",
	"title": "Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics",
	"topic_links": [
	"mathematics/combinatorics",
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize how a natural number can be expanded into its digits in some base and prove properties about functions that operate on digit expansions. This includes the formalization of concepts such as digit shifts and carries. For a base that is a power of 2 we formalize the binary AND, binary orthogonality and binary masking of two natural numbers. This library on digit expansions builds the basis for the formalization of the DPRM theorem.",
	"authors": [
	"Jonas Bayer",
	"Marco David",
	"Abhik Pal",
	"Benedikt Stock"
	],
	"date": "2022-04-20",
	- "id": 79,
	+ "id": 82,
	"link": "/entries/Digit_Expansions.html",
	"permalink": "/entries/Digit_Expansions.html",
	"shortname": "Digit_Expansions",
	"title": "Digit Expansions",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We consider the problem of comparing two multisets via the generalized multiset ordering. We show that the corresponding decision problem is NP-complete. To be more precise, we encode multiset-comparisons into propositional formulas or into conjunctive normal forms of quadratic size; we further prove that satisfiability of conjunctive normal forms can be encoded as multiset-comparison problems of linear size. As a corollary, we also show that the problem of deciding whether two terms are related by a recursive path order is NP-hard, provided the recursive path order is based on the generalized multiset ordering.",
	"authors": [
	"René Thiemann",
	"Lukas Schmidinger"
	],
	"date": "2022-04-20",
	- "id": 80,
	+ "id": 83,
	"link": "/entries/Multiset_Ordering_NPC.html",
	"permalink": "/entries/Multiset_Ordering_NPC.html",
	"shortname": "Multiset_Ordering_NPC",
	"title": "The Generalized Multiset Ordering is NP-Complete",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article provides a brief formalisation of the two equations known as the \u003cem\u003eSophomore's Dream\u003c/em\u003e, first discovered by Johann Bernoulli in 1697:\u003c/p\u003e \\[\\int_0^1 x^{-x}\\,\\text{d}x = \\sum_{n=1}^\\infty n^{-n} \\quad\\text{and}\\quad \\int_0^1 x^x\\,\\text{d}x = -\\sum_{n=1}^\\infty (-n)^{-n}\\]",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2022-04-10",
	- "id": 81,
	+ "id": 84,
	"link": "/entries/Sophomores_Dream.html",
	"permalink": "/entries/Sophomores_Dream.html",
	"shortname": "Sophomores_Dream",
	"title": "The Sophomore's Dream",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains a set of binary encodings for primitive data types, such as natural numbers, integers, floating-point numbers as well as combinators to construct encodings for products, lists, sets or functions of/between such types. For natural numbers and integers, the entry contains various encodings, such as Elias-Gamma-Codes and exponential Golomb Codes, which are efficient variable-length codes in use by current compression formats. A use-case for this library is measuring the persisted size of a complex data structure without having to hand-craft a dedicated encoding for it, independent of Isabelle's internal representation.",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-04-08",
	- "id": 82,
	+ "id": 85,
	"link": "/entries/Prefix_Free_Code_Combinators.html",
	"permalink": "/entries/Prefix_Free_Code_Combinators.html",
	"shortname": "Prefix_Free_Code_Combinators",
	"title": "A Combinator Library for Prefix-Free Codes",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "In 1999 Alon et. al. introduced the still active research topic of approximating the frequency moments of a data stream using randomized algorithms with minimal space usage. This includes the problem of estimating the cardinality of the stream elements - the zeroth frequency moment. But, also higher-order frequency moments that provide information about the skew of the data stream. (The \u003ci\u003ek\u003c/i\u003e-th frequency moment of a data stream is the sum of the \u003ci\u003ek\u003c/i\u003e-th powers of the occurrence counts of each element in the stream.) This entry formalizes three randomized algorithms for the approximation of \u003ci\u003eF\u003csub\u003e0\u003c/sub\u003e\u003c/i\u003e, \u003ci\u003eF\u003csub\u003e2\u003c/sub\u003e\u003c/i\u003e and \u003ci\u003eF\u003csub\u003ek\u003c/sub\u003e\u003c/i\u003e for \u003ci\u003ek ≥ 3\u003c/i\u003e based on [\u003ca href=\"https://doi.org/10.1006/jcss.1997.1545\"\u003e1\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/3-540-45726-7_1\"\u003e2\u003c/a\u003e] and verifies their expected accuracy, success probability and space usage.",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-04-08",
	- "id": 83,
	+ "id": 86,
	"link": "/entries/Frequency_Moments.html",
	"permalink": "/entries/Frequency_Moments.html",
	"shortname": "Frequency_Moments",
	"title": "Formalization of Randomized Approximation Algorithms for Frequency Moments",
	"topic_links": [
	"computer-science/algorithms/approximation",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Approximation",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 1
	},
	{
	"abstract": "The type of real numbers is constructed from the positive rationals using the method of Dedekind cuts. This development, briefly described in papers by the authors, follows the textbook presentation by Gleason. It's notable that the first formalisation of a significant piece of mathematics, by Jutting in 1977, involved a similar construction.",
	"authors": [
	"Jacques D. Fleuriot",
	"Lawrence C. Paulson"
	],
	"date": "2022-03-24",
	- "id": 84,
	+ "id": 87,
	"link": "/entries/Dedekind_Real.html",
	"permalink": "/entries/Dedekind_Real.html",
	"shortname": "Dedekind_Real",
	"title": "Constructing the Reals as Dedekind Cuts of Rationals",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Ackermann's function is defined in the usual way and a number of its elementary properties are proved. Then, the primitive recursive functions are defined inductively: as a predicate on the functions that map lists of numbers to numbers. It is shown that every primitive recursive function is strictly dominated by Ackermann's function. The formalisation follows an earlier one by Nora Szasz.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2022-03-23",
	- "id": 85,
	+ "id": 88,
	"link": "/entries/Ackermanns_not_PR.html",
	"permalink": "/entries/Ackermanns_not_PR.html",
	"shortname": "Ackermanns_not_PR",
	"title": "Ackermann's Function Is Not Primitive Recursive",
	"topic_links": [
	"logic/computability"
	],
	"topics": [
	"Logic/Computability"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e The AFP entry \u003ca href=\"https://www.isa-afp.org/entries/Abstract_Completeness.html\"\u003eAbstract Completeness\u003c/a\u003e by Blanchette, Popescu and Traytel formalizes the core of Beth/Hintikka-style completeness proofs for first-order logic and can be used to formalize executable sequent calculus provers. In the Journal of Automated Reasoning, the authors instantiate the framework with a sequent calculus for first-order logic and prove its completeness. Their use of an infinite set of proof rules indexed by formulas yields very direct arguments. A fair stream of these rules controls the prover, making its definition remarkably simple. The AFP entry, however, only contains a toy example for propositional logic. The AFP entry \u003ca href=\"https://www.isa-afp.org/entries/FOL_Seq_Calc2.html\"\u003eA Sequent Calculus Prover for First-Order Logic with Functions\u003c/a\u003e by From and Jacobsen also uses the framework, but uses a finite set of generic rules resulting in a more sophisticated prover with more complicated proofs. \u003c/p\u003e \u003cp\u003e This entry contains an executable sequent calculus prover for first-order logic with functions in the style presented by Blanchette et al. The prover can be exported to Haskell and this entry includes formalized proofs of its soundness and completeness. The proofs are simpler than those for the prover by From and Jacobsen but the performance of the prover is significantly worse. \u003c/p\u003e \u003cp\u003e The included theory \u003cem\u003eFair-Stream\u003c/em\u003e first proves that the sequence of natural numbers 0, 0, 1, 0, 1, 2, etc. is fair. It then proves that mapping any surjective function across the sequence preserves fairness. This method of obtaining a fair stream of rules is similar to the one given by Blanchette et al. The concrete functions from natural numbers to terms, formulas and rules are defined using the \u003cem\u003eNat-Bijection\u003c/em\u003e theory in the HOL-Library. \u003c/p\u003e",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2022-03-22",
	- "id": 86,
	+ "id": 89,
	"link": "/entries/FOL_Seq_Calc3.html",
	"permalink": "/entries/FOL_Seq_Calc3.html",
	"shortname": "FOL_Seq_Calc3",
	"title": "A Naive Prover for First-Order Logic",
	"topic_links": [
	"logic/general-logic/classical-first-order-logic",
	"logic/proof-theory",
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Classical first-order logic",
	"Logic/Proof theory",
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eIn this article, I formalise a proof from \u003ca href=\"https://dx.doi.org/10.1007/978-3-662-57265-8\"\u003eTHE BOOK\u003c/a\u003e; namely a formula that was called ‘one of the most beautiful formulas involving elementary functions’:\u003c/p\u003e \\[\\pi \\cot(\\pi z) = \\frac{1}{z} + \\sum_{n=1}^\\infty\\left(\\frac{1}{z+n} + \\frac{1}{z-n}\\right)\\] \u003cp\u003eThe proof uses Herglotz's trick to show the real case and analytic continuation for the complex case.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2022-03-15",
	- "id": 87,
	+ "id": 90,
	"link": "/entries/Cotangent_PFD_Formula.html",
	"permalink": "/entries/Cotangent_PFD_Formula.html",
	"shortname": "Cotangent_PFD_Formula",
	"title": "A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We redeveloped our formalization of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct proper generic extensions that satisfy the Continuum Hypothesis and its negation.",
	"authors": [
	"Emmanuel Gunther",
	"Miguel Pagano",
	"Pedro Sánchez Terraf",
	"Matías Steinberg"
	],
	"date": "2022-03-06",
	- "id": 88,
	+ "id": 91,
	"link": "/entries/Independence_CH.html",
	"permalink": "/entries/Independence_CH.html",
	"shortname": "Independence_CH",
	"title": "The Independence of the Continuum Hypothesis in Isabelle/ZF",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We extend the ZF-Constructibility library by relativizing theories of the Isabelle/ZF and Delta System Lemma sessions to a transitive class. We also relativize Paulson's work on Aleph and our former treatment of the Axiom of Dependent Choices. This work is a prerrequisite to our formalization of the independence of the Continuum Hypothesis.",
	"authors": [
	"Emmanuel Gunther",
	"Miguel Pagano",
	"Pedro Sánchez Terraf",
	"Matías Steinberg"
	],
	"date": "2022-03-03",
	- "id": 89,
	+ "id": 92,
	"link": "/entries/Transitive_Models.html",
	"permalink": "/entries/Transitive_Models.html",
	"shortname": "Transitive_Models",
	"title": "Transitive Models of Fragments of ZFC",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e A \u003cem\u003eresiduated transition system\u003c/em\u003e (RTS) is a transition system that is equipped with a certain partial binary operation, called \u003cem\u003eresiduation\u003c/em\u003e, on transitions. Using the residuation operation, one can express nuances, such as a distinction between nondeterministic and concurrent choice, as well as partial commutativity relationships between transitions, which are not captured by ordinary transition systems. A version of residuated transition systems was introduced in previous work by the author, in which they were called “concurrent transition systems” in view of the original motivation for their definition from the study of concurrency. In the first part of the present article, we give a formal development that generalizes and subsumes the original presentation. We give an axiomatic definition of residuated transition systems that assumes only a single partial binary operation as given structure. From the axioms, we derive notions of “arrow“ (transition), “source”, “target”, “identity”, as well as “composition” and “join” of transitions; thereby recovering structure that in the previous work was assumed as given. We formalize and generalize the result, that residuation extends from transitions to transition paths, and we systematically develop the properties of this extension. A significant generalization made in the present work is the identification of a general notion of congruence on RTS’s, along with an associated quotient construction. \u003c/p\u003e \u003cp\u003e In the second part of this article, we use the RTS framework to formalize several results in the theory of reduction in Church’s λ-calculus. Using a de Bruijn index-based syntax in which terms represent parallel reduction steps, we define residuation on terms and show that it satisfies the axioms for an RTS. An application of the results on paths from the first part of the article allows us to prove the classical Church-Rosser Theorem with little additional effort. We then use residuation to define the notion of “development” and we prove the Finite Developments Theorem, that every development is finite, formalizing and adapting to de Bruijn indices a proof by de Vrijer. We also use residuation to define the notion of a “standard reduction path”, and we prove the Standardization Theorem: that every reduction path is congruent to a standard one. As a corollary of the Standardization Theorem, we obtain the Leftmost Reduction Theorem: that leftmost reduction is a normalizing strategy. \u003c/p\u003e",
	"authors": [
	"Eugene W. Stark"
	],
	"date": "2022-02-28",
	- "id": 90,
	+ "id": 93,
	"link": "/entries/ResiduatedTransitionSystem.html",
	"permalink": "/entries/ResiduatedTransitionSystem.html",
	"shortname": "ResiduatedTransitionSystem",
	"title": "Residuated Transition Systems",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"computer-science/concurrency",
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Computer science/Concurrency",
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "A \u003ci\u003ek\u003c/i\u003e-universal hash family is a probability space of functions, which have uniform distribution and form \u003ci\u003ek\u003c/i\u003e-wise independent random variables. They can often be used in place of classic (or cryptographic) hash functions and allow the rigorous analysis of the performance of randomized algorithms and data structures that rely on hash functions. In 1981 \u003ca href=\"https://doi.org/10.1016/0022-0000(81)90033-7\"\u003eWegman and Carter\u003c/a\u003e introduced a generic construction for such families with arbitrary \u003ci\u003ek\u003c/i\u003e using polynomials over a finite field. This entry contains a formalization of them and establishes the property of \u003ci\u003ek\u003c/i\u003e-universality. To be useful the formalization also provides an explicit construction of finite fields using the factor ring of integers modulo a prime. Additionally, some generic results about independent families are shown that might be of independent interest.",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-02-20",
	- "id": 91,
	+ "id": 94,
	"link": "/entries/Universal_Hash_Families.html",
	"permalink": "/entries/Universal_Hash_Families.html",
	"shortname": "Universal_Hash_Families",
	"title": "Universal Hash Families",
	"topic_links": [
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 2
	},
	{
	"abstract": "Let $F$ be a set of analytic functions on the complex plane such that, for each $z\\in\\mathbb{C}$, the set $\\{f(z) \\mid f\\in F\\}$ is countable; must then $F$ itself be countable? The answer is yes if the Continuum Hypothesis is false, i.e., if the cardinality of $\\mathbb{R}$ exceeds $\\aleph_1$. But if CH is true then such an $F$, of cardinality $\\aleph_1$, can be constructed by transfinite recursion. The formal proof illustrates reasoning about complex analysis (analytic and homomorphic functions) and set theory (transfinite cardinalities) in a single setting. The mathematical text comes from \u003cem\u003eProofs from THE BOOK\u003c/em\u003e by Aigner and Ziegler.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2022-02-18",
	- "id": 92,
	+ "id": 95,
	"link": "/entries/Wetzels_Problem.html",
	"permalink": "/entries/Wetzels_Problem.html",
	"shortname": "Wetzels_Problem",
	"title": "Wetzel's Problem and the Continuum Hypothesis",
	"topic_links": [
	"mathematics/analysis",
	"logic/set-theory"
	],
	"topics": [
	"Mathematics/Analysis",
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize first-order query evaluation over an infinite domain with equality. We first define the syntax and semantics of first-order logic with equality. Next we define a locale \u003ci\u003eeval\u0026lowbar;fo\u003c/i\u003e abstracting a representation of a potentially infinite set of tuples satisfying a first-order query over finite relations. Inside the locale, we define a function \u003ci\u003eeval\u003c/i\u003e checking if the set of tuples satisfying a first-order query over a database (an interpretation of the query's predicates) is finite (i.e., deciding \u003ci\u003erelative safety\u003c/i\u003e) and computing the set of satisfying tuples if it is finite. Altogether the function \u003ci\u003eeval\u003c/i\u003e solves \u003ci\u003ecapturability\u003c/i\u003e (Avron and Hirshfeld, 1991) of first-order logic with equality. We also use the function \u003ci\u003eeval\u003c/i\u003e to prove a code equation for the semantics of first-order logic, i.e., the function checking if a first-order query over a database is satisfied by a variable assignment.\u003cbr/\u003e We provide an interpretation of the locale \u003ci\u003eeval\u0026lowbar;fo\u003c/i\u003e based on the approach by Ailamazyan et al. A core notion in the interpretation is the active domain of a query and a database that contains all domain elements that occur in the database or interpret the query's constants. We prove the main theorem of Ailamazyan et al. relating the satisfaction of a first-order query over an infinite domain to the satisfaction of this query over a finite domain consisting of the active domain and a few additional domain elements (outside the active domain) whose number only depends on the query. In our interpretation of the locale \u003ci\u003eeval\u0026lowbar;fo\u003c/i\u003e, we use a potentially higher number of the additional domain elements, but their number still only depends on the query and thus has no effect on the data complexity (Vardi, 1982) of query evaluation. Our interpretation yields an \u003ci\u003eexecutable\u003c/i\u003e function \u003ci\u003eeval\u003c/i\u003e. The time complexity of \u003ci\u003eeval\u003c/i\u003e on a query is linear in the total number of tuples in the intermediate relations for the subqueries. Specifically, we build a database index to evaluate a conjunction. We also optimize the case of a negated subquery in a conjunction. Finally, we export code for the infinite domain of natural numbers.",
	"authors": [
	"Martin Raszyk"
	],
	"date": "2022-02-15",
	- "id": 93,
	+ "id": 96,
	"link": "/entries/Eval_FO.html",
	"permalink": "/entries/Eval_FO.html",
	"shortname": "Eval_FO",
	"title": "First-Order Query Evaluation",
	"topic_links": [
	"computer-science/data-management-systems",
	"logic/general-logic/classical-first-order-logic"
	],
	"topics": [
	"Computer science/Data management systems",
	"Logic/General logic/Classical first-order logic"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eRuntime monitoring (or runtime verification) is an approach to checking compliance of a system's execution with a specification (e.g., a temporal formula). The system's execution is logged into a \u003ci\u003etrace\u003c/i\u003e\u0026mdash;a sequence of time-points, each consisting of a time-stamp and observed events. A \u003ci\u003emonitor\u003c/i\u003e is an algorithm that produces \u003ci\u003everdicts\u003c/i\u003e on the satisfaction of a temporal formula on a trace.\u003c/p\u003e \u003cp\u003eWe formalize the time-stamps as an abstract algebraic structure satisfying certain assumptions. Instances of this structure include natural numbers, real numbers, and lexicographic combinations of them. We also include the formalization of a conversion from the abstract time domain introduced by Koymans (1990) to our time-stamps.\u003c/p\u003e \u003cp\u003eWe formalize a monitoring algorithm for metric dynamic logic, an extension of metric temporal logic with regular expressions. The monitor computes whether a given formula is satisfied at every position in an input trace of time-stamped events. Our monitor follows the multi-head paradigm: it reads the input simultaneously at multiple positions and moves its reading heads asynchronously. This mode of operation results in unprecedented time and space complexity guarantees for metric dynamic logic: The monitor's amortized time complexity to process a time-point and the monitor's space complexity neither depends on the event-rate, i.e., the number of events within a fixed time-unit, nor on the numeric constants occurring in the quantitative temporal constraints in the given formula.\u003c/p\u003e \u003cp\u003eThe multi-head monitoring algorithm for metric dynamic logic is reported in our paper “Multi-Head Monitoring of Metric Dynamic Logic” published at ATVA 2020. We have also formalized unpublished specialized algorithms for the temporal operators of metric temporal logic.\u003c/p\u003e",
	"authors": [
	"Martin Raszyk"
	],
	"date": "2022-02-13",
	- "id": 94,
	+ "id": 97,
	"link": "/entries/VYDRA_MDL.html",
	"permalink": "/entries/VYDRA_MDL.html",
	"shortname": "VYDRA_MDL",
	"title": "Multi-Head Monitoring of Metric Dynamic Logic",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry contains a formalization of an algorithm enumerating all equivalence relations on an initial segment of the natural numbers. The approach follows the method described by Stanton and White \u003ca href=\"https://doi.org/10.1007/978-1-4612-4968-9\"\u003e[5,§ 1.5]\u003c/a\u003e using restricted growth functions.\u003c/p\u003e \u003cp\u003eThe algorithm internally enumerates restricted growth functions (as lists), whose equivalence kernels then form the equivalence relations. This has the advantage that the representation is compact and lookup of the relation reduces to a list lookup operation.\u003c/p\u003e \u003cp\u003eThe algorithm can also be used within a proof and an example application is included, where a sequence of variables is split by the possible partitions they can form.\u003c/p\u003e",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-02-04",
	- "id": 95,
	+ "id": 98,
	"link": "/entries/Equivalence_Relation_Enumeration.html",
	"permalink": "/entries/Equivalence_Relation_Enumeration.html",
	"shortname": "Equivalence_Relation_Enumeration",
	"title": "Enumeration of Equivalence Relations",
	"topic_links": [
	"mathematics/combinatorics",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize the weak and strong duality theorems of linear programming. For the strong duality theorem we provide three sufficient preconditions: both the primal problem and the dual problem are satisfiable, the primal problem is satisfiable and bounded, or the dual problem is satisfiable and bounded. The proofs are based on an existing formalization of Farkas' Lemma.",
	"authors": [
	"René Thiemann"
	],
	"date": "2022-02-03",
	- "id": 96,
	+ "id": 99,
	"link": "/entries/LP_Duality.html",
	"permalink": "/entries/LP_Duality.html",
	"shortname": "LP_Duality",
	"title": "Duality of Linear Programming",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "The notion of quasi-Borel spaces was introduced by \u003ca href=\"https://dl.acm.org/doi/10.5555/3329995.3330072\"\u003e Heunen et al\u003c/a\u003e. The theory provides a suitable denotational model for higher-order probabilistic programming languages with continuous distributions. This entry is a formalization of the theory of quasi-Borel spaces, including construction of quasi-Borel spaces (product, coproduct, function spaces), the adjunction between the category of measurable spaces and the category of quasi-Borel spaces, and the probability monad on quasi-Borel spaces. This entry also contains the formalization of the Bayesian regression presented in the work of Heunen et al. This work is a part of the work by same authors, \u003ci\u003eProgram Logic for Higher-Order Probabilistic Programs in Isabelle/HOL\u003c/i\u003e, which will be published in the proceedings of the 16th International Symposium on Functional and Logic Programming (FLOPS 2022).",
	"authors": [
	"Michikazu Hirata",
	"Yasuhiko Minamide",
	"Tetsuya Sato"
	],
	"date": "2022-02-03",
	- "id": 97,
	+ "id": 100,
	"link": "/entries/Quasi_Borel_Spaces.html",
	"permalink": "/entries/Quasi_Borel_Spaces.html",
	"shortname": "Quasi_Borel_Spaces",
	"title": "Quasi-Borel Spaces",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "The first-order theory of rewriting (FORT) is a decidable theory for linear variable-separated rewrite systems. The decision procedure is based on tree automata technique and an inference system presented in \"Certifying Proofs in the First-Order Theory of Rewriting\". This AFP entry provides a formalization of the underlying decision procedure. Moreover it allows to generate a function that can verify each inference step via the code generation facility of Isabelle/HOL. Additionally it contains the specification of a certificate language (that allows to state proofs in FORT) and a formalized function that allows to verify the validity of the proof. This gives software tool authors, that implement the decision procedure, the possibility to verify their output.",
	"authors": [
	"Alexander Lochmann",
	"Bertram Felgenhauer"
	],
	"date": "2022-02-02",
	- "id": 98,
	+ "id": 101,
	"link": "/entries/FO_Theory_Rewriting.html",
	"permalink": "/entries/FO_Theory_Rewriting.html",
	"shortname": "FO_Theory_Rewriting",
	"title": "First-Order Theory of Rewriting",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/rewriting",
	"logic/proof-theory"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/Rewriting",
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we verify its soundness and completeness using the Abstract Soundness and Abstract Completeness theories. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do so as well when building a countermodel from a failed proof. We formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover's output can be post-processed in Haskell to generate human-readable SeCaV proofs which are also machine-verifiable proof certificates. Paper: \u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2022.13\"\u003edoi.org/10.4230/LIPIcs.ITP.2022.13\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From",
	"Frederik Krogsdal Jacobsen"
	],
	"date": "2022-01-31",
	- "id": 99,
	+ "id": 102,
	"link": "/entries/FOL_Seq_Calc2.html",
	"permalink": "/entries/FOL_Seq_Calc2.html",
	"shortname": "FOL_Seq_Calc2",
	"title": "A Sequent Calculus Prover for First-Order Logic with Functions",
	"topic_links": [
	"logic/general-logic/classical-first-order-logic",
	"logic/proof-theory",
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Classical first-order logic",
	"Logic/Proof theory",
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "Young's inequality states that $$ ab \\leq \\int_0^a f(x)dx + \\int_0^b f^{-1}(y) dy $$ where $a\\geq 0$, $b\\geq 0$ and $f$ is strictly increasing and continuous. Its proof is formalised following \u003ca href=\"https://www.jstor.org/stable/2318018\"\u003ethe development\u003c/a\u003e by Cunningham and Grossman. Their idea is to make the intuitive, geometric folklore proof rigorous by reasoning about step functions. The lack of the Riemann integral makes the development longer than one would like, but their argument is reproduced faithfully.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2022-01-31",
	- "id": 100,
	+ "id": 103,
	"link": "/entries/Youngs_Inequality.html",
	"permalink": "/entries/Youngs_Inequality.html",
	"shortname": "Youngs_Inequality",
	"title": "Young's Inequality for Increasing Functions",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eA well known result from algebra is that, on any field, there is exactly one polynomial of degree less than n interpolating n points [\u003ca href=\"https://doi.org/10.1017/CBO9780511814549\"\u003e1\u003c/a\u003e, §7].\u003c/p\u003e \u003cp\u003eThis entry contains a formalization of the above result, as well as the following generalization in the case of finite fields \u003ci\u003eF\u003c/i\u003e: There are \u003ci\u003e\|F\|\u003csup\u003em-n\u003c/sup\u003e\u003c/i\u003e polynomials of degree less than \u003ci\u003em ≥ n\u003c/i\u003e interpolating the same n points, where \u003ci\u003e\|F\|\u003c/i\u003e denotes the size of the domain of the field. To establish the result the entry also includes a formalization of Lagrange interpolation, which might be of independent interest.\u003c/p\u003e \u003cp\u003eThe formalized results are defined on the algebraic structures from HOL-Algebra, which are distinct from the type-class based structures defined in HOL. Note that there is an existing formalization for polynomial interpolation and, in particular, Lagrange interpolation by Thiemann and Yamada [\u003ca href=\"https://www.isa-afp.org/entries/Polynomial_Interpolation.html\"\u003e2\u003c/a\u003e] on the type-class based structures in HOL.\u003c/p\u003e",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-01-29",
	- "id": 101,
	+ "id": 104,
	"link": "/entries/Interpolation_Polynomials_HOL_Algebra.html",
	"permalink": "/entries/Interpolation_Polynomials_HOL_Algebra.html",
	"shortname": "Interpolation_Polynomials_HOL_Algebra",
	"title": "Interpolation Polynomials (in HOL-Algebra)",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThe median method is an amplification result for randomized approximation algorithms described in [\u003ca href=\"https://doi.org/10.1006/jcss.1997.1545\"\u003e1\u003c/a\u003e]. Given an algorithm whose result is in a desired interval with a probability larger than \u003ci\u003e1/2\u003c/i\u003e, it is possible to improve the success probability, by running the algorithm multiple times independently and using the median. In contrast to using the mean, the amplification of the success probability grows exponentially with the number of independent runs.\u003c/p\u003e \u003cp\u003eThis entry contains a formalization of the underlying theorem: Given a sequence of n independent random variables, which are in a desired interval with a probability \u003ci\u003e1/2 + a\u003c/i\u003e. Then their median will be in the desired interval with a probability of \u003ci\u003e1 − exp(−2a\u003csup\u003e2\u003c/sup\u003e n)\u003c/i\u003e. In particular, the success probability approaches \u003ci\u003e1\u003c/i\u003e exponentially with the number of variables.\u003c/p\u003e \u003cp\u003eIn addition to that, this entry also contains a proof that order-statistics of Borel-measurable random variables are themselves measurable and that generalized intervals in linearly ordered Borel-spaces are measurable.\u003c/p\u003e",
	"authors": [
	"Emin Karayel"
	],
	"date": "2022-01-25",
	- "id": 102,
	+ "id": 105,
	"link": "/entries/Median_Method.html",
	"permalink": "/entries/Median_Method.html",
	"shortname": "Median_Method",
	"title": "Median Method",
	"topic_links": [
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 1
	},
	{
	"abstract": "Actuarial Mathematics is a theory in applied mathematics, which is mainly used for determining the prices of insurance products and evaluating the liability of a company associating with insurance contracts. It is related to calculus, probability theory and financial theory, etc. In this entry, I formalize the very basic part of Actuarial Mathematics in Isabelle/HOL. It includes the theory of interest, survival model, and life table. The theory of interest deals with interest rates, present value factors, an annuity certain, etc. The survival model is a probabilistic model that represents the human mortality. The life table is based on the survival model and used for practical calculations. I have already formalized the basic part of Actuarial Mathematics in Coq (https://github.com/Yosuke-Ito-345/Actuary) in a purely axiomatic manner. In contrast, Isabelle formalization is based on the probability theory and the survival model is developed as generally as possible. Such rigorous and general formulation seems very rare; at least I cannot find any similar documentation on the web. This formalization in Isabelle is still at an early stage, and I cannot guarantee the backward compatibility in the future development. If you heavily depend on this entry, please let me know. (Updated July 8th, 2023.) ",
	"authors": [
	"Yosuke Ito"
	],
	"date": "2022-01-23",
	- "id": 103,
	+ "id": 106,
	"link": "/entries/Actuarial_Mathematics.html",
	"permalink": "/entries/Actuarial_Mathematics.html",
	"shortname": "Actuarial_Mathematics",
	"title": "Actuarial Mathematics",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "An elementary proof is formalised: that \u003cem\u003eexp r\u003c/em\u003e is irrational for every nonzero rational number \u003cem\u003er\u003c/em\u003e. The mathematical development comes from the well-known volume \u003cem\u003eProofs from THE BOOK\u003c/em\u003e, by Aigner and Ziegler, who credit the idea to Hermite. The development illustrates a number of basic Isabelle techniques: the manipulation of summations, the calculation of quite complicated derivatives and the estimation of integrals. We also see how to import another AFP entry (Stirling's formula). As for the theorem itself, note that a much stronger and more general result (the Hermite--Lindemann--Weierstraß transcendence theorem) is already available in the AFP.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2022-01-08",
	- "id": 104,
	+ "id": 107,
	"link": "/entries/Irrationals_From_THEBOOK.html",
	"permalink": "/entries/Irrationals_From_THEBOOK.html",
	"shortname": "Irrationals_From_THEBOOK",
	"title": "Irrational numbers from THE BOOK",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a formalization of the article \u003ci\u003eKnight's Tour Revisited\u003c/i\u003e by Cull and De Curtins where they prove the existence of a Knight's path for arbitrary \u003ci\u003en \u0026times; m\u003c/i\u003e-boards with \u003ci\u003emin(n,m) \u0026ge; 5\u003c/i\u003e. If \u003ci\u003en \u0026middot; m\u003c/i\u003e is even, then there exists a Knight's circuit. A Knight's Path is a sequence of moves of a Knight on a chessboard s.t. the Knight visits every square of a chessboard exactly once. Finding a Knight's path is a an instance of the Hamiltonian path problem. A Knight's circuit is a Knight's path, where additionally the Knight can move from the last square to the first square of the path, forming a loop. During the formalization two mistakes in the original proof were discovered. These mistakes are corrected in this formalization.",
	"authors": [
	"Lukas Koller"
	],
	"date": "2022-01-04",
	- "id": 105,
	+ "id": 108,
	"link": "/entries/Knights_Tour.html",
	"permalink": "/entries/Knights_Tour.html",
	"shortname": "Knights_Tour",
	"title": "Knight's Tour Revisited Revisited",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eHyperdual numbers are ones with a real component and a number of infinitesimal components, usually written as $a_0 + a_1 \\cdot \\epsilon_1 + a_2 \\cdot \\epsilon_2 + a_3 \\cdot \\epsilon_1\\epsilon_2$. They have been proposed by \u003ca href=\"https://doi.org/10.2514/6.2011-886\"\u003eFike and Alonso\u003c/a\u003e in an approach to automatic differentiation.\u003c/p\u003e \u003cp\u003eIn this entry we formalise hyperdual numbers and their application to forward differentiation. We show them to be an instance of multiple algebraic structures and then, along with facts about twice-differentiability, we define what we call the hyperdual extensions of functions on real-normed fields. This extension formally represents the proposed way that the first and second derivatives of a function can be automatically calculated. We demonstrate it on the standard logistic function $f(x) = \\frac{1}{1 + e^{-x}}$ and also reproduce the example analytic function $f(x) = \\frac{e^x}{\\sqrt{sin(x)^3 + cos(x)^3}}$ used for demonstration by Fike and Alonso.\u003c/p\u003e",
	"authors": [
	"Filip Smola",
	"Jacques D. Fleuriot"
	],
	"date": "2021-12-31",
	- "id": 106,
	+ "id": 109,
	"link": "/entries/Hyperdual.html",
	"permalink": "/entries/Hyperdual.html",
	"shortname": "Hyperdual",
	"title": "Hyperdual Numbers and Forward Differentiation",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a stepwise refinement and proof of the Gale-Shapley stable matching (or marriage) algorithm down to executable code. Both a purely functional implementation based on lists and a functional implementation based on efficient arrays (provided by the Collections Framework in the AFP) are developed. The latter implementation runs in time \u003ci\u003eO(n\u003csup\u003e2\u003c/sup\u003e)\u003c/i\u003e where \u003ci\u003en\u003c/i\u003e is the cardinality of the two sets to be matched.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2021-12-29",
	- "id": 107,
	+ "id": 110,
	"link": "/entries/Gale_Shapley.html",
	"permalink": "/entries/Gale_Shapley.html",
	"shortname": "Gale_Shapley",
	"title": "Gale-Shapley Algorithm",
	"topic_links": [
	"computer-science/algorithms",
	"mathematics/games-and-economics"
	],
	"topics": [
	"Computer science/Algorithms",
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise a proof of Roth's Theorem on Arithmetic Progressions, a major result in additive combinatorics on the existence of 3-term arithmetic progressions in subsets of natural numbers. To this end, we follow a proof using graph regularity. We employ our recent formalisation of Szemerédi's Regularity Lemma, a major result in extremal graph theory, which we use here to prove the Triangle Counting Lemma and the Triangle Removal Lemma. Our sources are Yufei Zhao's MIT lecture notes \"\u003ca href=\"https://yufeizhao.com/gtac/gtac.pdf\"\u003eGraph Theory and Additive Combinatorics\u003c/a\u003e\" (latest version \u003ca href=\"https://yufeizhao.com/gtacbook/\"\u003ehere\u003c/a\u003e) and W.T. Gowers's Cambridge lecture notes \"\u003ca href=\"https://www.dpmms.cam.ac.uk/~par31/notes/tic.pdf\"\u003eTopics in Combinatorics\u003c/a\u003e\". We also refer to the University of Georgia notes by Stephanie Bell and Will Grodzicki, \"\u003ca href=\"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.432.327\"\u003eUsing Szemerédi's Regularity Lemma to Prove Roth's Theorem\u003c/a\u003e\".",
	"authors": [
	"Chelsea Edmonds",
	"Angeliki Koutsoukou-Argyraki",
	"Lawrence C. Paulson"
	],
	"date": "2021-12-28",
	- "id": 108,
	+ "id": 111,
	"link": "/entries/Roth_Arithmetic_Progressions.html",
	"permalink": "/entries/Roth_Arithmetic_Progressions.html",
	"shortname": "Roth_Arithmetic_Progressions",
	"title": "Roth's Theorem on Arithmetic Progressions",
	"topic_links": [
	"mathematics/graph-theory",
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Graph theory",
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of Markov Decision Processes with rewards. In particular we first build on Hölzl's formalization of MDPs (AFP entry: Markov_Models) and extend them with rewards. We proceed with an analysis of the expected total discounted reward criterion for infinite horizon MDPs. The central result is the construction of the iteration rule for the Bellman operator. We prove the optimality equations for this operator and show the existence of an optimal stationary deterministic solution. The analysis can be used to obtain dynamic programming algorithms such as value iteration and policy iteration to solve MDPs with formal guarantees. Our formalization is based on chapters 5 and 6 in Puterman's book \"Markov Decision Processes: Discrete Stochastic Dynamic Programming\".",
	"authors": [
	"Maximilian Schäffeler",
	"Mohammad Abdulaziz"
	],
	"date": "2021-12-16",
	- "id": 109,
	+ "id": 112,
	"link": "/entries/MDP-Rewards.html",
	"permalink": "/entries/MDP-Rewards.html",
	"shortname": "MDP-Rewards",
	"title": "Markov Decision Processes with Rewards",
	"topic_links": [
	"mathematics/probability-theory"
	],
	"topics": [
	"Mathematics/Probability theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a formalization of algorithms for solving Markov Decision Processes (MDPs) with formal guarantees on the optimality of their solutions. In particular we build on our analysis of the Bellman operator for discounted infinite horizon MDPs. From the iterator rule on the Bellman operator we directly derive executable value iteration and policy iteration algorithms to iteratively solve finite MDPs. We also prove correct optimized versions of value iteration that use matrix splittings to improve the convergence rate. In particular, we formally verify Gauss-Seidel value iteration and modified policy iteration. The algorithms are evaluated on two standard examples from the literature, namely, inventory management and gridworld. Our formalization covers most of chapter 6 in Puterman's book \"Markov Decision Processes: Discrete Stochastic Dynamic Programming\".",
	"authors": [
	"Maximilian Schäffeler",
	"Mohammad Abdulaziz"
	],
	"date": "2021-12-16",
	- "id": 110,
	+ "id": 113,
	"link": "/entries/MDP-Algorithms.html",
	"permalink": "/entries/MDP-Algorithms.html",
	"shortname": "MDP-Algorithms",
	"title": "Verified Algorithms for Solving Markov Decision Processes",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/algorithms"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "Tree automata have good closure properties and therefore a commonly used to prove/disprove properties. This formalization contains among other things the proofs of many closure properties of tree automata (anchored) ground tree transducers and regular relations. Additionally it includes the well known pumping lemma and a lifting of the Myhill Nerode theorem for regular languages to tree languages. We want to mention the existence of a \u003ca href=\"https://www.isa-afp.org/entries/Tree-Automata.html\"\u003etree automata APF-entry\u003c/a\u003e developed by Peter Lammich. His work is based on epsilon free top-down tree automata, while this entry builds on bottom-up tree auotamta with epsilon transitions. Moreover our formalization relies on the \u003ca href=\"https://www.isa-afp.org/entries/Collections.html\"\u003eCollections Framework\u003c/a\u003e, also by Peter Lammich, to obtain efficient code. All proven constructions of the closure properties are exportable using the Isabelle/HOL code generation facilities.",
	"authors": [
	"Alexander Lochmann",
	"Bertram Felgenhauer",
	"Christian Sternagel",
	"René Thiemann",
	"Thomas Sternagel"
	],
	"date": "2021-12-15",
	- "id": 111,
	+ "id": 114,
	"link": "/entries/Regular_Tree_Relations.html",
	"permalink": "/entries/Regular_Tree_Relations.html",
	"shortname": "Regular_Tree_Relations",
	"title": "Regular Tree Relations",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 2
	},
	{
	"abstract": "In this work we formalise the isomorphism between simplicial complexes of dimension $n$ and monotone Boolean functions in $n$ variables, mainly following the definitions and results as introduced by N. A. Scoville. We also take advantage of the AFP representation of \u003ca href=\"https://www.isa-afp.org/entries/ROBDD.html\"\u003eROBDD\u003c/a\u003e (Reduced Ordered Binary Decision Diagrams) to compute the ROBDD representation of a given simplicial complex (by means of the isomorphism to Boolean functions). Some examples of simplicial complexes and associated Boolean functions are also presented.",
	"authors": [
	"Jesús Aransay",
	"Alejandro del Campo",
	"Julius Michaelis"
	],
	"date": "2021-11-29",
	- "id": 112,
	+ "id": 115,
	"link": "/entries/Simplicial_complexes_and_boolean_functions.html",
	"permalink": "/entries/Simplicial_complexes_and_boolean_functions.html",
	"shortname": "Simplicial_complexes_and_boolean_functions",
	"title": "Simplicial Complexes and Boolean functions",
	"topic_links": [
	"mathematics/topology"
	],
	"topics": [
	"Mathematics/Topology"
	],
	"used_by": 0
	},
	{
	"abstract": "The \u003cem\u003evan Emde Boas tree\u003c/em\u003e or \u003cem\u003evan Emde Boas priority queue\u003c/em\u003e is a data structure supporting membership test, insertion, predecessor and successor search, minimum and maximum determination and deletion in \u003cem\u003eO(log log U)\u003c/em\u003e time, where \u003cem\u003eU = 0,...,2\u003csup\u003en-1\u003c/sup\u003e\u003c/em\u003e is the overall range to be considered. \u003cp/\u003e The presented formalization follows Chapter 20 of the popular \u003cem\u003eIntroduction to Algorithms (3rd ed.)\u003c/em\u003e by Cormen, Leiserson, Rivest and Stein (CLRS), extending the list of formally verified CLRS algorithms. Our current formalization is based on the first author's bachelor's thesis. \u003cp/\u003e First, we prove correct a \u003cem\u003efunctional\u003c/em\u003e implementation, w.r.t. an abstract data type for sets. Apart from functional correctness, we show a resource bound, and runtime bounds w.r.t. manually defined timing functions for the operations. \u003cp/\u003e Next, we refine the operations to Imperative HOL with time, and show correctness and complexity. This yields a practically more efficient implementation, and eliminates the manually defined timing functions from the trusted base of the proof.",
	"authors": [
	"Thomas Ammer",
	"Peter Lammich"
	],
	"date": "2021-11-23",
	- "id": 113,
	+ "id": 116,
	"link": "/entries/Van_Emde_Boas_Trees.html",
	"permalink": "/entries/Van_Emde_Boas_Trees.html",
	"shortname": "Van_Emde_Boas_Trees",
	"title": "van Emde Boas Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "\"Foundations of Geometry\" is a mathematical book written by Hilbert in 1899. This entry is a complete formalization of \"Incidence\" (excluding cubic axioms), \"Order\" and \"Congruence\" (excluding point sequences) of the axioms constructed in this book. In addition, the theorem of the problem about the part that is treated implicitly and is not clearly stated in it is being carried out in parallel.",
	"authors": [
	"Fumiya Iwama"
	],
	"date": "2021-11-22",
	- "id": 114,
	+ "id": 117,
	"link": "/entries/Foundation_of_geometry.html",
	"permalink": "/entries/Foundation_of_geometry.html",
	"shortname": "Foundation_of_geometry",
	"title": "Foundation of geometry in planes, and some complements: Excluding the parallel axioms",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work we formalize the Hahn decomposition theorem for signed measures, namely that any measure space for a signed measure can be decomposed into a positive and a negative set, where every measurable subset of the positive one has a positive measure, and every measurable subset of the negative one has a negative measure. We also formalize the Jordan decomposition theorem as a corollary, which states that the signed measure under consideration admits a unique decomposition into a difference of two positive measures, at least one of which is finite.",
	"authors": [
	"Marie Cousin",
	"Mnacho Echenim",
	"Hervé Guiol"
	],
	"date": "2021-11-19",
	- "id": 115,
	+ "id": 118,
	"link": "/entries/Hahn_Jordan_Decomposition.html",
	"permalink": "/entries/Hahn_Jordan_Decomposition.html",
	"shortname": "Hahn_Jordan_Decomposition",
	"title": "The Hahn and Jordan Decomposition Theorems",
	"topic_links": [
	"mathematics/measure-and-integration"
	],
	"topics": [
	"Mathematics/Measure and integration"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a shallow embedding of public announcement logic (PAL) with relativized general knowledge in HOL. We then use PAL to obtain an elegant encoding of the wise men puzzle, which we solve automatically using sledgehammer.",
	"authors": [
	"Christoph Benzmüller",
	"Sebastian Reiche"
	],
	"date": "2021-11-08",
	- "id": 116,
	+ "id": 119,
	"link": "/entries/PAL.html",
	"permalink": "/entries/PAL.html",
	"shortname": "PAL",
	"title": "Automating Public Announcement Logic and the Wise Men Puzzle in Isabelle/HOL",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eSimplified variants of Gödel's ontological argument are explored. Among those is a particularly interesting simplified argument which is (i) valid already in basic modal logics K or KT, (ii) which does not suffer from modal collapse, and (iii) which avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. \u003c/p\u003e\u003cp\u003e Whether the presented variants increase or decrease the attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and theology. \u003c/p\u003e",
	"authors": [
	"Christoph Benzmüller"
	],
	"date": "2021-11-08",
	- "id": 117,
	+ "id": 120,
	"link": "/entries/SimplifiedOntologicalArgument.html",
	"permalink": "/entries/SimplifiedOntologicalArgument.html",
	"shortname": "SimplifiedOntologicalArgument",
	"title": "Exploring Simplified Variants of Gödel’s Ontological Argument in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects",
	"logic/general-logic/modal-logic"
	],
	"topics": [
	"Logic/Philosophical aspects",
	"Logic/General logic/Modal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "The AFP already contains a verified implementation of algebraic numbers. However, it is has a severe limitation in its factorization algorithm of real and complex polynomials: the factorization is only guaranteed to succeed if the coefficients of the polynomial are rational numbers. In this work, we verify an algorithm to factor all real and complex polynomials whose coefficients are algebraic. The existence of such an algorithm proves in a constructive way that the set of complex algebraic numbers is algebraically closed. Internally, the algorithm is based on resultants of multivariate polynomials and an approximation algorithm using interval arithmetic.",
	"authors": [
	"Manuel Eberl",
	"René Thiemann"
	],
	"date": "2021-11-08",
	- "id": 118,
	+ "id": 121,
	"link": "/entries/Factor_Algebraic_Polynomial.html",
	"permalink": "/entries/Factor_Algebraic_Polynomial.html",
	"shortname": "Factor_Algebraic_Polynomial",
	"title": "Factorization of Polynomials with Algebraic Coefficients",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 4
	},
	{
	"abstract": "In this formalisation, we construct real exponents as the limits of sequences of rational exponents. In particular, if $a \\ge 1$ and $x \\in \\mathbb{R}$, we choose an increasing rational sequence $r_n$ such that $\\lim_{n\\to\\infty} {r_n} = x$. Then the sequence $a^{r_n}$ is increasing and if $r$ is any rational number such that $r \u003e x$, $a^{r_n}$ is bounded above by $a^r$. By the convergence criterion for monotone sequences, $a^{r_n}$ converges. We define $a^ x = \\lim_{n\\to\\infty} a^{r_n}$ and show that it has the expected properties (for $a \\ge 0$). This particular construction of real exponents is needed instead of the usual one using the natural logarithm and exponential functions (which already exists in Isabelle) to support our mechanical derivation of Euler's exponential series as an “infinite polynomial”. Aside from helping us avoid circular reasoning, this is, as far as we are aware, the first time real exponents are mechanised in this way within a proof assistant.",
	"authors": [
	"Jacques D. Fleuriot"
	],
	"date": "2021-11-08",
	- "id": 119,
	+ "id": 122,
	"link": "/entries/Real_Power.html",
	"permalink": "/entries/Real_Power.html",
	"shortname": "Real_Power",
	"title": "Real Exponents as the Limits of Sequences of Rational Exponents",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003ca href=\"https://en.wikipedia.org/wiki/Szemerédi_regularity_lemma\"\u003eSzemerédi's regularity lemma\u003c/a\u003e is a key result in the study of large graphs. It asserts the existence of an upper bound on the number of parts the vertices of a graph need to be partitioned into such that the edges between the parts are random in a certain sense. This bound depends only on the desired precision and not on the graph itself, in the spirit of Ramsey's theorem. The formalisation follows online course notes by \u003ca href=\"https://www.dpmms.cam.ac.uk/~par31/notes/tic.pdf\"\u003eTim Gowers\u003c/a\u003e and \u003ca href=\"https://yufeizhao.com/gtacbook/\"\u003eYufei Zhao\u003c/a\u003e.",
	"authors": [
	"Chelsea Edmonds",
	"Angeliki Koutsoukou-Argyraki",
	"Lawrence C. Paulson"
	],
	"date": "2021-11-05",
	- "id": 120,
	+ "id": 123,
	"link": "/entries/Szemeredi_Regularity.html",
	"permalink": "/entries/Szemeredi_Regularity.html",
	"shortname": "Szemeredi_Regularity",
	"title": "Szemerédi's Regularity Lemma",
	"topic_links": [
	"mathematics/graph-theory",
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Graph theory",
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "A formalization of the theory of quantum and classical registers as developed by (Unruh, Quantum and Classical Registers). In a nutshell, a register refers to a part of a larger memory or system that can be accessed independently. Registers can be constructed from other registers and several (compatible) registers can be composed. This formalization develops both the generic theory of registers as well as specific instantiations for classical and quantum registers.",
	"authors": [
	"Dominique Unruh"
	],
	"date": "2021-10-28",
	- "id": 121,
	+ "id": 124,
	"link": "/entries/Registers.html",
	"permalink": "/entries/Registers.html",
	"shortname": "Registers",
	"title": "Quantum and Classical Registers",
	"topic_links": [
	"computer-science/algorithms/quantum-computing",
	"computer-science/programming-languages/logics",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Algorithms/Quantum computing",
	"Computer science/Programming languages/Logics",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "The 1985 paper by Carlos Alchourrón, Peter Gärdenfors, and David Makinson (AGM), “On the Logic of Theory Change: Partial Meet Contraction and Revision Functions” launches a large and rapidly growing literature that employs formal models and logics to handle changing beliefs of a rational agent and to take into account new piece of information observed by this agent. In 2011, a review book titled \"AGM 25 Years: Twenty-Five Years of Research in Belief Change\" was edited to summarize the first twenty five years of works based on AGM. This HOL-based AFP entry is a faithful formalization of the AGM operators (e.g. contraction, revision, remainder ...) axiomatized in the original paper. It also contains the proofs of all the theorems stated in the paper that show how these operators combine. Both proofs of Harper and Levi identities are established.",
	"authors": [
	"Valentin Fouillard",
	"Safouan Taha",
	"Frédéric Boulanger",
	"Nicolas Sabouret"
	],
	"date": "2021-10-19",
	- "id": 122,
	+ "id": 125,
	"link": "/entries/Belief_Revision.html",
	"permalink": "/entries/Belief_Revision.html",
	"shortname": "Belief_Revision",
	"title": "Belief Revision Theory",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 0
	},
	{
	"abstract": "This AFP entry provides semantics for roughly 120 different X86-64 assembly instructions. These instructions include various moves, arithmetic/logical operations, jumps, call/return, SIMD extensions and others. External functions are supported by allowing a user to provide custom semantics for these calls. Floating-point operations are mapped to uninterpreted functions. The model provides semantics for register aliasing and a byte-level little-endian memory model. The semantics are purposefully incomplete, but overapproximative. For example, the precise effect of flags may be undefined for certain instructions, or instructions may simply have no semantics at all. In those cases, the semantics are mapped to universally quantified uninterpreted terms from a locale. Second, this entry provides a method to symbolic execution of basic blocks. The method, called “\u003ctt\u003ese_step\u003c/tt\u003e” (for: symbolic execution step) fetches an instruction and updates the current symbolic state while keeping track of assumptions made over the memory model. A key component is a set of theorems that prove how reads from memory resolve after writes have occurred. Thirdly, this entry provides a parser that allows the user to copy-paste the output of the standard disassembly tool objdump into Isabelle/HOL. A couple small and explanatory examples are included, including functions from the word count program. Several examples can be supplied upon request (they are not included due to the running time of verification): functions from the floating-point modulo function from FDLIBM, the GLIBC strlen function and the CoreUtils SHA256 implementation.",
	"authors": [
	"Freek Verbeek",
	"Abhijith Bharadwaj",
	"Joshua Bockenek",
	"Ian Roessle",
	"Timmy Weerwag",
	"Binoy Ravindran"
	],
	"date": "2021-10-13",
	- "id": 123,
	+ "id": 126,
	"link": "/entries/X86_Semantics.html",
	"permalink": "/entries/X86_Semantics.html",
	"shortname": "X86_Semantics",
	"title": "X86 instruction semantics and basic block symbolic execution",
	"topic_links": [
	"computer-science/hardware",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Hardware",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "We study models of state-based non-deterministic sequential computations and describe them using algebras. We propose algebras that describe iteration for strict and non-strict computations. They unify computation models which differ in the fixpoints used to represent iteration. We propose algebras that describe the infinite executions of a computation. They lead to a unified approximation order and results that connect fixpoints in the approximation and refinement orders. This unifies the semantics of recursion for a range of computation models. We propose algebras that describe preconditions and the effect of while-programs under postconditions. They unify correctness statements in two dimensions: one statement applies in various computation models to various correctness claims.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2021-10-12",
	- "id": 124,
	+ "id": 127,
	"link": "/entries/Correctness_Algebras.html",
	"permalink": "/entries/Correctness_Algebras.html",
	"shortname": "Correctness_Algebras",
	"title": "Algebras for Iteration, Infinite Executions and Correctness of Sequential Computations",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the first-order logic of real arithmetic is decidable by QE. However, in practice, QE algorithms are highly complicated and often combine multiple methods for performance. VS is a practically successful method for QE that targets formulas with low-degree polynomials. To our knowledge, this is the first work to formalize VS for quadratic real arithmetic including inequalities. The proofs necessitate various contributions to the existing multivariate polynomial libraries in Isabelle/HOL. Our framework is modularized and easily expandable (to facilitate integrating future optimizations), and could serve as a basis for developing practical general-purpose QE algorithms. Further, as our formalization is designed with practicality in mind, we export our development to SML and test the resulting code on 378 benchmarks from the literature, comparing to Redlog, Z3, Wolfram Engine, and SMT-RAT. This identified inconsistencies in some tools, underscoring the significance of a verified approach for the intricacies of real arithmetic.",
	"authors": [
	"Matias Scharager",
	"Katherine Kosaian",
	"Stefan Mitsch",
	"André Platzer"
	],
	"date": "2021-10-02",
	- "id": 125,
	+ "id": 128,
	"link": "/entries/Virtual_Substitution.html",
	"permalink": "/entries/Virtual_Substitution.html",
	"shortname": "Virtual_Substitution",
	"title": "Verified Quadratic Virtual Substitution for Real Arithmetic",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 1
	},
	{
	"abstract": "This work is a formalization of the soundness and completeness of an axiomatic system for first-order logic. The proof system is based on System Q1 by Smullyan and the completeness proof follows his textbook \"First-Order Logic\" (Springer-Verlag 1968). The completeness proof is in the Henkin style where a consistent set is extended to a maximal consistent set using Lindenbaum's construction and Henkin witnesses are added during the construction to ensure saturation as well. The resulting set is a Hintikka set which, by the model existence theorem, is satisfiable in the Herbrand universe. Paper: \u003ca href=\"https://doi.org/10.4230/LIPIcs.TYPES.2021.8\"\u003edoi.org/10.4230/LIPIcs.TYPES.2021.8\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2021-09-24",
	- "id": 126,
	+ "id": 129,
	"link": "/entries/FOL_Axiomatic.html",
	"permalink": "/entries/FOL_Axiomatic.html",
	"shortname": "FOL_Axiomatic",
	"title": "Soundness and Completeness of an Axiomatic System for First-Order Logic",
	"topic_links": [
	"logic/general-logic/classical-first-order-logic",
	"logic/proof-theory"
	],
	"topics": [
	"Logic/General logic/Classical first-order logic",
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of bounded operators on complex vector spaces. Our formalization contains material on complex vector spaces (normed spaces, Banach spaces, Hilbert spaces) that complements and goes beyond the developments of real vectors spaces in the Isabelle/HOL standard library. We define the type of bounded operators between complex vector spaces (\u003cem\u003ecblinfun\u003c/em\u003e) and develop the theory of unitaries, projectors, extension of bounded linear functions (BLT theorem), adjoints, Loewner order, closed subspaces and more. For the finite-dimensional case, we provide code generation support by identifying finite-dimensional operators with matrices as formalized in the \u003ca href=\"Jordan_Normal_Form.html\"\u003eJordan_Normal_Form\u003c/a\u003e AFP entry.",
	"authors": [
	"José Manuel Rodríguez Caballero",
	"Dominique Unruh"
	],
	"date": "2021-09-18",
	- "id": 127,
	+ "id": 130,
	"link": "/entries/Complex_Bounded_Operators.html",
	"permalink": "/entries/Complex_Bounded_Operators.html",
	"shortname": "Complex_Bounded_Operators",
	"title": "Complex Bounded Operators",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "We define the weighted path order (WPO) and formalize several properties such as strong normalization, the subterm property, and closure properties under substitutions and contexts. Our definition of WPO extends the original definition by also permitting multiset comparisons of arguments instead of just lexicographic extensions. Therefore, our WPO not only subsumes lexicographic path orders (LPO), but also recursive path orders (RPO). We formally prove these subsumptions and therefore all of the mentioned properties of WPO are automatically transferable to LPO and RPO as well. Such a transformation is not required for Knuth\u0026ndash;Bendix orders (KBO), since they have already been formalized. Nevertheless, we still provide a proof that WPO subsumes KBO and thereby underline the generality of WPO.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2021-09-16",
	- "id": 128,
	+ "id": 131,
	"link": "/entries/Weighted_Path_Order.html",
	"permalink": "/entries/Weighted_Path_Order.html",
	"shortname": "Weighted_Path_Order",
	"title": "A Formalization of Weighted Path Orders and Recursive Path Orders",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 3
	},
	{
	"abstract": "This article provides a foundational framework for the formalization of category theory in the object logic ZFC in HOL of the formal proof assistant Isabelle. More specifically, this article provides a formalization of canonical set-theoretic constructions internalized in the type \u003ci\u003eV\u003c/i\u003e associated with the ZFC in HOL, establishes a design pattern for the formalization of mathematical structures using sequences and locales, and showcases the developed infrastructure by providing formalizations of the elementary theories of digraphs and semicategories. The methodology chosen for the formalization of the theories of digraphs and semicategories (and categories in future articles) rests on the ideas that were originally expressed in the article \u003ci\u003eSet-Theoretical Foundations of Category Theory\u003c/i\u003e written by Solomon Feferman and Georg Kreisel. Thus, in the context of this work, each of the aforementioned mathematical structures is represented as a term of the type \u003ci\u003eV\u003c/i\u003e embedded into a stage of the von Neumann hierarchy.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 129,
	+ "id": 132,
	"link": "/entries/CZH_Foundations.html",
	"permalink": "/entries/CZH_Foundations.html",
	"shortname": "CZH_Foundations",
	"title": "Category Theory for ZFC in HOL I: Foundations: Design Patterns, Set Theory, Digraphs, Semicategories",
	"topic_links": [
	"mathematics/category-theory",
	"logic/set-theory"
	],
	"topics": [
	"Mathematics/Category theory",
	"Logic/Set theory"
	],
	"used_by": 1
	},
	{
	"abstract": "This article provides a formalization of the foundations of the theory of 1-categories in the object logic ZFC in HOL of the formal proof assistant Isabelle. The article builds upon the foundations that were established in the AFP entry \u003ci\u003eCategory Theory for ZFC in HOL I: Foundations: Design Patterns, Set Theory, Digraphs, Semicategories\u003c/i\u003e.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 130,
	+ "id": 133,
	"link": "/entries/CZH_Elementary_Categories.html",
	"permalink": "/entries/CZH_Elementary_Categories.html",
	"shortname": "CZH_Elementary_Categories",
	"title": "Category Theory for ZFC in HOL II: Elementary Theory of 1-Categories",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 1
	},
	{
	"abstract": "The article provides a formalization of elements of the theory of universal constructions for 1-categories (such as limits, adjoints and Kan extensions) in the object logic ZFC in HOL of the formal proof assistant Isabelle. The article builds upon the foundations established in the AFP entry \u003ci\u003eCategory Theory for ZFC in HOL II: Elementary Theory of 1-Categories\u003c/i\u003e.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 131,
	+ "id": 134,
	"link": "/entries/CZH_Universal_Constructions.html",
	"permalink": "/entries/CZH_Universal_Constructions.html",
	"shortname": "CZH_Universal_Constructions",
	"title": "Category Theory for ZFC in HOL III: Universal Constructions",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The article provides a collection of experimental general-purpose proof methods for the object logic Isabelle/HOL of the formal proof assistant Isabelle. The methods in the collection offer functionality that is similar to certain aspects of the functionality provided by the standard proof methods of Isabelle that combine classical reasoning and rewriting, such as the method \u003ci\u003eauto\u003c/i\u003e, but use a different approach for rewriting. More specifically, these methods allow for the side conditions of the rewrite rules to be solved via intro-resolution.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 132,
	+ "id": 135,
	"link": "/entries/Conditional_Simplification.html",
	"permalink": "/entries/Conditional_Simplification.html",
	"shortname": "Conditional_Simplification",
	"title": "Conditional Simplification",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "This article provides a collection of experimental utilities for unoverloading of definitions and synthesis of conditional transfer rules for the object logic Isabelle/HOL of the formal proof assistant Isabelle written in Isabelle/ML.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 133,
	+ "id": 136,
	"link": "/entries/Conditional_Transfer_Rule.html",
	"permalink": "/entries/Conditional_Transfer_Rule.html",
	"shortname": "Conditional_Transfer_Rule",
	"title": "Conditional Transfer Rule",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "In their article titled \u003ci\u003eFrom Types to Sets by Local Type Definitions in Higher-Order Logic\u003c/i\u003e and published in the proceedings of the conference \u003ci\u003eInteractive Theorem Proving\u003c/i\u003e in 2016, Ondřej Kunčar and Andrei Popescu propose an extension of the logic Isabelle/HOL and an associated algorithm for the relativization of the \u003ci\u003etype-based theorems\u003c/i\u003e to more flexible \u003ci\u003eset-based theorems\u003c/i\u003e, collectively referred to as \u003ci\u003eTypes-To-Sets\u003c/i\u003e. One of the aims of their work was to open an opportunity for the development of a software tool for applied relativization in the implementation of the logic Isabelle/HOL of the proof assistant Isabelle. In this article, we provide a prototype of a software framework for the interactive automated relativization of theorems in Isabelle/HOL, developed as an extension of the proof language Isabelle/Isar. The software framework incorporates the implementation of the proposed extension of the logic, and builds upon some of the ideas for further work expressed in the original article on Types-To-Sets by Ondřej Kunčar and Andrei Popescu and the subsequent article \u003ci\u003eSmooth Manifolds and Types to Sets for Linear Algebra in Isabelle/HOL\u003c/i\u003e that was written by Fabian Immler and Bohua Zhan and published in the proceedings of the \u003ci\u003eInternational Conference on Certified Programs and Proofs\u003c/i\u003e in 2019.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 134,
	+ "id": 137,
	"link": "/entries/Types_To_Sets_Extension.html",
	"permalink": "/entries/Types_To_Sets_Extension.html",
	"shortname": "Types_To_Sets_Extension",
	"title": "Extension of Types-To-Sets",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "The article provides the command \u003cb\u003emk_ide\u003c/b\u003e for the object logic Isabelle/HOL of the formal proof assistant Isabelle. The command \u003cb\u003emk_ide\u003c/b\u003e enables the automated synthesis of the introduction, destruction and elimination rules from arbitrary definitions of constant predicates stated in Isabelle/HOL.",
	"authors": [
	"Mihails Milehins"
	],
	"date": "2021-09-06",
	- "id": 135,
	+ "id": 138,
	"link": "/entries/Intro_Dest_Elim.html",
	"permalink": "/entries/Intro_Dest_Elim.html",
	"shortname": "Intro_Dest_Elim",
	"title": "IDE: Introduction, Destruction, Elimination",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry formalises the fast iterative algorithm for computing dominators due to Cooper, Harvey and Kennedy. It gives a specification of computing dominators on a control flow graph where each node refers to its reverse post order number. A semilattice of reversed-ordered list which represents dominators is built and a Kildall-style algorithm on the semilattice is defined for computing dominators. Finally the soundness and completeness of the algorithm are proved w.r.t. the specification.",
	"authors": [
	"Nan Jiang"
	],
	"date": "2021-09-05",
	- "id": 136,
	+ "id": 139,
	"link": "/entries/Dominance_CHK.html",
	"permalink": "/entries/Dominance_CHK.html",
	"shortname": "Dominance_CHK",
	"title": "A data flow analysis algorithm for computing dominators",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eWe formalize Cardano's formula to solve a cubic equation $$ax^3 + bx^2 + cx + d = 0,$$ as well as Ferrari's formula to solve a quartic equation. We further turn both formulas into executable algorithms based on the algebraic number implementation in the AFP. To this end we also slightly extended this library, namely by making the minimal polynomial of an algebraic number executable, and by defining and implementing $n$-th roots of complex numbers.\u003c/p\u003e",
	"authors": [
	"René Thiemann"
	],
	"date": "2021-09-03",
	- "id": 137,
	+ "id": 140,
	"link": "/entries/Cubic_Quartic_Equations.html",
	"permalink": "/entries/Cubic_Quartic_Equations.html",
	"shortname": "Cubic_Quartic_Equations",
	"title": "Solving Cubic and Quartic Equations",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "In the context of formal cryptographic protocol verification, logging-independent message anonymity is the property for a given message to remain anonymous despite the attacker's capability of mapping messages of that sort to agents based on some intrinsic feature of such messages, rather than by logging the messages exchanged by legitimate agents as with logging-dependent message anonymity. This paper illustrates how logging-independent message anonymity can be formalized according to the relational method for formal protocol verification by considering a real-world protocol, namely the Restricted Identification one by the BSI. This sample model is used to verify that the pseudonymous identifiers output by user identification tokens remain anonymous under the expected conditions.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2021-08-26",
	- "id": 138,
	+ "id": 141,
	"link": "/entries/Logging_Independent_Anonymity.html",
	"permalink": "/entries/Logging_Independent_Anonymity.html",
	"shortname": "Logging_Independent_Anonymity",
	"title": "Logging-independent Message Anonymity in the Relational Method",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "The Descartes test based on Bernstein coefficients and Descartes’ rule of signs effectively (over-)approximates the number of real roots of a univariate polynomial over an interval. In this entry we formalise the theorem of three circles, which gives sufficient conditions for when the Descartes test returns 0 or 1. This is the first step for efficient root isolation.",
	"authors": [
	"Fox Thomson",
	"Wenda Li"
	],
	"date": "2021-08-21",
	- "id": 139,
	+ "id": 142,
	"link": "/entries/Three_Circles.html",
	"permalink": "/entries/Three_Circles.html",
	"shortname": "Three_Circles",
	"title": "The Theorem of Three Circles",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains the confidentiality verification of the (functional kernel of) the CoCon conference management system [\u003ca href=\"https://doi.org/10.1007/978-3-319-08867-9_11\"\u003e1\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/s10817-020-09566-9\"\u003e2\u003c/a\u003e]. The confidentiality properties refer to the documents managed by the system, namely papers, reviews, discussion logs and acceptance/rejection decisions, and also to the assignment of reviewers to papers. They have all been formulated as instances of BD Security [\u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2021.3\"\u003e3\u003c/a\u003e, \u003ca href=\"https://www.isa-afp.org/entries/Bounded_Deducibility_Security.html\"\u003e4\u003c/a\u003e] and verified using the BD Security unwinding technique.",
	"authors": [
	"Andrei Popescu",
	"Peter Lammich",
	"Thomas Bauereiss"
	],
	"date": "2021-08-16",
	- "id": 140,
	+ "id": 143,
	"link": "/entries/CoCon.html",
	"permalink": "/entries/CoCon.html",
	"shortname": "CoCon",
	"title": "CoCon: A Confidentiality-Verified Conference Management System",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Building on a previous \u003ca href=\"https://www.isa-afp.org/entries/Bounded_Deducibility_Security.html\"\u003eAFP entry\u003c/a\u003e that formalizes the Bounded-Deducibility Security (BD Security) framework \u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2021.3\"\u003e[1]\u003c/a\u003e, we formalize compositionality and transport theorems for information flow security. These results allow lifting BD Security properties from individual components specified as transition systems, to a composition of systems specified as communicating products of transition systems. The underlying ideas of these results are presented in the papers \u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2021.3\"\u003e[1]\u003c/a\u003e and \u003ca href=\"https://doi.org/10.1109/SP.2017.24\"\u003e[2]\u003c/a\u003e. The latter paper also describes a major case study where these results have been used: on verifying the CoSMeDis distributed social media platform (itself formalized as an \u003ca href=\"https://www.isa-afp.org/entries/CoSMeDis.html\"\u003eAFP entry\u003c/a\u003e that builds on this entry).",
	"authors": [
	"Thomas Bauereiss",
	"Andrei Popescu"
	],
	"date": "2021-08-16",
	- "id": 141,
	+ "id": 144,
	"link": "/entries/BD_Security_Compositional.html",
	"permalink": "/entries/BD_Security_Compositional.html",
	"shortname": "BD_Security_Compositional",
	"title": "Compositional BD Security",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry contains the confidentiality verification of the (functional kernel of) the CoSMed social media platform. The confidentiality properties are formalized as instances of BD Security [\u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2021.3\"\u003e1\u003c/a\u003e, \u003ca href=\"https://www.isa-afp.org/entries/Bounded_Deducibility_Security.html\"\u003e2\u003c/a\u003e]. An innovation in the deployment of BD Security compared to previous work is the use of dynamic declassification triggers, incorporated as part of inductive bounds, for providing stronger guarantees that account for the repeated opening and closing of access windows. To further strengthen the confidentiality guarantees, we also prove \"traceback\" properties about the accessibility decisions affecting the information managed by the system.",
	"authors": [
	"Thomas Bauereiss",
	"Andrei Popescu"
	],
	"date": "2021-08-16",
	- "id": 142,
	+ "id": 145,
	"link": "/entries/CoSMed.html",
	"permalink": "/entries/CoSMed.html",
	"shortname": "CoSMed",
	"title": "CoSMed: A confidentiality-verified social media platform",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains the confidentiality verification of the (functional kernel of) the CoSMeDis distributed social media platform presented in [\u003ca href=\"https://doi.org/10.1109/SP.2017.24\"\u003e1\u003c/a\u003e]. CoSMeDis is a multi-node extension the CoSMed prototype social media platform [\u003ca href=\"https://doi.org/10.1007/978-3-319-43144-4_6\"\u003e2\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/s10817-017-9443-3\"\u003e3\u003c/a\u003e, \u003ca href=\"https://www.isa-afp.org/entries/CoSMed.html\"\u003e4\u003c/a\u003e]. The confidentiality properties are formalized as instances of BD Security [\u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2021.3\"\u003e5\u003c/a\u003e, \u003ca href=\"https://www.isa-afp.org/entries/Bounded_Deducibility_Security.html\"\u003e6\u003c/a\u003e]. The lifting of confidentiality properties from single nodes to the entire CoSMeDis network is performed using compositionality and transport theorems for BD Security, which are described in [\u003ca href=\"https://doi.org/10.1109/SP.2017.24\"\u003e1\u003c/a\u003e] and formalized in a separate \u003ca href=\"https://www.isa-afp.org/entries/BD_Security_Compositional.html\"\u003eAFP entry\u003c/a\u003e.",
	"authors": [
	"Thomas Bauereiss",
	"Andrei Popescu"
	],
	"date": "2021-08-16",
	- "id": 143,
	+ "id": 146,
	"link": "/entries/CoSMeDis.html",
	"permalink": "/entries/CoSMeDis.html",
	"shortname": "CoSMeDis",
	"title": "CoSMeDis: A confidentiality-verified distributed social media platform",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry defines a type class with an operator returning a fresh identifier, given a set of already used identifiers and a preferred identifier. The entry provides a default instantiation for any infinite type, as well as executable instantiations for natural numbers and strings.",
	"authors": [
	"Andrei Popescu",
	"Thomas Bauereiss"
	],
	"date": "2021-08-16",
	- "id": 144,
	+ "id": 147,
	"link": "/entries/Fresh_Identifiers.html",
	"permalink": "/entries/Fresh_Identifiers.html",
	"shortname": "Fresh_Identifiers",
	"title": "Fresh identifiers",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 5
	},
	{
	"abstract": "Combinatorial design theory studies incidence set systems with certain balance and symmetry properties. It is closely related to hypergraph theory. This formalisation presents a general library for formal reasoning on incidence set systems, designs and their applications, including formal definitions and proofs for many key properties, operations, and theorems on the construction and existence of designs. Notably, this includes formalising t-designs, balanced incomplete block designs (BIBD), group divisible designs (GDD), pairwise balanced designs (PBD), design isomorphisms, and the relationship between graphs and designs. A locale-centric approach has been used to manage the relationships between the many different types of designs. Theorems of particular interest include the necessary conditions for existence of a BIBD, Wilson's construction on GDDs, and Bose's inequality on resolvable designs. Parts of this formalisation are explored in the paper \"A Modular First Formalisation of Combinatorial Design Theory\", presented at CICM 2021.",
	"authors": [
	"Chelsea Edmonds",
	"Lawrence C. Paulson"
	],
	"date": "2021-08-13",
	- "id": 145,
	+ "id": 148,
	"link": "/entries/Design_Theory.html",
	"permalink": "/entries/Design_Theory.html",
	"shortname": "Design_Theory",
	"title": "Combinatorial Design Theory",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 2
	},
	{
	"abstract": "We study second-order formalisations of graph properties expressed as first-order formulas in relation algebras extended with a Kleene star. The formulas quantify over relations while still avoiding quantification over elements of the base set. We formalise the property of undirected graphs being acyclic this way. This involves a study of various kinds of orientation of graphs. We also verify basic algorithms to constructively prove several second-order properties.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2021-08-03",
	- "id": 146,
	+ "id": 149,
	"link": "/entries/Relational_Forests.html",
	"permalink": "/entries/Relational_Forests.html",
	"shortname": "Relational_Forests",
	"title": "Relational Forests",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a formalisation of Schutz' system of axioms for Minkowski spacetime published under the name \"Independent axioms for Minkowski space-time\" in 1997, as well as most of the results in the third chapter (\"Temporal Order on a Path\") of the above monograph. Many results are proven here that cannot be found in Schutz, either preceding the theorem they are needed for, or within their own thematic section.",
	"authors": [
	"Richard Schmoetten",
	"Jake Palmer",
	"Jacques D. Fleuriot"
	],
	"date": "2021-07-27",
	- "id": 147,
	+ "id": 150,
	"link": "/entries/Schutz_Spacetime.html",
	"permalink": "/entries/Schutz_Spacetime.html",
	"shortname": "Schutz_Spacetime",
	"title": "Schutz' Independent Axioms for Minkowski Spacetime",
	"topic_links": [
	"mathematics/physics",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Physics",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "This article deals with the formalisation of some group-theoretic results including the fundamental theorem of finitely generated abelian groups characterising the structure of these groups as a uniquely determined product of cyclic groups. Both the invariant factor decomposition and the primary decomposition are covered. Additional work includes results about the direct product, the internal direct product and more group-theoretic lemmas.",
	"authors": [
	"Joseph Thommes",
	"Manuel Eberl"
	],
	"date": "2021-07-07",
	- "id": 148,
	+ "id": 151,
	"link": "/entries/Finitely_Generated_Abelian_Groups.html",
	"permalink": "/entries/Finitely_Generated_Abelian_Groups.html",
	"shortname": "Finitely_Generated_Abelian_Groups",
	"title": "Finitely Generated Abelian Groups",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "SpecCheck is a \u003ca href=\"https://en.wikipedia.org/wiki/QuickCheck\"\u003eQuickCheck\u003c/a\u003e-like testing framework for Isabelle/ML. You can use it to write specifications for ML functions. SpecCheck then checks whether your specification holds by testing your function against a given number of generated inputs. It helps you to identify bugs by printing counterexamples on failure and provides you timing information. SpecCheck is customisable and allows you to specify your own input generators, test output formats, as well as pretty printers and shrinking functions for counterexamples among other things.",
	"authors": [
	"Kevin Kappelmann",
	"Lukas Bulwahn",
	"Sebastian Willenbrink"
	],
	"date": "2021-07-01",
	- "id": 149,
	+ "id": 152,
	"link": "/entries/SpecCheck.html",
	"permalink": "/entries/SpecCheck.html",
	"shortname": "SpecCheck",
	"title": "SpecCheck - Specification-Based Testing for Isabelle/ML",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 3
	},
	{
	"abstract": "This article formalises the proof of Van der Waerden's Theorem from Ramsey theory. Van der Waerden's Theorem states that for integers $k$ and $l$ there exists a number $N$ which guarantees that if an integer interval of length at least $N$ is coloured with $k$ colours, there will always be an arithmetic progression of length $l$ of the same colour in said interval. The proof goes along the lines of \\cite{Swan}. The smallest number $N_{k,l}$ fulfilling Van der Waerden's Theorem is then called the Van der Waerden Number. Finding the Van der Waerden Number is still an open problem for most values of $k$ and $l$.",
	"authors": [
	"Katharina Kreuzer",
	"Manuel Eberl"
	],
	"date": "2021-06-22",
	- "id": 150,
	+ "id": 153,
	"link": "/entries/Van_der_Waerden.html",
	"permalink": "/entries/Van_der_Waerden.html",
	"shortname": "Van_der_Waerden",
	"title": "Van der Waerden's Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "MiniSail is a kernel language for Sail, an instruction set architecture (ISA) specification language. Sail is an imperative language with a light-weight dependent type system similar to refinement type systems. From an ISA specification, the Sail compiler can generate theorem prover code and C (or OCaml) to give an executable emulator for an architecture. The idea behind MiniSail is to capture the key and novel features of Sail in terms of their syntax, typing rules and operational semantics, and to confirm that they work together by proving progress and preservation lemmas. We use the Nominal2 library to handle binding.",
	"authors": [
	"Mark Wassell"
	],
	"date": "2021-06-18",
	- "id": 151,
	+ "id": 154,
	"link": "/entries/MiniSail.html",
	"permalink": "/entries/MiniSail.html",
	"shortname": "MiniSail",
	"title": "MiniSail - A kernel language for the ISA specification language SAIL",
	"topic_links": [
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is a formalization of public announcement logic. It includes proofs of soundness and completeness for a variant of the axiom system PA + DIST! + NEC!. The completeness proof builds on the Epistemic Logic theory. Paper: \u003ca href=\"https://doi.org/10.1007/978-3-030-90138-7_2\"\u003edoi.org/10.1007/978-3-030-90138-7_2\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2021-06-17",
	- "id": 152,
	+ "id": 155,
	"link": "/entries/Public_Announcement_Logic.html",
	"permalink": "/entries/Public_Announcement_Logic.html",
	"shortname": "Public_Announcement_Logic",
	"title": "Public Announcement Logic",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 0
	},
	{
	"abstract": "This paper presents a compiler correctness proof for the didactic imperative programming language IMP, introduced in Nipkow and Klein's book on formal programming language semantics (version of March 2021), whose size is just two thirds of the book's proof in the number of formal text lines. As such, it promises to constitute a further enhanced reference for the formal verification of compilers meant for larger, real-world programming languages. The presented proof does not depend on language determinism, so that the proposed approach can be applied to non-deterministic languages as well. As a confirmation, this paper extends IMP with an additional non-deterministic choice command, and proves compiler correctness, viz. the simulation of compiled code execution by source code, for such extended language.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2021-06-04",
	- "id": 153,
	+ "id": 156,
	"link": "/entries/IMP_Compiler.html",
	"permalink": "/entries/IMP_Compiler.html",
	"shortname": "IMP_Compiler",
	"title": "A Shorter Compiler Correctness Proof for Language IMP",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize basics of Combinatorics on Words. This is an extension of existing theories on lists. We provide additional properties related to prefix, suffix, factor, length and rotation. The topics include prefix and suffix comparability, mismatch, word power, total and reversed morphisms, border, periods, primitivity and roots. We also formalize basic, mostly folklore results related to word equations: equidivisibility, commutation and conjugation. Slightly advanced properties include the Periodicity lemma (often cited as the Fine and Wilf theorem) and the variant of the Lyndon-Schützenberger theorem for words, including its full parametric solution. We support the algebraic point of view which sees words as generators of submonoids of a free monoid. This leads to the concepts of the (free) hull, the (free) basis (or code). We also provide relevant proof methods and a tool to generate reverse-symmetric claims.",
	"authors": [
	"Štěpán Holub",
	"Martin Raška",
	"Štěpán Starosta"
	],
	"date": "2021-05-24",
	- "id": 154,
	+ "id": 157,
	"link": "/entries/Combinatorics_Words.html",
	"permalink": "/entries/Combinatorics_Words.html",
	"shortname": "Combinatorics_Words",
	"title": "Combinatorics on Words Basics",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 4
	},
	{
	"abstract": "Graph lemma quantifies the defect effect of a system of word equations. That is, it provides an upper bound on the rank of the system. We formalize the proof based on the decomposition of a solution into its free basis. A direct application is an alternative proof of the fact that two noncommuting words form a code.",
	"authors": [
	"Štěpán Holub",
	"Štěpán Starosta"
	],
	"date": "2021-05-24",
	- "id": 155,
	+ "id": 158,
	"link": "/entries/Combinatorics_Words_Graph_Lemma.html",
	"permalink": "/entries/Combinatorics_Words_Graph_Lemma.html",
	"shortname": "Combinatorics_Words_Graph_Lemma",
	"title": "Graph Lemma",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 2
	},
	{
	"abstract": "Lyndon words are words lexicographically minimal in their conjugacy class. We formalize their basic properties and characterizations, in particular the concepts of the longest Lyndon suffix and the Lyndon factorization. Most of the work assumes a fixed lexicographical order. Nevertheless we also define the smallest relation guaranteeing lexicographical minimality of a given word (in its conjugacy class).",
	"authors": [
	"Štěpán Holub",
	"Štěpán Starosta"
	],
	"date": "2021-05-24",
	- "id": 156,
	+ "id": 159,
	"link": "/entries/Combinatorics_Words_Lyndon.html",
	"permalink": "/entries/Combinatorics_Words_Lyndon.html",
	"shortname": "Combinatorics_Words_Lyndon",
	"title": "Lyndon words",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This development provides a general definition for safe Regression Test Selection (RTS) algorithms. RTS algorithms select which tests to rerun on revised code, reducing the time required to check for newly introduced errors. An RTS algorithm is considered safe if and only if all deselected tests would have unchanged results. This definition is instantiated with two class-collection-based RTS algorithms run over the JVM as modeled by JinjaDCI. This is achieved with a general definition for Collection Semantics, small-step semantics instrumented to collect information during execution. As the RTS definition mandates safety, these instantiations include proofs of safety. This work is described in Mansky and Gunter's LSFA 2020 paper and Mansky's doctoral thesis (UIUC, 2020).",
	"authors": [
	"Susannah Mansky"
	],
	"date": "2021-04-30",
	- "id": 157,
	+ "id": 160,
	"link": "/entries/Regression_Test_Selection.html",
	"permalink": "/entries/Regression_Test_Selection.html",
	"shortname": "Regression_Test_Selection",
	"title": "Regression Test Selection",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "In this entry we formalize Isabelle's metalogic in Isabelle/HOL. Furthermore, we define a language of proof terms and an executable proof checker and prove its soundness wrt. the metalogic. The formalization is intentionally kept close to the Isabelle implementation(for example using de Brujin indices) to enable easy integration of generated code with the Isabelle system without a complicated translation layer. The formalization is described in our \u003ca href=\"https://arxiv.org/pdf/2104.12224.pdf\"\u003eCADE 28 paper\u003c/a\u003e.",
	"authors": [
	"Tobias Nipkow",
	"Simon Roßkopf"
	],
	"date": "2021-04-27",
	- "id": 158,
	+ "id": 161,
	"link": "/entries/Metalogic_ProofChecker.html",
	"permalink": "/entries/Metalogic_ProofChecker.html",
	"shortname": "Metalogic_ProofChecker",
	"title": "Isabelle's Metalogic: Formalization and Proof Checker",
	"topic_links": [
	"logic/general-logic"
	],
	"topics": [
	"Logic/General logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the \u003ci\u003eLifting the Exponent Lemma\u003c/i\u003e, which shows how to find the largest power of $p$ dividing $a^n \\pm b^n$, for a prime $p$ and positive integers $a$ and $b$. The proof follows \u003ca href=\"https://s3.amazonaws.com/aops-cdn.artofproblemsolving.com/resources/articles/lifting-the-exponent.pdf\"\u003eAmir Hossein Parvardi's\u003c/a\u003e.",
	"authors": [
	"Maya Kądziołka"
	],
	"date": "2021-04-27",
	- "id": 159,
	+ "id": 162,
	"link": "/entries/Lifting_the_Exponent.html",
	"permalink": "/entries/Lifting_the_Exponent.html",
	"shortname": "Lifting_the_Exponent",
	"title": "Lifting the Exponent",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the univariate case of Ben-Or, Kozen, and Reif's decision procedure for first-order real arithmetic (the BKR algorithm). We also formalize the univariate case of Renegar's variation of the BKR algorithm. The two formalizations differ mathematically in minor ways (that have significant impact on the multivariate case), but are quite similar in proof structure. Both rely on sign-determination (finding the set of consistent sign assignments for a set of polynomials). The method used for sign-determination is similar to Tarski's original quantifier elimination algorithm (it stores key information in a matrix equation), but with a reduction step to keep complexity low.",
	"authors": [
	"Katherine Kosaian",
	"Yong Kiam Tan",
	"André Platzer"
	],
	"date": "2021-04-24",
	- "id": 160,
	+ "id": 163,
	"link": "/entries/BenOr_Kozen_Reif.html",
	"permalink": "/entries/BenOr_Kozen_Reif.html",
	"shortname": "BenOr_Kozen_Reif",
	"title": "The BKR Decision Procedure for Univariate Real Arithmetic",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 3
	},
	{
	"abstract": "This is a formalisation of the main result of Gale and Stewart from 1953, showing that closed finite games are determined. This property is now known as the Gale Stewart Theorem. While the original paper shows some additional theorems as well, we only formalize this main result, but do so in a somewhat general way. We formalize games of a fixed arbitrary length, including infinite length, using co-inductive lists, and show that defensive strategies exist unless the other player is winning. For closed games, defensive strategies are winning for the closed player, proving that such games are determined. For finite games, which are a special case in our formalisation, all games are closed.",
	"authors": [
	"Sebastiaan J. C. Joosten"
	],
	"date": "2021-04-23",
	- "id": 161,
	+ "id": 164,
	"link": "/entries/GaleStewart_Games.html",
	"permalink": "/entries/GaleStewart_Games.html",
	"shortname": "GaleStewart_Games",
	"title": "Gale-Stewart Games",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "Large-scale stream processing systems often follow the dataflow paradigm, which enforces a program structure that exposes a high degree of parallelism. The Timely Dataflow distributed system supports expressive cyclic dataflows for which it offers low-latency data- and pipeline-parallel stream processing. To achieve high expressiveness and performance, Timely Dataflow uses an intricate distributed protocol for tracking the computation’s progress. We formalize this progress tracking protocol and verify its safety. Our formalization is described in detail in our forthcoming \u003ca href=\"https://traytel.bitbucket.io/papers/itp21-progress_tracking/safe.pdf\"\u003eITP'21 paper\u003c/a\u003e.",
	"authors": [
	"Matthias Brun",
	"Sára Decova",
	"Andrea Lattuada",
	"Dmitriy Traytel"
	],
	"date": "2021-04-13",
	- "id": 162,
	+ "id": 165,
	"link": "/entries/Progress_Tracking.html",
	"permalink": "/entries/Progress_Tracking.html",
	"shortname": "Progress_Tracking",
	"title": "Formalization of Timely Dataflow's Progress Tracking Protocol",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a characterisation of how information is propagated by program executions based on the tracking data and control dependencies within executions themselves. The characterisation might be used for deriving approximative safety properties to be targeted by static analyses or checked at runtime. We utilise a simple yet versatile control flow graph model as a program representation. As our model is not assumed to be finite it can be instantiated for a broad class of programs. The targeted security property is indistinguishable security where executions produce sequences of observations and only non-terminating executions are allowed to drop a tail of those. A very crude approximation of our characterisation is slicing based on program dependence graphs, which we use as a minimal example and derive a corresponding soundness result. For further details and applications refer to the authors upcoming dissertation.",
	"authors": [
	"Benedikt Nordhoff"
	],
	"date": "2021-04-01",
	- "id": 163,
	+ "id": 166,
	"link": "/entries/IFC_Tracking.html",
	"permalink": "/entries/IFC_Tracking.html",
	"shortname": "IFC_Tracking",
	"title": "Information Flow Control via Dependency Tracking",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize mainstream structures in algebraic geometry culminating in Grothendieck's schemes: presheaves of rings, sheaves of rings, ringed spaces, locally ringed spaces, affine schemes and schemes. We prove that the spectrum of a ring is a locally ringed space, hence an affine scheme. Finally, we prove that any affine scheme is a scheme.",
	"authors": [
	"Anthony Bordg",
	"Lawrence C. Paulson",
	"Wenda Li"
	],
	"date": "2021-03-29",
	- "id": 164,
	+ "id": 167,
	"link": "/entries/Grothendieck_Schemes.html",
	"permalink": "/entries/Grothendieck_Schemes.html",
	"shortname": "Grothendieck_Schemes",
	"title": "Grothendieck's Schemes in Algebraic Geometry",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the ring of \u003cem\u003ep\u003c/em\u003e-adic integers within the framework of the HOL-Algebra library. The carrier of the ring is formalized as the inverse limit of quotients of the integers by powers of a fixed prime \u003cem\u003ep\u003c/em\u003e. We define an integer-valued valuation, as well as an extended-integer valued valuation which sends 0 to the infinite element. Basic topological facts about the \u003cem\u003ep\u003c/em\u003e-adic integers are formalized, including completeness and sequential compactness. Taylor expansions of polynomials over a commutative ring are defined, culminating in the formalization of Hensel's Lemma based on a proof due to Keith Conrad.",
	"authors": [
	"Aaron Crighton"
	],
	"date": "2021-03-23",
	- "id": 165,
	+ "id": 168,
	"link": "/entries/Padic_Ints.html",
	"permalink": "/entries/Padic_Ints.html",
	"shortname": "Padic_Ints",
	"title": "Hensel's Lemma for the p-adic Integers",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "Constructive Cryptography (CC) [\u003ca href=\"https://conference.iiis.tsinghua.edu.cn/ICS2011/content/papers/14.html\"\u003eICS 2011\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/978-3-642-27375-9_3\"\u003eTOSCA 2011\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/978-3-662-53641-4_1\"\u003eTCC 2016\u003c/a\u003e] introduces an abstract approach to composable security statements that allows one to focus on a particular aspect of security proofs at a time. Instead of proving the properties of concrete systems, CC studies system classes, i.e., the shared behavior of similar systems, and their transformations. Modeling of systems communication plays a crucial role in composability and reusability of security statements; yet, this aspect has not been studied in any of the existing CC results. We extend our previous CC formalization [\u003ca href=\"https://isa-afp.org/entries/Constructive_Cryptography.html\"\u003eConstructive_Cryptography\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1109/CSF.2019.00018\"\u003eCSF 2019\u003c/a\u003e] with a new semantic domain called Fused Resource Templates (FRT) that abstracts over the systems communication patterns in CC proofs. This widens the scope of cryptography proof formalizations in the CryptHOL library [\u003ca href=\"https://isa-afp.org/entries/CryptHOL.html\"\u003eCryptHOL\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/978-3-662-49498-1_20\"\u003eESOP 2016\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/s00145-019-09341-z\"\u003eJ Cryptol 2020\u003c/a\u003e]. This formalization is described in \u003ca href=\"http://www.andreas-lochbihler.de/pub/basin2021.pdf\"\u003eAbstract Modeling of Systems Communication in Constructive Cryptography using CryptHOL\u003c/a\u003e.",
	"authors": [
	"Andreas Lochbihler",
	"S. Reza Sefidgar"
	],
	"date": "2021-03-17",
	- "id": 166,
	+ "id": 169,
	"link": "/entries/Constructive_Cryptography_CM.html",
	"permalink": "/entries/Constructive_Cryptography_CM.html",
	"shortname": "Constructive_Cryptography_CM",
	"title": "Constructive Cryptography in HOL: the Communication Modeling Aspect",
	"topic_links": [
	"computer-science/security/cryptography",
	"mathematics/probability-theory"
	],
	"topics": [
	"Computer science/Security/Cryptography",
	"Mathematics/Probability theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We verify two algorithms for which modular arithmetic plays an essential role: Storjohann's variant of the LLL lattice basis reduction algorithm and Kopparty's algorithm for computing the Hermite normal form of a matrix. To do this, we also formalize some facts about the modulo operation with symmetric range. Our implementations are based on the original papers, but are otherwise efficient. For basis reduction we formalize two versions: one that includes all of the optimizations/heuristics from Storjohann's paper, and one excluding a heuristic that we observed to often decrease efficiency. We also provide a fast, self-contained certifier for basis reduction, based on the efficient Hermite normal form algorithm.",
	"authors": [
	"Ralph Bottesch",
	"Jose Divasón",
	"René Thiemann"
	],
	"date": "2021-03-12",
	- "id": 167,
	+ "id": 170,
	"link": "/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html",
	"permalink": "/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html",
	"shortname": "Modular_arithmetic_LLL_and_HNF_algorithms",
	"title": "Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "This work contains a formalization of quantum projective measurements, also known as von Neumann measurements, which are based on elements of spectral theory. We also formalized the CHSH inequality, an inequality involving expectations in a probability space that is violated by quantum measurements, thus proving that quantum mechanics cannot be modeled with an underlying local hidden-variable theory.",
	"authors": [
	"Mnacho Echenim"
	],
	"date": "2021-03-03",
	- "id": 168,
	+ "id": 171,
	"link": "/entries/Projective_Measurements.html",
	"permalink": "/entries/Projective_Measurements.html",
	"shortname": "Projective_Measurements",
	"title": "Quantum projective measurements and the CHSH inequality",
	"topic_links": [
	"computer-science/algorithms/quantum-computing",
	"mathematics/physics/quantum-information"
	],
	"topics": [
	"Computer science/Algorithms/Quantum computing",
	"Mathematics/Physics/Quantum information"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of the Hermite-Lindemann-Weierstraß Theorem (also known as simply Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the crowning achievements of 19th century number theory.\u003c/p\u003e \u003cp\u003eThe theorem states that if $\\alpha_1, \\ldots, \\alpha_n\\in\\mathbb{C}$ are algebraic numbers that are linearly independent over $\\mathbb{Z}$, then $e^{\\alpha_1},\\ldots,e^{\\alpha_n}$ are algebraically independent over $\\mathbb{Q}$.\u003c/p\u003e \u003cp\u003eLike the \u003ca href=\"https://doi.org/10.1007/978-3-319-66107-0_5\"\u003eprevious formalisation in Coq by Bernard\u003c/a\u003e, I proceeded by formalising \u003ca href=\"https://doi.org/10.1017/CBO9780511565977\"\u003eBaker's version of the theorem and proof\u003c/a\u003e and then deriving the original one from that. Baker's version states that for any algebraic numbers $\\beta_1, \\ldots, \\beta_n\\in\\mathbb{C}$ and distinct algebraic numbers $\\alpha_i, \\ldots, \\alpha_n\\in\\mathbb{C}$, we have $\\beta_1 e^{\\alpha_1} + \\ldots + \\beta_n e^{\\alpha_n} = 0$ if and only if all the $\\beta_i$ are zero.\u003c/p\u003e \u003cp\u003eThis has a number of direct corollaries, e.g.:\u003c/p\u003e \u003cul\u003e \u003cli\u003e$e$ and $\\pi$ are transcendental\u003c/li\u003e \u003cli\u003e$e^z$, $\\sin z$, $\\tan z$, etc. are transcendental for algebraic $z\\in\\mathbb{C}\\setminus\\{0\\}$\u003c/li\u003e \u003cli\u003e$\\ln z$ is transcendental for algebraic $z\\in\\mathbb{C}\\setminus\\{0, 1\\}$\u003c/li\u003e \u003c/ul\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2021-03-03",
	- "id": 169,
	+ "id": 172,
	"link": "/entries/Hermite_Lindemann.html",
	"permalink": "/entries/Hermite_Lindemann.html",
	"shortname": "Hermite_Lindemann",
	"title": "The Hermite–Lindemann–Weierstraß Transcendence Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We use Isabelle/HOL to verify elementary theorems and alternative axiomatizations of classical extensional mereology.",
	"authors": [
	"Ben Blumson"
	],
	"date": "2021-03-01",
	- "id": 170,
	+ "id": 173,
	"link": "/entries/Mereology.html",
	"permalink": "/entries/Mereology.html",
	"shortname": "Mereology",
	"title": "Mereology",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "We formally define sunflowers and provide a formalization of the sunflower lemma of Erd\u0026odblac;s and Rado: whenever a set of size-\u003ci\u003ek\u003c/i\u003e-sets has a larger cardinality than \u003ci\u003e(r - 1)\u003csup\u003ek\u003c/sup\u003e \u0026middot; k!\u003c/i\u003e, then it contains a sunflower of cardinality \u003ci\u003er\u003c/i\u003e.",
	"authors": [
	"René Thiemann"
	],
	"date": "2021-02-25",
	- "id": 171,
	+ "id": 174,
	"link": "/entries/Sunflowers.html",
	"permalink": "/entries/Sunflowers.html",
	"shortname": "Sunflowers",
	"title": "The Sunflower Lemma of Erdős and Rado",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "In this work, we use the interactive theorem prover Isabelle/HOL to verify an imperative implementation of the classical B-tree data structure invented by Bayer and McCreight [ACM 1970]. The implementation supports set membership, insertion, deletion, iteration and range queries with efficient binary search for intra-node navigation. This is accomplished by first specifying the structure abstractly in the functional modeling language HOL and proving functional correctness. Using manual refinement, we derive an imperative implementation in Imperative/HOL. We show the validity of this refinement using the separation logic utilities from the \u003ca href=\"https://www.isa-afp.org/entries/Refine_Imperative_HOL.html\"\u003e Isabelle Refinement Framework \u003c/a\u003e . The code can be exported to the programming languages SML, OCaml and Scala. This entry contains two developments: \u003cdl\u003e \u003cdt\u003eB-Trees\u003c/dt\u003e \u003cdd\u003eThis formalisation is discussed in greater detail in the corresponding \u003ca href=\"https://mediatum.ub.tum.de/1596550\"\u003eBachelor's Thesis\u003c/a\u003e.\u003c/dd\u003e \u003cdt\u003eB+-Trees:\u003c/dt\u003e \u003cdd\u003eThis formalisation also supports range queries and is discussed in a paper published at ICTAC 2022.\u003c/dd\u003e \u003c/dl\u003e Change history: [2022-08-16]: Added formalisations of B+-Trees ",
	"authors": [
	"Niels Mündler"
	],
	"date": "2021-02-24",
	- "id": 172,
	+ "id": 175,
	"link": "/entries/BTree.html",
	"permalink": "/entries/BTree.html",
	"shortname": "BTree",
	"title": "A Verified Imperative Implementation of B-Trees",
	"topic_links": [
	"computer-science/data-management-systems",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data management systems",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eFormal Puiseux series are generalisations of formal power series and formal Laurent series that also allow for fractional exponents. They have the following general form: \\[\\sum_{i=N}^\\infty a_{i/d} X^{i/d}\\] where \u003cem\u003eN\u003c/em\u003e is an integer and \u003cem\u003ed\u003c/em\u003e is a positive integer.\u003c/p\u003e \u003cp\u003eThis entry defines these series including their basic algebraic properties. Furthermore, it proves the Newton–Puiseux Theorem, namely that the Puiseux series over an algebraically closed field of characteristic 0 are also algebraically closed.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2021-02-17",
	- "id": 173,
	+ "id": 176,
	"link": "/entries/Formal_Puiseux_Series.html",
	"permalink": "/entries/Formal_Puiseux_Series.html",
	"shortname": "Formal_Puiseux_Series",
	"title": "Formal Puiseux Series",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe Law of Large Numbers states that, informally, if one performs a random experiment $X$ many times and takes the average of the results, that average will be very close to the expected value $E[X]$.\u003c/p\u003e \u003cp\u003e More formally, let $(X_i)_{i\\in\\mathbb{N}}$ be a sequence of independently identically distributed random variables whose expected value $E[X_1]$ exists. Denote the running average of $X_1, \\ldots, X_n$ as $\\overline{X}_n$. Then:\u003c/p\u003e \u003cul\u003e \u003cli\u003eThe Weak Law of Large Numbers states that $\\overline{X}_{n} \\longrightarrow E[X_1]$ in probability for $n\\to\\infty$, i.e. $\\mathcal{P}(\|\\overline{X}_{n} - E[X_1]\| \u003e \\varepsilon) \\longrightarrow 0$ as $n\\to\\infty$ for any $\\varepsilon \u003e 0$.\u003c/li\u003e \u003cli\u003eThe Strong Law of Large Numbers states that $\\overline{X}_{n} \\longrightarrow E[X_1]$ almost surely for $n\\to\\infty$, i.e. $\\mathcal{P}(\\overline{X}_{n} \\longrightarrow E[X_1]) = 1$.\u003c/li\u003e \u003c/ul\u003e \u003cp\u003eIn this entry, I formally prove the strong law and from it the weak law. The approach used for the proof of the strong law is a particularly quick and slick one based on ergodic theory, which was formalised by Gouëzel in another AFP entry.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2021-02-10",
	- "id": 174,
	+ "id": 177,
	"link": "/entries/Laws_of_Large_Numbers.html",
	"permalink": "/entries/Laws_of_Large_Numbers.html",
	"shortname": "Laws_of_Large_Numbers",
	"title": "The Laws of Large Numbers",
	"topic_links": [
	"mathematics/probability-theory"
	],
	"topics": [
	"Mathematics/Probability theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe \u003ca href=\"https://geocoq.github.io/GeoCoq/\"\u003eGeoCoq library\u003c/a\u003e contains a formalization of geometry using the Coq proof assistant. It contains both proofs about the foundations of geometry and high-level proofs in the same style as in high school. We port a part of the GeoCoq 2.4.0 library to Isabelle/HOL: more precisely, the files Chap02.v to Chap13_3.v, suma.v as well as the associated definitions and some useful files for the demonstration of certain parallel postulates. The synthetic approach of the demonstrations is directly inspired by those contained in GeoCoq. The names of the lemmas and theorems used are kept as far as possible as well as the definitions. \u003c/p\u003e \u003cp\u003eIt should be noted that T.J.M. Makarios has done \u003ca href=\"https://www.isa-afp.org/entries/Tarskis_Geometry.html\"\u003esome proofs in Tarski's Geometry\u003c/a\u003e. It uses a definition that does not quite coincide with the definition used in Geocoq and here. Furthermore, corresponding definitions in the \u003ca href=\"https://www.isa-afp.org/entries/Poincare_Disc.html\"\u003ePoincaré Disc Model development\u003c/a\u003e are not identical to those defined in GeoCoq. \u003c/p\u003e \u003cp\u003eIn the last part, it is formalized that, in the neutral/absolute space, the axiom of the parallels of Tarski's system implies the Playfair axiom, the 5th postulate of Euclid and Euclid's original parallel postulate. These proofs, which are not constructive, are directly inspired by Pierre Boutry, Charly Gries, Julien Narboux and Pascal Schreck. \u003c/p\u003e",
	"authors": [
	"Roland Coghetto"
	],
	"date": "2021-01-31",
	- "id": 175,
	+ "id": 178,
	"link": "/entries/IsaGeoCoq.html",
	"permalink": "/entries/IsaGeoCoq.html",
	"shortname": "IsaGeoCoq",
	"title": "Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "In a \u003ca href=\"https://xkcd.com/blue_eyes.html\"\u003epuzzle published by Randall Munroe\u003c/a\u003e, perfect logicians forbidden from communicating are stranded on an island, and may only leave once they have figured out their own eye color. We present a method of modeling the behavior of perfect logicians and formalize a solution of the puzzle.",
	"authors": [
	"Maya Kądziołka"
	],
	"date": "2021-01-30",
	- "id": 176,
	+ "id": 179,
	"link": "/entries/Blue_Eyes.html",
	"permalink": "/entries/Blue_Eyes.html",
	"shortname": "Blue_Eyes",
	"title": "Solution to the xkcd Blue Eyes puzzle",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a verified implementation of a constant time queue. The original design is due to \u003ca href=\"https://doi.org/10.1016/0020-0190(81)90030-2\"\u003eHood and Melville\u003c/a\u003e. This formalization follows the presentation in \u003cem\u003ePurely Functional Data Structures\u003c/em\u003eby Okasaki.",
	"authors": [
	"Alejandro Gómez-Londoño"
	],
	"date": "2021-01-18",
	- "id": 177,
	+ "id": 180,
	"link": "/entries/Hood_Melville_Queue.html",
	"permalink": "/entries/Hood_Melville_Queue.html",
	"shortname": "Hood_Melville_Queue",
	"title": "Hood-Melville Queue",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "We extend Jinja to include static fields, methods, and instructions, and dynamic class initialization, based on the Java SE 8 specification. This includes extension of definitions and proofs. This work is partially described in Mansky and Gunter's paper at CPP 2019 and Mansky's doctoral thesis (UIUC, 2020).",
	"authors": [
	"Susannah Mansky"
	],
	"date": "2021-01-11",
	- "id": 178,
	+ "id": 181,
	"link": "/entries/JinjaDCI.html",
	"permalink": "/entries/JinjaDCI.html",
	"shortname": "JinjaDCI",
	"title": "JinjaDCI: a Java semantics with dynamic class initialization",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize the basic results on cofinality of linearly ordered sets and ordinals and Šanin’s Lemma for uncountable families of finite sets. This last result is used to prove the countable chain condition for Cohen posets. We work in the set theory framework of Isabelle/ZF, using the Axiom of Choice as needed.",
	"authors": [
	"Pedro Sánchez Terraf"
	],
	"date": "2020-12-27",
	- "id": 179,
	+ "id": 182,
	"link": "/entries/Delta_System_Lemma.html",
	"permalink": "/entries/Delta_System_Lemma.html",
	"shortname": "Delta_System_Lemma",
	"title": "Cofinality and the Delta System Lemma",
	"topic_links": [
	"mathematics/combinatorics",
	"logic/set-theory"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Logic/Set theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We introduce a generalized topological semantics for paraconsistent and paracomplete logics by drawing upon early works on topological Boolean algebras (cf. works by Kuratowski, Zarycki, McKinsey \u0026 Tarski, etc.). In particular, this work exemplarily illustrates the shallow semantical embeddings approach (\u003ca href=\"http://dx.doi.org/10.1007/s11787-012-0052-y\"\u003eSSE\u003c/a\u003e) employing the proof assistant Isabelle/HOL. By means of the SSE technique we can effectively harness theorem provers, model finders and 'hammers' for reasoning with quantified non-classical logics.",
	"authors": [
	"David Fuenmayor"
	],
	"date": "2020-12-17",
	- "id": 180,
	+ "id": 183,
	"link": "/entries/Topological_Semantics.html",
	"permalink": "/entries/Topological_Semantics.html",
	"shortname": "Topological_Semantics",
	"title": "Topological semantics for paraconsistent and paracomplete logics",
	"topic_links": [
	"logic/general-logic"
	],
	"topics": [
	"Logic/General logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We verify the correctness of Prim's, Kruskal's and Borůvka's minimum spanning tree algorithms based on algebras for aggregation and minimisation.",
	"authors": [
	"Walter Guttmann",
	"Nicolas Robinson-O'Brien"
	],
	"date": "2020-12-08",
	- "id": 181,
	+ "id": 184,
	"link": "/entries/Relational_Minimum_Spanning_Trees.html",
	"permalink": "/entries/Relational_Minimum_Spanning_Trees.html",
	"shortname": "Relational_Minimum_Spanning_Trees",
	"title": "Relational Minimum Spanning Tree Algorithms",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization builds on the \u003cem\u003eVeriComp\u003c/em\u003e entry of the \u003cem\u003eArchive of Formal Proofs\u003c/em\u003e to provide the following contributions: \u003cul\u003e \u003cli\u003ean operational semantics for a realistic virtual machine (Std) for dynamically typed programming languages;\u003c/li\u003e \u003cli\u003ethe formalization of an inline caching optimization (Inca), a proof of bisimulation with (Std), and a compilation function;\u003c/li\u003e \u003cli\u003ethe formalization of an unboxing optimization (Ubx), a proof of bisimulation with (Inca), and a simple compilation function.\u003c/li\u003e \u003c/ul\u003e This formalization was described in the CPP 2021 paper \u003cem\u003eTowards Efficient and Verified Virtual Machines for Dynamic Languages\u003c/em\u003e",
	"authors": [
	"Martin Desharnais"
	],
	"date": "2020-12-07",
	- "id": 182,
	+ "id": 185,
	"link": "/entries/Interpreter_Optimizations.html",
	"permalink": "/entries/Interpreter_Optimizations.html",
	"shortname": "Interpreter_Optimizations",
	"title": "Inline Caching and Unboxing Optimization for Interpreters",
	"topic_links": [
	"computer-science/programming-languages/misc"
	],
	"topics": [
	"Computer science/Programming languages/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "This paper introduces a new method for the formal verification of cryptographic protocols, the relational method, derived from Paulson's inductive method by means of some enhancements aimed at streamlining formal definitions and proofs, specially for protocols using public key cryptography. Moreover, this paper proposes a method to formalize a further security property, message anonymity, in addition to message confidentiality and authenticity. The relational method, including message anonymity, is then applied to the verification of a sample authentication protocol, comprising Password Authenticated Connection Establishment (PACE) with Chip Authentication Mapping followed by the explicit verification of an additional password over the PACE secure channel.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2020-12-05",
	- "id": 183,
	+ "id": 186,
	"link": "/entries/Relational_Method.html",
	"permalink": "/entries/Relational_Method.html",
	"shortname": "Relational_Method",
	"title": "The Relational Method with Message Anonymity for the Verification of Cryptographic Protocols",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is an effort to formalise some quantum algorithms and results in quantum information theory. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix representation for quantum circuits, successfully formalising the no-cloning theorem, quantum teleportation, Deutsch's algorithm, the Deutsch-Jozsa algorithm and the quantum Prisoner's Dilemma.",
	"authors": [
	"Anthony Bordg",
	"Hanna Lachnitt",
	"Yijun He"
	],
	"date": "2020-11-22",
	- "id": 184,
	+ "id": 187,
	"link": "/entries/Isabelle_Marries_Dirac.html",
	"permalink": "/entries/Isabelle_Marries_Dirac.html",
	"shortname": "Isabelle_Marries_Dirac",
	"title": "Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information",
	"topic_links": [
	"computer-science/algorithms/quantum-computing",
	"mathematics/physics/quantum-information"
	],
	"topics": [
	"Computer science/Algorithms/Quantum computing",
	"Mathematics/Physics/Quantum information"
	],
	"used_by": 1
	},
	{
	"abstract": "We use a formal development for CSP, called HOL-CSP2.0, to analyse a family of refinement notions, comprising classic and new ones. This analysis enables to derive a number of properties that allow to deepen the understanding of these notions, in particular with respect to specification decomposition principles for the case of infinite sets of events. The established relations between the refinement relations help to clarify some obscure points in the CSP literature, but also provide a weapon for shorter refinement proofs. Furthermore, we provide a framework for state-normalisation allowing to formally reason on parameterised process architectures. As a result, we have a modern environment for formal proofs of concurrent systems that allow for the combination of general infinite processes with locally finite ones in a logically safe way. We demonstrate these verification-techniques for classical, generalised examples: The CopyBuffer for arbitrary data and the Dijkstra's Dining Philosopher Problem of arbitrary size.",
	"authors": [
	"Safouan Taha",
	"Burkhart Wolff",
	"Lina Ye"
	],
	"date": "2020-11-19",
	- "id": 185,
	+ "id": 188,
	"link": "/entries/CSP_RefTK.html",
	"permalink": "/entries/CSP_RefTK.html",
	"shortname": "CSP_RefTK",
	"title": "The HOL-CSP Refinement Toolkit",
	"topic_links": [
	"computer-science/concurrency/process-calculi",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "This is an Isabelle/HOL formalisation of the semantics of the multi-valued planning tasks language that is used by the planning system Fast-Downward, the STRIPS fragment of the Planning Domain Definition Language (PDDL), and the STRIPS soundness meta-theory developed by Vladimir Lifschitz. It also contains formally verified checkers for checking the well-formedness of problems specified in either language as well the correctness of potential solutions. The formalisation in this entry was described in an earlier publication.",
	"authors": [
	"Mohammad Abdulaziz",
	"Peter Lammich"
	],
	"date": "2020-10-29",
	- "id": 186,
	+ "id": 189,
	"link": "/entries/AI_Planning_Languages_Semantics.html",
	"permalink": "/entries/AI_Planning_Languages_Semantics.html",
	"shortname": "AI_Planning_Languages_Semantics",
	"title": "AI Planning Languages Semantics",
	"topic_links": [
	"computer-science/artificial-intelligence"
	],
	"topics": [
	"Computer science/Artificial intelligence"
	],
	"used_by": 1
	},
	{
	"abstract": "We present an executable formally verified SAT encoding of classical AI planning that is based on the encodings by Kautz and Selman and the one by Rintanen et al. The encoding was experimentally tested and shown to be usable for reasonably sized standard AI planning benchmarks. We also use it as a reference to test a state-of-the-art SAT-based planner, showing that it sometimes falsely claims that problems have no solutions of certain lengths. The formalisation in this submission was described in an independent publication.",
	"authors": [
	"Mohammad Abdulaziz",
	"Friedrich Kurz"
	],
	"date": "2020-10-29",
	- "id": 187,
	+ "id": 190,
	"link": "/entries/Verified_SAT_Based_AI_Planning.html",
	"permalink": "/entries/Verified_SAT_Based_AI_Planning.html",
	"shortname": "Verified_SAT_Based_AI_Planning",
	"title": "Verified SAT-Based AI Planning",
	"topic_links": [
	"computer-science/artificial-intelligence"
	],
	"topics": [
	"Computer science/Artificial intelligence"
	],
	"used_by": 0
	},
	{
	"abstract": "The present Isabelle theory builds a formal model for both the International System of Quantities (ISQ) and the International System of Units (SI), which are both fundamental for physics and engineering. Both the ISQ and the SI are deeply integrated into Isabelle's type system. Quantities are parameterised by dimension types, which correspond to base vectors, and thus only quantities of the same dimension can be equated. Since the underlying \"algebra of quantities\" induces congruences on quantity and SI types, specific tactic support is developed to capture these. Our construction is validated by a test-set of known equivalences between both quantities and SI units. Moreover, the presented theory can be used for type-safe conversions between the SI system and others, like the British Imperial System (BIS).",
	"authors": [
	"Simon Foster",
	"Burkhart Wolff"
	],
	"date": "2020-10-20",
	- "id": 188,
	+ "id": 191,
	"link": "/entries/Physical_Quantities.html",
	"permalink": "/entries/Physical_Quantities.html",
	"shortname": "Physical_Quantities",
	"title": "A Sound Type System for Physical Quantities, Units, and Measurements",
	"topic_links": [
	"mathematics/physics",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Mathematics/Physics",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry includes useful syntactic sugar, new operators and functions, and their associated lemmas for finite maps which currently are not present in the standard Finite_Map theory.",
	"authors": [
	"Javier Díaz"
	],
	"date": "2020-10-12",
	- "id": 189,
	+ "id": 192,
	"link": "/entries/Finite-Map-Extras.html",
	"permalink": "/entries/Finite-Map-Extras.html",
	"shortname": "Finite-Map-Extras",
	"title": "Finite Map Extras",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "In this AFP entry, we extend our formalization of the core DOM with Shadow Roots. Shadow roots are a recent proposal of the web community to support a component-based development approach for client-side web applications. Shadow roots are a significant extension to the DOM standard and, as web standards are condemned to be backward compatible, such extensions often result in complex specification that may contain unwanted subtleties that can be detected by a formalization. Our Isabelle/HOL formalization is, in the sense of object-orientation, an extension of our formalization of the core DOM and enjoys the same basic properties, i.e., it is extensible, i.e., can be extended without the need of re-proving already proven properties and executable, i.e., we can generate executable code from our specification. We exploit the executability to show that our formalization complies to the official standard of the W3C, respectively, the WHATWG.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2020-09-28",
	- "id": 190,
	+ "id": 193,
	"link": "/entries/Shadow_DOM.html",
	"permalink": "/entries/Shadow_DOM.html",
	"shortname": "Shadow_DOM",
	"title": "A Formal Model of the Document Object Model with Shadow Roots",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "In this AFP entry, we extend our formalization of the safely composable DOM with Shadow Roots. This is a proposal for Shadow Roots with stricter safety guarantess than the standard compliant formalization (see \"Shadow DOM\"). Shadow Roots are a recent proposal of the web community to support a component-based development approach for client-side web applications. Shadow roots are a significant extension to the DOM standard and, as web standards are condemned to be backward compatible, such extensions often result in complex specification that may contain unwanted subtleties that can be detected by a formalization. Our Isabelle/HOL formalization is, in the sense of object-orientation, an extension of our formalization of the core DOM and enjoys the same basic properties, i.e., it is extensible, i.e., can be extended without the need of re-proving already proven properties and executable, i.e., we can generate executable code from our specification. We exploit the executability to show that our formalization complies to the official standard of the W3C, respectively, the WHATWG.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2020-09-28",
	- "id": 191,
	+ "id": 194,
	"link": "/entries/Shadow_SC_DOM.html",
	"permalink": "/entries/Shadow_SC_DOM.html",
	"shortname": "Shadow_SC_DOM",
	"title": "A Formal Model of the Safely Composable Document Object Model with Shadow Roots",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "While the (safely composable) DOM with shadow trees provide the technical basis for defining web components, it does neither defines the concept of web components nor specifies the safety properties that web components should guarantee. Consequently, the standard also does not discuss how or even if the methods for modifying the DOM respect component boundaries. In AFP entry, we present a formally verified model of safely composable web components and define safety properties which ensure that different web components can only interact with each other using well-defined interfaces. Moreover, our verification of the application programming interface (API) of the DOM revealed numerous invariants that implementations of the DOM API need to preserve to ensure the integrity of components. In comparison to the strict standard compliance formalization of Web Components in the AFP entry \"DOM_Components\", the notion of components in this entry (based on \"SC_DOM\" and \"Shadow_SC_DOM\") provides much stronger safety guarantees.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2020-09-28",
	- "id": 192,
	+ "id": 195,
	"link": "/entries/SC_DOM_Components.html",
	"permalink": "/entries/SC_DOM_Components.html",
	"shortname": "SC_DOM_Components",
	"title": "A Formalization of Safely Composable Web Components",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "While the DOM with shadow trees provide the technical basis for defining web components, the DOM standard neither defines the concept of web components nor specifies the safety properties that web components should guarantee. Consequently, the standard also does not discuss how or even if the methods for modifying the DOM respect component boundaries. In AFP entry, we present a formally verified model of web components and define safety properties which ensure that different web components can only interact with each other using well-defined interfaces. Moreover, our verification of the application programming interface (API) of the DOM revealed numerous invariants that implementations of the DOM API need to preserve to ensure the integrity of components.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2020-09-28",
	- "id": 193,
	+ "id": 196,
	"link": "/entries/DOM_Components.html",
	"permalink": "/entries/DOM_Components.html",
	"shortname": "DOM_Components",
	"title": "A Formalization of Web Components",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "In this AFP entry, we formalize the core of the Safely Composable Document Object Model (SC DOM). The SC DOM improve the standard DOM (as formalized in the AFP entry “Core DOM”) by strengthening the tree boundaries set by shadow roots: in the SC DOM, the shadow root is a sub-class of the document class (instead of a base class). This modifications also results in changes to some API methods (e.g., getOwnerDocument) to return the nearest shadow root rather than the document root. As a result, many API methods that, when called on a node inside a shadow tree, would previously “break out” and return or modify nodes that are possibly outside the shadow tree, now stay within its boundaries. This change in behavior makes programs that operate on shadow trees more predictable for the developer and allows them to make more assumptions about other code accessing the DOM.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2020-09-28",
	- "id": 194,
	+ "id": 197,
	"link": "/entries/Core_SC_DOM.html",
	"permalink": "/entries/Core_SC_DOM.html",
	"shortname": "Core_SC_DOM",
	"title": "The Safely Composable DOM",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "We present an abstract formalization of G\u0026ouml;del's incompleteness theorems. We analyze sufficient conditions for the theorems' applicability to a partially specified logic. Our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser's variation of the first theorem, Jeroslow's variation of the second theorem, and the Swierczkowski\u0026ndash;Paulson semantics-based approach. This AFP entry is the main entry point to the results described in our CADE-27 paper \u003ca href=\"https://dx.doi.org/10.1007/978-3-030-29436-6_26\"\u003eA Formally Verified Abstract Account of Gödel's Incompleteness Theorems\u003c/a\u003e. As part of our abstract formalization's validation, we instantiate our locales twice in the separate AFP entries \u003ca href=\"https://www.isa-afp.org/entries/Goedel_HFSet_Semantic.html\"\u003eGoedel_HFSet_Semantic\u003c/a\u003e and \u003ca href=\"https://www.isa-afp.org/entries/Goedel_HFSet_Semanticless.html\"\u003eGoedel_HFSet_Semanticless\u003c/a\u003e.",
	"authors": [
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2020-09-16",
	- "id": 195,
	+ "id": 198,
	"link": "/entries/Goedel_Incompleteness.html",
	"permalink": "/entries/Goedel_Incompleteness.html",
	"shortname": "Goedel_Incompleteness",
	"title": "An Abstract Formalization of G\u0026ouml;del's Incompleteness Theorems",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 2
	},
	{
	"abstract": "We validate an abstract formulation of G\u0026ouml;del's First and Second Incompleteness Theorems from a \u003ca href=\"https://www.isa-afp.org/entries/Goedel_Incompleteness.html\"\u003eseparate AFP entry\u003c/a\u003e by instantiating them to the case of \u003ci\u003efinite sound extensions of the Hereditarily Finite (HF) Set theory\u003c/i\u003e, i.e., FOL theories extending the HF Set theory with a finite set of axioms that are sound in the standard model. The concrete results had been previously formalised in an \u003ca href=\"https://www.isa-afp.org/entries/Incompleteness.html\"\u003eAFP entry by Larry Paulson\u003c/a\u003e; our instantiation reuses the infrastructure developed in that entry.",
	"authors": [
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2020-09-16",
	- "id": 196,
	+ "id": 199,
	"link": "/entries/Goedel_HFSet_Semantic.html",
	"permalink": "/entries/Goedel_HFSet_Semantic.html",
	"shortname": "Goedel_HFSet_Semantic",
	"title": "From Abstract to Concrete G\u0026ouml;del's Incompleteness Theorems\u0026mdash;Part I",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We validate an abstract formulation of G\u0026ouml;del's Second Incompleteness Theorem from a \u003ca href=\"https://www.isa-afp.org/entries/Goedel_Incompleteness.html\"\u003eseparate AFP entry\u003c/a\u003e by instantiating it to the case of \u003ci\u003efinite consistent extensions of the Hereditarily Finite (HF) Set theory\u003c/i\u003e, i.e., consistent FOL theories extending the HF Set theory with a finite set of axioms. The instantiation draws heavily on infrastructure previously developed by Larry Paulson in his \u003ca href=\"https://www.isa-afp.org/entries/Incompleteness.html\"\u003edirect formalisation of the concrete result\u003c/a\u003e. It strengthens Paulson's formalization of G\u0026ouml;del's Second from that entry by \u003ci\u003enot\u003c/i\u003e assuming soundness, and in fact not relying on any notion of model or semantic interpretation. The strengthening was obtained by first replacing some of Paulson’s semantic arguments with proofs within his HF calculus, and then plugging in some of Paulson's (modified) lemmas to instantiate our soundness-free G\u0026ouml;del's Second locale.",
	"authors": [
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2020-09-16",
	- "id": 197,
	+ "id": 200,
	"link": "/entries/Goedel_HFSet_Semanticless.html",
	"permalink": "/entries/Goedel_HFSet_Semanticless.html",
	"shortname": "Goedel_HFSet_Semanticless",
	"title": "From Abstract to Concrete G\u0026ouml;del's Incompleteness Theorems\u0026mdash;Part II",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We instantiate our syntax-independent logic infrastructure developed in \u003ca href=\"https://www.isa-afp.org/entries/Syntax_Independent_Logic.html\"\u003ea separate AFP entry\u003c/a\u003e to the FOL theory of Robinson arithmetic (also known as Q). The latter was formalised using Nominal Isabelle by adapting \u003ca href=\"https://www.isa-afp.org/entries/Incompleteness.html\"\u003eLarry Paulson’s formalization of the Hereditarily Finite Set theory\u003c/a\u003e.",
	"authors": [
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2020-09-16",
	- "id": 198,
	+ "id": 201,
	"link": "/entries/Robinson_Arithmetic.html",
	"permalink": "/entries/Robinson_Arithmetic.html",
	"shortname": "Robinson_Arithmetic",
	"title": "Robinson Arithmetic",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize a notion of logic whose terms and formulas are kept abstract. In particular, logical connectives, substitution, free variables, and provability are not defined, but characterized by their general properties as locale assumptions. Based on this abstract characterization, we develop further reusable reasoning infrastructure. For example, we define parallel substitution (along with proving its characterizing theorems) from single-point substitution. Similarly, we develop a natural deduction style proof system starting from the abstract Hilbert-style one. These one-time efforts benefit different concrete logics satisfying our locales' assumptions. We instantiate the syntax-independent logic infrastructure to Robinson arithmetic (also known as Q) in the AFP entry \u003ca href=\"https://www.isa-afp.org/entries/Robinson_Arithmetic.html\"\u003eRobinson_Arithmetic\u003c/a\u003e and to hereditarily finite set theory in the AFP entries \u003ca href=\"https://www.isa-afp.org/entries/Goedel_HFSet_Semantic.html\"\u003eGoedel_HFSet_Semantic\u003c/a\u003e and \u003ca href=\"https://www.isa-afp.org/entries/Goedel_HFSet_Semanticless.html\"\u003eGoedel_HFSet_Semanticless\u003c/a\u003e, which are part of our formalization of G\u0026ouml;del's Incompleteness Theorems described in our CADE-27 paper \u003ca href=\"https://dx.doi.org/10.1007/978-3-030-29436-6_26\"\u003eA Formally Verified Abstract Account of Gödel's Incompleteness Theorems\u003c/a\u003e.",
	"authors": [
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2020-09-16",
	- "id": 199,
	+ "id": 202,
	"link": "/entries/Syntax_Independent_Logic.html",
	"permalink": "/entries/Syntax_Independent_Logic.html",
	"shortname": "Syntax_Independent_Logic",
	"title": "Syntax-Independent Logic Infrastructure",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 2
	},
	{
	"abstract": "In this AFP entry, we provide a formalisation of extended finite state machines (EFSMs) where models are represented as finite sets of transitions between states. EFSMs execute traces to produce observable outputs. We also define various simulation and equality metrics for EFSMs in terms of traces and prove their strengths in relation to each other. Another key contribution is a framework of function definitions such that LTL properties can be phrased over EFSMs. Finally, we provide a simple example case study in the form of a drinks machine.",
	"authors": [
	"Michael Foster",
	"Achim D. Brucker",
	"Ramsay G. Taylor",
	"John Derrick"
	],
	"date": "2020-09-07",
	- "id": 200,
	+ "id": 203,
	"link": "/entries/Extended_Finite_State_Machines.html",
	"permalink": "/entries/Extended_Finite_State_Machines.html",
	"shortname": "Extended_Finite_State_Machines",
	"title": "A Formal Model of Extended Finite State Machines",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "In this AFP entry, we provide a formal implementation of a state-merging technique to infer extended finite state machines (EFSMs), complete with output and update functions, from black-box traces. In particular, we define the subsumption in context relation as a means of determining whether one transition is able to account for the behaviour of another. Building on this, we define the direct subsumption relation, which lifts the subsumption in context relation to EFSM level such that we can use it to determine whether it is safe to merge a given pair of transitions. Key proofs include the conditions necessary for subsumption to occur and that subsumption and direct subsumption are preorder relations. We also provide a number of different heuristics which can be used to abstract away concrete values into registers so that more states and transitions can be merged and provide proofs of the various conditions which must hold for these abstractions to subsume their ungeneralised counterparts. A Code Generator setup to create executable Scala code is also defined.",
	"authors": [
	"Michael Foster",
	"Achim D. Brucker",
	"Ramsay G. Taylor",
	"John Derrick"
	],
	"date": "2020-09-07",
	- "id": 201,
	+ "id": 204,
	"link": "/entries/Extended_Finite_State_Machine_Inference.html",
	"permalink": "/entries/Extended_Finite_State_Machine_Inference.html",
	"shortname": "Extended_Finite_State_Machine_Inference",
	"title": "Inference of Extended Finite State Machines",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Generating and checking proof certificates is important to increase the trust in automated reasoning tools. In recent years formal verification using computer algebra became more important and is heavily used in automated circuit verification. An existing proof format which covers algebraic reasoning and allows efficient proof checking is the practical algebraic calculus (PAC). In this development, we present the verified checker Pastèque that is obtained by synthesis via the Refinement Framework. This is the formalization going with our FMCAD'20 tool presentation.",
	"authors": [
	"Mathias Fleury",
	"Daniela Kaufmann"
	],
	"date": "2020-08-31",
	- "id": 202,
	+ "id": 205,
	"link": "/entries/PAC_Checker.html",
	"permalink": "/entries/PAC_Checker.html",
	"shortname": "PAC_Checker",
	"title": "Practical Algebraic Calculus Checker",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e This entry formalizes some classical concepts and results from inductive inference of recursive functions. In the basic setting a partial recursive function (\"strategy\") must identify (\"learn\") all functions from a set (\"class\") of recursive functions. To that end the strategy receives more and more values $f(0), f(1), f(2), \\ldots$ of some function $f$ from the given class and in turn outputs descriptions of partial recursive functions, for example, Gödel numbers. The strategy is considered successful if the sequence of outputs (\"hypotheses\") converges to a description of $f$. A class of functions learnable in this sense is called \"learnable in the limit\". The set of all these classes is denoted by LIM. \u003c/p\u003e \u003cp\u003e Other types of inference considered are finite learning (FIN), behaviorally correct learning in the limit (BC), and some variants of LIM with restrictions on the hypotheses: total learning (TOTAL), consistent learning (CONS), and class-preserving learning (CP). The main results formalized are the proper inclusions $\\mathrm{FIN} \\subset \\mathrm{CP} \\subset \\mathrm{TOTAL} \\subset \\mathrm{CONS} \\subset \\mathrm{LIM} \\subset \\mathrm{BC} \\subset 2^{\\mathcal{R}}$, where $\\mathcal{R}$ is the set of all total recursive functions. Further results show that for all these inference types except CONS, strategies can be assumed to be total recursive functions; that all inference types but CP are closed under the subset relation between classes; and that no inference type is closed under the union of classes. \u003c/p\u003e \u003cp\u003e The above is based on a formalization of recursive functions heavily inspired by the \u003ca href=\"https://www.isa-afp.org/entries/Universal_Turing_Machine.html\"\u003eUniversal Turing Machine\u003c/a\u003e entry by Xu et al., but different in that it models partial functions with codomain \u003cem\u003enat option\u003c/em\u003e. The formalization contains a construction of a universal partial recursive function, without resorting to Turing machines, introduces decidability and recursive enumerability, and proves some standard results: existence of a Kleene normal form, the \u003cem\u003es-m-n\u003c/em\u003e theorem, Rice's theorem, and assorted fixed-point theorems (recursion theorems) by Kleene, Rogers, and Smullyan. \u003c/p\u003e",
	"authors": [
	"Frank J. Balbach"
	],
	"date": "2020-08-31",
	- "id": 203,
	+ "id": 206,
	"link": "/entries/Inductive_Inference.html",
	"permalink": "/entries/Inductive_Inference.html",
	"shortname": "Inductive_Inference",
	"title": "Some classical results in inductive inference of recursive functions",
	"topic_links": [
	"logic/computability",
	"computer-science/machine-learning"
	],
	"topics": [
	"Logic/Computability",
	"Computer science/Machine learning"
	],
	"used_by": 0
	},
	{
	"abstract": "We give a simple relation-algebraic semantics of read and write operations on associative arrays. The array operations seamlessly integrate with assignments in the Hoare-logic library. Using relation algebras and Kleene algebras we verify the correctness of an array-based implementation of disjoint-set forests with a naive union operation and a find operation with path compression.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2020-08-26",
	- "id": 204,
	+ "id": 207,
	"link": "/entries/Relational_Disjoint_Set_Forests.html",
	"permalink": "/entries/Relational_Disjoint_Set_Forests.html",
	"shortname": "Relational_Disjoint_Set_Forests",
	"title": "Relational Disjoint-Set Forests",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "This Isabelle/HOL formalization extends the AFP entry \u003cem\u003eSaturation_Framework\u003c/em\u003e with the following contributions: \u003cul\u003e \u003cli\u003ean application of the framework to prove Bachmair and Ganzinger's resolution prover RP refutationally complete, which was formalized in a more ad hoc fashion by Schlichtkrull et al. in the AFP entry \u003cem\u003eOrdered_Resultion_Prover\u003c/em\u003e;\u003c/li\u003e \u003cli\u003egeneralizations of various basic concepts formalized by Schlichtkrull et al., which were needed to verify RP and could be useful to formalize other calculi, such as superposition;\u003c/li\u003e \u003cli\u003ealternative proofs of fairness (and hence saturation and ultimately refutational completeness) for the given clause procedures GC and LGC, based on invariance.\u003c/li\u003e \u003c/ul\u003e",
	"authors": [
	"Jasmin Christian Blanchette",
	"Sophie Tourret"
	],
	"date": "2020-08-25",
	- "id": 205,
	+ "id": 208,
	"link": "/entries/Saturation_Framework_Extensions.html",
	"permalink": "/entries/Saturation_Framework_Extensions.html",
	"shortname": "Saturation_Framework_Extensions",
	"title": "Extensions to the Comprehensive Framework for Saturation Theorem Proving",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 1
	},
	{
	"abstract": "Richard Bird and collaborators have proposed a derivation of an intricate cyclic program that implements the Morris-Pratt string matching algorithm. Here we provide a proof of total correctness for Bird's derivation and complete it by adding Knuth's optimisation.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2020-08-25",
	- "id": 206,
	+ "id": 209,
	"link": "/entries/BirdKMP.html",
	"permalink": "/entries/BirdKMP.html",
	"shortname": "BirdKMP",
	"title": "Putting the `K' into Bird's derivation of Knuth-Morris-Pratt string matching",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a formalisation of Amicable Numbers, involving some relevant material including Euler's sigma function, some relevant definitions, results and examples as well as rules such as Th\u0026#257;bit ibn Qurra's Rule, Euler's Rule, te Riele's Rule and Borho's Rule with breeders.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki"
	],
	"date": "2020-08-04",
	- "id": 207,
	+ "id": 210,
	"link": "/entries/Amicable_Numbers.html",
	"permalink": "/entries/Amicable_Numbers.html",
	"shortname": "Amicable_Numbers",
	"title": "Amicable Numbers",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The theory of partition relations concerns generalisations of Ramsey's theorem. For any ordinal $\\alpha$, write $\\alpha \\to (\\alpha, m)^2$ if for each function $f$ from unordered pairs of elements of $\\alpha$ into $\\{0,1\\}$, either there is a subset $X\\subseteq \\alpha$ order-isomorphic to $\\alpha$ such that $f\\{x,y\\}=0$ for all $\\{x,y\\}\\subseteq X$, or there is an $m$ element set $Y\\subseteq \\alpha$ such that $f\\{x,y\\}=1$ for all $\\{x,y\\}\\subseteq Y$. (In both cases, with $\\{x,y\\}$ we require $x\\not=y$.) In particular, the infinite Ramsey theorem can be written in this notation as $\\omega \\to (\\omega, \\omega)^2$, or if we restrict $m$ to the positive integers as above, then $\\omega \\to (\\omega, m)^2$ for all $m$. This entry formalises Larson's proof of $\\omega^\\omega \\to (\\omega^\\omega, m)^2$ along with a similar proof of a result due to Specker: $\\omega^2 \\to (\\omega^2, m)^2$. Also proved is a necessary result by Erdős and Milner: $\\omega^{1+\\alpha\\cdot n} \\to (\\omega^{1+\\alpha}, 2^n)^2$.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2020-08-03",
	- "id": 208,
	+ "id": 211,
	"link": "/entries/Ordinal_Partitions.html",
	"permalink": "/entries/Ordinal_Partitions.html",
	"shortname": "Ordinal_Partitions",
	"title": "Ordinal Partitions",
	"topic_links": [
	"mathematics/combinatorics",
	"logic/set-theory"
	],
	"topics": [
	"Mathematics/Combinatorics",
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a suitable distributed system model and implementation of the Chandy--Lamport distributed snapshot algorithm [ACM Transactions on Computer Systems, 3, 63-75, 1985]. Our main result is a formal termination and correctness proof of the Chandy--Lamport algorithm and its use in stable property detection.",
	"authors": [
	"Ben Fiedler",
	"Dmitriy Traytel"
	],
	"date": "2020-07-21",
	- "id": 209,
	+ "id": 212,
	"link": "/entries/Chandy_Lamport.html",
	"permalink": "/entries/Chandy_Lamport.html",
	"shortname": "Chandy_Lamport",
	"title": "A Formal Proof of The Chandy--Lamport Distributed Snapshot Algorithm",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in many areas of mathematics and computing, researchers usually fall back to point-wise reasoning when it comes to arguments about paths in a graph. We present a purely algebraic way to specify different kinds of paths in Kleene relation algebras, which are relation algebras equipped with an operation for reflexive transitive closure. We study the relationship between paths with a designated root vertex and paths without such a vertex. Since we stay in first-order logic this development helps with mechanising proofs. To demonstrate the applicability of the algebraic framework we verify the correctness of three basic graph algorithms.",
	"authors": [
	"Walter Guttmann",
	"Peter Höfner"
	],
	"date": "2020-07-13",
	- "id": 210,
	+ "id": 213,
	"link": "/entries/Relational_Paths.html",
	"permalink": "/entries/Relational_Paths.html",
	"shortname": "Relational_Paths",
	"title": "Relational Characterisations of Paths",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The Vienna Convention on Road Traffic defines the safe distance traffic rules informally. This could make autonomous vehicle liable for safe-distance-related accidents because there is no clear definition of how large a safe distance is. We provide a formally proven prescriptive definition of a safe distance, and checkers which can decide whether an autonomous vehicle is obeying the safe distance rule. Not only does our work apply to the domain of law, but it also serves as a specification for autonomous vehicle manufacturers and for online verification of path planners.",
	"authors": [
	"Albert Rizaldi",
	"Fabian Immler"
	],
	"date": "2020-06-01",
	- "id": 211,
	+ "id": 214,
	"link": "/entries/Safe_Distance.html",
	"permalink": "/entries/Safe_Distance.html",
	"shortname": "Safe_Distance",
	"title": "A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/physics"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Physics"
	],
	"used_by": 0
	},
	{
	"abstract": "This work presents a formal proof in Isabelle/HOL of an algorithm to transform a matrix into its Smith normal form, a canonical matrix form, in a general setting: the algorithm is parameterized by operations to prove its existence over elementary divisor rings, while execution is guaranteed over Euclidean domains. We also provide a formal proof on some results about the generality of this algorithm as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out switching conveniently between two different existing libraries: the Hermite normal form (based on HOL Analysis) and the Jordan normal form AFP entries. This permits to reuse results from both developments and it is done by means of the lifting and transfer package together with the use of local type definitions.",
	"authors": [
	"Jose Divasón"
	],
	"date": "2020-05-23",
	- "id": 212,
	+ "id": 215,
	"link": "/entries/Smith_Normal_Form.html",
	"permalink": "/entries/Smith_Normal_Form.html",
	"shortname": "Smith_Normal_Form",
	"title": "A verified algorithm for computing the Smith normal form of a matrix",
	"topic_links": [
	"mathematics/algebra",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Algebra",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 1
	},
	{
	"abstract": "In 1965, Nash-Williams discovered a generalisation of the infinite form of Ramsey's theorem. Where the latter concerns infinite sets of n-element sets for some fixed n, the Nash-Williams theorem concerns infinite sets of finite sets (or lists) subject to a “no initial segment” condition. The present formalisation follows a monograph on Ramsey Spaces by Todorčević.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2020-05-16",
	- "id": 213,
	+ "id": 216,
	"link": "/entries/Nash_Williams.html",
	"permalink": "/entries/Nash_Williams.html",
	"shortname": "Nash_Williams",
	"title": "The Nash-Williams Partition Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "We define a generalized version of Knuth\u0026ndash;Bendix orders, including subterm coefficient functions. For these orders we formalize several properties such as strong normalization, the subterm property, closure properties under substitutions and contexts, as well as ground totality.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2020-05-13",
	- "id": 214,
	+ "id": 217,
	"link": "/entries/Knuth_Bendix_Order.html",
	"permalink": "/entries/Knuth_Bendix_Order.html",
	"shortname": "Knuth_Bendix_Order",
	"title": "A Formalization of Knuth–Bendix Orders",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 3
	},
	{
	"abstract": "We formalise certain irrationality criteria for infinite series of the form: \\[\\sum_{n=1}^\\infty \\frac{b_n}{\\prod_{i=1}^n a_i} \\] where $\\{b_n\\}$ is a sequence of integers and $\\{a_n\\}$ a sequence of positive integers with $a_n \u003e1$ for all large n. The results are due to P. Erdős and E. G. Straus \u003ca href=\"https://projecteuclid.org/euclid.pjm/1102911140\"\u003e[1]\u003c/a\u003e. In particular, we formalise Theorem 2.1, Corollary 2.10 and Theorem 3.1. The latter is an application of Theorem 2.1 involving the prime numbers.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Wenda Li"
	],
	"date": "2020-05-12",
	- "id": 215,
	+ "id": 218,
	"link": "/entries/Irrational_Series_Erdos_Straus.html",
	"permalink": "/entries/Irrational_Series_Erdos_Straus.html",
	"shortname": "Irrational_Series_Erdos_Straus",
	"title": "Irrationality Criteria for Series by Erdős and Straus",
	"topic_links": [
	"mathematics/number-theory",
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Number theory",
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "This document contains a proof of the recursion theorem. This is a mechanization of the proof of the recursion theorem from the text \u003ci\u003eIntroduction to Set Theory\u003c/i\u003e, by Karel Hrbacek and Thomas Jech. This implementation may be used as the basis for a model of Peano arithmetic in ZF. While recursion and the natural numbers are already available in Isabelle/ZF, this clean development is much easier to follow.",
	"authors": [
	"Georgy Dunaev"
	],
	"date": "2020-05-11",
	- "id": 216,
	+ "id": 219,
	"link": "/entries/Recursion-Addition.html",
	"permalink": "/entries/Recursion-Addition.html",
	"shortname": "Recursion-Addition",
	"title": "Recursion Theorem in ZF",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\\bigwedge_{i=1}^n \\mathbf{G}\\mathbf{F} \\varphi_i \\vee \\mathbf{F}\\mathbf{G} \\psi_i$, where $\\varphi_i$ and $\\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present an executable formalisation of a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up.",
	"authors": [
	"Salomon Sickert"
	],
	"date": "2020-05-08",
	- "id": 217,
	+ "id": 220,
	"link": "/entries/LTL_Normal_Form.html",
	"permalink": "/entries/LTL_Normal_Form.html",
	"shortname": "LTL_Normal_Form",
	"title": "An Efficient Normalisation Procedure for Linear Temporal Logic: Isabelle/HOL Formalisation",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/general-logic/temporal-logic"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/General logic/Temporal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies ZFC.",
	"authors": [
	"Emmanuel Gunther",
	"Miguel Pagano",
	"Pedro Sánchez Terraf"
	],
	"date": "2020-05-06",
	- "id": 218,
	+ "id": 221,
	"link": "/entries/Forcing.html",
	"permalink": "/entries/Forcing.html",
	"shortname": "Forcing",
	"title": "Formalization of Forcing in Isabelle/ZF",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize in Isabelle/HOL a result due to S. Banach and H. Steinhaus known as the Banach-Steinhaus theorem or Uniform boundedness principle: a pointwise-bounded family of continuous linear operators from a Banach space to a normed space is uniformly bounded. Our approach is an adaptation to Isabelle/HOL of a proof due to A. Sokal.",
	"authors": [
	"Dominique Unruh",
	"José Manuel Rodríguez Caballero"
	],
	"date": "2020-05-02",
	- "id": 219,
	+ "id": 222,
	"link": "/entries/Banach_Steinhaus.html",
	"permalink": "/entries/Banach_Steinhaus.html",
	"shortname": "Banach_Steinhaus",
	"title": "Banach-Steinhaus Theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "In this article, we present a proof theory for Attack Trees. Attack Trees are a well established and useful model for the construction of attacks on systems since they allow a stepwise exploration of high level attacks in application scenarios. Using the expressiveness of Higher Order Logic in Isabelle, we develop a generic theory of Attack Trees with a state-based semantics based on Kripke structures and CTL. The resulting framework allows mechanically supported logic analysis of the meta-theory of the proof calculus of Attack Trees and at the same time the developed proof theory enables application to case studies. A central correctness and completeness result proved in Isabelle establishes a connection between the notion of Attack Tree validity and CTL. The application is illustrated on the example of a healthcare IoT system and GDPR compliance verification.",
	"authors": [
	"Florian Kammüller"
	],
	"date": "2020-04-27",
	- "id": 220,
	+ "id": 223,
	"link": "/entries/Attack_Trees.html",
	"permalink": "/entries/Attack_Trees.html",
	"shortname": "Attack_Trees",
	"title": "Attack Trees in Isabelle for GDPR compliance of IoT healthcare systems",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe Gaussian integers are the subring \u0026#8484;[i] of the complex numbers, i. e. the ring of all complex numbers with integral real and imaginary part. This article provides a definition of this ring as well as proofs of various basic properties, such as that they form a Euclidean ring and a full classification of their primes. An executable (albeit not very efficient) factorisation algorithm is also provided.\u003c/p\u003e \u003cp\u003eLastly, this Gaussian integer formalisation is used in two short applications:\u003c/p\u003e \u003col\u003e \u003cli\u003e The characterisation of all positive integers that can be written as sums of two squares\u003c/li\u003e \u003cli\u003e Euclid's formula for primitive Pythagorean triples\u003c/li\u003e \u003c/ol\u003e \u003cp\u003eWhile elementary proofs for both of these are already available in the AFP, the theory of Gaussian integers provides more concise proofs and a more high-level view.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2020-04-24",
	- "id": 221,
	+ "id": 224,
	"link": "/entries/Gaussian_Integers.html",
	"permalink": "/entries/Gaussian_Integers.html",
	"shortname": "Gaussian_Integers",
	"title": "Gaussian Integers",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of the symmetric multivariate polynomials known as \u003cem\u003epower sum polynomials\u003c/em\u003e. These are of the form p\u003csub\u003en\u003c/sub\u003e(\u003cem\u003eX\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\u0026hellip;, \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e) = \u003cem\u003eX\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003csup\u003en\u003c/sup\u003e + \u0026hellip; + X\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e\u003csup\u003en\u003c/sup\u003e. A formal proof of the Girard–Newton Theorem is also given. This theorem relates the power sum polynomials to the elementary symmetric polynomials s\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e in the form of a recurrence relation (-1)\u003csup\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sup\u003e \u003cem\u003ek\u003c/em\u003e s\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e = \u0026sum;\u003csub\u003ei\u0026isinv;[0,\u003cem\u003ek\u003c/em\u003e)\u003c/sub\u003e (-1)\u003csup\u003ei\u003c/sup\u003e s\u003csub\u003ei\u003c/sub\u003e p\u003csub\u003e\u003cem\u003ek\u003c/em\u003e-\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;.\u003c/p\u003e \u003cp\u003eAs an application, this is then used to solve a generalised form of a puzzle given as an exercise in Dummit and Foote's \u003cem\u003eAbstract Algebra\u003c/em\u003e: For \u003cem\u003ek\u003c/em\u003e complex unknowns \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e, define p\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e := \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003csup\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sup\u003e + \u0026hellip; + \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sup\u003e. Then for each vector \u003cem\u003ea\u003c/em\u003e \u0026isinv; \u0026#x2102;\u003csup\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sup\u003e, show that there is exactly one solution to the system p\u003csub\u003e1\u003c/sub\u003e = a\u003csub\u003e1\u003c/sub\u003e, \u0026hellip;, p\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e = a\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e up to permutation of the \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e and determine the value of p\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e for i\u0026gt;k.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2020-04-24",
	- "id": 222,
	+ "id": 225,
	"link": "/entries/Power_Sum_Polynomials.html",
	"permalink": "/entries/Power_Sum_Polynomials.html",
	"shortname": "Power_Sum_Polynomials",
	"title": "Power Sum Polynomials",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThe Lambert \u003cem\u003eW\u003c/em\u003e function is a multi-valued function defined as the inverse function of \u003cem\u003ex\u003c/em\u003e \u0026#x21A6; \u003cem\u003ex\u003c/em\u003e e\u003csup\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sup\u003e. Besides numerous applications in combinatorics, physics, and engineering, it also frequently occurs when solving equations containing both e\u003csup\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sup\u003e and \u003cem\u003ex\u003c/em\u003e, or both \u003cem\u003ex\u003c/em\u003e and log \u003cem\u003ex\u003c/em\u003e.\u003c/p\u003e \u003cp\u003eThis article provides a definition of the two real-valued branches \u003cem\u003eW\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e(\u003cem\u003ex\u003c/em\u003e) and \u003cem\u003eW\u003c/em\u003e\u003csub\u003e-1\u003c/sub\u003e(\u003cem\u003ex\u003c/em\u003e) and proves various properties such as basic identities and inequalities, monotonicity, differentiability, asymptotic expansions, and the MacLaurin series of \u003cem\u003eW\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e(\u003cem\u003ex\u003c/em\u003e) at \u003cem\u003ex\u003c/em\u003e = 0.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2020-04-24",
	- "id": 223,
	+ "id": 226,
	"link": "/entries/Lambert_W.html",
	"permalink": "/entries/Lambert_W.html",
	"shortname": "Lambert_W",
	"title": "The Lambert W Function on the Reals",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Our theories formalise various matrix properties that serve to establish existence, uniqueness and characterisation of the solution to affine systems of ordinary differential equations (ODEs). In particular, we formalise the operator and maximum norm of matrices. Then we use them to prove that square matrices form a Banach space, and in this setting, we show an instance of Picard-Lindelöf’s theorem for affine systems of ODEs. Finally, we use this formalisation to verify three simple hybrid programs.",
	"authors": [
	"Jonathan Julian Huerta y Munive"
	],
	"date": "2020-04-19",
	- "id": 224,
	+ "id": 227,
	"link": "/entries/Matrices_for_ODEs.html",
	"permalink": "/entries/Matrices_for_ODEs.html",
	"shortname": "Matrices_for_ODEs",
	"title": "Matrices for ODEs",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "Authenticated data structures allow several systems to convince each other that they are referring to the same data structure, even if each of them knows only a part of the data structure. Using inclusion proofs, knowledgeable systems can selectively share their knowledge with other systems and the latter can verify the authenticity of what is being shared. In this article, we show how to modularly define authenticated data structures, their inclusion proofs, and operations thereon as datatypes in Isabelle/HOL, using a shallow embedding. Modularity allows us to construct complicated trees from reusable building blocks, which we call Merkle functors. Merkle functors include sums, products, and function spaces and are closed under composition and least fixpoints. As a practical application, we model the hierarchical transactions of \u003ca href=\"https://www.canton.io\"\u003eCanton\u003c/a\u003e, a practical interoperability protocol for distributed ledgers, as authenticated data structures. This is a first step towards formalizing the Canton protocol and verifying its integrity and security guarantees.",
	"authors": [
	"Andreas Lochbihler",
	"Ognjen Marić"
	],
	"date": "2020-04-16",
	- "id": 225,
	+ "id": 228,
	"link": "/entries/ADS_Functor.html",
	"permalink": "/entries/ADS_Functor.html",
	"shortname": "ADS_Functor",
	"title": "Authenticated Data Structures As Functors",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Basin et al.'s \u003ca href=\"https://doi.org/10.1016/j.ipl.2014.09.009\"\u003esliding window algorithm (SWA)\u003c/a\u003e is an algorithm for combining the elements of subsequences of a sequence with an associative operator. It is greedy and minimizes the number of operator applications. We formalize the algorithm and verify its functional correctness. We extend the algorithm with additional operations and provide an alternative interface to the slide operation that does not require the entire input sequence.",
	"authors": [
	"Lukas Heimes",
	"Dmitriy Traytel",
	"Joshua Schneider"
	],
	"date": "2020-04-10",
	- "id": 226,
	+ "id": 229,
	"link": "/entries/Sliding_Window_Algorithm.html",
	"permalink": "/entries/Sliding_Window_Algorithm.html",
	"shortname": "Sliding_Window_Algorithm",
	"title": "Formalization of an Algorithm for Greedily Computing Associative Aggregations on Sliding Windows",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization is the companion of the technical report “A comprehensive framework for saturation theorem proving”, itself companion of the eponym IJCAR 2020 paper, written by Uwe Waldmann, Sophie Tourret, Simon Robillard and Jasmin Blanchette. It verifies a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition, and allows to model entire prover architectures in such a way that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus using a variant of the given clause loop. The technical report “A comprehensive framework for saturation theorem proving” is available \u003ca href=\"http://matryoshka.gforge.inria.fr/pubs/satur_report.pdf\"\u003eon the Matryoshka website\u003c/a\u003e. The names of the Isabelle lemmas and theorems corresponding to the results in the report are indicated in the margin of the report.",
	"authors": [
	"Sophie Tourret"
	],
	"date": "2020-04-09",
	- "id": 227,
	+ "id": 230,
	"link": "/entries/Saturation_Framework.html",
	"permalink": "/entries/Saturation_Framework.html",
	"shortname": "Saturation_Framework",
	"title": "A Comprehensive Framework for Saturation Theorem Proving",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 3
	},
	{
	"abstract": "A monitor is a runtime verification tool that solves the following problem: Given a stream of time-stamped events and a policy formulated in a specification language, decide whether the policy is satisfied at every point in the stream. We verify the correctness of an executable monitor for specifications given as formulas in metric first-order dynamic logic (MFODL), which combines the features of metric first-order temporal logic (MFOTL) and metric dynamic logic. Thus, MFODL supports real-time constraints, first-order parameters, and regular expressions. Additionally, the monitor supports aggregation operations such as count and sum. This formalization, which is described in a \u003ca href=\"http://people.inf.ethz.ch/trayteld/papers/ijcar20-verimonplus/verimonplus.pdf\"\u003e forthcoming paper at IJCAR 2020\u003c/a\u003e, significantly extends \u003ca href=\"https://www.isa-afp.org/entries/MFOTL_Monitor.html\"\u003eprevious work on a verified monitor\u003c/a\u003e for MFOTL. Apart from the addition of regular expressions and aggregations, we implemented \u003ca href=\"https://www.isa-afp.org/entries/Generic_Join.html\"\u003emulti-way joins\u003c/a\u003e and a specialized sliding window algorithm to further optimize the monitor.",
	"authors": [
	"Thibault Dardinier",
	"Lukas Heimes",
	"Martin Raszyk",
	"Joshua Schneider",
	"Dmitriy Traytel"
	],
	"date": "2020-04-09",
	- "id": 228,
	+ "id": 231,
	"link": "/entries/MFODL_Monitor_Optimized.html",
	"permalink": "/entries/MFODL_Monitor_Optimized.html",
	"shortname": "MFODL_Monitor_Optimized",
	"title": "Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations",
	"topic_links": [
	"computer-science/algorithms",
	"logic/general-logic/modal-logic",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Algorithms",
	"Logic/General logic/Modal logic",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "In protocol verification we observe a wide spectrum from fully automated methods to interactive theorem proving with proof assistants like Isabelle/HOL. In this AFP entry, we present a fully-automated approach for verifying stateful security protocols, i.e., protocols with mutable state that may span several sessions. The approach supports reachability goals like secrecy and authentication. We also include a simple user-friendly transaction-based protocol specification language that is embedded into Isabelle.",
	"authors": [
	"Andreas V. Hess",
	"Sebastian Mödersheim",
	"Achim D. Brucker",
	"Anders Schlichtkrull"
	],
	"date": "2020-04-08",
	- "id": 229,
	+ "id": 232,
	"link": "/entries/Automated_Stateful_Protocol_Verification.html",
	"permalink": "/entries/Automated_Stateful_Protocol_Verification.html",
	"shortname": "Automated_Stateful_Protocol_Verification",
	"title": "Automated Stateful Protocol Verification",
	"topic_links": [
	"computer-science/security",
	"tools"
	],
	"topics": [
	"Computer science/Security",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide in this AFP entry several relative soundness results for security protocols. In particular, we prove typing and compositionality results for stateful protocols (i.e., protocols with mutable state that may span several sessions), and that focuses on reachability properties. Such results are useful to simplify protocol verification by reducing it to a simpler problem: Typing results give conditions under which it is safe to verify a protocol in a typed model where only \"well-typed\" attacks can occur whereas compositionality results allow us to verify a composed protocol by only verifying the component protocols in isolation. The conditions on the protocols under which the results hold are furthermore syntactic in nature allowing for full automation. The foundation presented here is used in another entry to provide fully automated and formalized security proofs of stateful protocols.",
	"authors": [
	"Andreas V. Hess",
	"Sebastian Mödersheim",
	"Achim D. Brucker"
	],
	"date": "2020-04-08",
	- "id": 230,
	+ "id": 233,
	"link": "/entries/Stateful_Protocol_Composition_and_Typing.html",
	"permalink": "/entries/Stateful_Protocol_Composition_and_Typing.html",
	"shortname": "Stateful_Protocol_Composition_and_Typing",
	"title": "Stateful Protocol Composition and Typing",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 1
	},
	{
	"abstract": "This work presents a formalisation of a generating function proof for Lucas's theorem. We first outline extensions to the existing Formal Power Series (FPS) library, including an equivalence relation for coefficients modulo \u003cem\u003en\u003c/em\u003e, an alternate binomial theorem statement, and a formalised proof of the Freshman's dream (mod \u003cem\u003ep\u003c/em\u003e) lemma. The second part of the work presents the formal proof of Lucas's Theorem. Working backwards, the formalisation first proves a well known corollary of the theorem which is easier to formalise, and then applies induction to prove the original theorem statement. The proof of the corollary aims to provide a good example of a formalised generating function equivalence proof using the FPS library. The final theorem statement is intended to be integrated into the formalised proof of Hilbert's 10th Problem.",
	"authors": [
	"Chelsea Edmonds"
	],
	"date": "2020-04-07",
	- "id": 231,
	+ "id": 234,
	"link": "/entries/Lucas_Theorem.html",
	"permalink": "/entries/Lucas_Theorem.html",
	"shortname": "Lucas_Theorem",
	"title": "Lucas's Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "Commutative Replicated Data Types (CRDTs) are a promising new class of data structures for large-scale shared mutable content in applications that only require eventual consistency. The WithOut Operational Transforms (WOOT) framework is a CRDT for collaborative text editing introduced by Oster et al. (CSCW 2006) for which the eventual consistency property was verified only for a bounded model to date. We contribute a formal proof for WOOTs strong eventual consistency.",
	"authors": [
	"Emin Karayel",
	"Edgar Gonzàlez"
	],
	"date": "2020-03-25",
	- "id": 232,
	+ "id": 235,
	"link": "/entries/WOOT_Strong_Eventual_Consistency.html",
	"permalink": "/entries/WOOT_Strong_Eventual_Consistency.html",
	"shortname": "WOOT_Strong_Eventual_Consistency",
	"title": "Strong Eventual Consistency of the Collaborative Editing Framework WOOT",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article gives a formal version of Furstenberg's topological proof of the infinitude of primes. He defines a topology on the integers based on arithmetic progressions (or, equivalently, residue classes). Using some fairly obvious properties of this topology, the infinitude of primes is then easily obtained.\u003c/p\u003e \u003cp\u003eApart from this, this topology is also fairly ‘nice’ in general: it is second countable, metrizable, and perfect. All of these (well-known) facts are formally proven, including an explicit metric for the topology given by Zulfeqarr.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2020-03-22",
	- "id": 233,
	+ "id": 236,
	"link": "/entries/Furstenberg_Topology.html",
	"permalink": "/entries/Furstenberg_Topology.html",
	"shortname": "Furstenberg_Topology",
	"title": "Furstenberg's topology and his proof of the infinitude of primes",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Recently, authors have proposed under-approximate logics for reasoning about programs. So far, all such logics have been confined to reasoning about individual program behaviours. Yet there exist many over-approximate relational logics for reasoning about pairs of programs and relating their behaviours. We present the first under-approximate relational logic, for the simple imperative language IMP. We prove our logic is both sound and complete. Additionally, we show how reasoning in this logic can be decomposed into non-relational reasoning in an under-approximate Hoare logic, mirroring Beringer’s result for over-approximate relational logics. We illustrate the application of our logic on some small examples in which we provably demonstrate the presence of insecurity.",
	"authors": [
	"Toby Murray"
	],
	"date": "2020-03-12",
	- "id": 234,
	+ "id": 237,
	"link": "/entries/Relational-Incorrectness-Logic.html",
	"permalink": "/entries/Relational-Incorrectness-Logic.html",
	"shortname": "Relational-Incorrectness-Logic",
	"title": "An Under-Approximate Relational Logic",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/security"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "In this article, we present a formalization of the well-known \"Hello, World!\" code, including a formal framework for reasoning about IO. Our model is inspired by the handling of IO in Haskell. We start by formalizing the 🌍 and embrace the IO monad afterwards. Then we present a sample main :: IO (), followed by its proof of correctness.",
	"authors": [
	"Cornelius Diekmann",
	"Lars Hupel"
	],
	"date": "2020-03-07",
	- "id": 235,
	+ "id": 238,
	"link": "/entries/Hello_World.html",
	"permalink": "/entries/Hello_World.html",
	"shortname": "Hello_World",
	"title": "Hello World",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "In this formalization, we develop an implementation of the Goodstein function G in plain \u0026lambda;-calculus, linked to a concise, self-contained specification. The implementation works on a Church-encoded representation of countable ordinals. The initial conversion to hereditary base 2 is not covered, but the material is sufficient to compute the particular value G(16), and easily extends to other fixed arguments.",
	"authors": [
	"Bertram Felgenhauer"
	],
	"date": "2020-02-21",
	- "id": 236,
	+ "id": 239,
	"link": "/entries/Goodstein_Lambda.html",
	"permalink": "/entries/Goodstein_Lambda.html",
	"shortname": "Goodstein_Lambda",
	"title": "Implementing the Goodstein Function in \u0026lambda;-Calculus",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a generic framework for formalizing compiler transformations. It leverages Isabelle/HOL’s locales to abstract over concrete languages and transformations. It states common definitions for language semantics, program behaviours, forward and backward simulations, and compilers. We provide generic operations, such as simulation and compiler composition, and prove general (partial) correctness theorems, resulting in reusable proof components.",
	"authors": [
	"Martin Desharnais"
	],
	"date": "2020-02-10",
	- "id": 237,
	+ "id": 240,
	"link": "/entries/VeriComp.html",
	"permalink": "/entries/VeriComp.html",
	"shortname": "VeriComp",
	"title": "A Generic Framework for Verified Compilers",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 1
	},
	{
	"abstract": "This article provides a formalization of the solution obtained by the author of the Problem “ARITHMETIC PROGRESSIONS” from the \u003ca href=\"https://www.ocf.berkeley.edu/~wwu/riddles/putnam.shtml\"\u003e Putnam exam problems of 2002\u003c/a\u003e. The statement of the problem is as follows: For which integers \u003cem\u003en\u003c/em\u003e \u003e 1 does the set of positive integers less than and relatively prime to \u003cem\u003en\u003c/em\u003e constitute an arithmetic progression?",
	"authors": [
	"José Manuel Rodríguez Caballero"
	],
	"date": "2020-02-01",
	- "id": 238,
	+ "id": 241,
	"link": "/entries/Arith_Prog_Rel_Primes.html",
	"permalink": "/entries/Arith_Prog_Rel_Primes.html",
	"shortname": "Arith_Prog_Rel_Primes",
	"title": "Arithmetic progressions and relative primes",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a collection of axiom systems for the construction of Boolean subalgebras of larger overall algebras. The subalgebras are defined as the range of a complement-like operation on a semilattice. This technique has been used, for example, with the antidomain operation, dynamic negation and Stone algebras. We present a common ground for these constructions based on a new equational axiomatisation of Boolean algebras.",
	"authors": [
	"Walter Guttmann",
	"Bernhard Möller"
	],
	"date": "2020-01-31",
	- "id": 239,
	+ "id": 242,
	"link": "/entries/Subset_Boolean_Algebras.html",
	"permalink": "/entries/Subset_Boolean_Algebras.html",
	"shortname": "Subset_Boolean_Algebras",
	"title": "A Hierarchy of Algebras for Boolean Subsets",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis article provides formal proofs of basic properties of Mersenne numbers, i. e. numbers of the form 2\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e - 1, and especially of Mersenne primes.\u003c/p\u003e \u003cp\u003eIn particular, an efficient, verified, and executable version of the Lucas\u0026ndash;Lehmer test is developed. This test decides primality for Mersenne numbers in time polynomial in \u003cem\u003en\u003c/em\u003e.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2020-01-17",
	- "id": 240,
	+ "id": 243,
	"link": "/entries/Mersenne_Primes.html",
	"permalink": "/entries/Mersenne_Primes.html",
	"shortname": "Mersenne_Primes",
	"title": "Mersenne primes and the Lucas–Lehmer test",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We present the first formal verification of approximation algorithms for NP-complete optimization problems: vertex cover, set cover, independent set, center selection, load balancing, and bin packing. The proofs correct incompletenesses in existing proofs and improve the approximation ratio in one case.",
	"authors": [
	"Robin Eßmann",
	"Tobias Nipkow",
	"Simon Robillard",
	"Ujkan Sulejmani"
	],
	"date": "2020-01-16",
	- "id": 241,
	+ "id": 244,
	"link": "/entries/Approximation_Algorithms.html",
	"permalink": "/entries/Approximation_Algorithms.html",
	"shortname": "Approximation_Algorithms",
	"title": "Verified Approximation Algorithms",
	"topic_links": [
	"computer-science/algorithms/approximation"
	],
	"topics": [
	"Computer science/Algorithms/Approximation"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides two related verified divide-and-conquer algorithms solving the fundamental \u003cem\u003eClosest Pair of Points\u003c/em\u003e problem in Computational Geometry. Functional correctness and the optimal running time of \u003cem\u003eO\u003c/em\u003e(\u003cem\u003en\u003c/em\u003e log \u003cem\u003en\u003c/em\u003e) are proved. Executable code is generated which is empirically competitive with handwritten reference implementations.",
	"authors": [
	"Martin Rau",
	"Tobias Nipkow"
	],
	"date": "2020-01-13",
	- "id": 242,
	+ "id": 245,
	"link": "/entries/Closest_Pair_Points.html",
	"permalink": "/entries/Closest_Pair_Points.html",
	"shortname": "Closest_Pair_Points",
	"title": "Closest Pair of Points Algorithms",
	"topic_links": [
	"computer-science/algorithms/geometry"
	],
	"topics": [
	"Computer science/Algorithms/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e Skip lists are sorted linked lists enhanced with shortcuts and are an alternative to binary search trees. A skip lists consists of multiple levels of sorted linked lists where a list on level n is a subsequence of the list on level n − 1. In the ideal case, elements are skipped in such a way that a lookup in a skip lists takes O(log n) time. In a randomised skip list the skipped elements are choosen randomly. \u003c/p\u003e \u003cp\u003e This entry contains formalized proofs of the textbook results about the expected height and the expected length of a search path in a randomised skip list. \u003c/p\u003e",
	"authors": [
	"Max W. Haslbeck",
	"Manuel Eberl"
	],
	"date": "2020-01-09",
	- "id": 243,
	+ "id": 246,
	"link": "/entries/Skip_Lists.html",
	"permalink": "/entries/Skip_Lists.html",
	"shortname": "Skip_Lists",
	"title": "Skip Lists",
	"topic_links": [
	"computer-science/data-structures",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Data structures",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e Taking as a starting point the author's previous work on developing aspects of category theory in Isabelle/HOL, this article gives a compatible formalization of the notion of \"bicategory\" and develops a framework within which formal proofs of facts about bicategories can be given. The framework includes a number of basic results, including the Coherence Theorem, the Strictness Theorem, pseudofunctors and biequivalence, and facts about internal equivalences and adjunctions in a bicategory. As a driving application and demonstration of the utility of the framework, it is used to give a formal proof of a theorem, due to Carboni, Kasangian, and Street, that characterizes up to biequivalence the bicategories of spans in a category with pullbacks. The formalization effort necessitated the filling-in of many details that were not evident from the brief presentation in the original paper, as well as identifying a few minor corrections along the way. \u003c/p\u003e\u003cp\u003e Revisions made subsequent to the first version of this article added additional material on pseudofunctors, pseudonatural transformations, modifications, and equivalence of bicategories; the main thrust being to give a proof that a pseudofunctor is a biequivalence if and only if it can be extended to an equivalence of bicategories. \u003c/p\u003e",
	"authors": [
	"Eugene W. Stark"
	],
	"date": "2020-01-06",
	- "id": 244,
	+ "id": 247,
	"link": "/entries/Bicategory.html",
	"permalink": "/entries/Bicategory.html",
	"shortname": "Bicategory",
	"title": "Bicategories",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of Beukers's straightforward analytic proof that ζ(3) is irrational. This was first proven by Apéry (which is why this result is also often called ‘Apéry's Theorem’) using a more algebraic approach. This formalisation follows \u003ca href=\"http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes4.pdf\"\u003eFilaseta's presentation\u003c/a\u003e of Beukers's proof.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2019-12-27",
	- "id": 245,
	+ "id": 248,
	"link": "/entries/Zeta_3_Irrational.html",
	"permalink": "/entries/Zeta_3_Irrational.html",
	"shortname": "Zeta_3_Irrational",
	"title": "The Irrationality of ζ(3)",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is a formalization of soundness and completeness proofs for a Seligman-style tableau system for hybrid logic. The completeness result is obtained via a synthetic approach using maximally consistent sets of tableau blocks. The formalization differs from previous work in a few ways. First, to avoid the need to backtrack in the construction of a tableau, the formalized system has no unnamed initial segment, and therefore no Name rule. Second, I show that the full Bridge rule is admissible in the system. Third, I start from rules restricted to only extend the branch with new formulas, including only witnessing diamonds that are not already witnessed, and show that the unrestricted rules are admissible. Similarly, I start from simpler versions of the @-rules and show that these are sufficient. The GoTo rule is restricted using a notion of potential such that each application consumes potential and potential is earned through applications of the remaining rules. I show that if a branch can be closed then it can be closed starting from a single unit. Finally, Nom is restricted by a fixed set of allowed nominals. The resulting system should be terminating. Paper: \u003ca href=\"https://doi.org/10.4230/LIPIcs.TYPES.2020.5\"\u003edoi.org/10.4230/LIPIcs.TYPES.2020.5\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2019-12-20",
	- "id": 246,
	+ "id": 249,
	"link": "/entries/Hybrid_Logic.html",
	"permalink": "/entries/Hybrid_Logic.html",
	"shortname": "Hybrid_Logic",
	"title": "Formalizing a Seligman-Style Tableau System for Hybrid Logic",
	"topic_links": [
	"logic/general-logic/modal-logic"
	],
	"topics": [
	"Logic/General logic/Modal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "The Poincaré-Bendixson theorem is a classical result in the study of (continuous) dynamical systems. Colloquially, it restricts the possible behaviors of planar dynamical systems: such systems cannot be chaotic. In practice, it is a useful tool for proving the existence of (limiting) periodic behavior in planar systems. The theorem is an interesting and challenging benchmark for formalized mathematics because proofs in the literature rely on geometric sketches and only hint at symmetric cases. It also requires a substantial background of mathematical theories, e.g., the Jordan curve theorem, real analysis, ordinary differential equations, and limiting (long-term) behavior of dynamical systems.",
	"authors": [
	"Fabian Immler",
	"Yong Kiam Tan"
	],
	"date": "2019-12-18",
	- "id": 247,
	+ "id": 250,
	"link": "/entries/Poincare_Bendixson.html",
	"permalink": "/entries/Poincare_Bendixson.html",
	"shortname": "Poincare_Bendixson",
	"title": "The Poincaré-Bendixson Theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "A formalization of geometry of complex numbers is presented. Fundamental objects that are investigated are the complex plane extended by a single infinite point, its objects (points, lines and circles), and groups of transformations that act on them (e.g., inversions and Möbius transformations). Most objects are defined algebraically, but correspondence with classical geometric definitions is shown.",
	"authors": [
	"Filip Marić",
	"Danijela Simić"
	],
	"date": "2019-12-16",
	- "id": 248,
	+ "id": 251,
	"link": "/entries/Complex_Geometry.html",
	"permalink": "/entries/Complex_Geometry.html",
	"shortname": "Complex_Geometry",
	"title": "Complex Geometry",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 2
	},
	{
	"abstract": "We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the extended complex plane (one dimensional complex projectives space \u0026#8450;P1), formalized in the AFP entry “Complex Geometry”. Points, lines, congruence of pairs of points, betweenness of triples of points, circles, and isometries are defined within the model. It is shown that the model satisfies all Tarski's axioms except the Euclid's axiom. It is shown that it satisfies its negation and the limiting parallels axiom (which proves it to be a model of hyperbolic geometry).",
	"authors": [
	"Danijela Simić",
	"Filip Marić",
	"Pierre Boutry"
	],
	"date": "2019-12-16",
	- "id": 249,
	+ "id": 252,
	"link": "/entries/Poincare_Disc.html",
	"permalink": "/entries/Poincare_Disc.html",
	"shortname": "Poincare_Disc",
	"title": "Poincaré Disc Model",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article provides a full formalisation of Chapter 8 of Apostol's \u003cem\u003e\u003ca href=\"https://www.springer.com/de/book/9780387901633\"\u003eIntroduction to Analytic Number Theory\u003c/a\u003e\u003c/em\u003e. Subjects that are covered are:\u003c/p\u003e \u003cul\u003e \u003cli\u003eperiodic arithmetic functions and their finite Fourier series\u003c/li\u003e \u003cli\u003e(generalised) Ramanujan sums\u003c/li\u003e \u003cli\u003eGauss sums and separable characters\u003c/li\u003e \u003cli\u003einduced moduli and primitive characters\u003c/li\u003e \u003cli\u003ethe Pólya\u0026mdash;Vinogradov inequality\u003c/li\u003e \u003c/ul\u003e",
	"authors": [
	"Rodrigo Raya",
	"Manuel Eberl"
	],
	"date": "2019-12-10",
	- "id": 250,
	+ "id": 253,
	"link": "/entries/Gauss_Sums.html",
	"permalink": "/entries/Gauss_Sums.html",
	"shortname": "Gauss_Sums",
	"title": "Gauss Sums and the Pólya–Vinogradov Inequality",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Counting sort is a well-known algorithm that sorts objects of any kind mapped to integer keys, or else to keys in one-to-one correspondence with some subset of the integers (e.g. alphabet letters). However, it is suitable for direct use, viz. not just as a subroutine of another sorting algorithm (e.g. radix sort), only if the key range is not significantly larger than the number of the objects to be sorted. This paper describes a tail-recursive generalization of counting sort making use of a bounded number of counters, suitable for direct use in case of a large, or even infinite key range of any kind, subject to the only constraint of being a subset of an arbitrary linear order. After performing a pen-and-paper analysis of how such algorithm has to be designed to maximize its efficiency, this paper formalizes the resulting generalized counting sort (GCsort) algorithm and then formally proves its correctness properties, namely that (a) the counters' number is maximized never exceeding the fixed upper bound, (b) objects are conserved, (c) objects get sorted, and (d) the algorithm is stable.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2019-12-04",
	- "id": 251,
	+ "id": 254,
	"link": "/entries/Generalized_Counting_Sort.html",
	"permalink": "/entries/Generalized_Counting_Sort.html",
	"shortname": "Generalized_Counting_Sort",
	"title": "An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "Interval_Arithmetic implements conservative interval arithmetic computations, then uses this interval arithmetic to implement a simple programming language where all terms have 32-bit signed word values, with explicit infinities for terms outside the representable bounds. Our target use case is interpreters for languages that must have a well-understood low-level behavior. We include a formalization of bounded-length strings which are used for the identifiers of our language. Bounded-length identifiers are useful in some applications, for example the \u003ca href=\"https://www.isa-afp.org/entries/Differential_Dynamic_Logic.html\"\u003eDifferential_Dynamic_Logic\u003c/a\u003e article, where a Euclidean space indexed by identifiers demands that identifiers are finitely many.",
	"authors": [
	"Rose Bohrer"
	],
	"date": "2019-11-27",
	- "id": 252,
	+ "id": 255,
	"link": "/entries/Interval_Arithmetic_Word32.html",
	"permalink": "/entries/Interval_Arithmetic_Word32.html",
	"shortname": "Interval_Arithmetic_Word32",
	"title": "Interval Arithmetic on 32-bit Words",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry is a new formalisation of ZFC set theory in Isabelle/HOL. It is logically equivalent to Obua's HOLZF; the point is to have the closest possible integration with the rest of Isabelle/HOL, minimising the amount of new notations and exploiting type classes.\u003c/p\u003e \u003cp\u003eThere is a type \u003cem\u003eV\u003c/em\u003e of sets and a function \u003cem\u003eelts :: V =\u0026gt; V set\u003c/em\u003e mapping a set to its elements. Classes simply have type \u003cem\u003eV set\u003c/em\u003e, and a predicate identifies the small classes: those that correspond to actual sets. Type classes connected with orders and lattices are used to minimise the amount of new notation for concepts such as the subset relation, union and intersection. Basic concepts — Cartesian products, disjoint sums, natural numbers, functions, etc. — are formalised.\u003c/p\u003e \u003cp\u003eMore advanced set-theoretic concepts, such as transfinite induction, ordinals, cardinals and the transitive closure of a set, are also provided. The definition of addition and multiplication for general sets (not just ordinals) follows Kirby.\u003c/p\u003e \u003cp\u003eThe theory provides two type classes with the aim of facilitating developments that combine \u003cem\u003eV\u003c/em\u003e with other Isabelle/HOL types: \u003cem\u003eembeddable\u003c/em\u003e, the class of types that can be injected into \u003cem\u003eV\u003c/em\u003e (including \u003cem\u003eV\u003c/em\u003e itself as well as \u003cem\u003eV*V\u003c/em\u003e, etc.), and \u003cem\u003esmall\u003c/em\u003e, the class of types that correspond to some ZF set.\u003c/p\u003e extra-history = Change history: [2020-01-28]: Generalisation of the \"small\" predicate and order types to arbitrary sets; ordinal exponentiation; introduction of the coercion ord_of_nat :: \"nat =\u003e V\"; numerous new lemmas. (revision 6081d5be8d08)",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2019-10-24",
	- "id": 253,
	+ "id": 256,
	"link": "/entries/ZFC_in_HOL.html",
	"permalink": "/entries/ZFC_in_HOL.html",
	"shortname": "ZFC_in_HOL",
	"title": "Zermelo Fraenkel Set Theory in Higher-Order Logic",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 4
	},
	{
	"abstract": "We present a framework for C code in C11 syntax deeply integrated into the Isabelle/PIDE development environment. Our framework provides an abstract interface for verification back-ends to be plugged-in independently. Thus, various techniques such as deductive program verification or white-box testing can be applied to the same source, which is part of an integrated PIDE document model. Semantic back-ends are free to choose the supported C fragment and its semantics. In particular, they can differ on the chosen memory model or the specification mechanism for framing conditions. Our framework supports semantic annotations of C sources in the form of comments. Annotations serve to locally control back-end settings, and can express the term focus to which an annotation refers. Both the logical and the syntactic context are available when semantic annotations are evaluated. As a consequence, a formula in an annotation can refer both to HOL or C variables. Our approach demonstrates the degree of maturity and expressive power the Isabelle/PIDE sub-system has achieved in recent years. Our integration technique employs Lex and Yacc style grammars to ensure efficient deterministic parsing. This is the core-module of Isabelle/C; the AFP package for Clean and Clean_wrapper as well as AutoCorres and AutoCorres_wrapper (available via git) are applications of this front-end.",
	"authors": [
	"Frédéric Tuong",
	"Burkhart Wolff"
	],
	"date": "2019-10-22",
	- "id": 254,
	+ "id": 257,
	"link": "/entries/Isabelle_C.html",
	"permalink": "/entries/Isabelle_C.html",
	"shortname": "Isabelle_C",
	"title": "Isabelle/C",
	"topic_links": [
	"computer-science/programming-languages/language-definitions",
	"computer-science/semantics-and-reasoning",
	"tools"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions",
	"Computer science/Semantics and reasoning",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "VerifyThis 2019 (http://www.pm.inf.ethz.ch/research/verifythis.html) was a program verification competition associated with ETAPS 2019. It was the 8th event in the VerifyThis competition series. In this entry, we present polished and completed versions of our solutions that we created during the competition.",
	"authors": [
	"Peter Lammich",
	"Simon Wimmer"
	],
	"date": "2019-10-16",
	- "id": 255,
	+ "id": 258,
	"link": "/entries/VerifyThis2019.html",
	"permalink": "/entries/VerifyThis2019.html",
	"shortname": "VerifyThis2019",
	"title": "VerifyThis 2019 -- Polished Isabelle Solutions",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise with Isabelle/HOL some basic elements of Aristotle's assertoric syllogistic following the \u003ca href=\"https://plato.stanford.edu/entries/aristotle-logic/\"\u003earticle from the Stanford Encyclopedia of Philosophy by Robin Smith.\u003c/a\u003e To this end, we use a set theoretic formulation (covering both individual and general predication). In particular, we formalise the deductions in the Figures and after that we present Aristotle's metatheoretical observation that all deductions in the Figures can in fact be reduced to either Barbara or Celarent. As the formal proofs prove to be straightforward, the interest of this entry lies in illustrating the functionality of Isabelle and high efficiency of Sledgehammer for simple exercises in philosophy.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki"
	],
	"date": "2019-10-08",
	- "id": 256,
	+ "id": 259,
	"link": "/entries/Aristotles_Assertoric_Syllogistic.html",
	"permalink": "/entries/Aristotles_Assertoric_Syllogistic.html",
	"shortname": "Aristotles_Assertoric_Syllogistic",
	"title": "Aristotle's Assertoric Syllogistic",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "We use CryptHOL to formalise commitment schemes and Sigma-protocols. Both are widely used fundamental two party cryptographic primitives. Security for commitment schemes is considered using game-based definitions whereas the security of Sigma-protocols is considered using both the game-based and simulation-based security paradigms. In this work, we first define security for both primitives and then prove secure multiple case studies: the Schnorr, Chaum-Pedersen and Okamoto Sigma-protocols as well as a construction that allows for compound (AND and OR statements) Sigma-protocols and the Pedersen and Rivest commitment schemes. We also prove that commitment schemes can be constructed from Sigma-protocols. We formalise this proof at an abstract level, only assuming the existence of a Sigma-protocol; consequently, the instantiations of this result for the concrete Sigma-protocols we consider come for free.",
	"authors": [
	"David Butler",
	"Andreas Lochbihler"
	],
	"date": "2019-10-07",
	- "id": 257,
	+ "id": 260,
	"link": "/entries/Sigma_Commit_Crypto.html",
	"permalink": "/entries/Sigma_Commit_Crypto.html",
	"shortname": "Sigma_Commit_Crypto",
	"title": "Sigma Protocols and Commitment Schemes",
	"topic_links": [
	"computer-science/security/cryptography"
	],
	"topics": [
	"Computer science/Security/Cryptography"
	],
	"used_by": 1
	},
	{
	"abstract": "Clean is based on a simple, abstract execution model for an imperative target language. “Abstract” is understood in contrast to “Concrete Semantics”; alternatively, the term “shallow-style embedding” could be used. It strives for a type-safe notion of program-variables, an incremental construction of the typed state-space, support of incremental verification, and open-world extensibility of new type definitions being intertwined with the program definitions. Clean is based on a “no-frills” state-exception monad with the usual definitions of bind and unit for the compositional glue of state-based computations. Clean offers conditionals and loops supporting C-like control-flow operators such as break and return. The state-space construction is based on the extensible record package. Direct recursion of procedures is supported. Clean’s design strives for extreme simplicity. It is geared towards symbolic execution and proven correct verification tools. The underlying libraries of this package, however, deliberately restrict themselves to the most elementary infrastructure for these tasks. The package is intended to serve as demonstrator semantic backend for Isabelle/C, or for the test-generation techniques.",
	"authors": [
	"Frédéric Tuong",
	"Burkhart Wolff"
	],
	"date": "2019-10-04",
	- "id": 258,
	+ "id": 261,
	"link": "/entries/Clean.html",
	"permalink": "/entries/Clean.html",
	"shortname": "Clean",
	"title": "Clean - An Abstract Imperative Programming Language and its Theory",
	"topic_links": [
	"computer-science/programming-languages",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Programming languages",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 0
	},
	{
	"abstract": "Worst-case optimal multiway-join algorithms are recent seminal achievement of the database community. These algorithms compute the natural join of multiple relational databases and improve in the worst case over traditional query plan optimizations of nested binary joins. In 2014, \u003ca href=\"https://doi.org/10.1145/2590989.2590991\"\u003eNgo, Ré, and Rudra\u003c/a\u003e gave a unified presentation of different multi-way join algorithms. We formalized and proved correct their \"Generic Join\" algorithm and extended it to support negative joins.",
	"authors": [
	"Thibault Dardinier"
	],
	"date": "2019-09-16",
	- "id": 259,
	+ "id": 262,
	"link": "/entries/Generic_Join.html",
	"permalink": "/entries/Generic_Join.html",
	"shortname": "Generic_Join",
	"title": "Formalization of Multiway-Join Algorithms",
	"topic_links": [
	"computer-science/data-management-systems",
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Data management systems",
	"Computer science/Algorithms"
	],
	"used_by": 1
	},
	{
	"abstract": "These components formalise a semantic framework for the deductive verification of hybrid systems. They support reasoning about continuous evolutions of hybrid programs in the style of differential dynamics logic. Vector fields or flows model these evolutions, and their verification is done with invariants for the former or orbits for the latter. Laws of modal Kleene algebra or categorical predicate transformers implement the verification condition generation. Examples show the approach at work.",
	"authors": [
	"Jonathan Julian Huerta y Munive"
	],
	"date": "2019-09-10",
	- "id": 260,
	+ "id": 263,
	"link": "/entries/Hybrid_Systems_VCs.html",
	"permalink": "/entries/Hybrid_Systems_VCs.html",
	"shortname": "Hybrid_Systems_VCs",
	"title": "Verification Components for Hybrid Systems",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "This development formalises the square integrable functions over the reals and the basics of Fourier series. It culminates with a proof that every well-behaved periodic function can be approximated by a Fourier series. The material is ported from HOL Light: https://github.com/jrh13/hol-light/blob/master/100/fourier.ml",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2019-09-06",
	- "id": 261,
	+ "id": 264,
	"link": "/entries/Fourier.html",
	"permalink": "/entries/Fourier.html",
	"shortname": "Fourier",
	"title": "Fourier Series",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "The focus of this case study is re-use in abstract algebra. It contains locale-based formalisations of selected parts of set, group and ring theory from Jacobson's \u003ci\u003eBasic Algebra\u003c/i\u003e leading to the respective fundamental homomorphism theorems. The study is not intended as a library base for abstract algebra. It rather explores an approach towards abstract algebra in Isabelle.",
	"authors": [
	"Clemens Ballarin"
	],
	"date": "2019-08-30",
	- "id": 262,
	+ "id": 265,
	"link": "/entries/Jacobson_Basic_Algebra.html",
	"permalink": "/entries/Jacobson_Basic_Algebra.html",
	"shortname": "Jacobson_Basic_Algebra",
	"title": "A Case Study in Basic Algebra",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 4
	},
	{
	"abstract": "This entry provides a formalisation of a refinement of an adaptive state counting algorithm, used to test for reduction between finite state machines. The algorithm has been originally presented by Hierons in the paper \u003ca href=\"https://doi.org/10.1109/TC.2004.85\"\u003eTesting from a Non-Deterministic Finite State Machine Using Adaptive State Counting\u003c/a\u003e. Definitions for finite state machines and adaptive test cases are given and many useful theorems are derived from these. The algorithm is formalised using mutually recursive functions, for which it is proven that the generated test suite is sufficient to test for reduction against finite state machines of a certain fault domain. Additionally, the algorithm is specified in a simple WHILE-language and its correctness is shown using Hoare-logic.",
	"authors": [
	"Robert Sachtleben"
	],
	"date": "2019-08-16",
	- "id": 263,
	+ "id": 266,
	"link": "/entries/Adaptive_State_Counting.html",
	"permalink": "/entries/Adaptive_State_Counting.html",
	"shortname": "Adaptive_State_Counting",
	"title": "Formalisation of an Adaptive State Counting Algorithm",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry formalizes the Laplace transform and concrete Laplace transforms for arithmetic functions, frequency shift, integration and (higher) differentiation in the time domain. It proves Lerch's lemma and uniqueness of the Laplace transform for continuous functions. In order to formalize the foundational assumptions, this entry contains a formalization of piecewise continuous functions and functions of exponential order.",
	"authors": [
	"Fabian Immler"
	],
	"date": "2019-08-14",
	- "id": 264,
	+ "id": 267,
	"link": "/entries/Laplace_Transform.html",
	"permalink": "/entries/Laplace_Transform.html",
	"shortname": "Laplace_Transform",
	"title": "Laplace Transform",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Communicating Concurrent Kleene Algebra (C²KA) is a mathematical framework for capturing the communicating and concurrent behaviour of agents in distributed systems. It extends Hoare et al.'s Concurrent Kleene Algebra (CKA) with communication actions through the notions of stimuli and shared environments. C²KA has applications in studying system-level properties of distributed systems such as safety, security, and reliability. In this work, we formalize results about C²KA and its application for distributed systems specification. We first formalize the stimulus structure and behaviour structure (CKA). Next, we combine them to formalize C²KA and its properties. Then, we formalize notions and properties related to the topology of distributed systems and the potential for communication via stimuli and via shared environments of agents, all within the algebraic setting of C²KA.",
	"authors": [
	"Maxime Buyse",
	"Jason Jaskolka"
	],
	"date": "2019-08-06",
	- "id": 265,
	+ "id": 268,
	"link": "/entries/C2KA_DistributedSystems.html",
	"permalink": "/entries/C2KA_DistributedSystems.html",
	"shortname": "C2KA_DistributedSystems",
	"title": "Communicating Concurrent Kleene Algebra for Distributed Systems Specification",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "We use the previous formalization of the general simplex algorithm to formulate an algorithm for solving linear programs. We encode the linear programs using only linear constraints. Solving these constraints also solves the original linear program. This algorithm is proven to be sound by applying the weak duality theorem which is also part of this formalization.",
	"authors": [
	"Julian Parsert",
	"Cezary Kaliszyk"
	],
	"date": "2019-08-06",
	- "id": 266,
	+ "id": 269,
	"link": "/entries/Linear_Programming.html",
	"permalink": "/entries/Linear_Programming.html",
	"shortname": "Linear_Programming",
	"title": "Linear Programming",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry contains formalisations of the answers to three of the six problem of the International Mathematical Olympiad 2019, namely Q1, Q4, and Q5.\u003c/p\u003e \u003cp\u003eThe reason why these problems were chosen is that they are particularly amenable to formalisation: they can be solved with minimal use of libraries. The remaining three concern geometry and graph theory, which, in the author's opinion, are more difficult to formalise resp. require a more complex library.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2019-08-05",
	- "id": 267,
	+ "id": 270,
	"link": "/entries/IMO2019.html",
	"permalink": "/entries/IMO2019.html",
	"shortname": "IMO2019",
	"title": "Selected Problems from the International Mathematical Olympiad 2019",
	"topic_links": [
	"mathematics/misc"
	],
	"topics": [
	"Mathematics/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the static properties of personal Byzantine quorum systems (PBQSs) and Stellar quorum systems, as described in the paper “Stellar Consensus by Reduction” (to appear at DISC 2019).",
	"authors": [
	"Giuliano Losa"
	],
	"date": "2019-08-01",
	- "id": 268,
	+ "id": 271,
	"link": "/entries/Stellar_Quorums.html",
	"permalink": "/entries/Stellar_Quorums.html",
	"shortname": "Stellar_Quorums",
	"title": "Stellar Quorum Systems",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "The design of complex systems involves different formalisms for modeling their different parts or aspects. The global model of a system may therefore consist of a coordination of concurrent sub-models that use different paradigms. We develop here a theory for a language used to specify the timed coordination of such heterogeneous subsystems by addressing the following issues: \u003cul\u003e\u003cli\u003ethe behavior of the sub-systems is observed only at a series of discrete instants,\u003c/li\u003e\u003cli\u003eevents may occur in different sub-systems at unrelated times, leading to polychronous systems, which do not necessarily have a common base clock,\u003c/li\u003e\u003cli\u003ecoordination between subsystems involves causality, so the occurrence of an event may enforce the occurrence of other events, possibly after a certain duration has elapsed or an event has occurred a given number of times,\u003c/li\u003e\u003cli\u003ethe domain of time (discrete, rational, continuous...) may be different in the subsystems, leading to polytimed systems,\u003c/li\u003e\u003cli\u003ethe time frames of different sub-systems may be related (for instance, time in a GPS satellite and in a GPS receiver on Earth are related although they are not the same).\u003c/li\u003e\u003c/ul\u003e Firstly, a denotational semantics of the language is defined. Then, in order to be able to incrementally check the behavior of systems, an operational semantics is given, with proofs of progress, soundness and completeness with regard to the denotational semantics. These proofs are made according to a setup that can scale up when new operators are added to the language. In order for specifications to be composed in a clean way, the language should be invariant by stuttering (i.e., adding observation instants at which nothing happens). The proof of this invariance is also given.",
	"authors": [
	"Hai Nguyen Van",
	"Frédéric Boulanger",
	"Burkhart Wolff"
	],
	"date": "2019-07-30",
	- "id": 269,
	+ "id": 272,
	"link": "/entries/TESL_Language.html",
	"permalink": "/entries/TESL_Language.html",
	"shortname": "TESL_Language",
	"title": "A Formal Development of a Polychronous Polytimed Coordination Language",
	"topic_links": [
	"computer-science/system-description-languages",
	"computer-science/semantics-and-reasoning",
	"computer-science/concurrency"
	],
	"topics": [
	"Computer science/System description languages",
	"Computer science/Semantics and reasoning",
	"Computer science/Concurrency"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry is concerned with the principle of order extension, i.e. the extension of an order relation to a total order relation. To this end, we prove a more general version of Szpilrajn's extension theorem employing terminology from the book \"Consistency, Choice, and Rationality\" by Bossert and Suzumura. We also formalize theorem 2.7 of their book.",
	"authors": [
	"Peter Zeller",
	"Lukas Stevens"
	],
	"date": "2019-07-27",
	- "id": 270,
	+ "id": 273,
	"link": "/entries/Szpilrajn.html",
	"permalink": "/entries/Szpilrajn.html",
	"shortname": "Szpilrajn",
	"title": "Order Extension and Szpilrajn's Extension Theorem",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 1
	},
	{
	"abstract": "This work formalizes soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a complete semantic tableau calculus, the proof of which is based on the First-Order Logic According to Fitting theory. The calculi and proof techniques are taken from Ben-Ari's Mathematical Logic for Computer Science. Papers: \u003ca href=\"http://ceur-ws.org/Vol-3002/paper7.pdf\"\u003eceur-ws.org/Vol-3002/paper7.pdf\u003c/a\u003e and \u003ca href=\"https://doi.org/10.1093/logcom/exad013\"\u003edoi.org/10.1093/logcom/exad013\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2019-07-18",
	- "id": 271,
	+ "id": 274,
	"link": "/entries/FOL_Seq_Calc1.html",
	"permalink": "/entries/FOL_Seq_Calc1.html",
	"shortname": "FOL_Seq_Calc1",
	"title": "A Sequent Calculus for First-Order Logic",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry contains the formalization that accompanies my PhD thesis (see https://lars.hupel.info/research/codegen/). I develop a verified compilation toolchain from executable specifications in Isabelle/HOL to CakeML abstract syntax trees. This improves over the state-of-the-art in Isabelle by providing a trustworthy procedure for code generation.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2019-07-08",
	- "id": 272,
	+ "id": 275,
	"link": "/entries/CakeML_Codegen.html",
	"permalink": "/entries/CakeML_Codegen.html",
	"shortname": "CakeML_Codegen",
	"title": "A Verified Code Generator from Isabelle/HOL to CakeML",
	"topic_links": [
	"computer-science/programming-languages/compiling",
	"logic/rewriting"
	],
	"topics": [
	"Computer science/Programming languages/Compiling",
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "A monitor is a runtime verification tool that solves the following problem: Given a stream of time-stamped events and a policy formulated in a specification language, decide whether the policy is satisfied at every point in the stream. We verify the correctness of an executable monitor for specifications given as formulas in metric first-order temporal logic (MFOTL), an expressive extension of linear temporal logic with real-time constraints and first-order quantification. The verified monitor implements a simplified variant of the algorithm used in the efficient MonPoly monitoring tool. The formalization is presented in a \u003ca href=\"https://doi.org/10.1007/978-3-030-32079-9_18\"\u003eRV 2019 paper\u003c/a\u003e, which also compares the output of the verified monitor to that of other monitoring tools on randomly generated inputs. This case study revealed several errors in the optimized but unverified tools.",
	"authors": [
	"Joshua Schneider",
	"Dmitriy Traytel"
	],
	"date": "2019-07-04",
	- "id": 273,
	+ "id": 276,
	"link": "/entries/MFOTL_Monitor.html",
	"permalink": "/entries/MFOTL_Monitor.html",
	"shortname": "MFOTL_Monitor",
	"title": "Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic",
	"topic_links": [
	"computer-science/algorithms",
	"logic/general-logic/temporal-logic",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Algorithms",
	"Logic/General logic/Temporal logic",
	"Computer science/Automata and formal languages"
	],
	"used_by": 2
	},
	{
	"abstract": "We develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any properties of ordering, thus complete non-orders. In particular, we generalize the Knaster–Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition—attractivity—which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene’s fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene’s fixed points are least quasi-fixed points.",
	"authors": [
	"Akihisa Yamada",
	"Jérémy Dubut"
	],
	"date": "2019-06-27",
	- "id": 274,
	+ "id": 277,
	"link": "/entries/Complete_Non_Orders.html",
	"permalink": "/entries/Complete_Non_Orders.html",
	"shortname": "Complete_Non_Orders",
	"title": "Complete Non-Orders and Fixed Points",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a new, purely functional, simple and efficient data structure combining a search tree and a priority queue, which we call a \u003cem\u003epriority search tree\u003c/em\u003e. The salient feature of priority search trees is that they offer a decrease-key operation, something that is missing from other simple, purely functional priority queue implementations. Priority search trees can be implemented on top of any search tree. This entry does the implementation for red-black trees. This entry formalizes the first part of our ITP-2019 proof pearl \u003cem\u003ePurely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra\u003c/em\u003e.",
	"authors": [
	"Peter Lammich",
	"Tobias Nipkow"
	],
	"date": "2019-06-25",
	- "id": 275,
	+ "id": 278,
	"link": "/entries/Priority_Search_Trees.html",
	"permalink": "/entries/Priority_Search_Trees.html",
	"shortname": "Priority_Search_Trees",
	"title": "Priority Search Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "We verify purely functional, simple and efficient implementations of Prim's and Dijkstra's algorithms. This constitutes the first verification of an executable and even efficient version of Prim's algorithm. This entry formalizes the second part of our ITP-2019 proof pearl \u003cem\u003ePurely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra\u003c/em\u003e.",
	"authors": [
	"Peter Lammich",
	"Tobias Nipkow"
	],
	"date": "2019-06-25",
	- "id": 276,
	+ "id": 279,
	"link": "/entries/Prim_Dijkstra_Simple.html",
	"permalink": "/entries/Prim_Dijkstra_Simple.html",
	"shortname": "Prim_Dijkstra_Simple",
	"title": "Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize results about linear inqualities, mainly from Schrijver's book. The main results are the proof of the fundamental theorem on linear inequalities, Farkas' lemma, Carathéodory's theorem, the Farkas-Minkowsky-Weyl theorem, the decomposition theorem of polyhedra, and Meyer's result that the integer hull of a polyhedron is a polyhedron itself. Several theorems include bounds on the appearing numbers, and in particular we provide an a-priori bound on mixed-integer solutions of linear inequalities.",
	"authors": [
	"Ralph Bottesch",
	"Alban Reynaud",
	"René Thiemann"
	],
	"date": "2019-06-21",
	- "id": 277,
	+ "id": 280,
	"link": "/entries/Linear_Inequalities.html",
	"permalink": "/entries/Linear_Inequalities.html",
	"shortname": "Linear_Inequalities",
	"title": "Linear Inequalities",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "This entry formalizes Hilbert's Nullstellensatz, an important theorem in algebraic geometry that can be viewed as the generalization of the Fundamental Theorem of Algebra to multivariate polynomials: If a set of (multivariate) polynomials over an algebraically closed field has no common zero, then the ideal it generates is the entire polynomial ring. The formalization proves several equivalent versions of this celebrated theorem: the weak Nullstellensatz, the strong Nullstellensatz (connecting algebraic varieties and radical ideals), and the field-theoretic Nullstellensatz. The formalization follows Chapter 4.1. of \u003ca href=\"https://link.springer.com/book/10.1007/978-0-387-35651-8\"\u003eIdeals, Varieties, and Algorithms\u003c/a\u003e by Cox, Little and O'Shea.",
	"authors": [
	"Alexander Maletzky"
	],
	"date": "2019-06-16",
	- "id": 278,
	+ "id": 281,
	"link": "/entries/Nullstellensatz.html",
	"permalink": "/entries/Nullstellensatz.html",
	"shortname": "Nullstellensatz",
	"title": "Hilbert's Nullstellensatz",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry formalizes the connection between Gröbner bases and Macaulay matrices (sometimes also referred to as `generalized Sylvester matrices'). In particular, it contains a method for computing Gröbner bases, which proceeds by first constructing some Macaulay matrix of the initial set of polynomials, then row-reducing this matrix, and finally converting the result back into a set of polynomials. The output is shown to be a Gröbner basis if the Macaulay matrix constructed in the first step is sufficiently large. In order to obtain concrete upper bounds on the size of the matrix (and hence turn the method into an effectively executable algorithm), Dubé's degree bounds on Gröbner bases are utilized; consequently, they are also part of the formalization.",
	"authors": [
	"Alexander Maletzky"
	],
	"date": "2019-06-15",
	- "id": 279,
	+ "id": 282,
	"link": "/entries/Groebner_Macaulay.html",
	"permalink": "/entries/Groebner_Macaulay.html",
	"shortname": "Groebner_Macaulay",
	"title": "Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "In this submission array-based binary minimum heaps are formalized. The correctness of the following heap operations is proved: insert, get-min, delete-min and make-heap. These are then used to verify an in-place heapsort. The formalization is based on IMP2, an imperative program verification framework implemented in Isabelle/HOL. The verified heap functions are iterative versions of the partly recursive functions found in \"Algorithms and Data Structures – The Basic Toolbox\" by K. Mehlhorn and P. Sanders and \"Introduction to Algorithms\" by T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein.",
	"authors": [
	"Simon Griebel"
	],
	"date": "2019-06-13",
	- "id": 280,
	+ "id": 283,
	"link": "/entries/IMP2_Binary_Heap.html",
	"permalink": "/entries/IMP2_Binary_Heap.html",
	"shortname": "IMP2_Binary_Heap",
	"title": "Binary Heaps for IMP2",
	"topic_links": [
	"computer-science/data-structures",
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Data structures",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalization provides differential game logic (dGL), a logic for proving properties of hybrid game. In addition to the syntax and semantics, it formalizes a uniform substitution calculus for dGL. Church's uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. The uniform substitutions for dGL also substitute hybrid games for a game symbol everywhere. We prove soundness of one-pass uniform substitutions and the axioms of differential game logic with respect to their denotational semantics. One-pass uniform substitutions are faster by postponing soundness-critical admissibility checks with a linear pass homomorphic application and regain soundness by a variable condition at the replacements. The formalization is based on prior non-mechanized soundness proofs for dGL.",
	"authors": [
	"André Platzer"
	],
	"date": "2019-06-03",
	- "id": 281,
	+ "id": 284,
	"link": "/entries/Differential_Game_Logic.html",
	"permalink": "/entries/Differential_Game_Logic.html",
	"shortname": "Differential_Game_Logic",
	"title": "Differential Game Logic",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides a formalization of multidimensional binary trees, also known as k-d trees. It includes a balanced build algorithm as well as the nearest neighbor algorithm and the range search algorithm. It is based on the papers \u003ca href=\"https://dl.acm.org/citation.cfm?doid=361002.361007\"\u003eMultidimensional binary search trees used for associative searching\u003c/a\u003e and \u003ca href=\"https://dl.acm.org/citation.cfm?doid=355744.355745\"\u003e An Algorithm for Finding Best Matches in Logarithmic Expected Time\u003c/a\u003e.",
	"authors": [
	"Martin Rau"
	],
	"date": "2019-05-30",
	- "id": 282,
	+ "id": 285,
	"link": "/entries/KD_Tree.html",
	"permalink": "/entries/KD_Tree.html",
	"shortname": "KD_Tree",
	"title": "Multidimensional Binary Search Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Authenticated data structures are a technique for outsourcing data storage and maintenance to an untrusted server. The server is required to produce an efficiently checkable and cryptographically secure proof that it carried out precisely the requested computation. \u003ca href=\"https://doi.org/10.1145/2535838.2535851\"\u003eMiller et al.\u003c/a\u003e introduced \u0026lambda;\u0026bull; (pronounced \u003ci\u003elambda auth\u003c/i\u003e)\u0026mdash;a functional programming language with a built-in primitive authentication construct, which supports a wide range of user-specified authenticated data structures while guaranteeing certain correctness and security properties for all well-typed programs. We formalize \u0026lambda;\u0026bull; and prove its correctness and security properties. With Isabelle's help, we uncover and repair several mistakes in the informal proofs and lemma statements. Our findings are summarized in an \u003ca href=\"https://doi.org/10.4230/LIPIcs.ITP.2019.10\"\u003eITP'19 paper\u003c/a\u003e.",
	"authors": [
	"Matthias Brun",
	"Dmitriy Traytel"
	],
	"date": "2019-05-14",
	- "id": 283,
	+ "id": 286,
	"link": "/entries/LambdaAuth.html",
	"permalink": "/entries/LambdaAuth.html",
	"shortname": "LambdaAuth",
	"title": "Formalization of Generic Authenticated Data Structures",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "We use CryptHOL to consider Multi-Party Computation (MPC) protocols. MPC was first considered by Yao in 1983 and recent advances in efficiency and an increased demand mean it is now deployed in the real world. Security is considered using the real/ideal world paradigm. We first define security in the semi-honest security setting where parties are assumed not to deviate from the protocol transcript. In this setting we prove multiple Oblivious Transfer (OT) protocols secure and then show security for the gates of the GMW protocol. We then define malicious security, this is a stronger notion of security where parties are assumed to be fully corrupted by an adversary. In this setting we again consider OT, as it is a fundamental building block of almost all MPC protocols.",
	"authors": [
	"David Aspinall",
	"David Butler"
	],
	"date": "2019-05-09",
	- "id": 284,
	+ "id": 287,
	"link": "/entries/Multi_Party_Computation.html",
	"permalink": "/entries/Multi_Party_Computation.html",
	"shortname": "Multi_Party_Computation",
	"title": "Multi-Party Computation",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a complete formalization of the work of Hoare and Roscoe on the denotational semantics of the Failure/Divergence Model of CSP. It follows essentially the presentation of CSP in Roscoe’s Book ”Theory and Practice of Concurrency” [8] and the semantic details in a joint Paper of Roscoe and Brooks ”An improved failures model for communicating processes\". The present work is based on a prior formalization attempt, called HOL-CSP 1.0, done in 1997 by H. Tej and B. Wolff with the Isabelle proof technology available at that time. This work revealed minor, but omnipresent foundational errors in key concepts like the process invariant. The present version HOL-CSP profits from substantially improved libraries (notably HOLCF), improved automated proof techniques, and structured proof techniques in Isar and is substantially shorter but more complete.",
	"authors": [
	"Safouan Taha",
	"Lina Ye",
	"Burkhart Wolff"
	],
	"date": "2019-04-26",
	- "id": 285,
	+ "id": 288,
	"link": "/entries/HOL-CSP.html",
	"permalink": "/entries/HOL-CSP.html",
	"shortname": "HOL-CSP",
	"title": "HOL-CSP Version 2.0",
	"topic_links": [
	"computer-science/concurrency/process-calculi",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a formalisation of the unified translation approach of linear temporal logic (LTL) into ω-automata from [1]. This approach decomposes LTL formulas into “simple” languages and allows a clear separation of concerns: first, we formalise the purely logical result yielding this decomposition; second, we instantiate this generic theory to obtain a construction for deterministic (state-based) Rabin automata (DRA). We extract from this particular instantiation an executable tool translating LTL to DRAs. To the best of our knowledge this is the first verified translation from LTL to DRAs that is proven to be double exponential in the worst case which asymptotically matches the known lower bound. \u003cp\u003e [1] Javier Esparza, Jan Kretínský, Salomon Sickert. One Theorem to Rule Them All: A Unified Translation of LTL into ω-Automata. LICS 2018",
	"authors": [
	"Benedikt Seidl",
	"Salomon Sickert"
	],
	"date": "2019-04-16",
	- "id": 286,
	+ "id": 289,
	"link": "/entries/LTL_Master_Theorem.html",
	"permalink": "/entries/LTL_Master_Theorem.html",
	"shortname": "LTL_Master_Theorem",
	"title": "A Compositional and Unified Translation of LTL into ω-Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory includes many properties of the standard operators on terms: substitution, swapping and freshness. It also includes bindings-aware induction and recursion principles and support for semantic interpretation. This work has been presented in the ITP 2017 paper “A Formalized General Theory of Syntax with Bindings”.",
	"authors": [
	"Lorenzo Gheri",
	"Andrei Popescu"
	],
	"date": "2019-04-06",
	- "id": 287,
	+ "id": 290,
	"link": "/entries/Binding_Syntax_Theory.html",
	"permalink": "/entries/Binding_Syntax_Theory.html",
	"shortname": "Binding_Syntax_Theory",
	"title": "A General Theory of Syntax with Bindings",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi",
	"computer-science/functional-programming",
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi",
	"Computer science/Functional programming",
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the proofs of two transcendence criteria by J. Hančl and P. Rucki that assert the transcendence of the sums of certain infinite series built up by sequences that fulfil certain properties. Both proofs make use of Roth's celebrated theorem on diophantine approximations to algebraic numbers from 1955 which we implement as an assumption without having formalised its proof.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Wenda Li"
	],
	"date": "2019-03-27",
	- "id": 288,
	+ "id": 291,
	"link": "/entries/Transcendence_Series_Hancl_Rucki.html",
	"permalink": "/entries/Transcendence_Series_Hancl_Rucki.html",
	"shortname": "Transcendence_Series_Hancl_Rucki",
	"title": "The Transcendence of Certain Infinite Series",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize quantum Hoare logic as given in [1]. In particular, we specify the syntax and denotational semantics of a simple model of quantum programs. Then, we write down the rules of quantum Hoare logic for partial correctness, and show the soundness and completeness of the resulting proof system. As an application, we verify the correctness of Grover’s algorithm.",
	"authors": [
	"Junyi Liu",
	"Bohua Zhan",
	"Shuling Wang",
	"Shenggang Ying",
	"Tao Liu",
	"Yangjia Li",
	"Mingsheng Ying",
	"Naijun Zhan"
	],
	"date": "2019-03-24",
	- "id": 289,
	+ "id": 292,
	"link": "/entries/QHLProver.html",
	"permalink": "/entries/QHLProver.html",
	"shortname": "QHLProver",
	"title": "Quantum Hoare Logic",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThe theory is a formalization of the \u003ca href=\"https://www.omg.org/spec/OCL/\"\u003eOCL\u003c/a\u003e type system, its abstract syntax and expression typing rules. The theory does not define a concrete syntax and a semantics. In contrast to \u003ca href=\"https://www.isa-afp.org/entries/Featherweight_OCL.html\"\u003eFeatherweight OCL\u003c/a\u003e, it is based on a deep embedding approach. The type system is defined from scratch, it is not based on the Isabelle HOL type system.\u003c/p\u003e \u003cp\u003eThe Safe OCL distincts nullable and non-nullable types. Also the theory gives a formal definition of \u003ca href=\"http://ceur-ws.org/Vol-1512/paper07.pdf\"\u003esafe navigation operations\u003c/a\u003e. The Safe OCL typing rules are much stricter than rules given in the OCL specification. It allows one to catch more errors on a type checking phase.\u003c/p\u003e \u003cp\u003eThe type theory presented is four-layered: classes, basic types, generic types, errorable types. We introduce the following new types: non-nullable types (T[1]), nullable types (T[?]), OclSuper. OclSuper is a supertype of all other types (basic types, collections, tuples). This type allows us to define a total supremum function, so types form an upper semilattice. It allows us to define rich expression typing rules in an elegant manner.\u003c/p\u003e \u003cp\u003eThe Preliminaries Chapter of the theory defines a number of helper lemmas for transitive closures and tuples. It defines also a generic object model independent from OCL. It allows one to use the theory as a reference for formalization of analogous languages.\u003c/p\u003e",
	"authors": [
	"Denis Nikiforov"
	],
	"date": "2019-03-09",
	- "id": 290,
	+ "id": 293,
	"link": "/entries/Safe_OCL.html",
	"permalink": "/entries/Safe_OCL.html",
	"shortname": "Safe_OCL",
	"title": "Safe OCL",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry is a formalisation of Chapter 4 (and parts of Chapter 3) of Apostol's \u003ca href=\"https://www.springer.com/de/book/9780387901633\"\u003e\u003cem\u003eIntroduction to Analytic Number Theory\u003c/em\u003e\u003c/a\u003e. The main topics that are addressed are properties of the distribution of prime numbers that can be shown in an elementary way (i.\u0026thinsp;e. without the Prime Number Theorem), the various equivalent forms of the PNT (which imply each other in elementary ways), and consequences that follow from the PNT in elementary ways. The latter include, most notably, asymptotic bounds for the number of distinct prime factors of \u003cem\u003en\u003c/em\u003e, the divisor function \u003cem\u003ed(n)\u003c/em\u003e, Euler's totient function \u003cem\u003e\u0026phi;(n)\u003c/em\u003e, and lcm(1,\u0026hellip;,\u003cem\u003en\u003c/em\u003e).\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2019-02-21",
	- "id": 291,
	+ "id": 294,
	"link": "/entries/Prime_Distribution_Elementary.html",
	"permalink": "/entries/Prime_Distribution_Elementary.html",
	"shortname": "Prime_Distribution_Elementary",
	"title": "Elementary Facts About the Distribution of Primes",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 3
	},
	{
	"abstract": "This Isabelle/HOL formalization defines a greedy algorithm for finding a minimum weight basis on a weighted matroid and proves its correctness. This algorithm is an abstract version of Kruskal's algorithm. We interpret the abstract algorithm for the cycle matroid (i.e. forests in a graph) and refine it to imperative executable code using an efficient union-find data structure. Our formalization can be instantiated for different graph representations. We provide instantiations for undirected graphs and symmetric directed graphs.",
	"authors": [
	"Maximilian P. L. Haslbeck",
	"Peter Lammich",
	"Julian Biendarra"
	],
	"date": "2019-02-14",
	- "id": 292,
	+ "id": 295,
	"link": "/entries/Kruskal.html",
	"permalink": "/entries/Kruskal.html",
	"shortname": "Kruskal",
	"title": "Kruskal's Algorithm for Minimum Spanning Forest",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe most efficient known primality tests are \u003cem\u003eprobabilistic\u003c/em\u003e in the sense that they use randomness and may, with some probability, mistakenly classify a composite number as prime \u0026ndash; but never a prime number as composite. Examples of this are the Miller\u0026ndash;Rabin test, the Solovay\u0026ndash;Strassen test, and (in most cases) Fermat's test.\u003c/p\u003e \u003cp\u003eThis entry defines these three tests and proves their correctness. It also develops some of the number-theoretic foundations, such as Carmichael numbers and the Jacobi symbol with an efficient executable algorithm to compute it.\u003c/p\u003e",
	"authors": [
	"Daniel Stüwe",
	"Manuel Eberl"
	],
	"date": "2019-02-11",
	- "id": 293,
	+ "id": 296,
	"link": "/entries/Probabilistic_Prime_Tests.html",
	"permalink": "/entries/Probabilistic_Prime_Tests.html",
	"shortname": "Probabilistic_Prime_Tests",
	"title": "Probabilistic Primality Testing",
	"topic_links": [
	"computer-science/algorithms/randomized",
	"mathematics/number-theory"
	],
	"topics": [
	"Computer science/Algorithms/Randomized",
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis entry formalises results from computability theory, for example recursive functions, undecidability of the halting problem, the existence of a universal Turing machine and so on. This formalisation is the AFP entry corresponding to the paper Mechanising Turing Machines and Computability Theory in Isabelle/HOL from ITP 2013. The main book used for this formalisation is by Boolos, Burgess, and Jeffrey on \u003ci\u003eComputability and Logic\u003c/i\u003e.\u003c/p\u003e \u003cp\u003eJoosten contributed by making the files ready for the AFP in 2019. His need for a formalisation of Turing Machines arose from realising that the current formalisation of saturation graphs (also in the AFP) was missing a key undecidability result present in his paper on \u003ci\u003eFinding models through graph saturation\u003c/i\u003e.\u003c/p\u003e \u003cp\u003eRegensburger contributed in 2022 by adding definitions for concepts like Turing Decidability, Turing Computability and Turing Reducibility for problem reduction. He also enhanced the result about the undecidability of the General Halting Problem given in the original AFP entry by first proving the undecidability of the Special Halting Problem and then proving its reducibility to the general problem. The original version of this AFP entry did only prove a weak form of the undecidability theorem. The main motivation behind this contribution is to make the AFP entry accessible for bachelor and master students.\u003c/p\u003e ",
	"authors": [
	"Jian Xu",
	"Xingyuan Zhang",
	"Christian Urban",
	"Sebastiaan J. C. Joosten",
	"Franz Regensburger"
	],
	"date": "2019-02-08",
	- "id": 294,
	+ "id": 297,
	"link": "/entries/Universal_Turing_Machine.html",
	"permalink": "/entries/Universal_Turing_Machine.html",
	"shortname": "Universal_Turing_Machine",
	"title": "Universal Turing Machine",
	"topic_links": [
	"logic/computability",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Logic/Computability",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Isabelle/UTP is a mechanised theory engineering toolkit based on Hoare and He’s Unifying Theories of Programming (UTP). UTP enables the creation of denotational, algebraic, and operational semantics for different programming languages using an alphabetised relational calculus. We provide a semantic embedding of the alphabetised relational calculus in Isabelle/HOL, including new type definitions, relational constructors, automated proof tactics, and accompanying algebraic laws. Isabelle/UTP can be used to both capture laws of programming for different languages, and put these fundamental theorems to work in the creation of associated verification tools, using calculi like Hoare logics. This document describes the relational core of the UTP in Isabelle/HOL.",
	"authors": [
	"Simon Foster",
	"Frank Zeyda",
	"Yakoub Nemouchi",
	"Pedro Ribeiro",
	"Burkhart Wolff"
	],
	"date": "2019-02-01",
	- "id": 295,
	+ "id": 298,
	"link": "/entries/UTP.html",
	"permalink": "/entries/UTP.html",
	"shortname": "UTP",
	"title": "Isabelle/UTP: Mechanised Theory Engineering for Unifying Theories of Programming",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry defines the set of \u003cem\u003einversions\u003c/em\u003e of a list, i.e. the pairs of indices that violate sortedness. It also proves the correctness of the well-known \u003cem\u003eO\u003c/em\u003e(\u003cem\u003en log n\u003c/em\u003e) divide-and-conquer algorithm to compute the number of inversions.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2019-02-01",
	- "id": 296,
	+ "id": 299,
	"link": "/entries/List_Inversions.html",
	"permalink": "/entries/List_Inversions.html",
	"shortname": "List_Inversions",
	"title": "The Inversions of a List",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize a proof of Motzkin's transposition theorem and Farkas' lemma in Isabelle/HOL. Our proof is based on the formalization of the simplex algorithm which, given a set of linear constraints, either returns a satisfying assignment to the problem or detects unsatisfiability. By reusing facts about the simplex algorithm we show that a set of linear constraints is unsatisfiable if and only if there is a linear combination of the constraints which evaluates to a trivially unsatisfiable inequality.",
	"authors": [
	"Ralph Bottesch",
	"Max W. Haslbeck",
	"René Thiemann"
	],
	"date": "2019-01-17",
	- "id": 297,
	+ "id": 300,
	"link": "/entries/Farkas.html",
	"permalink": "/entries/Farkas.html",
	"shortname": "Farkas",
	"title": "Farkas' Lemma and Motzkin's Transposition Theorem",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "In this formalization, I introduce a higher-order term algebra, generalizing the notions of free variables, matching, and substitution. The need arose from the work on a \u003ca href=\"http://dx.doi.org/10.1007/978-3-319-89884-1_35\"\u003everified compiler from Isabelle to CakeML\u003c/a\u003e. Terms can be thought of as consisting of a generic (free variables, constants, application) and a specific part. As example applications, this entry provides instantiations for de-Bruijn terms, terms with named variables, and \u003ca href=\"https://www.isa-afp.org/entries/Lambda_Free_RPOs.html\"\u003eBlanchette’s \u0026lambda;-free higher-order terms\u003c/a\u003e. Furthermore, I implement translation functions between de-Bruijn terms and named terms and prove their correctness.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2019-01-15",
	- "id": 298,
	+ "id": 301,
	"link": "/entries/Higher_Order_Terms.html",
	"permalink": "/entries/Higher_Order_Terms.html",
	"shortname": "Higher_Order_Terms",
	"title": "An Algebra for Higher-Order Terms",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 1
	},
	{
	"abstract": "IMP2 is a simple imperative language together with Isabelle tooling to create a program verification environment in Isabelle/HOL. The tools include a C-like syntax, a verification condition generator, and Isabelle commands for the specification of programs. The framework is modular, i.e., it allows easy reuse of already proved programs within larger programs. This entry comes with a quickstart guide and a large collection of examples, spanning basic algorithms with simple proofs to more advanced algorithms and proof techniques like data refinement. Some highlights from the examples are: \u003cul\u003e \u003cli\u003eBisection Square Root, \u003c/li\u003e \u003cli\u003eExtended Euclid, \u003c/li\u003e \u003cli\u003eExponentiation by Squaring, \u003c/li\u003e \u003cli\u003eBinary Search, \u003c/li\u003e \u003cli\u003eInsertion Sort, \u003c/li\u003e \u003cli\u003eQuicksort, \u003c/li\u003e \u003cli\u003eDepth First Search. \u003c/li\u003e \u003c/ul\u003e The abstract syntax and semantics are very simple and well-documented. They are suitable to be used in a course, as extension to the IMP language which comes with the Isabelle distribution. While this entry is limited to a simple imperative language, the ideas could be extended to more sophisticated languages.",
	"authors": [
	"Peter Lammich",
	"Simon Wimmer"
	],
	"date": "2019-01-15",
	- "id": 299,
	+ "id": 302,
	"link": "/entries/IMP2.html",
	"permalink": "/entries/IMP2.html",
	"shortname": "IMP2",
	"title": "IMP2 – Simple Program Verification in Isabelle/HOL",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Algorithms"
	],
	"used_by": 1
	},
	{
	"abstract": "When verifying a concurrent program, it is usual to assume that memory is sequentially consistent. However, most modern multiprocessors depend on store buffering for efficiency, and provide native sequential consistency only at a substantial performance penalty. To regain sequential consistency, a programmer has to follow an appropriate programming discipline. However, na\u0026iuml;ve disciplines, such as protecting all shared accesses with locks, are not flexible enough for building high-performance multiprocessor software. We present a new discipline for concurrent programming under TSO (total store order, with store buffer forwarding). It does not depend on concurrency primitives, such as locks. Instead, threads use ghost operations to acquire and release ownership of memory addresses. A thread can write to an address only if no other thread owns it, and can read from an address only if it owns it or it is shared and the thread has flushed its store buffer since it last wrote to an address it did not own. This discipline covers both coarse-grained concurrency (where data is protected by locks) as well as fine-grained concurrency (where atomic operations race to memory). We formalize this discipline in Isabelle/HOL, and prove that if every execution of a program in a system without store buffers follows the discipline, then every execution of the program with store buffers is sequentially consistent. Thus, we can show sequential consistency under TSO by ordinary assertional reasoning about the program, without having to consider store buffers at all.",
	"authors": [
	"Ernie Cohen",
	"Norbert Schirmer"
	],
	"date": "2019-01-07",
	- "id": 300,
	+ "id": 303,
	"link": "/entries/Store_Buffer_Reduction.html",
	"permalink": "/entries/Store_Buffer_Reduction.html",
	"shortname": "Store_Buffer_Reduction",
	"title": "A Reduction Theorem for Store Buffers",
	"topic_links": [
	"computer-science/concurrency"
	],
	"topics": [
	"Computer science/Concurrency"
	],
	"used_by": 0
	},
	{
	"abstract": "In this AFP entry, we formalize the core of the Document Object Model (DOM). At its core, the DOM defines a tree-like data structure for representing documents in general and HTML documents in particular. It is the heart of any modern web browser. Formalizing the key concepts of the DOM is a prerequisite for the formal reasoning over client-side JavaScript programs and for the analysis of security concepts in modern web browsers. We present a formalization of the core DOM, with focus on the node-tree and the operations defined on node-trees, in Isabelle/HOL. We use the formalization to verify the functional correctness of the most important functions defined in the DOM standard. Moreover, our formalization is 1) extensible, i.e., can be extended without the need of re-proving already proven properties and 2) executable, i.e., we can generate executable code from our specification.",
	"authors": [
	"Achim D. Brucker",
	"Michael Herzberg"
	],
	"date": "2018-12-26",
	- "id": 301,
	+ "id": 304,
	"link": "/entries/Core_DOM.html",
	"permalink": "/entries/Core_DOM.html",
	"shortname": "Core_DOM",
	"title": "A Formal Model of the Document Object Model",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "Concurrent revisions is a concurrency control model developed by Microsoft Research. It has many interesting properties that distinguish it from other well-known models such as transactional memory. One of these properties is \u003cem\u003edeterminacy\u003c/em\u003e: programs written within the model always produce the same outcome, independent of scheduling activity. The concurrent revisions model has an operational semantics, with an informal proof of determinacy. This document contains an Isabelle/HOL formalization of this semantics and the proof of determinacy.",
	"authors": [
	"Roy Overbeek"
	],
	"date": "2018-12-25",
	- "id": 302,
	+ "id": 305,
	"link": "/entries/Concurrent_Revisions.html",
	"permalink": "/entries/Concurrent_Revisions.html",
	"shortname": "Concurrent_Revisions",
	"title": "Formalization of Concurrent Revisions",
	"topic_links": [
	"computer-science/concurrency"
	],
	"topics": [
	"Computer science/Concurrency"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains the application of auto2 to verifying functional and imperative programs. Algorithms and data structures that are verified include linked lists, binary search trees, red-black trees, interval trees, priority queue, quicksort, union-find, Dijkstra's algorithm, and a sweep-line algorithm for detecting rectangle intersection. The imperative verification is based on Imperative HOL and its separation logic framework. A major goal of this work is to set up automation in order to reduce the length of proof that the user needs to provide, both for verifying functional programs and for working with separation logic.",
	"authors": [
	"Bohua Zhan"
	],
	"date": "2018-12-21",
	- "id": 303,
	+ "id": 306,
	"link": "/entries/Auto2_Imperative_HOL.html",
	"permalink": "/entries/Auto2_Imperative_HOL.html",
	"shortname": "Auto2_Imperative_HOL",
	"title": "Verifying Imperative Programs using Auto2",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Inspired by Abstract Cryptography, we extend CryptHOL, a framework for formalizing game-based proofs, with an abstract model of Random Systems and provide proof rules about their composition and equality. This foundation facilitates the formalization of Constructive Cryptography proofs, where the security of a cryptographic scheme is realized as a special form of construction in which a complex random system is built from simpler ones. This is a first step towards a fully-featured compositional framework, similar to Universal Composability framework, that supports formalization of simulation-based proofs.",
	"authors": [
	"Andreas Lochbihler",
	"S. Reza Sefidgar"
	],
	"date": "2018-12-17",
	- "id": 304,
	+ "id": 307,
	"link": "/entries/Constructive_Cryptography.html",
	"permalink": "/entries/Constructive_Cryptography.html",
	"shortname": "Constructive_Cryptography",
	"title": "Constructive Cryptography in HOL",
	"topic_links": [
	"computer-science/security/cryptography",
	"mathematics/probability-theory"
	],
	"topics": [
	"Computer science/Security/Cryptography",
	"Mathematics/Probability theory"
	],
	"used_by": 1
	},
	{
	"abstract": "These components add further fundamental order and lattice-theoretic concepts and properties to Isabelle's libraries. They follow by and large the introductory sections of the Compendium of Continuous Lattices, covering directed and filtered sets, down-closed and up-closed sets, ideals and filters, Galois connections, closure and co-closure operators. Some emphasis is on duality and morphisms between structures, as in the Compendium. To this end, three ad-hoc approaches to duality are compared.",
	"authors": [
	"Georg Struth"
	],
	"date": "2018-12-11",
	- "id": 305,
	+ "id": 308,
	"link": "/entries/Order_Lattice_Props.html",
	"permalink": "/entries/Order_Lattice_Props.html",
	"shortname": "Order_Lattice_Props",
	"title": "Properties of Orderings and Lattices",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 2
	},
	{
	"abstract": "These mathematical components formalise basic properties of quantales, together with some important models, constructions, and concepts, including quantic nuclei and conuclei.",
	"authors": [
	"Georg Struth"
	],
	"date": "2018-12-11",
	- "id": 306,
	+ "id": 309,
	"link": "/entries/Quantales.html",
	"permalink": "/entries/Quantales.html",
	"shortname": "Quantales",
	"title": "Quantales",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	- "used_by": 1
	+ "used_by": 2
	},
	{
	"abstract": "These mathematical components formalise predicate transformer semantics for programs, yet currently only for partial correctness and in the absence of faults. A first part for isotone (or monotone), Sup-preserving and Inf-preserving transformers follows Back and von Wright's approach, with additional emphasis on the quantalic structure of algebras of transformers. The second part develops Sup-preserving and Inf-preserving predicate transformers from the powerset monad, via its Kleisli category and Eilenberg-Moore algebras, with emphasis on adjunctions and dualities, as well as isomorphisms between relations, state transformers and predicate transformers.",
	"authors": [
	"Georg Struth"
	],
	"date": "2018-12-11",
	- "id": 307,
	+ "id": 310,
	"link": "/entries/Transformer_Semantics.html",
	"permalink": "/entries/Transformer_Semantics.html",
	"shortname": "Transformer_Semantics",
	"title": "Transformer Semantics",
	"topic_links": [
	"mathematics/algebra",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Mathematics/Algebra",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 1
	},
	{
	"abstract": "This Isabelle/HOL formalization refines the abstract ordered resolution prover presented in Section 4.3 of Bachmair and Ganzinger's \"Resolution Theorem Proving\" chapter in the \u003ci\u003eHandbook of Automated Reasoning\u003c/i\u003e. The result is a functional implementation of a first-order prover.",
	"authors": [
	"Anders Schlichtkrull",
	"Jasmin Christian Blanchette",
	"Dmitriy Traytel"
	],
	"date": "2018-11-23",
	- "id": 308,
	+ "id": 311,
	"link": "/entries/Functional_Ordered_Resolution_Prover.html",
	"permalink": "/entries/Functional_Ordered_Resolution_Prover.html",
	"shortname": "Functional_Ordered_Resolution_Prover",
	"title": "A Verified Functional Implementation of Bachmair and Ganzinger's Ordered Resolution Prover",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 1
	},
	{
	"abstract": "This is an Isabelle/HOL formalisation of graph saturation, closely following a \u003ca href=\"https://doi.org/10.1016/j.jlamp.2018.06.005\"\u003epaper by the author\u003c/a\u003e on graph saturation. Nine out of ten lemmas of the original paper are proven in this formalisation. The formalisation additionally includes two theorems that show the main premise of the paper: that consistency and entailment are decided through graph saturation. This formalisation does not give executable code, and it did not implement any of the optimisations suggested in the paper.",
	"authors": [
	"Sebastiaan J. C. Joosten"
	],
	"date": "2018-11-23",
	- "id": 309,
	+ "id": 312,
	"link": "/entries/Graph_Saturation.html",
	"permalink": "/entries/Graph_Saturation.html",
	"shortname": "Graph_Saturation",
	"title": "Graph Saturation",
	"topic_links": [
	"logic/rewriting",
	"mathematics/graph-theory"
	],
	"topics": [
	"Logic/Rewriting",
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Auto2 is a saturation-based heuristic prover for higher-order logic, implemented as a tactic in Isabelle. This entry contains the instantiation of auto2 for Isabelle/HOL, along with two basic examples: solutions to some of the Pelletier’s problems, and elementary number theory of primes.",
	"authors": [
	"Bohua Zhan"
	],
	"date": "2018-11-20",
	- "id": 310,
	+ "id": 313,
	"link": "/entries/Auto2_HOL.html",
	"permalink": "/entries/Auto2_HOL.html",
	"shortname": "Auto2_HOL",
	"title": "Auto2 Prover",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis article defines the combinatorial structures known as \u003cem\u003eIndependence Systems\u003c/em\u003e and \u003cem\u003eMatroids\u003c/em\u003e and provides basic concepts and theorems related to them. These structures play an important role in combinatorial optimisation, e. g. greedy algorithms such as Kruskal's algorithm. The development is based on Oxley's \u003ca href=\"http://www.math.lsu.edu/~oxley/survey4.pdf\"\u003e`What is a Matroid?'\u003c/a\u003e.\u003c/p\u003e",
	"authors": [
	"Jonas Keinholz"
	],
	"date": "2018-11-16",
	- "id": 311,
	+ "id": 314,
	"link": "/entries/Matroids.html",
	"permalink": "/entries/Matroids.html",
	"shortname": "Matroids",
	"title": "Matroids",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eWe provide a framework for automatically deriving instances for generic type classes. Our approach is inspired by Haskell's \u003ci\u003egeneric-deriving\u003c/i\u003e package and Scala's \u003ci\u003eshapeless\u003c/i\u003e library. In addition to generating the code for type class functions, we also attempt to automatically prove type class laws for these instances. As of now, however, some manual proofs are still required for recursive datatypes.\u003c/p\u003e \u003cp\u003eNote: There are already articles in the AFP that provide automatic instantiation for a number of classes. Concretely, \u003ca href=\"https://www.isa-afp.org/entries/Deriving.html\"\u003eDeriving\u003c/a\u003e allows the automatic instantiation of comparators, linear orders, equality, and hashing. \u003ca href=\"https://www.isa-afp.org/entries/Show.html\"\u003eShow\u003c/a\u003e instantiates a Haskell-style \u003ci\u003eshow\u003c/i\u003e class.\u003c/p\u003e\u003cp\u003eOur approach works for arbitrary classes (with some Isabelle/HOL overhead for each class), but a smaller set of datatypes.\u003c/p\u003e",
	"authors": [
	"Jonas Rädle",
	"Lars Hupel"
	],
	"date": "2018-11-06",
	- "id": 312,
	+ "id": 315,
	"link": "/entries/Generic_Deriving.html",
	"permalink": "/entries/Generic_Deriving.html",
	"shortname": "Generic_Deriving",
	"title": "Deriving generic class instances for datatypes",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "An ambitious ethical theory ---Alan Gewirth's \"Principle of Generic Consistency\"--- is encoded and analysed in Isabelle/HOL. Gewirth's theory has stirred much attention in philosophy and ethics and has been proposed as a potential means to bound the impact of artificial general intelligence.",
	"authors": [
	"David Fuenmayor",
	"Christoph Benzmüller"
	],
	"date": "2018-10-30",
	- "id": 313,
	+ "id": 316,
	"link": "/entries/GewirthPGCProof.html",
	"permalink": "/entries/GewirthPGCProof.html",
	"shortname": "GewirthPGCProof",
	"title": "Formalisation and Evaluation of Alan Gewirth's Proof for the Principle of Generic Consistency in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "This work is a formalization of epistemic logic. It includes proofs of soundness and completeness for the axiom system K. The completeness proof is based on the textbook \"Reasoning About Knowledge\" by Fagin, Halpern, Moses and Vardi (MIT Press 1995). The extensions of system K (T, KB, K4, S4, S5) and their completeness proofs are based on the textbook \"Modal Logic\" by Blackburn, de Rijke and Venema (Cambridge University Press 2001). Papers: \u003ca href=\"https://doi.org/10.1007/978-3-030-88853-4_1\"\u003edoi.org/10.1007/978-3-030-88853-4_1\u003c/a\u003e, \u003ca href=\"https://doi.org/10.1007/978-3-030-90138-7_2\"\u003edoi.org/10.1007/978-3-030-90138-7_2\u003c/a\u003e.",
	"authors": [
	"Asta Halkjær From"
	],
	"date": "2018-10-29",
	- "id": 314,
	+ "id": 317,
	"link": "/entries/Epistemic_Logic.html",
	"permalink": "/entries/Epistemic_Logic.html",
	"shortname": "Epistemic_Logic",
	"title": "Epistemic Logic: Completeness of Modal Logics",
	"topic_links": [
	"logic/general-logic/logics-of-knowledge-and-belief"
	],
	"topics": [
	"Logic/General logic/Logics of knowledge and belief"
	],
	"used_by": 2
	},
	{
	"abstract": "We formalize the definition and basic properties of smooth manifolds in Isabelle/HOL. Concepts covered include partition of unity, tangent and cotangent spaces, and the fundamental theorem of path integrals. We also examine some concrete manifolds such as spheres and projective spaces. The formalization makes extensive use of the analysis and linear algebra libraries in Isabelle/HOL, in particular its “types-to-sets” mechanism.",
	"authors": [
	"Fabian Immler",
	"Bohua Zhan"
	],
	"date": "2018-10-22",
	- "id": 315,
	+ "id": 318,
	"link": "/entries/Smooth_Manifolds.html",
	"permalink": "/entries/Smooth_Manifolds.html",
	"shortname": "Smooth_Manifolds",
	"title": "Smooth Manifolds",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/topology"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Topology"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization defines the Embedding Path Order (EPO) for higher-order terms without lambda-abstraction and proves many useful properties about it. In contrast to the lambda-free recursive path orders, it does not fully coincide with RPO on first-order terms, but it is compatible with arbitrary higher-order contexts.",
	"authors": [
	"Alexander Bentkamp"
	],
	"date": "2018-10-19",
	- "id": 316,
	+ "id": 319,
	"link": "/entries/Lambda_Free_EPO.html",
	"permalink": "/entries/Lambda_Free_EPO.html",
	"shortname": "Lambda_Free_EPO",
	"title": "Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis work is a formalisation of the Randomised Binary Search Trees introduced by Martínez and Roura, including definitions and correctness proofs.\u003c/p\u003e \u003cp\u003eLike randomised treaps, they are a probabilistic data structure that behaves exactly as if elements were inserted into a non-balancing BST in random order. However, unlike treaps, they only use discrete probability distributions, but their use of randomness is more complicated.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2018-10-19",
	- "id": 317,
	+ "id": 320,
	"link": "/entries/Randomised_BSTs.html",
	"permalink": "/entries/Randomised_BSTs.html",
	"shortname": "Randomised_BSTs",
	"title": "Randomised Binary Search Trees",
	"topic_links": [
	"computer-science/data-structures",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Data structures",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 0
	},
	{
	"abstract": "A completeness threshold is required to guarantee the completeness of planning as satisfiability, and bounded model checking of safety properties. One valid completeness threshold is the diameter of the underlying transition system. The diameter is the maximum element in the set of lengths of all shortest paths between pairs of states. The diameter is not calculated exactly in our setting, where the transition system is succinctly described using a (propositionally) factored representation. Rather, an upper bound on the diameter is calculated compositionally, by bounding the diameters of small abstract subsystems, and then composing those. We port a HOL4 formalisation of a compositional algorithm for computing a relatively tight upper bound on the system diameter. This compositional algorithm exploits acyclicity in the state space to achieve compositionality, and it was introduced by Abdulaziz et. al. The formalisation that we port is described as a part of another paper by Abdulaziz et. al. As a part of this porting we developed a libray about transition systems, which shall be of use in future related mechanisation efforts.",
	"authors": [
	"Friedrich Kurz",
	"Mohammad Abdulaziz"
	],
	"date": "2018-10-12",
	- "id": 318,
	+ "id": 321,
	"link": "/entries/Factored_Transition_System_Bounding.html",
	"permalink": "/entries/Factored_Transition_System_Bounding.html",
	"shortname": "Factored_Transition_System_Bounding",
	"title": "Upper Bounding Diameters of State Spaces of Factored Transition Systems",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"mathematics/graph-theory"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry shows the transcendence of \u0026pi; based on the classic proof using the fundamental theorem of symmetric polynomials first given by von Lindemann in 1882, but the formalisation mostly follows the version by Niven. The proof reuses much of the machinery developed in the AFP entry on the transcendence of \u003cem\u003ee\u003c/em\u003e.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2018-09-28",
	- "id": 319,
	+ "id": 322,
	"link": "/entries/Pi_Transcendental.html",
	"permalink": "/entries/Pi_Transcendental.html",
	"shortname": "Pi_Transcendental",
	"title": "The Transcendence of π",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eA symmetric polynomial is a polynomial in variables \u003cem\u003eX\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\u0026hellip;,\u003cem\u003eX\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e that does not discriminate between its variables, i.\u0026thinsp;e. it is invariant under any permutation of them. These polynomials are important in the study of the relationship between the coefficients of a univariate polynomial and its roots in its algebraic closure.\u003c/p\u003e \u003cp\u003eThis article provides a definition of symmetric polynomials and the elementary symmetric polynomials e\u003csub\u003e1\u003c/sub\u003e,\u0026hellip;,e\u003csub\u003en\u003c/sub\u003e and proofs of their basic properties, including three notable ones:\u003c/p\u003e \u003cul\u003e \u003cli\u003e Vieta's formula, which gives an explicit expression for the \u003cem\u003ek\u003c/em\u003e-th coefficient of a univariate monic polynomial in terms of its roots \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\u0026hellip;,\u003cem\u003ex\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e, namely \u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e = (-1)\u003csup\u003e\u003cem\u003en\u003c/em\u003e-\u003cem\u003ek\u003c/em\u003e\u003c/sup\u003e\u0026thinsp;e\u003csub\u003e\u003cem\u003en\u003c/em\u003e-\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\u0026hellip;,\u003cem\u003ex\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e).\u003c/li\u003e \u003cli\u003eSecond, the Fundamental Theorem of Symmetric Polynomials, which states that any symmetric polynomial is itself a uniquely determined polynomial combination of the elementary symmetric polynomials.\u003c/li\u003e \u003cli\u003eThird, as a corollary of the previous two, that given a polynomial over some ring \u003cem\u003eR\u003c/em\u003e, any symmetric polynomial combination of its roots is also in \u003cem\u003eR\u003c/em\u003e even when the roots are not. \u003c/ul\u003e \u003cp\u003e Both the symmetry property itself and the witness for the Fundamental Theorem are executable. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2018-09-25",
	- "id": 320,
	+ "id": 323,
	"link": "/entries/Symmetric_Polynomials.html",
	"permalink": "/entries/Symmetric_Polynomials.html",
	"shortname": "Symmetric_Polynomials",
	"title": "Symmetric Polynomials",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis article formalizes signature-based algorithms for computing Gr\u0026ouml;bner bases. Such algorithms are, in general, superior to other algorithms in terms of efficiency, and have not been formalized in any proof assistant so far. The present development is both generic, in the sense that most known variants of signature-based algorithms are covered by it, and effectively executable on concrete input thanks to Isabelle's code generator. Sample computations of benchmark problems show that the verified implementation of signature-based algorithms indeed outperforms the existing implementation of Buchberger's algorithm in Isabelle/HOL.\u003c/p\u003e \u003cp\u003eBesides total correctness of the algorithms, the article also proves that under certain conditions they a-priori detect and avoid all useless zero-reductions, and always return 'minimal' (in some sense) Gr\u0026ouml;bner bases if an input parameter is chosen in the right way.\u003c/p\u003e\u003cp\u003eThe formalization follows the recent survey article by Eder and Faug\u0026egrave;re.\u003c/p\u003e",
	"authors": [
	"Alexander Maletzky"
	],
	"date": "2018-09-20",
	- "id": 321,
	+ "id": 324,
	"link": "/entries/Signature_Groebner.html",
	"permalink": "/entries/Signature_Groebner.html",
	"shortname": "Signature_Groebner",
	"title": "Signature-Based Gröbner Basis Algorithms",
	"topic_links": [
	"mathematics/algebra",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Algebra",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis article provides a short proof of the Prime Number Theorem in several equivalent forms, most notably \u0026pi;(\u003cem\u003ex\u003c/em\u003e) ~ \u003cem\u003ex\u003c/em\u003e/ln \u003cem\u003ex\u003c/em\u003e where \u0026pi;(\u003cem\u003ex\u003c/em\u003e) is the number of primes no larger than \u003cem\u003ex\u003c/em\u003e. It also defines other basic number-theoretic functions related to primes like Chebyshev's functions \u0026thetasym; and \u0026psi; and the \u0026ldquo;\u003cem\u003en\u003c/em\u003e-th prime number\u0026rdquo; function p\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e. We also show various bounds and relationship between these functions are shown. Lastly, we derive Mertens' First and Second Theorem, i.\u0026thinsp;e. \u0026sum;\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u0026le;\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e ln \u003cem\u003ep\u003c/em\u003e/\u003cem\u003ep\u003c/em\u003e = ln \u003cem\u003ex\u003c/em\u003e + \u003cem\u003eO\u003c/em\u003e(1) and \u0026sum;\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u0026le;\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e 1/\u003cem\u003ep\u003c/em\u003e = ln ln \u003cem\u003ex\u003c/em\u003e + M + \u003cem\u003eO\u003c/em\u003e(1/ln \u003cem\u003ex\u003c/em\u003e). We also give explicit bounds for the remainder terms.\u003c/p\u003e \u003cp\u003eThe proof of the Prime Number Theorem builds on a library of Dirichlet series and analytic combinatorics. We essentially follow the presentation by Newman. The core part of the proof is a Tauberian theorem for Dirichlet series, which is proven using complex analysis and then used to strengthen Mertens' First Theorem to \u0026sum;\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u0026le;\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e ln \u003cem\u003ep\u003c/em\u003e/\u003cem\u003ep\u003c/em\u003e = ln \u003cem\u003ex\u003c/em\u003e + c + \u003cem\u003eo\u003c/em\u003e(1).\u003c/p\u003e \u003cp\u003eA variant of this proof has been formalised before by Harrison in HOL Light, and formalisations of Selberg's elementary proof exist both by Avigad \u003cem\u003eet al.\u003c/em\u003e in Isabelle and by Carneiro in Metamath. The advantage of the analytic proof is that, while it requires more powerful mathematical tools, it is considerably shorter and clearer. This article attempts to provide a short and clear formalisation of all components of that proof using the full range of mathematical machinery available in Isabelle, staying as close as possible to Newman's simple paper proof.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl",
	"Lawrence C. Paulson"
	],
	"date": "2018-09-19",
	- "id": 322,
	+ "id": 325,
	"link": "/entries/Prime_Number_Theorem.html",
	"permalink": "/entries/Prime_Number_Theorem.html",
	"shortname": "Prime_Number_Theorem",
	"title": "The Prime Number Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 3
	},
	{
	"abstract": "We develop algebras for aggregation and minimisation for weight matrices and for edge weights in graphs. We verify the correctness of Prim's and Kruskal's minimum spanning tree algorithms based on these algebras. We also show numerous instances of these algebras based on linearly ordered commutative semigroups.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2018-09-15",
	- "id": 323,
	+ "id": 326,
	"link": "/entries/Aggregation_Algebras.html",
	"permalink": "/entries/Aggregation_Algebras.html",
	"shortname": "Aggregation_Algebras",
	"title": "Aggregation Algebras",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "We develop the basic theory of Octonions, including various identities and properties of the octonions and of the octonionic product, a description of 7D isometries and representations of orthogonal transformations. To this end we first develop the theory of the vector cross product in 7 dimensions. The development of the theory of Octonions is inspired by that of the theory of Quaternions by Lawrence Paulson. However, we do not work within the type class real_algebra_1 because the octonionic product is not associative.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki"
	],
	"date": "2018-09-14",
	- "id": 324,
	+ "id": 327,
	"link": "/entries/Octonions.html",
	"permalink": "/entries/Octonions.html",
	"shortname": "Octonions",
	"title": "Octonions",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory is inspired by the HOL Light development of quaternions, but follows its own route. Quaternions are developed coinductively, as in the existing formalisation of the complex numbers. Quaternions are quickly shown to belong to the type classes of real normed division algebras and real inner product spaces. And therefore they inherit a great body of facts involving algebraic laws, limits, continuity, etc., which must be proved explicitly in the HOL Light version. The development concludes with the geometric interpretation of the product of imaginary quaternions.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2018-09-05",
	- "id": 325,
	+ "id": 328,
	"link": "/entries/Quaternions.html",
	"permalink": "/entries/Quaternions.html",
	"shortname": "Quaternions",
	"title": "Quaternions",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry is mainly about counting and approximating real roots (of a polynomial) with multiplicity. We have first formalised the Budan–Fourier theorem: given a polynomial with real coefficients, we can calculate sign variations on Fourier sequences to over-approximate the number of real roots (counting multiplicity) within an interval. When all roots are known to be real, the over-approximation becomes tight: we can utilise this theorem to count real roots exactly. It is also worth noting that Descartes' rule of sign is a direct consequence of the Budan–Fourier theorem, and has been included in this entry. In addition, we have extended previous formalised Sturm's theorem to count real roots with multiplicity, while the original Sturm's theorem only counts distinct real roots. Compared to the Budan–Fourier theorem, our extended Sturm's theorem always counts roots exactly but may suffer from greater computational cost.",
	"authors": [
	"Wenda Li"
	],
	"date": "2018-09-02",
	- "id": 326,
	+ "id": 329,
	"link": "/entries/Budan_Fourier.html",
	"permalink": "/entries/Budan_Fourier.html",
	"shortname": "Budan_Fourier",
	"title": "The Budan–Fourier Theorem and Counting Real Roots with Multiplicity",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "We present an Isabelle/HOL formalization and total correctness proof for the incremental version of the Simplex algorithm which is used in most state-of-the-art SMT solvers. It supports extraction of satisfying assignments, extraction of minimal unsatisfiable cores, incremental assertion of constraints and backtracking. The formalization relies on stepwise program refinement, starting from a simple specification, going through a number of refinement steps, and ending up in a fully executable functional implementation. Symmetries present in the algorithm are handled with special care.",
	"authors": [
	"Filip Marić",
	"Mirko Spasić",
	"René Thiemann"
	],
	"date": "2018-08-24",
	- "id": 327,
	+ "id": 330,
	"link": "/entries/Simplex.html",
	"permalink": "/entries/Simplex.html",
	"shortname": "Simplex",
	"title": "An Incremental Simplex Algorithm with Unsatisfiable Core Generation",
	"topic_links": [
	"computer-science/algorithms/optimization"
	],
	"topics": [
	"Computer science/Algorithms/Optimization"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e We formalize undecidablity results for Minsky machines. To this end, we also formalize recursive inseparability. \u003c/p\u003e\u003cp\u003e We start by proving that Minsky machines can compute arbitrary primitive recursive and recursive functions. We then show that there is a deterministic Minsky machine with one argument and two final states such that the set of inputs that are accepted in one state is recursively inseparable from the set of inputs that are accepted in the other state. \u003c/p\u003e\u003cp\u003e As a corollary, the set of Minsky configurations that reach the first state but not the second recursively inseparable from the set of Minsky configurations that reach the second state but not the first. In particular both these sets are undecidable. \u003c/p\u003e\u003cp\u003e We do \u003cem\u003enot\u003c/em\u003e prove that recursive functions can simulate Minsky machines. \u003c/p\u003e",
	"authors": [
	"Bertram Felgenhauer"
	],
	"date": "2018-08-14",
	- "id": 328,
	+ "id": 331,
	"link": "/entries/Minsky_Machines.html",
	"permalink": "/entries/Minsky_Machines.html",
	"shortname": "Minsky_Machines",
	"title": "Minsky Machines",
	"topic_links": [
	"logic/computability"
	],
	"topics": [
	"Logic/Computability"
	],
	"used_by": 0
	},
	{
	"abstract": "We have formalized the computation of fair prices for derivative products in discrete financial models. As an application, we derive a way to compute fair prices of derivative products in the Cox-Ross-Rubinstein model of a financial market, thus completing the work that was presented in this \u003ca href=\"https://hal.archives-ouvertes.fr/hal-01562944\"\u003epaper\u003c/a\u003e.",
	"authors": [
	"Mnacho Echenim"
	],
	"date": "2018-07-16",
	- "id": 329,
	+ "id": 332,
	"link": "/entries/DiscretePricing.html",
	"permalink": "/entries/DiscretePricing.html",
	"shortname": "DiscretePricing",
	"title": "Pricing in discrete financial models",
	"topic_links": [
	"mathematics/probability-theory",
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "Utility functions form an essential part of game theory and economics. In order to guarantee the existence of utility functions most of the time sufficient properties are assumed in an axiomatic manner. One famous and very common set of such assumptions is that of expected utility theory. Here, the rationality, continuity, and independence of preferences is assumed. The von-Neumann-Morgenstern Utility theorem shows that these assumptions are necessary and sufficient for an expected utility function to exists. This theorem was proven by Neumann and Morgenstern in “Theory of Games and Economic Behavior” which is regarded as one of the most influential works in game theory. The formalization includes formal definitions of the underlying concepts including continuity and independence of preferences.",
	"authors": [
	"Julian Parsert",
	"Cezary Kaliszyk"
	],
	"date": "2018-07-04",
	- "id": 330,
	+ "id": 333,
	"link": "/entries/Neumann_Morgenstern_Utility.html",
	"permalink": "/entries/Neumann_Morgenstern_Utility.html",
	"shortname": "Neumann_Morgenstern_Utility",
	"title": "Von-Neumann-Morgenstern Utility Theorem",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e This article gives the basic theory of Pell's equation \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e = 1 + \u003cem\u003eD\u003c/em\u003e\u0026thinsp;\u003cem\u003ey\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e, where \u003cem\u003eD\u003c/em\u003e\u0026thinsp;\u0026isin;\u0026thinsp;\u0026#8469; is a parameter and \u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e are integer variables. \u003c/p\u003e \u003cp\u003e The main result that is proven is the following: If \u003cem\u003eD\u003c/em\u003e is not a perfect square, then there exists a \u003cem\u003efundamental solution\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) that is not the trivial solution (1, 0) and which generates all other solutions (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) in the sense that there exists some \u003cem\u003en\u003c/em\u003e\u0026thinsp;\u0026isin;\u0026thinsp;\u0026#8469; such that \|\u003cem\u003ex\u003c/em\u003e\| + \|\u003cem\u003ey\u003c/em\u003e\|\u0026thinsp;\u0026radic;\u003cspan style=\"text-decoration: overline\"\u003e\u003cem\u003eD\u003c/em\u003e\u003c/span\u003e = (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e + \u003cem\u003ey\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;\u0026radic;\u003cspan style=\"text-decoration: overline\"\u003e\u003cem\u003eD\u003c/em\u003e\u003c/span\u003e)\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e. This also implies that the set of solutions is infinite, and it gives us an explicit and executable characterisation of all the solutions. \u003c/p\u003e \u003cp\u003e Based on this, simple executable algorithms for computing the fundamental solution and the infinite sequence of all non-negative solutions are also provided. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2018-06-23",
	- "id": 331,
	+ "id": 334,
	"link": "/entries/Pell.html",
	"permalink": "/entries/Pell.html",
	"shortname": "Pell",
	"title": "Pell's Equation",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize the basics of projective geometry. In particular, we give a proof of the so-called Hessenberg's theorem in projective plane geometry. We also provide a proof of the so-called Desargues's theorem based on an axiomatization of (higher) projective space geometry using the notion of rank of a matroid. This last approach allows to handle incidence relations in an homogeneous way dealing only with points and without the need of talking explicitly about lines, planes or any higher entity.",
	"authors": [
	"Anthony Bordg"
	],
	"date": "2018-06-14",
	- "id": 332,
	+ "id": 335,
	"link": "/entries/Projective_Geometry.html",
	"permalink": "/entries/Projective_Geometry.html",
	"shortname": "Projective_Geometry",
	"title": "Projective Geometry",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the localization of a commutative ring R with respect to a multiplicative subset (i.e. a submonoid of R seen as a multiplicative monoid). This localization is itself a commutative ring and we build the natural homomorphism of rings from R to its localization.",
	"authors": [
	"Anthony Bordg"
	],
	"date": "2018-06-14",
	- "id": 333,
	+ "id": 336,
	"link": "/entries/Localization_Ring.html",
	"permalink": "/entries/Localization_Ring.html",
	"shortname": "Localization_Ring",
	"title": "The Localization of a Commutative Ring",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry provides a formalization of the abstract theory of ample set partial order reduction. The formalization includes transition systems with actions, trace theory, as well as basics on finite, infinite, and lazy sequences. We also provide a basic framework for static analysis on concurrent systems with respect to the ample set condition.",
	"authors": [
	"Julian Brunner"
	],
	"date": "2018-06-05",
	- "id": 334,
	+ "id": 337,
	"link": "/entries/Partial_Order_Reduction.html",
	"permalink": "/entries/Partial_Order_Reduction.html",
	"shortname": "Partial_Order_Reduction",
	"title": "Partial Order Reduction",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "This article formalizes recursive algorithms for the construction of optimal binary search trees given fixed access frequencies. We follow Knuth (1971), Yao (1980) and Mehlhorn (1984). The algorithms are memoized with the help of the AFP article \u003ca href=\"Monad_Memo_DP.html\"\u003eMonadification, Memoization and Dynamic Programming\u003c/a\u003e, thus yielding dynamic programming algorithms.",
	"authors": [
	"Tobias Nipkow",
	"Dániel Somogyi"
	],
	"date": "2018-05-27",
	- "id": 335,
	+ "id": 338,
	"link": "/entries/Optimal_BST.html",
	"permalink": "/entries/Optimal_BST.html",
	"shortname": "Optimal_BST",
	"title": "Optimal Binary Search Trees",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains a formalization of hidden Markov models [3] based on Johannes Hölzl's formalization of discrete time Markov chains [1]. The basic definitions are provided and the correctness of two main (dynamic programming) algorithms for hidden Markov models is proved: the forward algorithm for computing the likelihood of an observed sequence, and the Viterbi algorithm for decoding the most probable hidden state sequence. The Viterbi algorithm is made executable including memoization. Hidden markov models have various applications in natural language processing. For an introduction see Jurafsky and Martin [2].",
	"authors": [
	"Simon Wimmer"
	],
	"date": "2018-05-25",
	- "id": 336,
	+ "id": 339,
	"link": "/entries/Hidden_Markov_Models.html",
	"permalink": "/entries/Hidden_Markov_Models.html",
	"shortname": "Hidden_Markov_Models",
	"title": "Hidden Markov Models",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/algorithms"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of probabilistic timed automata (PTA) for which we try to follow the formula MDP + TA = PTA as far as possible: our work starts from our existing formalizations of Markov decision processes (MDP) and timed automata (TA) and combines them modularly. We prove the fundamental result for probabilistic timed automata: the region construction that is known from timed automata carries over to the probabilistic setting. In particular, this allows us to prove that minimum and maximum reachability probabilities can be computed via a reduction to MDP model checking, including the case where one wants to disregard unrealizable behavior. Further information can be found in our ITP paper [2].",
	"authors": [
	"Simon Wimmer",
	"Johannes Hölzl"
	],
	"date": "2018-05-24",
	- "id": 337,
	+ "id": 340,
	"link": "/entries/Probabilistic_Timed_Automata.html",
	"permalink": "/entries/Probabilistic_Timed_Automata.html",
	"shortname": "Probabilistic_Timed_Automata",
	"title": "Probabilistic Timed Automata",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This document provides a concise overview on the core results of our previous work on the exploration of axioms systems for category theory. Extending the previous studies (http://arxiv.org/abs/1609.01493) we include one further axiomatic theory in our experiments. This additional theory has been suggested by Mac Lane in 1948. We show that the axioms proposed by Mac Lane are equivalent to the ones we studied before, which includes an axioms set suggested by Scott in the 1970s and another axioms set proposed by Freyd and Scedrov in 1990, which we slightly modified to remedy a minor technical issue.",
	"authors": [
	"Christoph Benzmüller",
	"Dana Scott"
	],
	"date": "2018-05-23",
	- "id": 338,
	+ "id": 341,
	"link": "/entries/AxiomaticCategoryTheory.html",
	"permalink": "/entries/AxiomaticCategoryTheory.html",
	"shortname": "AxiomaticCategoryTheory",
	"title": "Axiom Systems for Category Theory in Free Logic",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize with Isabelle/HOL a proof of a theorem by J. Hancl asserting the irrationality of the sum of a series consisting of rational numbers, built up by sequences that fulfill certain properties. Even though the criterion is a number theoretic result, the proof makes use only of analytical arguments. We also formalize a corollary of the theorem for a specific series fulfilling the assumptions of the theorem.",
	"authors": [
	"Angeliki Koutsoukou-Argyraki",
	"Wenda Li"
	],
	"date": "2018-05-23",
	- "id": 339,
	+ "id": 342,
	"link": "/entries/Irrationality_J_Hancl.html",
	"permalink": "/entries/Irrationality_J_Hancl.html",
	"shortname": "Irrationality_J_Hancl",
	"title": "Irrational Rapidly Convergent Series",
	"topic_links": [
	"mathematics/number-theory",
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Number theory",
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a lightweight framework for the automatic verified (functional or imperative) memoization of recursive functions. Our tool can turn a pure Isabelle/HOL function definition into a monadified version in a state monad or the Imperative HOL heap monad, and prove a correspondence theorem. We provide a variety of memory implementations for the two types of monads. A number of simple techniques allow us to achieve bottom-up computation and space-efficient memoization. The framework’s utility is demonstrated on a number of representative dynamic programming problems. A detailed description of our work can be found in the accompanying paper [2].",
	"authors": [
	"Simon Wimmer",
	"Shuwei Hu",
	"Tobias Nipkow"
	],
	"date": "2018-05-22",
	- "id": 340,
	+ "id": 343,
	"link": "/entries/Monad_Memo_DP.html",
	"permalink": "/entries/Monad_Memo_DP.html",
	"shortname": "Monad_Memo_DP",
	"title": "Monadification, Memoization and Dynamic Programming",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Functional programming"
	],
	"used_by": 2
	},
	{
	"abstract": "We introduce OpSets, an executable framework for specifying and reasoning about the semantics of replicated datatypes that provide eventual consistency in a distributed system, and for mechanically verifying algorithms that implement these datatypes. Our approach is simple but expressive, allowing us to succinctly specify a variety of abstract datatypes, including maps, sets, lists, text, graphs, trees, and registers. Our datatypes are also composable, enabling the construction of complex data structures. To demonstrate the utility of OpSets for analysing replication algorithms, we highlight an important correctness property for collaborative text editing that has traditionally been overlooked; algorithms that do not satisfy this property can exhibit awkward interleaving of text. We use OpSets to specify this correctness property and prove that although one existing replication algorithm satisfies this property, several other published algorithms do not.",
	"authors": [
	"Martin Kleppmann",
	"Victor B. F. Gomes",
	"Dominic P. Mulligan",
	"Alastair R. Beresford"
	],
	"date": "2018-05-10",
	- "id": 341,
	+ "id": 344,
	"link": "/entries/OpSets.html",
	"permalink": "/entries/OpSets.html",
	"shortname": "OpSets",
	"title": "OpSets: Sequential Specifications for Replicated Datatypes",
	"topic_links": [
	"computer-science/algorithms/distributed",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms/Distributed",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "The \"Modular Assembly Kit for Security Properties\" (MAKS) is a framework for both the definition and verification of possibilistic information-flow security properties at the specification-level. MAKS supports the uniform representation of a wide range of possibilistic information-flow properties and provides support for the verification of such properties via unwinding results and compositionality results. We provide a formalization of this framework in Isabelle/HOL.",
	"authors": [
	"Oliver Bračevac",
	"Richard Gay",
	"Sylvia Grewe",
	"Heiko Mantel",
	"Henning Sudbrock",
	"Markus Tasch"
	],
	"date": "2018-05-07",
	- "id": 342,
	+ "id": 345,
	"link": "/entries/Modular_Assembly_Kit_Security.html",
	"permalink": "/entries/Modular_Assembly_Kit_Security.html",
	"shortname": "Modular_Assembly_Kit_Security",
	"title": "An Isabelle/HOL Formalization of the Modular Assembly Kit for Security Properties",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a mechanised specification of the WebAssembly language, drawn mainly from the previously published paper formalisation of Haas et al. Also included is a full proof of soundness of the type system, together with a verified type checker and interpreter. We include only a partial procedure for the extraction of the type checker and interpreter here. For more details, please see our paper in CPP 2018.",
	"authors": [
	"Conrad Watt"
	],
	"date": "2018-04-29",
	- "id": 343,
	+ "id": 346,
	"link": "/entries/WebAssembly.html",
	"permalink": "/entries/WebAssembly.html",
	"shortname": "WebAssembly",
	"title": "WebAssembly",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003ca href=\"http://www.pm.inf.ethz.ch/research/verifythis.html\"\u003eVerifyThis 2018\u003c/a\u003e was a program verification competition associated with ETAPS 2018. It was the 7th event in the VerifyThis competition series. In this entry, we present polished and completed versions of our solutions that we created during the competition.",
	"authors": [
	"Peter Lammich",
	"Simon Wimmer"
	],
	"date": "2018-04-27",
	- "id": 344,
	+ "id": 347,
	"link": "/entries/VerifyThis2018.html",
	"permalink": "/entries/VerifyThis2018.html",
	"shortname": "VerifyThis2018",
	"title": "VerifyThis 2018 - Polished Isabelle Solutions",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "Bounded natural functors (BNFs) provide a modular framework for the construction of (co)datatypes in higher-order logic. Their functorial operations, the mapper and relator, are restricted to a subset of the parameters, namely those where recursion can take place. For certain applications, such as free theorems, data refinement, quotients, and generalised rewriting, it is desirable that these operations do not ignore the other parameters. In this article, we formalise the generalisation BNF\u003csub\u003eCC\u003c/sub\u003e that extends the mapper and relator to covariant and contravariant parameters. We show that \u003col\u003e \u003cli\u003e BNF\u003csub\u003eCC\u003c/sub\u003es are closed under functor composition and least and greatest fixpoints,\u003c/li\u003e \u003cli\u003e subtypes inherit the BNF\u003csub\u003eCC\u003c/sub\u003e structure under conditions that generalise those for the BNF case, and\u003c/li\u003e \u003cli\u003e BNF\u003csub\u003eCC\u003c/sub\u003es preserve quotients under mild conditions.\u003c/li\u003e \u003c/ol\u003e These proofs are carried out for abstract BNF\u003csub\u003eCC\u003c/sub\u003es similar to the AFP entry BNF Operations. In addition, we apply the BNF\u003csub\u003eCC\u003c/sub\u003e theory to several concrete functors.",
	"authors": [
	"Andreas Lochbihler",
	"Joshua Schneider"
	],
	"date": "2018-04-24",
	- "id": 345,
	+ "id": 348,
	"link": "/entries/BNF_CC.html",
	"permalink": "/entries/BNF_CC.html",
	"shortname": "BNF_CC",
	"title": "Bounded Natural Functors with Covariance and Contravariance",
	"topic_links": [
	"computer-science/functional-programming",
	"tools"
	],
	"topics": [
	"Computer science/Functional programming",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis formalisation contains the proof that there is no anonymous Social Choice Function for at least three agents and alternatives that fulfils both Pareto-Efficiency and Fishburn-Strategyproofness. It was derived from a proof of \u003ca href=\"http://dss.in.tum.de/files/brandt-research/stratset.pdf\"\u003eBrandt \u003cem\u003eet al.\u003c/em\u003e\u003c/a\u003e, which relies on an unverified translation of a fixed finite instance of the original problem to SAT. This Isabelle proof contains a machine-checked version of both the statement for exactly three agents and alternatives and the lifting to the general case.\u003c/p\u003e",
	"authors": [
	"Felix Brandt",
	"Manuel Eberl",
	"Christian Saile",
	"Christian Stricker"
	],
	"date": "2018-03-22",
	- "id": 346,
	+ "id": 349,
	"link": "/entries/Fishburn_Impossibility.html",
	"permalink": "/entries/Fishburn_Impossibility.html",
	"shortname": "Fishburn_Impossibility",
	"title": "The Incompatibility of Fishburn-Strategyproofness and Pareto-Efficiency",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory provides a verified implementation of weight-balanced trees following the work of \u003ca href=\"https://doi.org/10.1017/S0956796811000104\"\u003eHirai and Yamamoto\u003c/a\u003e who proved that all parameters in a certain range are valid, i.e. guarantee that insertion and deletion preserve weight-balance. Instead of a general theorem we provide parameterized proofs of preservation of the invariant that work for many (all?) valid parameters.",
	"authors": [
	"Tobias Nipkow",
	"Stefan Dirix"
	],
	"date": "2018-03-13",
	- "id": 347,
	+ "id": 350,
	"link": "/entries/Weight_Balanced_Trees.html",
	"permalink": "/entries/Weight_Balanced_Trees.html",
	"shortname": "Weight_Balanced_Trees",
	"title": "Weight-Balanced Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "CakeML is a functional programming language with a proven-correct compiler and runtime system. This entry contains an unofficial version of the CakeML semantics that has been exported from the Lem specifications to Isabelle. Additionally, there are some hand-written theory files that adapt the exported code to Isabelle and port proofs from the HOL4 formalization, e.g. termination and equivalence proofs.",
	"authors": [
	"Lars Hupel",
	"Yu Zhang"
	],
	"date": "2018-03-12",
	- "id": 348,
	+ "id": 351,
	"link": "/entries/CakeML.html",
	"permalink": "/entries/CakeML.html",
	"shortname": "CakeML",
	"title": "CakeML",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 1
	},
	{
	"abstract": "The following document formalizes and verifies several architectural design patterns. Each pattern specification is formalized in terms of a locale where the locale assumptions correspond to the assumptions which a pattern poses on an architecture. Thus, pattern specifications may build on top of each other by interpreting the corresponding locale. A pattern is verified using the framework provided by the AFP entry Dynamic Architectures. Currently, the document consists of formalizations of 4 different patterns: the singleton, the publisher subscriber, the blackboard pattern, and the blockchain pattern. Thereby, the publisher component of the publisher subscriber pattern is modeled as an instance of the singleton pattern and the blackboard pattern is modeled as an instance of the publisher subscriber pattern. In general, this entry provides the first steps towards an overall theory of architectural design patterns.",
	"authors": [
	"Diego Marmsoler"
	],
	"date": "2018-03-01",
	- "id": 349,
	+ "id": 352,
	"link": "/entries/Architectural_Design_Patterns.html",
	"permalink": "/entries/Architectural_Design_Patterns.html",
	"shortname": "Architectural_Design_Patterns",
	"title": "A Theory of Architectural Design Patterns",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We study three different Hoare logics for reasoning about time bounds of imperative programs and formalize them in Isabelle/HOL: a classical Hoare like logic due to Nielson, a logic with potentials due to Carbonneaux \u003ci\u003eet al.\u003c/i\u003e and a \u003ci\u003eseparation logic\u003c/i\u003e following work by Atkey, Chaguérand and Pottier. These logics are formally shown to be sound and complete. Verification condition generators are developed and are shown sound and complete too. We also consider variants of the systems where we abstract from multiplicative constants in the running time bounds, thus supporting a big-O style of reasoning. Finally we compare the expressive power of the three systems.",
	"authors": [
	"Maximilian P. L. Haslbeck",
	"Tobias Nipkow"
	],
	"date": "2018-02-26",
	- "id": 350,
	+ "id": 353,
	"link": "/entries/Hoare_Time.html",
	"permalink": "/entries/Hoare_Time.html",
	"shortname": "Hoare_Time",
	"title": "Hoare Logics for Time Bounds",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "Short vectors in lattices and factors of integer polynomials are related. Each factor of an integer polynomial belongs to a certain lattice. When factoring polynomials, the condition that we are looking for an irreducible polynomial means that we must look for a small element in a lattice, which can be done by a basis reduction algorithm. In this development we formalize this connection and thereby one main application of the LLL basis reduction algorithm: an algorithm to factor square-free integer polynomials which runs in polynomial time. The work is based on our previous Berlekamp–Zassenhaus development, where the exponential reconstruction phase has been replaced by the polynomial-time basis reduction algorithm. Thanks to this formalization we found a serious flaw in a textbook.",
	"authors": [
	"Jose Divasón",
	"Sebastiaan J. C. Joosten",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2018-02-06",
	- "id": 351,
	+ "id": 354,
	"link": "/entries/LLL_Factorization.html",
	"permalink": "/entries/LLL_Factorization.html",
	"shortname": "LLL_Factorization",
	"title": "A verified factorization algorithm for integer polynomials with polynomial complexity",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize basic results on first-order terms, including matching and a first-order unification algorithm, as well as well-foundedness of the subsumption order. This entry is part of the \u003ci\u003eIsabelle Formalization of Rewriting\u003c/i\u003e \u003ca href=\"http://cl-informatik.uibk.ac.at/isafor\"\u003eIsaFoR\u003c/a\u003e, where first-order terms are omni-present: the unification algorithm is used to certify several confluence and termination techniques, like critical-pair computation and dependency graph approximations; and the subsumption order is a crucial ingredient for completion.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2018-02-06",
	- "id": 352,
	+ "id": 355,
	"link": "/entries/First_Order_Terms.html",
	"permalink": "/entries/First_Order_Terms.html",
	"shortname": "First_Order_Terms",
	"title": "First-Order Terms",
	"topic_links": [
	"logic/rewriting",
	"computer-science/algorithms"
	],
	"topics": [
	"Logic/Rewriting",
	"Computer science/Algorithms"
	],
	"used_by": 6
	},
	{
	"abstract": "\u003cp\u003e This entry provides the definitions and basic properties of the complex and real error function erf and the complementary error function erfc. Additionally, it gives their full asymptotic expansions. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2018-02-06",
	- "id": 353,
	+ "id": 356,
	"link": "/entries/Error_Function.html",
	"permalink": "/entries/Error_Function.html",
	"shortname": "Error_Function",
	"title": "The Error Function",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e A Treap is a binary tree whose nodes contain pairs consisting of some payload and an associated priority. It must have the search-tree property w.r.t. the payloads and the heap property w.r.t. the priorities. Treaps are an interesting data structure that is related to binary search trees (BSTs) in the following way: if one forgets all the priorities of a treap, the resulting BST is exactly the same as if one had inserted the elements into an empty BST in order of ascending priority. This means that a treap behaves like a BST where we can pretend the elements were inserted in a different order from the one in which they were actually inserted. \u003c/p\u003e \u003cp\u003e In particular, by choosing these priorities at random upon insertion of an element, we can pretend that we inserted the elements in \u003cem\u003erandom order\u003c/em\u003e, so that the shape of the resulting tree is that of a random BST no matter in what order we insert the elements. This is the main result of this formalisation.\u003c/p\u003e",
	"authors": [
	"Max W. Haslbeck",
	"Manuel Eberl",
	"Tobias Nipkow"
	],
	"date": "2018-02-06",
	- "id": 354,
	+ "id": 357,
	"link": "/entries/Treaps.html",
	"permalink": "/entries/Treaps.html",
	"shortname": "Treaps",
	"title": "Treaps",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "The Lenstra-Lenstra-Lovász basis reduction algorithm, also known as LLL algorithm, is an algorithm to find a basis with short, nearly orthogonal vectors of an integer lattice. Thereby, it can also be seen as an approximation to solve the shortest vector problem (SVP), which is an NP-hard problem, where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm also possesses many applications in diverse fields of computer science, from cryptanalysis to number theory, but it is specially well-known since it was used to implement the first polynomial-time algorithm to factor polynomials. In this work we present the first mechanized soundness proof of the LLL algorithm to compute short vectors in lattices. The formalization follows a textbook by von zur Gathen and Gerhard.",
	"authors": [
	"Ralph Bottesch",
	"Jose Divasón",
	"Max W. Haslbeck",
	"Sebastiaan J. C. Joosten",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2018-02-02",
	- "id": 355,
	+ "id": 358,
	"link": "/entries/LLL_Basis_Reduction.html",
	"permalink": "/entries/LLL_Basis_Reduction.html",
	"shortname": "LLL_Basis_Reduction",
	"title": "A verified LLL algorithm",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Algebra"
	],
	"used_by": 4
	},
	{
	"abstract": "This Isabelle/HOL formalization covers Sections 2 to 4 of Bachmair and Ganzinger's \"Resolution Theorem Proving\" chapter in the \u003cem\u003eHandbook of Automated Reasoning\u003c/em\u003e. This includes soundness and completeness of unordered and ordered variants of ground resolution with and without literal selection, the standard redundancy criterion, a general framework for refutational theorem proving, and soundness and completeness of an abstract first-order prover.",
	"authors": [
	"Anders Schlichtkrull",
	"Jasmin Christian Blanchette",
	"Dmitriy Traytel",
	"Uwe Waldmann"
	],
	"date": "2018-01-18",
	- "id": 356,
	+ "id": 359,
	"link": "/entries/Ordered_Resolution_Prover.html",
	"permalink": "/entries/Ordered_Resolution_Prover.html",
	"shortname": "Ordered_Resolution_Prover",
	"title": "Formalization of Bachmair and Ganzinger's Ordered Resolution Prover",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 5
	},
	{
	"abstract": "A geodesic metric space is Gromov hyperbolic if all its geodesic triangles are thin, i.e., every side is contained in a fixed thickening of the two other sides. While this definition looks innocuous, it has proved extremely important and versatile in modern geometry since its introduction by Gromov. We formalize the basic classical properties of Gromov hyperbolic spaces, notably the Morse lemma asserting that quasigeodesics are close to geodesics, the invariance of hyperbolicity under quasi-isometries, we define and study the Gromov boundary and its associated distance, and prove that a quasi-isometry between Gromov hyperbolic spaces extends to a homeomorphism of the boundaries. We also prove a less classical theorem, by Bonk and Schramm, asserting that a Gromov hyperbolic space embeds isometrically in a geodesic Gromov-hyperbolic space. As the original proof uses a transfinite sequence of Cauchy completions, this is an interesting formalization exercise. Along the way, we introduce basic material on isometries, quasi-isometries, Lipschitz maps, geodesic spaces, the Hausdorff distance, the Cauchy completion of a metric space, and the exponential on extended real numbers.",
	"authors": [
	"Sebastien Gouezel"
	],
	"date": "2018-01-16",
	- "id": 357,
	+ "id": 360,
	"link": "/entries/Gromov_Hyperbolicity.html",
	"permalink": "/entries/Gromov_Hyperbolicity.html",
	"shortname": "Gromov_Hyperbolicity",
	"title": "Gromov Hyperbolicity",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise a statement of Green’s theorem—the first formalisation to our knowledge—in Isabelle/HOL. The theorem statement that we formalise is enough for most applications, especially in physics and engineering. Our formalisation is made possible by a novel proof that avoids the ubiquitous line integral cancellation argument. This eliminates the need to formalise orientations and region boundaries explicitly with respect to the outwards-pointing normal vector. Instead we appeal to a homological argument about equivalences between paths.",
	"authors": [
	"Mohammad Abdulaziz",
	"Lawrence C. Paulson"
	],
	"date": "2018-01-11",
	- "id": 358,
	+ "id": 361,
	"link": "/entries/Green.html",
	"permalink": "/entries/Green.html",
	"shortname": "Green",
	"title": "An Isabelle/HOL formalisation of Green's Theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formally verified implementation of multivariate Taylor models. Taylor models are a form of rigorous polynomial approximation, consisting of an approximation polynomial based on Taylor expansions, combined with a rigorous bound on the approximation error. Taylor models were introduced as a tool to mitigate the dependency problem of interval arithmetic. Our implementation automatically computes Taylor models for the class of elementary functions, expressed by composition of arithmetic operations and basic functions like exp, sin, or square root.",
	"authors": [
	"Christoph Traut",
	"Fabian Immler"
	],
	"date": "2018-01-08",
	- "id": 359,
	+ "id": 362,
	"link": "/entries/Taylor_Models.html",
	"permalink": "/entries/Taylor_Models.html",
	"shortname": "Taylor_Models",
	"title": "Taylor Models",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"computer-science/data-structures",
	"mathematics/analysis",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Computer science/Data structures",
	"Mathematics/Analysis",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry shows that the falling factorial of a sum can be computed with an expression using binomial coefficients and the falling factorial of its summands. The entry provides three different proofs: a combinatorial proof, an induction proof and an algebraic proof using the Vandermonde identity. The three formalizations try to follow their informal presentations from a Mathematics Stack Exchange page as close as possible. The induction and algebraic formalization end up to be very close to their informal presentation, whereas the combinatorial proof first requires the introduction of list interleavings, and significant more detail than its informal presentation.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2017-12-22",
	- "id": 360,
	+ "id": 363,
	"link": "/entries/Falling_Factorial_Sum.html",
	"permalink": "/entries/Falling_Factorial_Sum.html",
	"shortname": "Falling_Factorial_Sum",
	"title": "The Falling Factorial of a Sum",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of Dirichlet characters and Dirichlet \u003cem\u003eL\u003c/em\u003e-functions including proofs of their basic properties \u0026ndash; most notably their analyticity, their areas of convergence, and their non-vanishing for \u0026Re;(s) \u0026ge; 1. All of this is built in a very high-level style using Dirichlet series. The proof of the non-vanishing follows a very short and elegant proof by Newman, which we attempt to reproduce faithfully in a similar level of abstraction in Isabelle.\u003c/p\u003e \u003cp\u003eThis also leads to a relatively short proof of Dirichlet’s Theorem, which states that, if \u003cem\u003eh\u003c/em\u003e and \u003cem\u003en\u003c/em\u003e are coprime, there are infinitely many primes \u003cem\u003ep\u003c/em\u003e with \u003cem\u003ep\u003c/em\u003e \u0026equiv; \u003cem\u003eh\u003c/em\u003e (mod \u003cem\u003en\u003c/em\u003e).\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-12-21",
	- "id": 361,
	+ "id": 364,
	"link": "/entries/Dirichlet_L.html",
	"permalink": "/entries/Dirichlet_L.html",
	"shortname": "Dirichlet_L",
	"title": "Dirichlet L-Functions and Dirichlet's Theorem",
	"topic_links": [
	"mathematics/number-theory",
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Number theory",
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis article provides a formalisation of Snyder’s simple and elegant proof of the Mason\u0026ndash;Stothers theorem, which is the polynomial analogue of the famous abc Conjecture for integers. Remarkably, Snyder found this very elegant proof when he was still a high-school student.\u003c/p\u003e \u003cp\u003eIn short, the statement of the theorem is that three non-zero coprime polynomials \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e over a field which sum to 0 and do not all have vanishing derivatives fulfil max{deg(\u003cem\u003eA\u003c/em\u003e), deg(\u003cem\u003eB\u003c/em\u003e), deg(\u003cem\u003eC\u003c/em\u003e)} \u003c deg(rad(\u003cem\u003eABC\u003c/em\u003e)) where the rad(\u003cem\u003eP\u003c/em\u003e) denotes the \u003cem\u003eradical\u003c/em\u003e of \u003cem\u003eP\u003c/em\u003e, i.\u0026thinsp;e. the product of all unique irreducible factors of \u003cem\u003eP\u003c/em\u003e.\u003c/p\u003e \u003cp\u003eThis theorem also implies a kind of polynomial analogue of Fermat’s Last Theorem for polynomials: except for trivial cases, \u003cem\u003eA\u003csup\u003en\u003c/sup\u003e\u003c/em\u003e + \u003cem\u003eB\u003csup\u003en\u003c/sup\u003e\u003c/em\u003e + \u003cem\u003eC\u003csup\u003en\u003c/sup\u003e\u003c/em\u003e = 0 implies n\u0026nbsp;\u0026le;\u0026nbsp;2 for coprime polynomials \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e over a field.\u003c/em\u003e\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-12-21",
	- "id": 362,
	+ "id": 365,
	"link": "/entries/Mason_Stothers.html",
	"permalink": "/entries/Mason_Stothers.html",
	"shortname": "Mason_Stothers",
	"title": "The Mason–Stothers Theorem",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis entry provides an executable functional implementation of the Median-of-Medians algorithm for selecting the \u003cem\u003ek\u003c/em\u003e-th smallest element of an unsorted list deterministically in linear time. The size bounds for the recursive call that lead to the linear upper bound on the run-time of the algorithm are also proven. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-12-21",
	- "id": 363,
	+ "id": 366,
	"link": "/entries/Median_Of_Medians_Selection.html",
	"permalink": "/entries/Median_Of_Medians_Selection.html",
	"shortname": "Median_Of_Medians_Selection",
	"title": "The Median-of-Medians Selection Algorithm",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry formalizes the closure property of bounded natural functors (BNFs) under seven operations. These operations and the corresponding proofs constitute the core of Isabelle's (co)datatype package. To be close to the implemented tactics, the proofs are deliberately formulated as detailed apply scripts. The (co)datatypes together with (co)induction principles and (co)recursors are byproducts of the fixpoint operations LFP and GFP. Composition of BNFs is subdivided into four simpler operations: Compose, Kill, Lift, and Permute. The N2M operation provides mutual (co)induction principles and (co)recursors for nested (co)datatypes.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2017-12-19",
	- "id": 364,
	+ "id": 367,
	"link": "/entries/BNF_Operations.html",
	"permalink": "/entries/BNF_Operations.html",
	"shortname": "BNF_Operations",
	"title": "Operations on Bounded Natural Functors",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "The Knuth-Morris-Pratt algorithm is often used to show that the problem of finding a string \u003ci\u003es\u003c/i\u003e in a text \u003ci\u003et\u003c/i\u003e can be solved deterministically in \u003ci\u003eO(\|s\| + \|t\|)\u003c/i\u003e time. We use the Isabelle Refinement Framework to formulate and verify the algorithm. Via refinement, we apply some optimisations and finally use the \u003cem\u003eSepref\u003c/em\u003e tool to obtain executable code in \u003cem\u003eImperative/HOL\u003c/em\u003e.",
	"authors": [
	"Fabian Hellauer",
	"Peter Lammich"
	],
	"date": "2017-12-18",
	- "id": 365,
	+ "id": 368,
	"link": "/entries/Knuth_Morris_Pratt.html",
	"permalink": "/entries/Knuth_Morris_Pratt.html",
	"shortname": "Knuth_Morris_Pratt",
	"title": "The string search algorithm by Knuth, Morris and Pratt",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "Stochastic matrices are a convenient way to model discrete-time and finite state Markov chains. The Perron\u0026ndash;Frobenius theorem tells us something about the existence and uniqueness of non-negative eigenvectors of a stochastic matrix. In this entry, we formalize stochastic matrices, link the formalization to the existing AFP-entry on Markov chains, and apply the Perron\u0026ndash;Frobenius theorem to prove that stationary distributions always exist, and they are unique if the stochastic matrix is irreducible.",
	"authors": [
	"René Thiemann"
	],
	"date": "2017-11-22",
	- "id": 366,
	+ "id": 369,
	"link": "/entries/Stochastic_Matrices.html",
	"permalink": "/entries/Stochastic_Matrices.html",
	"shortname": "Stochastic_Matrices",
	"title": "Stochastic Matrices and the Perron-Frobenius Theorem",
	"topic_links": [
	"mathematics/algebra",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Mathematics/Algebra",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide our Isabelle/HOL formalization of a Conflict-free Replicated Datatype for Internet Message Access Protocol commands. We show that Strong Eventual Consistency (SEC) is guaranteed by proving the commutativity of concurrent operations. We base our formalization on the recently proposed \"framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes\" (AFP.CRDT) from Gomes et al. Hence, we provide an additional example of how the recently proposed framework can be used to design and prove CRDTs.",
	"authors": [
	"Tim Jungnickel",
	"Lennart Oldenburg",
	"Matthias Loibl"
	],
	"date": "2017-11-09",
	- "id": 367,
	+ "id": 370,
	"link": "/entries/IMAP-CRDT.html",
	"permalink": "/entries/IMAP-CRDT.html",
	"shortname": "IMAP-CRDT",
	"title": "The IMAP CmRDT",
	"topic_links": [
	"computer-science/algorithms/distributed",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms/Distributed",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a semantic embedding of a spatio-temporal multi-modal logic, specifically defined to reason about motorway traffic, into Isabelle/HOL. The semantic model is an abstraction of a motorway, emphasising local spatial properties, and parameterised by the types of sensors deployed in the vehicles. We use the logic to define controller constraints to ensure safety, i.e., the absence of collisions on the motorway. After proving safety with a restrictive definition of sensors, we relax these assumptions and show how to amend the controller constraints to still guarantee safety.",
	"authors": [
	"Sven Linker"
	],
	"date": "2017-11-06",
	- "id": 368,
	+ "id": 371,
	"link": "/entries/Hybrid_Multi_Lane_Spatial_Logic.html",
	"permalink": "/entries/Hybrid_Multi_Lane_Spatial_Logic.html",
	"shortname": "Hybrid_Multi_Lane_Spatial_Logic",
	"title": "Hybrid Multi-Lane Spatial Logic",
	"topic_links": [
	"logic/general-logic/modal-logic"
	],
	"topics": [
	"Logic/General logic/Modal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We discuss a topological curiosity discovered by Kuratowski (1922): the fact that the number of distinct operators on a topological space generated by compositions of closure and complement never exceeds 14, and is exactly 14 in the case of R. In addition, we prove a theorem due to Chagrov (1982) that classifies topological spaces according to the number of such operators they support.",
	"authors": [
	"Peter Gammie",
	"Gianpaolo Gioiosa"
	],
	"date": "2017-10-26",
	- "id": 369,
	+ "id": 372,
	"link": "/entries/Kuratowski_Closure_Complement.html",
	"permalink": "/entries/Kuratowski_Closure_Complement.html",
	"shortname": "Kuratowski_Closure_Complement",
	"title": "The Kuratowski Closure-Complement Theorem",
	"topic_links": [
	"mathematics/topology"
	],
	"topics": [
	"Mathematics/Topology"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides a verified implementation of rank-based Büchi Complementation. The verification is done in three steps: \u003col\u003e \u003cli\u003eDefinition of odd rankings and proof that an automaton rejects a word iff there exists an odd ranking for it.\u003c/li\u003e \u003cli\u003eDefinition of the complement automaton and proof that it accepts exactly those words for which there is an odd ranking.\u003c/li\u003e \u003cli\u003eVerified implementation of the complement automaton using the Isabelle Collections Framework.\u003c/li\u003e \u003c/ol\u003e",
	"authors": [
	"Julian Brunner"
	],
	"date": "2017-10-19",
	- "id": 370,
	+ "id": 373,
	"link": "/entries/Buchi_Complementation.html",
	"permalink": "/entries/Buchi_Complementation.html",
	"shortname": "Buchi_Complementation",
	"title": "Büchi Complementation",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides a very abstract theory of transition systems that can be instantiated to express various types of automata. A transition system is typically instantiated by providing a set of initial states, a predicate for enabled transitions, and a transition execution function. From this, it defines the concepts of finite and infinite paths as well as the set of reachable states, among other things. Many useful theorems, from basic path manipulation rules to coinduction and run construction rules, are proven in this abstract transition system context. The library comes with instantiations for DFAs, NFAs, and Büchi automata.",
	"authors": [
	"Julian Brunner"
	],
	"date": "2017-10-19",
	- "id": 371,
	+ "id": 374,
	"link": "/entries/Transition_Systems_and_Automata.html",
	"permalink": "/entries/Transition_Systems_and_Automata.html",
	"shortname": "Transition_Systems_and_Automata",
	"title": "Transition Systems and Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 4
	},
	{
	"abstract": "Based on evaluating Cauchy indices through remainder sequences, this entry provides an effective procedure to count the number of complex roots (with multiplicity) of a polynomial within various shapes (e.g., rectangle, circle and half-plane). Potential applications of this entry include certified complex root isolation (of a polynomial) and testing the Routh-Hurwitz stability criterion (i.e., to check whether all the roots of some characteristic polynomial have negative real parts).",
	"authors": [
	"Wenda Li"
	],
	"date": "2017-10-17",
	- "id": 372,
	+ "id": 375,
	"link": "/entries/Count_Complex_Roots.html",
	"permalink": "/entries/Count_Complex_Roots.html",
	"shortname": "Count_Complex_Roots",
	"title": "Count the Number of Complex Roots",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "In complex analysis, the winding number measures the number of times a path (counterclockwise) winds around a point, while the Cauchy index can approximate how the path winds. This entry provides a formalisation of the Cauchy index, which is then shown to be related to the winding number. In addition, this entry also offers a tactic that enables users to evaluate the winding number by calculating Cauchy indices.",
	"authors": [
	"Wenda Li"
	],
	"date": "2017-10-17",
	- "id": 373,
	+ "id": 376,
	"link": "/entries/Winding_Number_Eval.html",
	"permalink": "/entries/Winding_Number_Eval.html",
	"shortname": "Winding_Number_Eval",
	"title": "Evaluate Winding Numbers through Cauchy Indices",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "We formalize the theory of homogeneous linear diophantine equations, focusing on two main results: (1) an abstract characterization of minimal complete sets of solutions, and (2) an algorithm computing them. Both, the characterization and the algorithm are based on previous work by Huet. Our starting point is a simple but inefficient variant of Huet's lexicographic algorithm incorporating improved bounds due to Clausen and Fortenbacher. We proceed by proving its soundness and completeness. Finally, we employ code equations to obtain a reasonably efficient implementation. Thus, we provide a formally verified solver for homogeneous linear diophantine equations.",
	"authors": [
	"Florian Messner",
	"Julian Parsert",
	"Jonas Schöpf",
	"Christian Sternagel"
	],
	"date": "2017-10-14",
	- "id": 374,
	+ "id": 377,
	"link": "/entries/Diophantine_Eqns_Lin_Hom.html",
	"permalink": "/entries/Diophantine_Eqns_Lin_Hom.html",
	"shortname": "Diophantine_Eqns_Lin_Hom",
	"title": "Homogeneous Linear Diophantine Equations",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/number-theory",
	"tools"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Number theory",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry is a formalisation of much of Chapters 2, 3, and 11 of Apostol's \u0026ldquo;Introduction to Analytic Number Theory\u0026rdquo;. This includes: \u003cul\u003e \u003cli\u003eDefinitions and basic properties for several number-theoretic functions (Euler's \u0026phi;, M\u0026ouml;bius \u0026mu;, Liouville's \u0026lambda;, the divisor function \u0026sigma;, von Mangoldt's \u0026Lambda;)\u003c/li\u003e \u003cli\u003eExecutable code for most of these functions, the most efficient implementations using the factoring algorithm by Thiemann \u003ci\u003eet al.\u003c/i\u003e\u003c/li\u003e \u003cli\u003eDirichlet products and formal Dirichlet series\u003c/li\u003e \u003cli\u003eAnalytic results connecting convergent formal Dirichlet series to complex functions\u003c/li\u003e \u003cli\u003eEuler product expansions\u003c/li\u003e \u003cli\u003eAsymptotic estimates of number-theoretic functions including the density of squarefree integers and the average number of divisors of a natural number\u003c/li\u003e \u003c/ul\u003e These results are useful as a basis for developing more number-theoretic results, such as the Prime Number Theorem.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-10-12",
	- "id": 375,
	+ "id": 378,
	"link": "/entries/Dirichlet_Series.html",
	"permalink": "/entries/Dirichlet_Series.html",
	"shortname": "Dirichlet_Series",
	"title": "Dirichlet Series",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 4
	},
	{
	"abstract": "\u003cp\u003e Linear recurrences with constant coefficients are an interesting class of recurrence equations that can be solved explicitly. The most famous example are certainly the Fibonacci numbers with the equation \u003ci\u003ef\u003c/i\u003e(\u003ci\u003en\u003c/i\u003e) = \u003ci\u003ef\u003c/i\u003e(\u003ci\u003en\u003c/i\u003e-1) + \u003ci\u003ef\u003c/i\u003e(\u003ci\u003en\u003c/i\u003e - 2) and the quite non-obvious closed form (\u003ci\u003e\u0026phi;\u003c/i\u003e\u003csup\u003e\u003ci\u003en\u003c/i\u003e\u003c/sup\u003e - (-\u003ci\u003e\u0026phi;\u003c/i\u003e)\u003csup\u003e-\u003ci\u003en\u003c/i\u003e\u003c/sup\u003e) / \u0026radic;\u003cspan style=\"text-decoration: overline\"\u003e5\u003c/span\u003e where \u0026phi; is the golden ratio. \u003c/p\u003e \u003cp\u003e In this work, I build on existing tools in Isabelle \u0026ndash; such as formal power series and polynomial factorisation algorithms \u0026ndash; to develop a theory of these recurrences and derive a fully executable solver for them that can be exported to programming languages like Haskell. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-10-12",
	- "id": 376,
	+ "id": 379,
	"link": "/entries/Linear_Recurrences.html",
	"permalink": "/entries/Linear_Recurrences.html",
	"shortname": "Linear_Recurrences",
	"title": "Linear Recurrences",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis entry builds upon the results about formal and analytic Dirichlet series to define the Hurwitz \u0026zeta; function \u0026zeta;(\u003cem\u003ea\u003c/em\u003e,\u003cem\u003es\u003c/em\u003e) and, based on that, the Riemann \u0026zeta; function \u0026zeta;(\u003cem\u003es\u003c/em\u003e). This is done by first defining them for \u0026real;(\u003cem\u003ez\u003c/em\u003e) \u003e 1 and then successively extending the domain to the left using the Euler\u0026ndash;MacLaurin formula.\u003c/p\u003e \u003cp\u003eApart from the most basic facts such as analyticity, the following results are provided:\u003c/p\u003e \u003cul\u003e \u003cli\u003ethe Stieltjes constants and the Laurent expansion of \u0026zeta;(\u003cem\u003es\u003c/em\u003e) at \u003cem\u003es\u003c/em\u003e = 1\u003c/li\u003e \u003cli\u003ethe non-vanishing of \u0026zeta;(\u003cem\u003es\u003c/em\u003e) for \u0026real;(\u003cem\u003ez\u003c/em\u003e) \u0026ge; 1\u003c/li\u003e \u003cli\u003ethe relationship between \u0026zeta;(\u003cem\u003ea\u003c/em\u003e,\u003cem\u003es\u003c/em\u003e) and \u0026Gamma;\u003c/li\u003e \u003cli\u003ethe special values at negative integers and positive even integers\u003c/li\u003e \u003cli\u003eHurwitz's formula and the reflection formula for \u0026zeta;(\u003cem\u003es\u003c/em\u003e)\u003c/li\u003e \u003cli\u003ethe \u003ca href=\"https://arxiv.org/abs/math/0405478\"\u003e Hadjicostas\u0026ndash;Chapman formula\u003c/a\u003e\u003c/li\u003e \u003c/ul\u003e \u003cp\u003eThe entry also contains Euler's analytic proof of the infinitude of primes, based on the fact that \u0026zeta;(\u003ci\u003es\u003c/i\u003e) has a pole at \u003ci\u003es\u003c/i\u003e = 1.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-10-12",
	- "id": 377,
	+ "id": 380,
	"link": "/entries/Zeta_Function.html",
	"permalink": "/entries/Zeta_Function.html",
	"shortname": "Zeta_Function",
	"title": "The Hurwitz and Riemann ζ Functions",
	"topic_links": [
	"mathematics/number-theory",
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Number theory",
	"Mathematics/Analysis"
	],
	"used_by": 4
	},
	{
	"abstract": "Computers may help us to understand --not just verify-- philosophical arguments. By utilizing modern proof assistants in an iterative interpretive process, we can reconstruct and assess an argument by fully formal means. Through the mechanization of a variant of St. Anselm's ontological argument by E. J. Lowe, which is a paradigmatic example of a natural-language argument with strong ties to metaphysics and religion, we offer an ideal showcase for our computer-assisted interpretive method.",
	"authors": [
	"David Fuenmayor",
	"Christoph Benzmüller"
	],
	"date": "2017-09-21",
	- "id": 378,
	+ "id": 381,
	"link": "/entries/Lowe_Ontological_Argument.html",
	"permalink": "/entries/Lowe_Ontological_Argument.html",
	"shortname": "Lowe_Ontological_Argument",
	"title": "Computer-assisted Reconstruction and Assessment of E. J. Lowe's Modal Ontological Argument",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e We present an embedding of the second-order fragment of the Theory of Abstract Objects as described in Edward Zalta's upcoming work \u003ca href=\"https://mally.stanford.edu/principia.pdf\"\u003ePrincipia Logico-Metaphysica (PLM)\u003c/a\u003e in the automated reasoning framework Isabelle/HOL. The Theory of Abstract Objects is a metaphysical theory that reifies property patterns, as they for example occur in the abstract reasoning of mathematics, as \u003cb\u003eabstract objects\u003c/b\u003e and provides an axiomatic framework that allows to reason about these objects. It thereby serves as a fundamental metaphysical theory that can be used to axiomatize and describe a wide range of philosophical objects, such as Platonic forms or Leibniz' concepts, and has the ambition to function as a foundational theory of mathematics. The target theory of our embedding as described in chapters 7-9 of PLM employs a modal relational type theory as logical foundation for which a representation in functional type theory is \u003ca href=\"https://mally.stanford.edu/Papers/rtt.pdf\"\u003eknown to be challenging\u003c/a\u003e. \u003c/p\u003e \u003cp\u003e Nevertheless we arrive at a functioning representation of the theory in the functional logic of Isabelle/HOL based on a semantical representation of an Aczel-model of the theory. Based on this representation we construct an implementation of the deductive system of PLM which allows to automatically and interactively find and verify theorems of PLM. \u003c/p\u003e \u003cp\u003e Our work thereby supports the concept of shallow semantical embeddings of logical systems in HOL as a universal tool for logical reasoning \u003ca href=\"http://www.mi.fu-berlin.de/inf/groups/ag-ki/publications/Universal-Reasoning/1703_09620_pd.pdf\"\u003eas promoted by Christoph Benzm\u0026uuml;ller\u003c/a\u003e. \u003c/p\u003e \u003cp\u003e The most notable result of the presented work is the discovery of a previously unknown paradox in the formulation of the Theory of Abstract Objects. The embedding of the theory in Isabelle/HOL played a vital part in this discovery. Furthermore it was possible to immediately offer several options to modify the theory to guarantee its consistency. Thereby our work could provide a significant contribution to the development of a proper grounding for object theory. \u003c/p\u003e",
	"authors": [
	"Daniel Kirchner"
	],
	"date": "2017-09-17",
	- "id": 379,
	+ "id": 382,
	"link": "/entries/PLM.html",
	"permalink": "/entries/PLM.html",
	"shortname": "PLM",
	"title": "Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "Paul Oppenheimer and Edward Zalta's formalisation of Anselm's ontological argument for the existence of God is automated by embedding a free logic for definite descriptions within Isabelle/HOL.",
	"authors": [
	"Ben Blumson"
	],
	"date": "2017-09-06",
	- "id": 380,
	+ "id": 383,
	"link": "/entries/AnselmGod.html",
	"permalink": "/entries/AnselmGod.html",
	"shortname": "AnselmGod",
	"title": "Anselm's God in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "Economic activity has always been a fundamental part of society. Due to modern day politics, economic theory has gained even more influence on our lives. Thus we want models and theories to be as precise as possible. This can be achieved using certification with the help of formal proof technology. Hence we will use Isabelle/HOL to construct two economic models, that of the the pure exchange economy and a version of the Arrow-Debreu Model. We will prove that the \u003ci\u003eFirst Theorem of Welfare Economics\u003c/i\u003e holds within both. The theorem is the mathematical formulation of Adam Smith's famous \u003ci\u003einvisible hand\u003c/i\u003e and states that a group of self-interested and rational actors will eventually achieve an efficient allocation of goods and services.",
	"authors": [
	"Julian Parsert",
	"Cezary Kaliszyk"
	],
	"date": "2017-09-01",
	- "id": 381,
	+ "id": 384,
	"link": "/entries/First_Welfare_Theorem.html",
	"permalink": "/entries/First_Welfare_Theorem.html",
	"shortname": "First_Welfare_Theorem",
	"title": "Microeconomics and the First Welfare Theorem",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 1
	},
	{
	"abstract": "The Orbit-Stabiliser theorem is a basic result in the algebra of groups that factors the order of a group into the sizes of its orbits and stabilisers. We formalize the notion of a group action and the related concepts of orbits and stabilisers. This allows us to prove the orbit-stabiliser theorem. In the second part of this work, we formalize the tetrahedral group and use the orbit-stabiliser theorem to prove that there are twelve (orientation-preserving) rotations of the tetrahedron.",
	"authors": [
	"Jonas Rädle"
	],
	"date": "2017-08-20",
	- "id": 382,
	+ "id": 385,
	"link": "/entries/Orbit_Stabiliser.html",
	"permalink": "/entries/Orbit_Stabiliser.html",
	"shortname": "Orbit_Stabiliser",
	"title": "Orbit-Stabiliser Theorem with Application to Rotational Symmetries",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e Andersson introduced \u003cem\u003egeneral balanced trees\u003c/em\u003e, search trees based on the design principle of partial rebuilding: perform update operations naively until the tree becomes too unbalanced, at which point a whole subtree is rebalanced. This article defines and analyzes a functional version of general balanced trees, which we call \u003cem\u003eroot-balanced trees\u003c/em\u003e. Using a lightweight model of execution time, amortized logarithmic complexity is verified in the theorem prover Isabelle. \u003c/p\u003e \u003cp\u003e This is the Isabelle formalization of the material decribed in the APLAS 2017 article \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs/aplas17.html\"\u003eVerified Root-Balanced Trees\u003c/a\u003e by the same author, which also presents experimental results that show competitiveness of root-balanced with AVL and red-black trees. \u003c/p\u003e",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2017-08-20",
	- "id": 383,
	+ "id": 386,
	"link": "/entries/Root_Balanced_Tree.html",
	"permalink": "/entries/Root_Balanced_Tree.html",
	"shortname": "Root_Balanced_Tree",
	"title": "Root-Balanced Tree",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 2
	},
	{
	"abstract": "The propositions-as-types correspondence is ordinarily presented as linking the metatheory of typed λ-calculi and the proof theory of intuitionistic logic. Griffin observed that this correspondence could be extended to classical logic through the use of control operators. This observation set off a flurry of further research, leading to the development of Parigots λμ-calculus. In this work, we formalise λμ- calculus in Isabelle/HOL and prove several metatheoretical properties such as type preservation and progress.",
	"authors": [
	"Cristina Matache",
	"Victor B. F. Gomes",
	"Dominic P. Mulligan"
	],
	"date": "2017-08-16",
	- "id": 384,
	+ "id": 387,
	"link": "/entries/LambdaMu.html",
	"permalink": "/entries/LambdaMu.html",
	"shortname": "LambdaMu",
	"title": "The LambdaMu-calculus",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry formalizes the two geometric theorems, Stewart's and Apollonius' theorem. Stewart's Theorem relates the length of a triangle's cevian to the lengths of the triangle's two sides. Apollonius' Theorem is a specialisation of Stewart's theorem, restricting the cevian to be the median. The proof applies the law of cosines, some basic geometric facts about triangles and then simply transforms the terms algebraically to yield the conjectured relation. The formalization in Isabelle can closely follow the informal proofs described in the Wikipedia articles of those two theorems.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2017-07-31",
	- "id": 385,
	+ "id": 388,
	"link": "/entries/Stewart_Apollonius.html",
	"permalink": "/entries/Stewart_Apollonius.html",
	"shortname": "Stewart_Apollonius",
	"title": "Stewart's Theorem and Apollonius' Theorem",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "The architecture of a system describes the system's overall organization into components and connections between those components. With the emergence of mobile computing, dynamic architectures have become increasingly important. In such architectures, components may appear or disappear, and connections may change over time. In the following we mechanize a theory of dynamic architectures and verify the soundness of a corresponding calculus. Therefore, we first formalize the notion of configuration traces as a model for dynamic architectures. Then, the behavior of single components is formalized in terms of behavior traces and an operator is introduced and studied to extract the behavior of a single component out of a given configuration trace. Then, behavior trace assertions are introduced as a temporal specification technique to specify behavior of components. Reasoning about component behavior in a dynamic context is formalized in terms of a calculus for dynamic architectures. Finally, the soundness of the calculus is verified by introducing an alternative interpretation for behavior trace assertions over configuration traces and proving the rules of the calculus. Since projection may lead to finite as well as infinite behavior traces, they are formalized in terms of coinductive lists. Thus, our theory is based on Lochbihler's formalization of coinductive lists. The theory may be applied to verify properties for dynamic architectures.",
	"authors": [
	"Diego Marmsoler"
	],
	"date": "2017-07-28",
	- "id": 386,
	+ "id": 389,
	"link": "/entries/DynamicArchitectures.html",
	"permalink": "/entries/DynamicArchitectures.html",
	"shortname": "DynamicArchitectures",
	"title": "Dynamic Architectures",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a semantics for an applied call-by-value lambda-calculus that is compositional, extensional, and elementary. We present four different views of the semantics: 1) as a relational (big-step) semantics that is not operational but instead declarative, 2) as a denotational semantics that does not use domain theory, 3) as a non-deterministic interpreter, and 4) as a variant of the intersection type systems of the Torino group. We prove that the semantics is correct by showing that it is sound and complete with respect to operational semantics on programs and that is sound with respect to contextual equivalence. We have not yet investigated whether it is fully abstract. We demonstrate that this approach to semantics is useful with three case studies. First, we use the semantics to prove correctness of a compiler optimization that inlines function application. Second, we adapt the semantics to the polymorphic lambda-calculus extended with general recursion and prove semantic type soundness. Third, we adapt the semantics to the call-by-value lambda-calculus with mutable references. \u003cbr\u003e The paper that accompanies these Isabelle theories is \u003ca href=\"https://arxiv.org/abs/1707.03762\"\u003eavailable on arXiv\u003c/a\u003e.",
	"authors": [
	"Jeremy Siek"
	],
	"date": "2017-07-21",
	- "id": 387,
	+ "id": 390,
	"link": "/entries/Decl_Sem_Fun_PL.html",
	"permalink": "/entries/Decl_Sem_Fun_PL.html",
	"shortname": "Decl_Sem_Fun_PL",
	"title": "Declarative Semantics for Functional Languages",
	"topic_links": [
	"computer-science/programming-languages"
	],
	"topics": [
	"Computer science/Programming languages"
	],
	"used_by": 0
	},
	{
	"abstract": "The Isabelle/HOLCF-Prelude is a formalization of a large part of Haskell's standard prelude in Isabelle/HOLCF. We use it to prove the correctness of the Eratosthenes' Sieve, in its self-referential implementation commonly used to showcase Haskell's laziness; prove correctness of GHC's \"fold/build\" rule and related rewrite rules; and certify a number of hints suggested by HLint.",
	"authors": [
	"Joachim Breitner",
	"Brian Huffman",
	"Neil Mitchell",
	"Christian Sternagel"
	],
	"date": "2017-07-15",
	- "id": 388,
	+ "id": 391,
	"link": "/entries/HOLCF-Prelude.html",
	"permalink": "/entries/HOLCF-Prelude.html",
	"shortname": "HOLCF-Prelude",
	"title": "HOLCF-Prelude",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eMinkowski's theorem relates a subset of \u0026#8477;\u003csup\u003en\u003c/sup\u003e, the Lebesgue measure, and the integer lattice \u0026#8484;\u003csup\u003en\u003c/sup\u003e: It states that any convex subset of \u0026#8477;\u003csup\u003en\u003c/sup\u003e with volume greater than 2\u003csup\u003en\u003c/sup\u003e contains at least one lattice point from \u0026#8484;\u003csup\u003en\u003c/sup\u003e\\{0}, i.\u0026thinsp;e. a non-zero point with integer coefficients.\u003c/p\u003e \u003cp\u003eA related theorem which directly implies this is Blichfeldt's theorem, which states that any subset of \u0026#8477;\u003csup\u003en\u003c/sup\u003e with a volume greater than 1 contains two different points whose difference vector has integer components.\u003c/p\u003e \u003cp\u003eThe entry contains a proof of both theorems.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-07-13",
	- "id": 389,
	+ "id": 392,
	"link": "/entries/Minkowskis_Theorem.html",
	"permalink": "/entries/Minkowskis_Theorem.html",
	"shortname": "Minkowskis_Theorem",
	"title": "Minkowski's Theorem",
	"topic_links": [
	"mathematics/geometry",
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Geometry",
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "I formalise a Church-style simply-typed \$\\lambda\$-calculus, extended with pairs, a unit value, and projection functions, and show some metatheory of the calculus, such as the subject reduction property. Particular attention is paid to the treatment of names in the calculus. A nominal style of binding is used, but I use a manual approach over Nominal Isabelle in order to extract an executable type inference algorithm. More information can be found in my \u003ca href=\"http://www.openthesis.org/documents/Verified-Metatheory-Type-Inference-Simply-603182.html\"\u003eundergraduate dissertation\u003c/a\u003e.",
	"authors": [
	"Michael Rawson"
	],
	"date": "2017-07-09",
	- "id": 390,
	+ "id": 393,
	"link": "/entries/Name_Carrying_Type_Inference.html",
	"permalink": "/entries/Name_Carrying_Type_Inference.html",
	"shortname": "Name_Carrying_Type_Inference",
	"title": "Verified Metatheory and Type Inference for a Name-Carrying Simply-Typed Lambda Calculus",
	"topic_links": [
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work, we focus on the correctness of Conflict-free Replicated Data Types (CRDTs), a class of algorithm that provides strong eventual consistency guarantees for replicated data. We develop a modular and reusable framework for verifying the correctness of CRDT algorithms. We avoid correctness issues that have dogged previous mechanised proofs in this area by including a network model in our formalisation, and proving that our theorems hold in all possible network behaviours. Our axiomatic network model is a standard abstraction that accurately reflects the behaviour of real-world computer networks. Moreover, we identify an abstract convergence theorem, a property of order relations, which provides a formal definition of strong eventual consistency. We then obtain the first machine-checked correctness theorems for three concrete CRDTs: the Replicated Growable Array, the Observed-Remove Set, and an Increment-Decrement Counter.",
	"authors": [
	"Victor B. F. Gomes",
	"Martin Kleppmann",
	"Dominic P. Mulligan",
	"Alastair R. Beresford"
	],
	"date": "2017-07-07",
	- "id": 391,
	+ "id": 394,
	"link": "/entries/CRDT.html",
	"permalink": "/entries/CRDT.html",
	"shortname": "CRDT",
	"title": "A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes",
	"topic_links": [
	"computer-science/algorithms/distributed",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms/Distributed",
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "We develop Stone-Kleene relation algebras, which expand Stone relation algebras with a Kleene star operation to describe reachability in weighted graphs. Many properties of the Kleene star arise as a special case of a more general theory of iteration based on Conway semirings extended by simulation axioms. This includes several theorems representing complex program transformations. We formally prove the correctness of Conway's automata-based construction of the Kleene star of a matrix. We prove numerous results useful for reasoning about weighted graphs.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2017-07-06",
	- "id": 392,
	+ "id": 395,
	"link": "/entries/Stone_Kleene_Relation_Algebras.html",
	"permalink": "/entries/Stone_Kleene_Relation_Algebras.html",
	"shortname": "Stone_Kleene_Relation_Algebras",
	"title": "Stone-Kleene Relation Algebras",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 3
	},
	{
	"abstract": "We formalize a range of proof systems for classical propositional logic (sequent calculus, natural deduction, Hilbert systems, resolution) and prove the most important meta-theoretic results about semantics and proofs: compactness, soundness, completeness, translations between proof systems, cut-elimination, interpolation and model existence.",
	"authors": [
	"Julius Michaelis",
	"Tobias Nipkow"
	],
	"date": "2017-06-21",
	- "id": 393,
	+ "id": 396,
	"link": "/entries/Propositional_Proof_Systems.html",
	"permalink": "/entries/Propositional_Proof_Systems.html",
	"shortname": "Propositional_Proof_Systems",
	"title": "Propositional Proof Systems",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 2
	},
	{
	"abstract": "Partial Semigroups are relevant to the foundations of quantum mechanics and combinatorics as well as to interval and separation logics. Convolution algebras can be understood either as algebras of generalised binary modalities over ternary Kripke frames, in particular over partial semigroups, or as algebras of quantale-valued functions which are equipped with a convolution-style operation of multiplication that is parametrised by a ternary relation. Convolution algebras provide algebraic semantics for various substructural logics, including categorial, relevance and linear logics, for separation logic and for interval logics; they cover quantitative and qualitative applications. These mathematical components for partial semigroups and convolution algebras provide uniform foundations from which models of computation based on relations, program traces or pomsets, and verification components for separation or interval temporal logics can be built with little effort.",
	"authors": [
	"Brijesh Dongol",
	"Victor B. F. Gomes",
	"Ian J. Hayes",
	"Georg Struth"
	],
	"date": "2017-06-13",
	- "id": 394,
	+ "id": 397,
	"link": "/entries/PSemigroupsConvolution.html",
	"permalink": "/entries/PSemigroupsConvolution.html",
	"shortname": "PSemigroupsConvolution",
	"title": "Partial Semigroups and Convolution Algebras",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "In the 18th century, Georges-Louis Leclerc, Comte de Buffon posed and later solved the following problem, which is often called the first problem ever solved in geometric probability: Given a floor divided into vertical strips of the same width, what is the probability that a needle thrown onto the floor randomly will cross two strips? This entry formally defines the problem in the case where the needle's position is chosen uniformly at random in a single strip around the origin (which is equivalent to larger arrangements due to symmetry). It then provides proofs of the simple solution in the case where the needle's length is no greater than the width of the strips and the more complicated solution in the opposite case.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-06-06",
	- "id": 395,
	+ "id": 398,
	"link": "/entries/Buffons_Needle.html",
	"permalink": "/entries/Buffons_Needle.html",
	"shortname": "Buffons_Needle",
	"title": "Buffon's Needle Problem",
	"topic_links": [
	"mathematics/probability-theory",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of flow networks and the Min-Cut-Max-Flow theorem. Our formal proof closely follows a standard textbook proof, and is accessible even without being an expert in Isabelle/HOL, the interactive theorem prover used for the formalization.",
	"authors": [
	"Peter Lammich",
	"S. Reza Sefidgar"
	],
	"date": "2017-06-01",
	- "id": 396,
	+ "id": 399,
	"link": "/entries/Flow_Networks.html",
	"permalink": "/entries/Flow_Networks.html",
	"shortname": "Flow_Networks",
	"title": "Flow Networks and the Min-Cut-Max-Flow Theorem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 2
	},
	{
	"abstract": "We present a formalization of push-relabel algorithms for computing the maximum flow in a network. We start with Goldberg's et al.~generic push-relabel algorithm, for which we show correctness and the time complexity bound of O(V^2E). We then derive the relabel-to-front and FIFO implementation. Using stepwise refinement techniques, we derive an efficient verified implementation. Our formal proof of the abstract algorithms closely follows a standard textbook proof. It is accessible even without being an expert in Isabelle/HOL, the interactive theorem prover used for the formalization.",
	"authors": [
	"Peter Lammich",
	"S. Reza Sefidgar"
	],
	"date": "2017-06-01",
	- "id": 397,
	+ "id": 400,
	"link": "/entries/Prpu_Maxflow.html",
	"permalink": "/entries/Prpu_Maxflow.html",
	"shortname": "Prpu_Maxflow",
	"title": "Formalizing Push-Relabel Algorithms",
	"topic_links": [
	"computer-science/algorithms/graph",
	"mathematics/graph-theory"
	],
	"topics": [
	"Computer science/Algorithms/Graph",
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Lenses provide an abstract interface for manipulating data types through spatially-separated views. They are defined abstractly in terms of two functions, \u003cem\u003eget\u003c/em\u003e, the return a value from the source type, and \u003cem\u003eput\u003c/em\u003e that updates the value. We mechanise the underlying theory of lenses, in terms of an algebraic hierarchy of lenses, including well-behaved and very well-behaved lenses, each lens class being characterised by a set of lens laws. We also mechanise a lens algebra in Isabelle that enables their composition and comparison, so as to allow construction of complex lenses. This is accompanied by a large library of algebraic laws. Moreover we also show how the lens classes can be applied by instantiating them with a number of Isabelle data types.",
	"authors": [
	"Simon Foster",
	"Christian Pardillo-Laursen",
	"Frank Zeyda"
	],
	"date": "2017-05-25",
	- "id": 398,
	+ "id": 401,
	"link": "/entries/Optics.html",
	"permalink": "/entries/Optics.html",
	"shortname": "Optics",
	"title": "Optics",
	"topic_links": [
	"computer-science/functional-programming",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Functional programming",
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "We propose a development method for security protocols based on stepwise refinement. Our refinement strategy transforms abstract security goals into protocols that are secure when operating over an insecure channel controlled by a Dolev-Yao-style intruder. As intermediate levels of abstraction, we employ messageless guard protocols and channel protocols communicating over channels with security properties. These abstractions provide insights on why protocols are secure and foster the development of families of protocols sharing common structure and properties. We have implemented our method in Isabelle/HOL and used it to develop different entity authentication and key establishment protocols, including realistic features such as key confirmation, replay caches, and encrypted tickets. Our development highlights that guard protocols and channel protocols provide fundamental abstractions for bridging the gap between security properties and standard protocol descriptions based on cryptographic messages. It also shows that our refinement approach scales to protocols of nontrivial size and complexity.",
	"authors": [
	"Christoph Sprenger",
	"Ivano Somaini"
	],
	"date": "2017-05-24",
	- "id": 399,
	+ "id": 402,
	"link": "/entries/Security_Protocol_Refinement.html",
	"permalink": "/entries/Security_Protocol_Refinement.html",
	"shortname": "Security_Protocol_Refinement",
	"title": "Developing Security Protocols by Refinement",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Isabelle's code generator natively supports type classes. For targets that do not have language support for classes and instances, it performs the well-known dictionary translation, as described by Haftmann and Nipkow. This translation happens outside the logic, i.e., there is no guarantee that it is correct, besides the pen-and-paper proof. This work implements a certified dictionary translation that produces new class-free constants and derives equality theorems.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2017-05-24",
	- "id": 400,
	+ "id": 403,
	"link": "/entries/Dict_Construction.html",
	"permalink": "/entries/Dict_Construction.html",
	"shortname": "Dict_Construction",
	"title": "Dictionary Construction",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "The Floyd-Warshall algorithm [Flo62, Roy59, War62] is a classic dynamic programming algorithm to compute the length of all shortest paths between any two vertices in a graph (i.e. to solve the all-pairs shortest path problem, or APSP for short). Given a representation of the graph as a matrix of weights M, it computes another matrix M' which represents a graph with the same path lengths and contains the length of the shortest path between any two vertices i and j. This is only possible if the graph does not contain any negative cycles. However, in this case the Floyd-Warshall algorithm will detect the situation by calculating a negative diagonal entry. This entry includes a formalization of the algorithm and of these key properties. The algorithm is refined to an efficient imperative version using the Imperative Refinement Framework.",
	"authors": [
	"Simon Wimmer",
	"Peter Lammich"
	],
	"date": "2017-05-08",
	- "id": 401,
	+ "id": 404,
	"link": "/entries/Floyd_Warshall.html",
	"permalink": "/entries/Floyd_Warshall.html",
	"shortname": "Floyd_Warshall",
	"title": "The Floyd-Warshall Algorithm for Shortest Paths",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eCryptHOL provides a framework for formalising cryptographic arguments in Isabelle/HOL. It shallowly embeds a probabilistic functional programming language in higher order logic. The language features monadic sequencing, recursion, random sampling, failures and failure handling, and black-box access to oracles. Oracles are probabilistic functions which maintain hidden state between different invocations. All operators are defined in the new semantic domain of generative probabilistic values, a codatatype. We derive proof rules for the operators and establish a connection with the theory of relational parametricity. Thus, the resuting proofs are trustworthy and comprehensible, and the framework is extensible and widely applicable. \u003c/p\u003e\u003cp\u003e The framework is used in the accompanying AFP entry \"Game-based Cryptography in HOL\". There, we show-case our framework by formalizing different game-based proofs from the literature. This formalisation continues the work described in the author's ESOP 2016 paper.\u003c/p\u003e",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2017-05-05",
	- "id": 402,
	+ "id": 405,
	"link": "/entries/CryptHOL.html",
	"permalink": "/entries/CryptHOL.html",
	"shortname": "CryptHOL",
	"title": "CryptHOL",
	"topic_links": [
	"computer-science/security/cryptography",
	"computer-science/functional-programming",
	"mathematics/probability-theory"
	],
	"topics": [
	"Computer science/Security/Cryptography",
	"Computer science/Functional programming",
	"Mathematics/Probability theory"
	],
	"used_by": 4
	},
	{
	"abstract": "The notion of a monad cannot be expressed within higher-order logic (HOL) due to type system restrictions. We show that if a monad is used with values of only one type, this notion can be formalised in HOL. Based on this idea, we develop a library of effect specifications and implementations of monads and monad transformers. Hence, we can abstract over the concrete monad in HOL definitions and thus use the same definition for different (combinations of) effects. We illustrate the usefulness of effect polymorphism with a monadic interpreter for a simple language.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2017-05-05",
	- "id": 403,
	+ "id": 406,
	"link": "/entries/Monomorphic_Monad.html",
	"permalink": "/entries/Monomorphic_Monad.html",
	"shortname": "Monomorphic_Monad",
	"title": "Effect polymorphism in higher-order logic",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eIn this AFP entry, we show how to specify game-based cryptographic security notions and formally prove secure several cryptographic constructions from the literature using the CryptHOL framework. Among others, we formalise the notions of a random oracle, a pseudo-random function, an unpredictable function, and of encryption schemes that are indistinguishable under chosen plaintext and/or ciphertext attacks. We prove the random-permutation/random-function switching lemma, security of the Elgamal and hashed Elgamal public-key encryption scheme and correctness and security of several constructions with pseudo-random functions. \u003c/p\u003e\u003cp\u003eOur proofs follow the game-hopping style advocated by Shoup and Bellare and Rogaway, from which most of the examples have been taken. We generalise some of their results such that they can be reused in other proofs. Thanks to CryptHOL's integration with Isabelle's parametricity infrastructure, many simple hops are easily justified using the theory of representation independence.\u003c/p\u003e",
	"authors": [
	"Andreas Lochbihler",
	"S. Reza Sefidgar",
	"Bhargav Bhatt"
	],
	"date": "2017-05-05",
	- "id": 404,
	+ "id": 407,
	"link": "/entries/Game_Based_Crypto.html",
	"permalink": "/entries/Game_Based_Crypto.html",
	"shortname": "Game_Based_Crypto",
	"title": "Game-based cryptography in HOL",
	"topic_links": [
	"computer-science/security/cryptography"
	],
	"topics": [
	"Computer science/Security/Cryptography"
	],
	"used_by": 2
	},
	{
	"abstract": "The usual monad laws can directly be used as rewrite rules for Isabelle’s simplifier to normalise monadic HOL terms and decide equivalences. In a commutative monad, however, the commutativity law is a higher-order permutative rewrite rule that makes the simplifier loop. This AFP entry implements a simproc that normalises monadic expressions in commutative monads using ordered rewriting. The simproc can also permute computations across control operators like if and case.",
	"authors": [
	"Joshua Schneider",
	"Manuel Eberl",
	"Andreas Lochbihler"
	],
	"date": "2017-05-05",
	- "id": 405,
	+ "id": 408,
	"link": "/entries/Monad_Normalisation.html",
	"permalink": "/entries/Monad_Normalisation.html",
	"shortname": "Monad_Normalisation",
	"title": "Monad normalisation",
	"topic_links": [
	"tools",
	"computer-science/functional-programming",
	"logic/rewriting"
	],
	"topics": [
	"Tools",
	"Computer science/Functional programming",
	"Logic/Rewriting"
	],
	"used_by": 3
	},
	{
	"abstract": "This AFP entry defines a probabilistic while operator based on sub-probability mass functions and formalises zero-one laws and variant rules for probabilistic loop termination. As applications, we implement probabilistic algorithms for the Bernoulli, geometric and arbitrary uniform distributions that only use fair coin flips, and prove them correct and terminating with probability 1.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2017-05-05",
	- "id": 406,
	+ "id": 409,
	"link": "/entries/Probabilistic_While.html",
	"permalink": "/entries/Probabilistic_While.html",
	"shortname": "Probabilistic_While",
	"title": "Probabilistic while loop",
	"topic_links": [
	"computer-science/functional-programming",
	"mathematics/probability-theory",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Functional programming",
	"Mathematics/Probability theory",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003e Building on the formalization of basic category theory set out in the author's previous AFP article, the present article formalizes some basic aspects of the theory of monoidal categories. Among the notions defined here are monoidal category, monoidal functor, and equivalence of monoidal categories. The main theorems formalized are MacLane's coherence theorem and the constructions of the free monoidal category and free strict monoidal category generated by a given category. The coherence theorem is proved syntactically, using a structurally recursive approach to reduction of terms that might have some novel aspects. We also give proofs of some results given by Etingof et al, which may prove useful in a formal setting. In particular, we show that the left and right unitors need not be taken as given data in the definition of monoidal category, nor does the definition of monoidal functor need to take as given a specific isomorphism expressing the preservation of the unit object. Our definitions of monoidal category and monoidal functor are stated so as to take advantage of the economy afforded by these facts. \u003c/p\u003e\u003cp\u003e Revisions made subsequent to the first version of this article added material on cartesian monoidal categories; showing that the underlying category of a cartesian monoidal category is a cartesian category, and that every cartesian category extends to a cartesian monoidal category. \u003c/p\u003e",
	"authors": [
	"Eugene W. Stark"
	],
	"date": "2017-05-04",
	- "id": 407,
	+ "id": 410,
	"link": "/entries/MonoidalCategory.html",
	"permalink": "/entries/MonoidalCategory.html",
	"shortname": "MonoidalCategory",
	"title": "Monoidal Categories",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 1
	},
	{
	"abstract": "A computer-formalisation of the essential parts of Fitting's textbook \"Types, Tableaus and Gödel's God\" in Isabelle/HOL is presented. In particular, Fitting's (and Anderson's) variant of the ontological argument is verified and confirmed. This variant avoids the modal collapse, which has been criticised as an undesirable side-effect of Kurt Gödel's (and Dana Scott's) versions of the ontological argument. Fitting's work is employing an intensional higher-order modal logic, which we shallowly embed here in classical higher-order logic. We then utilize the embedded logic for the formalisation of Fitting's argument. (See also the earlier AFP entry “Gödel's God in Isabelle/HOL”.)",
	"authors": [
	"David Fuenmayor",
	"Christoph Benzmüller"
	],
	"date": "2017-05-01",
	- "id": 408,
	+ "id": 411,
	"link": "/entries/Types_Tableaus_and_Goedels_God.html",
	"permalink": "/entries/Types_Tableaus_and_Goedels_God.html",
	"shortname": "Types_Tableaus_and_Goedels_God",
	"title": "Types, Tableaus and Gödel’s God in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalisation accompanies the paper \u003ca href=\"https://arxiv.org/abs/1702.03277\"\u003eLocal Lexing\u003c/a\u003e which introduces a novel parsing concept of the same name. The paper also gives a high-level algorithm for local lexing as an extension of Earley's algorithm. This formalisation proves the algorithm to be correct with respect to its local lexing semantics. As a special case, this formalisation thus also contains a proof of the correctness of Earley's algorithm. The paper contains a short outline of how this formalisation is organised.",
	"authors": [
	"Steven Obua"
	],
	"date": "2017-04-28",
	- "id": 409,
	+ "id": 412,
	"link": "/entries/LocalLexing.html",
	"permalink": "/entries/LocalLexing.html",
	"shortname": "LocalLexing",
	"title": "Local Lexing",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Isabelle's code generator performs various adaptations for target languages. Among others, constructor applications have to be fully saturated. That means that for constructor calls occuring as arguments to higher-order functions, synthetic lambdas have to be inserted. This entry provides tooling to avoid this construction altogether by introducing constructor functions.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2017-04-19",
	- "id": 410,
	+ "id": 413,
	"link": "/entries/Constructor_Funs.html",
	"permalink": "/entries/Constructor_Funs.html",
	"shortname": "Constructor_Funs",
	"title": "Constructor Functions",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "Isabelle's code generator performs various adaptations for target languages. Among others, case statements are printed as match expressions. Internally, this is a sophisticated procedure, because in HOL, case statements are represented as nested calls to the case combinators as generated by the datatype package. Furthermore, the procedure relies on laziness of match expressions in the target language, i.e., that branches guarded by patterns that fail to match are not evaluated. Similarly, \u003ctt\u003eif-then-else\u003c/tt\u003e is printed to the corresponding construct in the target language. This entry provides tooling to replace these special cases in the code generator by ignoring these target language features, instead printing case expressions and \u003ctt\u003eif-then-else\u003c/tt\u003e as functions.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2017-04-18",
	- "id": 411,
	+ "id": 414,
	"link": "/entries/Lazy_Case.html",
	"permalink": "/entries/Lazy_Case.html",
	"shortname": "Lazy_Case",
	"title": "Lazifying case constants",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize the theory of subresultants and the subresultant polynomial remainder sequence as described by Brown and Traub. As a result, we obtain efficient certified algorithms for computing the resultant and the greatest common divisor of polynomials.",
	"authors": [
	"Sebastiaan J. C. Joosten",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2017-04-06",
	- "id": 412,
	+ "id": 415,
	"link": "/entries/Subresultants.html",
	"permalink": "/entries/Subresultants.html",
	"shortname": "Subresultants",
	"title": "Subresultants",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis entry contains proofs for the textbook results about the distributions of the height and internal path length of random binary search trees (BSTs), i.\u0026thinsp;e. BSTs that are formed by taking an empty BST and inserting elements from a fixed set in random order.\u003c/p\u003e \u003cp\u003eIn particular, we prove a logarithmic upper bound on the expected height and the \u003cem\u003eΘ(n log n)\u003c/em\u003e closed-form solution for the expected internal path length in terms of the harmonic numbers. We also show how the internal path length relates to the average-case cost of a lookup in a BST.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-04-04",
	- "id": 413,
	+ "id": 416,
	"link": "/entries/Random_BSTs.html",
	"permalink": "/entries/Random_BSTs.html",
	"shortname": "Random_BSTs",
	"title": "Expected Shape of Random Binary Search Trees",
	"topic_links": [
	"computer-science/data-structures",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Data structures",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis article contains a formal proof of the well-known fact that number of comparisons that a comparison-based sorting algorithm needs to perform to sort a list of length \u003cem\u003en\u003c/em\u003e is at least \u003cem\u003elog\u003csub\u003e2\u003c/sub\u003e\u0026nbsp;(n!)\u003c/em\u003e in the worst case, i.\u0026thinsp;e.\u0026nbsp;\u003cem\u003eΩ(n log n)\u003c/em\u003e.\u003c/p\u003e \u003cp\u003eFor this purpose, a shallow embedding for comparison-based sorting algorithms is defined: a sorting algorithm is a recursive datatype containing either a HOL function or a query of a comparison oracle with a continuation containing the remaining computation. This makes it possible to force the algorithm to use only comparisons and to track the number of comparisons made.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-03-15",
	- "id": 414,
	+ "id": 417,
	"link": "/entries/Comparison_Sort_Lower_Bound.html",
	"permalink": "/entries/Comparison_Sort_Lower_Bound.html",
	"shortname": "Comparison_Sort_Lower_Bound",
	"title": "Lower bound on comparison-based sorting algorithms",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eWe give a formal proof of the well-known results about the number of comparisons performed by two variants of QuickSort: first, the expected number of comparisons of randomised QuickSort (i.\u0026thinsp;e.\u0026nbsp;QuickSort with random pivot choice) is \u003cem\u003e2\u0026thinsp;(n+1)\u0026thinsp;H\u003csub\u003en\u003c/sub\u003e - 4\u0026thinsp;n\u003c/em\u003e, which is asymptotically equivalent to \u003cem\u003e2\u0026thinsp;n ln n\u003c/em\u003e; second, the number of comparisons performed by the classic non-randomised QuickSort has the same distribution in the average case as the randomised one.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-03-15",
	- "id": 415,
	+ "id": 418,
	"link": "/entries/Quick_Sort_Cost.html",
	"permalink": "/entries/Quick_Sort_Cost.html",
	"shortname": "Quick_Sort_Cost",
	"title": "The number of comparisons in QuickSort",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThe Euler-MacLaurin formula relates the value of a discrete sum to that of the corresponding integral in terms of the derivatives at the borders of the summation and a remainder term. Since the remainder term is often very small as the summation bounds grow, this can be used to compute asymptotic expansions for sums.\u003c/p\u003e \u003cp\u003eThis entry contains a proof of this formula for functions from the reals to an arbitrary Banach space. Two variants of the formula are given: the standard textbook version and a variant outlined in \u003cem\u003eConcrete Mathematics\u003c/em\u003e that is more useful for deriving asymptotic estimates.\u003c/p\u003e \u003cp\u003eAs example applications, we use that formula to derive the full asymptotic expansion of the harmonic numbers and the sum of inverse squares.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-03-10",
	- "id": 416,
	+ "id": 419,
	"link": "/entries/Euler_MacLaurin.html",
	"permalink": "/entries/Euler_MacLaurin.html",
	"shortname": "Euler_MacLaurin",
	"title": "The Euler–MacLaurin Formula",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "We prove the group law for elliptic curves in Weierstrass form over fields of characteristic greater than 2. In addition to affine coordinates, we also formalize projective coordinates, which allow for more efficient computations. By specializing the abstract formalization to prime fields, we can apply the curve operations to parameters used in standard security protocols.",
	"authors": [
	"Stefan Berghofer"
	],
	"date": "2017-02-28",
	- "id": 417,
	+ "id": 420,
	"link": "/entries/Elliptic_Curves_Group_Law.html",
	"permalink": "/entries/Elliptic_Curves_Group_Law.html",
	"shortname": "Elliptic_Curves_Group_Law",
	"title": "The Group Law for Elliptic Curves",
	"topic_links": [
	"computer-science/security/cryptography"
	],
	"topics": [
	"Computer science/Security/Cryptography"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a formalization of Menger's Theorem for directed and undirected graphs in Isabelle/HOL. This well-known result shows that if two non-adjacent distinct vertices u, v in a directed graph have no separator smaller than n, then there exist n internally vertex-disjoint paths from u to v. The version for undirected graphs follows immediately because undirected graphs are a special case of directed graphs.",
	"authors": [
	"Christoph Dittmann"
	],
	"date": "2017-02-26",
	- "id": 418,
	+ "id": 421,
	"link": "/entries/Menger.html",
	"permalink": "/entries/Menger.html",
	"shortname": "Menger",
	"title": "Menger's Theorem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize differential dynamic logic, a logic for proving properties of hybrid systems. The proof calculus in this formalization is based on the uniform substitution principle. We show it is sound with respect to our denotational semantics, which provides increased confidence in the correctness of the KeYmaera X theorem prover based on this calculus. As an application, we include a proof term checker embedded in Isabelle/HOL with several example proofs. Published in: Rose Bohrer, Vincent Rahli, Ivana Vukotic, Marcus Völp, André Platzer: Formally verified differential dynamic logic. CPP 2017.",
	"authors": [
	"Rose Bohrer"
	],
	"date": "2017-02-13",
	- "id": 419,
	+ "id": 422,
	"link": "/entries/Differential_Dynamic_Logic.html",
	"permalink": "/entries/Differential_Dynamic_Logic.html",
	"shortname": "Differential_Dynamic_Logic",
	"title": "Differential Dynamic Logic",
	"topic_links": [
	"logic/general-logic/modal-logic",
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Logic/General logic/Modal logic",
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "A formalized coinductive account of the abstract development of Brotherston, Gorogiannis, and Petersen [APLAS 2012], in a slightly more general form since we work with arbitrary infinite proofs, which may be acyclic. This work is described in detail in an article by the authors, published in 2017 in the \u003cem\u003eJournal of Automated Reasoning\u003c/em\u003e. The abstract proof can be instantiated for various formalisms, including first-order logic with inductive predicates.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2017-02-10",
	- "id": 420,
	+ "id": 423,
	"link": "/entries/Abstract_Soundness.html",
	"permalink": "/entries/Abstract_Soundness.html",
	"shortname": "Abstract_Soundness",
	"title": "Abstract Soundness",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 2
	},
	{
	"abstract": "We develop Stone relation algebras, which generalise relation algebras by replacing the underlying Boolean algebra structure with a Stone algebra. We show that finite matrices over extended real numbers form an instance. As a consequence, relation-algebraic concepts and methods can be used for reasoning about weighted graphs. We also develop a fixpoint calculus and apply it to compare different definitions of reflexive-transitive closures in semirings.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2017-02-07",
	- "id": 421,
	+ "id": 424,
	"link": "/entries/Stone_Relation_Algebras.html",
	"permalink": "/entries/Stone_Relation_Algebras.html",
	"shortname": "Stone_Relation_Algebras",
	"title": "Stone Relation Algebras",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "We develop a family of key agreement protocols that are correct by construction. Our work substantially extends prior work on developing security protocols by refinement. First, we strengthen the adversary by allowing him to compromise different resources of protocol participants, such as their long-term keys or their session keys. This enables the systematic development of protocols that ensure strong properties such as perfect forward secrecy. Second, we broaden the class of protocols supported to include those with non-atomic keys and equationally defined cryptographic operators. We use these extensions to develop key agreement protocols including signed Diffie-Hellman and the core of IKEv1 and SKEME.",
	"authors": [
	"Joseph Lallemand",
	"Christoph Sprenger"
	],
	"date": "2017-01-31",
	- "id": 422,
	+ "id": 425,
	"link": "/entries/Key_Agreement_Strong_Adversaries.html",
	"permalink": "/entries/Key_Agreement_Strong_Adversaries.html",
	"shortname": "Key_Agreement_Strong_Adversaries",
	"title": "Refining Authenticated Key Agreement with Strong Adversaries",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eBernoulli numbers were first discovered in the closed-form expansion of the sum 1\u003csup\u003em\u003c/sup\u003e + 2\u003csup\u003em\u003c/sup\u003e + \u0026hellip; + n\u003csup\u003em\u003c/sup\u003e for a fixed m and appear in many other places. This entry provides three different definitions for them: a recursive one, an explicit one, and one through their exponential generating function.\u003c/p\u003e \u003cp\u003eIn addition, we prove some basic facts, e.g. their relation to sums of powers of integers and that all odd Bernoulli numbers except the first are zero, and some advanced facts like their relationship to the Riemann zeta function on positive even integers.\u003c/p\u003e \u003cp\u003eWe also prove the correctness of the Akiyama\u0026ndash;Tanigawa algorithm for computing Bernoulli numbers with reasonable efficiency, and we define the periodic Bernoulli polynomials (which appear e.g. in the Euler\u0026ndash;MacLaurin summation formula and the expansion of the log-Gamma function) and prove their basic properties.\u003c/p\u003e",
	"authors": [
	"Lukas Bulwahn",
	"Manuel Eberl"
	],
	"date": "2017-01-24",
	- "id": 423,
	+ "id": 426,
	"link": "/entries/Bernoulli.html",
	"permalink": "/entries/Bernoulli.html",
	"shortname": "Bernoulli",
	"title": "Bernoulli Numbers",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Number theory"
	],
	"used_by": 5
	},
	{
	"abstract": "\u003cp\u003eBertrand's postulate is an early result on the distribution of prime numbers: For every positive integer n, there exists a prime number that lies strictly between n and 2n. The proof is ported from John Harrison's formalisation in HOL Light. It proceeds by first showing that the property is true for all n greater than or equal to 600 and then showing that it also holds for all n below 600 by case distinction. \u003c/p\u003e",
	"authors": [
	"Julian Biendarra",
	"Manuel Eberl"
	],
	"date": "2017-01-17",
	- "id": 424,
	+ "id": 427,
	"link": "/entries/Bertrands_Postulate.html",
	"permalink": "/entries/Bertrands_Postulate.html",
	"shortname": "Bertrands_Postulate",
	"title": "Bertrand's postulate",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis formalization is an extension to \u003ca href=\"https://www.isa-afp.org/entries/Formal_SSA.html\"\u003e\"Verified Construction of Static Single Assignment Form\"\u003c/a\u003e. In their work, the authors have shown that \u003ca href=\"https://doi.org/10.1007/978-3-642-37051-9_6\"\u003eBraun et al.'s static single assignment (SSA) construction algorithm\u003c/a\u003e produces minimal SSA form for input programs with a reducible control flow graph (CFG). However Braun et al. also proposed an extension to their algorithm that they claim produces minimal SSA form even for irreducible CFGs.\u003cbr\u003e In this formalization we support that claim by giving a mechanized proof. \u003c/p\u003e \u003cp\u003eAs the extension of Braun et al.'s algorithm aims for removing so-called redundant strongly connected components of phi functions, we show that this suffices to guarantee minimality according to \u003ca href=\"https://doi.org/10.1145/115372.115320\"\u003eCytron et al.\u003c/a\u003e.\u003c/p\u003e",
	"authors": [
	"Max Wagner",
	"Denis Lohner"
	],
	"date": "2017-01-17",
	- "id": 425,
	+ "id": 428,
	"link": "/entries/Minimal_SSA.html",
	"permalink": "/entries/Minimal_SSA.html",
	"shortname": "Minimal_SSA",
	"title": "Minimal Static Single Assignment Form",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThis work contains a proof that Euler's number e is transcendental. The proof follows the standard approach of assuming that e is algebraic and then using a specific integer polynomial to derive two inconsistent bounds, leading to a contradiction.\u003c/p\u003e \u003cp\u003eThis kind of approach can be found in many different sources; this formalisation mostly follows a \u003ca href=\"http://planetmath.org/proofoflindemannweierstrasstheoremandthateandpiaretranscendental\"\u003ePlanetMath article\u003c/a\u003e by Roger Lipsett.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2017-01-12",
	- "id": 426,
	+ "id": 429,
	"link": "/entries/E_Transcendental.html",
	"permalink": "/entries/E_Transcendental.html",
	"shortname": "E_Transcendental",
	"title": "The Transcendence of e",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Number theory"
	],
	"used_by": 2
	},
	{
	"abstract": "We present a formal model of network protocols and their application to modeling firewall policies. The formalization is based on the Unified Policy Framework (UPF). The formalization was originally developed with for generating test cases for testing the security configuration actual firewall and router (middle-boxes) using HOL-TestGen. Our work focuses on modeling application level protocols on top of tcp/ip.",
	"authors": [
	"Achim D. Brucker",
	"Lukas Brügger",
	"Burkhart Wolff"
	],
	"date": "2017-01-08",
	- "id": 427,
	+ "id": 430,
	"link": "/entries/UPF_Firewall.html",
	"permalink": "/entries/UPF_Firewall.html",
	"shortname": "UPF_Firewall",
	"title": "Formal Network Models and Their Application to Firewall Policies",
	"topic_links": [
	"computer-science/security",
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Networks"
	],
	"used_by": 0
	},
	{
	"abstract": "This paper constructs a formal model of a Diffie-Hellman password-based authentication protocol between a user and a smart card, and proves its security. The protocol provides for the dispatch of the user's password to the smart card on a secure messaging channel established by means of Password Authenticated Connection Establishment (PACE), where the mapping method being used is Chip Authentication Mapping. By applying and suitably extending Paulson's Inductive Method, this paper proves that the protocol establishes trustworthy secure messaging channels, preserves the secrecy of users' passwords, and provides an effective mutual authentication service. What is more, these security properties turn out to hold independently of the secrecy of the PACE authentication key.",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2017-01-03",
	- "id": 428,
	+ "id": 431,
	"link": "/entries/Password_Authentication_Protocol.html",
	"permalink": "/entries/Password_Authentication_Protocol.html",
	"shortname": "Password_Authentication_Protocol",
	"title": "Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eWe present a certified declarative first-order prover with equality based on John Harrison's Handbook of Practical Logic and Automated Reasoning, Cambridge University Press, 2009. ML code reflection is used such that the entire prover can be executed within Isabelle as a very simple interactive proof assistant. As examples we consider Pelletier's problems 1-46.\u003c/p\u003e \u003cp\u003eReference: Programming and Verifying a Declarative First-Order Prover in Isabelle/HOL. Alexander Birch Jensen, John Bruntse Larsen, Anders Schlichtkrull \u0026 Jørgen Villadsen. AI Communications 31:281-299 2018. \u003ca href=\"https://content.iospress.com/articles/ai-communications/aic764\"\u003e https://content.iospress.com/articles/ai-communications/aic764\u003c/a\u003e\u003c/p\u003e \u003cp\u003eSee also: Students' Proof Assistant (SPA). \u003ca href=https://github.com/logic-tools/spa\u003e https://github.com/logic-tools/spa\u003c/a\u003e\u003c/p\u003e",
	"authors": [
	"Alexander Birch Jensen",
	"Anders Schlichtkrull",
	"Jørgen Villadsen"
	],
	"date": "2017-01-01",
	- "id": 429,
	+ "id": 432,
	"link": "/entries/FOL_Harrison.html",
	"permalink": "/entries/FOL_Harrison.html",
	"shortname": "FOL_Harrison",
	"title": "First-Order Logic According to Harrison",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "The concurrent refinement algebra developed here is designed to provide a foundation for rely/guarantee reasoning about concurrent programs. The algebra builds on a complete lattice of commands by providing sequential composition, parallel composition and a novel weak conjunction operator. The weak conjunction operator coincides with the lattice supremum providing its arguments are non-aborting, but aborts if either of its arguments do. Weak conjunction provides an abstract version of a guarantee condition as a guarantee process. We distinguish between models that distribute sequential composition over non-deterministic choice from the left (referred to as being conjunctive in the refinement calculus literature) and those that don't. Least and greatest fixed points of monotone functions are provided to allow recursion and iteration operators to be added to the language. Additional iteration laws are available for conjunctive models. The rely quotient of processes \u003ci\u003ec\u003c/i\u003e and \u003ci\u003ei\u003c/i\u003e is the process that, if executed in parallel with \u003ci\u003ei\u003c/i\u003e implements \u003ci\u003ec\u003c/i\u003e. It represents an abstract version of a rely condition generalised to a process.",
	"authors": [
	"Julian Fell",
	"Ian J. Hayes",
	"Andrius Velykis"
	],
	"date": "2016-12-30",
	- "id": 430,
	+ "id": 433,
	"link": "/entries/Concurrent_Ref_Alg.html",
	"permalink": "/entries/Concurrent_Ref_Alg.html",
	"shortname": "Concurrent_Ref_Alg",
	"title": "Concurrent Refinement Algebra and Rely Quotients",
	"topic_links": [
	"computer-science/concurrency"
	],
	"topics": [
	"Computer science/Concurrency"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides all cardinality theorems of the Twelvefold Way. The Twelvefold Way systematically classifies twelve related combinatorial problems concerning two finite sets, which include counting permutations, combinations, multisets, set partitions and number partitions. This development builds upon the existing formal developments with cardinality theorems for those structures. It provides twelve bijections from the various structures to different equivalence classes on finite functions, and hence, proves cardinality formulae for these equivalence classes on finite functions.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-12-29",
	- "id": 431,
	+ "id": 434,
	"link": "/entries/Twelvefold_Way.html",
	"permalink": "/entries/Twelvefold_Way.html",
	"shortname": "Twelvefold_Way",
	"title": "The Twelvefold Way",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "Isabelle includes various automatic tools for finding proofs under certain conditions. However, for each conjecture, knowing which automation to use, and how to tweak its parameters, is currently labour intensive. We have developed a language, PSL, designed to capture high level proof strategies. PSL offloads the construction of human-readable fast-to-replay proof scripts to automatic search, making use of search-time information about each conjecture. Our preliminary evaluations show that PSL reduces the labour cost of interactive theorem proving. This submission contains the implementation of PSL and an example theory file, Example.thy, showing how to write poof strategies in PSL.",
	"authors": [
	"Yutaka Nagashima"
	],
	"date": "2016-12-20",
	- "id": 432,
	+ "id": 435,
	"link": "/entries/Proof_Strategy_Language.html",
	"permalink": "/entries/Proof_Strategy_Language.html",
	"shortname": "Proof_Strategy_Language",
	"title": "Proof Strategy Language",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "Paraconsistency is about handling inconsistency in a coherent way. In classical and intuitionistic logic everything follows from an inconsistent theory. A paraconsistent logic avoids the explosion. Quite a few applications in computer science and engineering are discussed in the Intelligent Systems Reference Library Volume 110: Towards Paraconsistent Engineering (Springer 2016). We formalize a paraconsistent many-valued logic that we motivated and described in a special issue on logical approaches to paraconsistency (Journal of Applied Non-Classical Logics 2005). We limit ourselves to the propositional fragment of the higher-order logic. The logic is based on so-called key equalities and has a countably infinite number of truth values. We prove theorems in the logic using the definition of validity. We verify truth tables and also counterexamples for non-theorems. We prove meta-theorems about the logic and finally we investigate a case study.",
	"authors": [
	"Anders Schlichtkrull",
	"Jørgen Villadsen"
	],
	"date": "2016-12-07",
	- "id": 433,
	+ "id": 436,
	"link": "/entries/Paraconsistency.html",
	"permalink": "/entries/Paraconsistency.html",
	"shortname": "Paraconsistency",
	"title": "Paraconsistency",
	"topic_links": [
	"logic/general-logic/paraconsistent-logics"
	],
	"topics": [
	"Logic/General logic/Paraconsistent logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We propose a concurrency reasoning framework for imperative programs, based on the Owicki-Gries (OG) foundational shared-variable concurrency method. Our framework combines the approaches of Hoare-Parallel, a formalisation of OG in Isabelle/HOL for a simple while-language, and Simpl, a generic imperative language embedded in Isabelle/HOL, allowing formal reasoning on C programs. We define the Complx language, extending the syntax and semantics of Simpl with support for parallel composition and synchronisation. We additionally define an OG logic, which we prove sound w.r.t. the semantics, and a verification condition generator, both supporting involved low-level imperative constructs such as function calls and abrupt termination. We illustrate our framework on an example that features exceptions, guards and function calls. We aim to then target concurrent operating systems, such as the interruptible eChronos embedded operating system for which we already have a model-level OG proof using Hoare-Parallel.",
	"authors": [
	"Sidney Amani",
	"June Andronick",
	"Maksym Bortin",
	"Corey Lewis",
	"Christine Rizkallah",
	"Joseph Tuong"
	],
	"date": "2016-11-29",
	- "id": 434,
	+ "id": 437,
	"link": "/entries/Complx.html",
	"permalink": "/entries/Complx.html",
	"shortname": "Complx",
	"title": "COMPLX: A Verification Framework for Concurrent Imperative Programs",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "This is the Isabelle formalization of the material decribed in the eponymous \u003ca href=\"https://doi.org/10.1007/978-3-642-32347-8_9\"\u003eITP 2012 paper\u003c/a\u003e. It develops a generic abstract interpreter for a while-language, including widening and narrowing. The collecting semantics and the abstract interpreter operate on annotated commands: the program is represented as a syntax tree with the semantic information directly embedded, without auxiliary labels. The aim of the formalization is simplicity, not efficiency or precision. This is motivated by the inclusion of the material in a theorem prover based course on semantics. A similar (but more polished) development is covered in the book \u003ca href=\"https://doi.org/10.1007/978-3-319-10542-0\"\u003eConcrete Semantics\u003c/a\u003e.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2016-11-23",
	- "id": 435,
	+ "id": 438,
	"link": "/entries/Abs_Int_ITP2012.html",
	"permalink": "/entries/Abs_Int_ITP2012.html",
	"shortname": "Abs_Int_ITP2012",
	"title": "Abstract Interpretation of Annotated Commands",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We bring the labelled sequent calculus $LS_{PASL}$ for propositional abstract separation logic to Isabelle. The tactics given here are directly applied on an extension of the Separation Algebra in the AFP. In addition to the cancellative separation algebra, we further consider some useful properties in the heap model of separation logic, such as indivisible unit, disjointness, and cross-split. The tactics are essentially a proof search procedure for the calculus $LS_{PASL}$. We wrap the tactics in an Isabelle method called separata, and give a few examples of separation logic formulae which are provable by separata.",
	"authors": [
	"Zhe Hou",
	"David Sanan",
	"Alwen Tiu",
	"Rajeev Gore",
	"Ranald Clouston"
	],
	"date": "2016-11-16",
	- "id": 436,
	+ "id": 439,
	"link": "/entries/Separata.html",
	"permalink": "/entries/Separata.html",
	"shortname": "Separata",
	"title": "Separata: Isabelle tactics for Separation Algebra",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"tools"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization defines Knuth–Bendix orders for higher-order terms without lambda-abstraction and proves many useful properties about them. The main order fully coincides with the standard transfinite KBO with subterm coefficients on first-order terms. It appears promising as the basis of a higher-order superposition calculus.",
	"authors": [
	"Heiko Becker",
	"Jasmin Christian Blanchette",
	"Uwe Waldmann",
	"Daniel Wand"
	],
	"date": "2016-11-12",
	- "id": 437,
	+ "id": 440,
	"link": "/entries/Lambda_Free_KBOs.html",
	"permalink": "/entries/Lambda_Free_KBOs.html",
	"shortname": "Lambda_Free_KBOs",
	"title": "Formalization of Knuth–Bendix Orders for Lambda-Free Higher-Order Terms",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization introduces a nested multiset datatype and defines Dershowitz and Manna's nested multiset order. The order is proved well founded and linear. By removing one constructor, we transform the nested multisets into hereditary multisets. These are isomorphic to the syntactic ordinals—the ordinals can be recursively expressed in Cantor normal form. Addition, subtraction, multiplication, and linear orders are provided on this type.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Mathias Fleury",
	"Dmitriy Traytel"
	],
	"date": "2016-11-12",
	- "id": 438,
	+ "id": 441,
	"link": "/entries/Nested_Multisets_Ordinals.html",
	"permalink": "/entries/Nested_Multisets_Ordinals.html",
	"shortname": "Nested_Multisets_Ordinals",
	"title": "Formalization of Nested Multisets, Hereditary Multisets, and Syntactic Ordinals",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 7
	},
	{
	"abstract": "Deep learning has had a profound impact on computer science in recent years, with applications to search engines, image recognition and language processing, bioinformatics, and more. Recently, Cohen et al. provided theoretical evidence for the superiority of deep learning over shallow learning. This formalization of their work simplifies and generalizes the original proof, while working around the limitations of the Isabelle type system. To support the formalization, I developed reusable libraries of formalized mathematics, including results about the matrix rank, the Lebesgue measure, and multivariate polynomials, as well as a library for tensor analysis.",
	"authors": [
	"Alexander Bentkamp"
	],
	"date": "2016-11-10",
	- "id": 439,
	+ "id": 442,
	"link": "/entries/Deep_Learning.html",
	"permalink": "/entries/Deep_Learning.html",
	"shortname": "Deep_Learning",
	"title": "Expressiveness of Deep Learning",
	"topic_links": [
	"computer-science/machine-learning",
	"mathematics/analysis"
	],
	"topics": [
	"Computer science/Machine learning",
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalize a uniform semantic substrate for a wide variety of process calculi where states and action labels can be from arbitrary nominal sets. A Hennessy-Milner logic for these systems is defined, and proved adequate for bisimulation equivalence. A main novelty is the construction of an infinitary nominal data type to model formulas with (finitely supported) infinite conjunctions and actions that may contain binding names. The logic is generalized to treat different bisimulation variants such as early, late and open in a systematic way.",
	"authors": [
	"Tjark Weber",
	"Lars-Henrik Eriksson",
	"Joachim Parrow",
	"Johannes Borgström",
	"Ramunas Gutkovas"
	],
	"date": "2016-10-25",
	- "id": 440,
	+ "id": 443,
	"link": "/entries/Modal_Logics_for_NTS.html",
	"permalink": "/entries/Modal_Logics_for_NTS.html",
	"shortname": "Modal_Logics_for_NTS",
	"title": "Modal Logics for Nominal Transition Systems",
	"topic_links": [
	"computer-science/concurrency/process-calculi",
	"logic/general-logic/modal-logic"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi",
	"Logic/General logic/Modal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "We mechanize proofs of several results from the matching with contracts literature, which generalize those of the classical two-sided matching scenarios that go by the name of stable marriage. Our focus is on game theoretic issues. Along the way we develop executable algorithms for computing optimal stable matches.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2016-10-24",
	- "id": 441,
	+ "id": 444,
	"link": "/entries/Stable_Matching.html",
	"permalink": "/entries/Stable_Matching.html",
	"shortname": "Stable_Matching",
	"title": "Stable Matching",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "We present LOFT — Linux firewall OpenFlow Translator, a system that transforms the main routing table and FORWARD chain of iptables of a Linux-based firewall into a set of static OpenFlow rules. Our implementation is verified against a model of a simplified Linux-based router and we can directly show how much of the original functionality is preserved.",
	"authors": [
	"Julius Michaelis",
	"Cornelius Diekmann"
	],
	"date": "2016-10-21",
	- "id": 442,
	+ "id": 445,
	"link": "/entries/LOFT.html",
	"permalink": "/entries/LOFT.html",
	"shortname": "LOFT",
	"title": "LOFT — Verified Migration of Linux Firewalls to SDN",
	"topic_links": [
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Networks"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise the SPARCv8 instruction set architecture (ISA) which is used in processors such as LEON3. Our formalisation can be specialised to any SPARCv8 CPU, here we use LEON3 as a running example. Our model covers the operational semantics for all the instructions in the integer unit of the SPARCv8 architecture and it supports Isabelle code export, which effectively turns the Isabelle model into a SPARCv8 CPU simulator. We prove the language-based non-interference property for the LEON3 processor. Our model is based on deterministic monad, which is a modified version of the non-deterministic monad from NICTA/l4v.",
	"authors": [
	"Zhe Hou",
	"David Sanan",
	"Alwen Tiu",
	"Yang Liu"
	],
	"date": "2016-10-19",
	- "id": 443,
	+ "id": 446,
	"link": "/entries/SPARCv8.html",
	"permalink": "/entries/SPARCv8.html",
	"shortname": "SPARCv8",
	"title": "A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor",
	"topic_links": [
	"computer-science/security",
	"computer-science/hardware"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Hardware"
	],
	"used_by": 0
	},
	{
	"abstract": "This document contains a proof of the necessary condition on the code rate of a source code, namely that this code rate is bounded by the entropy of the source. This represents one half of Shannon's source coding theorem, which is itself an equivalence.",
	"authors": [
	"Quentin Hibon",
	"Lawrence C. Paulson"
	],
	"date": "2016-10-19",
	- "id": 444,
	+ "id": 447,
	"link": "/entries/Source_Coding_Theorem.html",
	"permalink": "/entries/Source_Coding_Theorem.html",
	"shortname": "Source_Coding_Theorem",
	"title": "Source Coding Theorem",
	"topic_links": [
	"mathematics/probability-theory"
	],
	"topics": [
	"Mathematics/Probability theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eWe formalize the Berlekamp-Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. \u003c/p\u003e \u003cp\u003eThe algorithm first performs a factorization in the prime field GF(p) and then performs computations in the integer ring modulo p^k, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using Isabelle’s recent addition of local type definitions. \u003c/p\u003e \u003cp\u003eThrough experiments we verify that our algorithm factors polynomials of degree 100 within seconds. \u003c/p\u003e",
	"authors": [
	"Jose Divasón",
	"Sebastiaan J. C. Joosten",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2016-10-14",
	- "id": 445,
	+ "id": 448,
	"link": "/entries/Berlekamp_Zassenhaus.html",
	"permalink": "/entries/Berlekamp_Zassenhaus.html",
	"shortname": "Berlekamp_Zassenhaus",
	"title": "The Factorization Algorithm of Berlekamp and Zassenhaus",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 8
	},
	{
	"abstract": "This entry provides a geometric proof of the intersecting chords theorem. The theorem states that when two chords intersect each other inside a circle, the products of their segments are equal. After a short review of existing proofs in the literature, I decided to use a proof approach that employs reasoning about lengths of line segments, the orthogonality of two lines and the Pythagoras Law. Hence, one can understand the formalized proof easily with the knowledge of a few general geometric facts that are commonly taught in high-school. This theorem is the 55th theorem of the Top 100 Theorems list.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-10-11",
	- "id": 446,
	+ "id": 449,
	"link": "/entries/Chord_Segments.html",
	"permalink": "/entries/Chord_Segments.html",
	"shortname": "Chord_Segments",
	"title": "Intersecting Chords Theorem",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "Lp is the space of functions whose p-th power is integrable. It is one of the most fundamental Banach spaces that is used in analysis and probability. We develop a framework for function spaces, and then implement the Lp spaces in this framework using the existing integration theory in Isabelle/HOL. Our development contains most fundamental properties of Lp spaces, notably the Hölder and Minkowski inequalities, completeness of Lp, duality, stability under almost sure convergence, multiplication of functions in Lp and Lq, stability under conditional expectation.",
	"authors": [
	"Sebastien Gouezel"
	],
	"date": "2016-10-05",
	- "id": 447,
	+ "id": 450,
	"link": "/entries/Lp.html",
	"permalink": "/entries/Lp.html",
	"shortname": "Lp",
	"title": "Lp spaces",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis work defines and proves the correctness of the Fisher–Yates algorithm for shuffling – i.e. producing a random permutation – of a list. The algorithm proceeds by traversing the list and in each step swapping the current element with a random element from the remaining list.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2016-09-30",
	- "id": 448,
	+ "id": 451,
	"link": "/entries/Fisher_Yates.html",
	"permalink": "/entries/Fisher_Yates.html",
	"shortname": "Fisher_Yates",
	"title": "Fisher–Yates shuffle",
	"topic_links": [
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 0
	},
	{
	"abstract": "Allen’s interval calculus is a qualitative temporal representation of time events. Allen introduced 13 binary relations that describe all the possible arrangements between two events, i.e. intervals with non-zero finite length. The compositions are pertinent to reasoning about knowledge of time. In particular, a consistency problem of relation constraints is commonly solved with a guideline from these compositions. We formalize the relations together with an axiomatic system. We proof the validity of the 169 compositions of these relations. We also define nests as the sets of intervals that share a meeting point. We prove that nests give the ordering properties of points without introducing a new datatype for points. [1] J.F. Allen. Maintaining Knowledge about Temporal Intervals. In Commun. ACM, volume 26, pages 832–843, 1983. [2] J. F. Allen and P. J. Hayes. A Common-sense Theory of Time. In Proceedings of the 9th International Joint Conference on Artificial Intelligence (IJCAI’85), pages 528–531, 1985.",
	"authors": [
	"Fadoua Ghourabi"
	],
	"date": "2016-09-29",
	- "id": 449,
	+ "id": 452,
	"link": "/entries/Allen_Calculus.html",
	"permalink": "/entries/Allen_Calculus.html",
	"shortname": "Allen_Calculus",
	"title": "Allen's Interval Calculus",
	"topic_links": [
	"logic/general-logic/temporal-logic",
	"mathematics/order"
	],
	"topics": [
	"Logic/General logic/Temporal logic",
	"Mathematics/Order"
	],
	"used_by": 0
	},
	{
	"abstract": "This Isabelle/HOL formalization defines recursive path orders (RPOs) for higher-order terms without lambda-abstraction and proves many useful properties about them. The main order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work. An optimized variant is formalized as well. It appears promising as the basis of a higher-order superposition calculus.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Uwe Waldmann",
	"Daniel Wand"
	],
	"date": "2016-09-23",
	- "id": 450,
	+ "id": 453,
	"link": "/entries/Lambda_Free_RPOs.html",
	"permalink": "/entries/Lambda_Free_RPOs.html",
	"shortname": "Lambda_Free_RPOs",
	"title": "Formalization of Recursive Path Orders for Lambda-Free Higher-Order Terms",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 5
	},
	{
	"abstract": "We present a big step semantics of the filtering behavior of the Linux/netfilter iptables firewall. We provide algorithms to simplify complex iptables rulests to a simple firewall model (c.f. AFP entry \u003ca href=\"https://www.isa-afp.org/entries/Simple_Firewall.html\"\u003eSimple_Firewall\u003c/a\u003e) and to verify spoofing protection of a ruleset. Internally, we embed our semantics into ternary logic, ultimately supporting every iptables match condition by abstracting over unknowns. Using this AFP entry and all entries it depends on, we created an easy-to-use, stand-alone haskell tool called \u003ca href=\"http://iptables.isabelle.systems\"\u003efffuu\u003c/a\u003e. The tool does not require any input \u0026mdash;except for the \u003ctt\u003eiptables-save\u003c/tt\u003e dump of the analyzed firewall\u0026mdash; and presents interesting results about the user's ruleset. Real-Word firewall errors have been uncovered, and the correctness of rulesets has been proved, with the help of our tool.",
	"authors": [
	"Cornelius Diekmann",
	"Lars Hupel"
	],
	"date": "2016-09-09",
	- "id": 451,
	+ "id": 454,
	"link": "/entries/Iptables_Semantics.html",
	"permalink": "/entries/Iptables_Semantics.html",
	"shortname": "Iptables_Semantics",
	"title": "Iptables Semantics",
	"topic_links": [
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Networks"
	],
	"used_by": 2
	},
	{
	"abstract": "We provide a formalization of a variant of the superposition calculus, together with formal proofs of soundness and refutational completeness (w.r.t. the usual redundancy criteria based on clause ordering). This version of the calculus uses all the standard restrictions of the superposition rules, together with the following refinement, inspired by the basic superposition calculus: each clause is associated with a set of terms which are assumed to be in normal form -- thus any application of the replacement rule on these terms is blocked. The set is initially empty and terms may be added or removed at each inference step. The set of terms that are assumed to be in normal form includes any term introduced by previous unifiers as well as any term occurring in the parent clauses at a position that is smaller (according to some given ordering on positions) than a previously replaced term. The standard superposition calculus corresponds to the case where the set of irreducible terms is always empty.",
	"authors": [
	"Nicolas Peltier"
	],
	"date": "2016-09-06",
	- "id": 452,
	+ "id": 455,
	"link": "/entries/SuperCalc.html",
	"permalink": "/entries/SuperCalc.html",
	"shortname": "SuperCalc",
	"title": "A Variant of the Superposition Calculus",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "A range of algebras between lattices and Boolean algebras generalise the notion of a complement. We develop a hierarchy of these pseudo-complemented algebras that includes Stone algebras. Independently of this theory we study filters based on partial orders. Both theories are combined to prove Chen and Grätzer's construction theorem for Stone algebras. The latter involves extensive reasoning about algebraic structures in addition to reasoning in algebraic structures.",
	"authors": [
	"Walter Guttmann"
	],
	"date": "2016-09-06",
	- "id": 453,
	+ "id": 456,
	"link": "/entries/Stone_Algebras.html",
	"permalink": "/entries/Stone_Algebras.html",
	"shortname": "Stone_Algebras",
	"title": "Stone Algebras",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003eThis work contains a proof of Stirling's formula both for the factorial $n! \\sim \\sqrt{2\\pi n} (n/e)^n$ on natural numbers and the real Gamma function $\\Gamma(x)\\sim \\sqrt{2\\pi/x} (x/e)^x$. The proof is based on work by \u003ca href=\"http://www.maths.lancs.ac.uk/~jameson/stirlgamma.pdf\"\u003eGraham Jameson\u003c/a\u003e.\u003c/p\u003e \u003cp\u003eThis is then extended to the full asymptotic expansion $$\\log\\Gamma(z) = \\big(z - \\tfrac{1}{2}\\big)\\log z - z + \\tfrac{1}{2}\\log(2\\pi) + \\sum_{k=1}^{n-1} \\frac{B_{k+1}}{k(k+1)} z^{-k}\\\\ {} - \\frac{1}{n} \\int_0^\\infty B_n([t])(t + z)^{-n}\\,\\text{d}t$$ uniformly for all complex $z\\neq 0$ in the cone $\\text{arg}(z)\\leq \\alpha$ for any $\\alpha\\in(0,\\pi)$, with which the above asymptotic relation for \u0026Gamma; is also extended to complex arguments.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2016-09-01",
	- "id": 454,
	+ "id": 457,
	"link": "/entries/Stirling_Formula.html",
	"permalink": "/entries/Stirling_Formula.html",
	"shortname": "Stirling_Formula",
	"title": "Stirling's formula",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 6
	},
	{
	"abstract": "This entry contains definitions for routing with routing tables/longest prefix matching. A routing table entry is modelled as a record of a prefix match, a metric, an output port, and an optional next hop. A routing table is a list of entries, sorted by prefix length and metric. Additionally, a parser and serializer for the output of the ip-route command, a function to create a relation from output port to corresponding destination IP space, and a model of a Linux-style router are included.",
	"authors": [
	"Julius Michaelis",
	"Cornelius Diekmann"
	],
	"date": "2016-08-31",
	- "id": 455,
	+ "id": 458,
	"link": "/entries/Routing.html",
	"permalink": "/entries/Routing.html",
	"shortname": "Routing",
	"title": "Routing",
	"topic_links": [
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Networks"
	],
	"used_by": 1
	},
	{
	"abstract": "We present a simple model of a firewall. The firewall can accept or drop a packet and can match on interfaces, IP addresses, protocol, and ports. It was designed to feature nice mathematical properties: The type of match expressions was carefully crafted such that the conjunction of two match expressions is only one match expression. This model is too simplistic to mirror all aspects of the real world. In the upcoming entry \"Iptables Semantics\", we will translate the Linux firewall iptables to this model. For a fixed service (e.g. ssh, http), we provide an algorithm to compute an overview of the firewall's filtering behavior. The algorithm computes minimal service matrices, i.e. graphs which partition the complete IPv4 and IPv6 address space and visualize the allowed accesses between partitions. For a detailed description, see \u003ca href=\"http://dl.ifip.org/db/conf/networking/networking2016/1570232858.pdf\"\u003eVerified iptables Firewall Analysis\u003c/a\u003e, IFIP Networking 2016.",
	"authors": [
	"Cornelius Diekmann",
	"Julius Michaelis",
	"Max W. Haslbeck"
	],
	"date": "2016-08-24",
	- "id": 456,
	+ "id": 459,
	"link": "/entries/Simple_Firewall.html",
	"permalink": "/entries/Simple_Firewall.html",
	"shortname": "Simple_Firewall",
	"title": "Simple Firewall",
	"topic_links": [
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Networks"
	],
	"used_by": 1
	},
	{
	"abstract": "TRACER is a tool for verifying safety properties of sequential C programs. TRACER attempts at building a finite symbolic execution graph which over-approximates the set of all concrete reachable states and the set of feasible paths. We present an abstract framework for TRACER and similar CEGAR-like systems. The framework provides 1) a graph- transformation based method for reducing the feasible paths in control-flow graphs, 2) a model for symbolic execution, subsumption, predicate abstraction and invariant generation. In this framework we formally prove two key properties: correct construction of the symbolic states and preservation of feasible paths. The framework focuses on core operations, leaving to concrete prototypes to “fit in” heuristics for combining them. The accompanying paper (published in ITP 2016) can be found at https://www.lri.fr/∼wolff/papers/conf/2016-itp-InfPathsNSE.pdf.",
	"authors": [
	"Romain Aissat",
	"Frederic Voisin",
	"Burkhart Wolff"
	],
	"date": "2016-08-18",
	- "id": 457,
	+ "id": 460,
	"link": "/entries/InfPathElimination.html",
	"permalink": "/entries/InfPathElimination.html",
	"shortname": "InfPathElimination",
	"title": "Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of the Ford-Fulkerson method for computing the maximum flow in a network. Our formal proof closely follows a standard textbook proof, and is accessible even without being an expert in Isabelle/HOL--- the interactive theorem prover used for the formalization. We then use stepwise refinement to obtain the Edmonds-Karp algorithm, and formally prove a bound on its complexity. Further refinement yields a verified implementation, whose execution time compares well to an unverified reference implementation in Java. This entry is based on our ITP-2016 paper with the same title.",
	"authors": [
	"Peter Lammich",
	"S. Reza Sefidgar"
	],
	"date": "2016-08-12",
	- "id": 458,
	+ "id": 461,
	"link": "/entries/EdmondsKarp_Maxflow.html",
	"permalink": "/entries/EdmondsKarp_Maxflow.html",
	"shortname": "EdmondsKarp_Maxflow",
	"title": "Formalizing the Edmonds-Karp Algorithm",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 1
	},
	{
	"abstract": "We present the Imperative Refinement Framework (IRF), a tool that supports a stepwise refinement based approach to imperative programs. This entry is based on the material we presented in [ITP-2015, CPP-2016]. It uses the Monadic Refinement Framework as a frontend for the specification of the abstract programs, and Imperative/HOL as a backend to generate executable imperative programs. The IRF comes with tool support to synthesize imperative programs from more abstract, functional ones, using efficient imperative implementations for the abstract data structures. This entry also includes the Imperative Isabelle Collection Framework (IICF), which provides a library of re-usable imperative collection data structures. Moreover, this entry contains a quickstart guide and a reference manual, which provide an introduction to using the IRF for Isabelle/HOL experts. It also provids a collection of (partly commented) practical examples, some highlights being Dijkstra's Algorithm, Nested-DFS, and a generic worklist algorithm with subsumption. Finally, this entry contains benchmark scripts that compare the runtime of some examples against reference implementations of the algorithms in Java and C++. [ITP-2015] Peter Lammich: Refinement to Imperative/HOL. ITP 2015: 253--269 [CPP-2016] Peter Lammich: Refinement based verification of imperative data structures. CPP 2016: 27--36",
	"authors": [
	"Peter Lammich"
	],
	"date": "2016-08-08",
	- "id": 459,
	+ "id": 462,
	"link": "/entries/Refine_Imperative_HOL.html",
	"permalink": "/entries/Refine_Imperative_HOL.html",
	"shortname": "Refine_Imperative_HOL",
	"title": "The Imperative Refinement Framework",
	"topic_links": [
	"computer-science/semantics-and-reasoning",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Semantics and reasoning",
	"Computer science/Data structures"
	],
	"used_by": 5
	},
	{
	"abstract": "This entry provides an analytic proof to Ptolemy's Theorem using polar form transformation and trigonometric identities. In this formalization, we use ideas from John Harrison's HOL Light formalization and the proof sketch on the Wikipedia entry of Ptolemy's Theorem. This theorem is the 95th theorem of the Top 100 Theorems list.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-08-07",
	- "id": 460,
	+ "id": 463,
	"link": "/entries/Ptolemys_Theorem.html",
	"permalink": "/entries/Ptolemys_Theorem.html",
	"shortname": "Ptolemys_Theorem",
	"title": "Ptolemy's Theorem",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "In 1964, Fitch showed that the paradox of the surprise hanging can be resolved by showing that the judge’s verdict is inconsistent. His formalization builds on Gödel’s coding of provability. In this theory, we reproduce his proof in Isabelle, building on Paulson’s formalisation of Gödel’s incompleteness theorems.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2016-07-17",
	- "id": 461,
	+ "id": 464,
	"link": "/entries/Surprise_Paradox.html",
	"permalink": "/entries/Surprise_Paradox.html",
	"shortname": "Surprise_Paradox",
	"title": "Surprise Paradox",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This library defines three different versions of pairing heaps: a functional version of the original design based on binary trees [Fredman et al. 1986], the version by Okasaki [1998] and a modified version of the latter that is free of structural invariants. \u003cp\u003e The amortized complexity of pairing heaps is analyzed in the AFP article \u003ca href=\"http://isa-afp.org/entries/Amortized_Complexity.html\"\u003eAmortized Complexity\u003c/a\u003e.",
	"authors": [
	"Hauke Brinkop",
	"Tobias Nipkow"
	],
	"date": "2016-07-14",
	- "id": 462,
	+ "id": 465,
	"link": "/entries/Pairing_Heap.html",
	"permalink": "/entries/Pairing_Heap.html",
	"shortname": "Pairing_Heap",
	"title": "Pairing Heap",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003e This entry presents a framework for the modular verification of DFS-based algorithms, which is described in our [CPP-2015] paper. It provides a generic DFS algorithm framework, that can be parameterized with user-defined actions on certain events (e.g. discovery of new node). It comes with an extensible library of invariants, which can be used to derive invariants of a specific parameterization. Using refinement techniques, efficient implementations of the algorithms can easily be derived. Here, the framework comes with templates for a recursive and a tail-recursive implementation, and also with several templates for implementing the data structures required by the DFS algorithm. Finally, this entry contains a set of re-usable DFS-based algorithms, which illustrate the application of the framework. \u003c/p\u003e\u003cp\u003e [CPP-2015] Peter Lammich, René Neumann: A Framework for Verifying Depth-First Search Algorithms. CPP 2015: 137-146\u003c/p\u003e",
	"authors": [
	"Peter Lammich",
	"René Neumann"
	],
	"date": "2016-07-05",
	- "id": 463,
	+ "id": 466,
	"link": "/entries/DFS_Framework.html",
	"permalink": "/entries/DFS_Framework.html",
	"shortname": "DFS_Framework",
	"title": "A Framework for Verifying Depth-First Search Algorithms",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 4
	},
	{
	"abstract": "We provide a basic formal framework for the theory of chamber complexes and Coxeter systems, and for buildings as thick chamber complexes endowed with a system of apartments. Along the way, we develop some of the general theory of abstract simplicial complexes and of groups (relying on the \u003ci\u003egroup_add\u003c/i\u003e class for the basics), including free groups and group presentations, and their universal properties. The main results verified are that the deletion condition is both necessary and sufficient for a group with a set of generators of order two to be a Coxeter system, and that the apartments in a (thick) building are all uniformly Coxeter.",
	"authors": [
	"Jeremy Sylvestre"
	],
	"date": "2016-07-01",
	- "id": 464,
	+ "id": 467,
	"link": "/entries/Buildings.html",
	"permalink": "/entries/Buildings.html",
	"shortname": "Buildings",
	"title": "Chamber Complexes, Coxeter Systems, and Buildings",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory is a formalization of the resolution calculus for first-order logic. It is proven sound and complete. The soundness proof uses the substitution lemma, which shows a correspondence between substitutions and updates to an environment. The completeness proof uses semantic trees, i.e. trees whose paths are partial Herbrand interpretations. It employs Herbrand's theorem in a formulation which states that an unsatisfiable set of clauses has a finite closed semantic tree. It also uses the lifting lemma which lifts resolution derivation steps from the ground world up to the first-order world. The theory is presented in a paper in the Journal of Automated Reasoning [Sch18] which extends a paper presented at the International Conference on Interactive Theorem Proving [Sch16]. An earlier version was presented in an MSc thesis [Sch15]. The formalization mostly follows textbooks by Ben-Ari [BA12], Chang and Lee [CL73], and Leitsch [Lei97]. The theory is part of the IsaFoL project [IsaFoL]. \u003cp\u003e \u003ca name=\"Sch18\"\u003e\u003c/a\u003e[Sch18] Anders Schlichtkrull. \"Formalization of the Resolution Calculus for First-Order Logic\". Journal of Automated Reasoning, 2018.\u003cbr\u003e \u003ca name=\"Sch16\"\u003e\u003c/a\u003e[Sch16] Anders Schlichtkrull. \"Formalization of the Resolution Calculus for First-Order Logic\". In: ITP 2016. Vol. 9807. LNCS. Springer, 2016.\u003cbr\u003e \u003ca name=\"Sch15\"\u003e\u003c/a\u003e[Sch15] Anders Schlichtkrull. \u003ca href=\"https://people.compute.dtu.dk/andschl/Thesis.pdf\"\u003e \"Formalization of Resolution Calculus in Isabelle\"\u003c/a\u003e. \u003ca href=\"https://people.compute.dtu.dk/andschl/Thesis.pdf\"\u003ehttps://people.compute.dtu.dk/andschl/Thesis.pdf\u003c/a\u003e. MSc thesis. Technical University of Denmark, 2015.\u003cbr\u003e \u003ca name=\"BA12\"\u003e\u003c/a\u003e[BA12] Mordechai Ben-Ari. \u003ci\u003eMathematical Logic for Computer Science\u003c/i\u003e. 3rd. Springer, 2012.\u003cbr\u003e \u003ca name=\"CL73\"\u003e\u003c/a\u003e[CL73] Chin-Liang Chang and Richard Char-Tung Lee. \u003ci\u003eSymbolic Logic and Mechanical Theorem Proving\u003c/i\u003e. 1st. Academic Press, Inc., 1973.\u003cbr\u003e \u003ca name=\"Lei97\"\u003e\u003c/a\u003e[Lei97] Alexander Leitsch. \u003ci\u003eThe Resolution Calculus\u003c/i\u003e. Texts in theoretical computer science. Springer, 1997.\u003cbr\u003e \u003ca name=\"IsaFoL\"\u003e\u003c/a\u003e[IsaFoL] IsaFoL authors. \u003ca href=\"https://bitbucket.org/jasmin_blanchette/isafol\"\u003e IsaFoL: Isabelle Formalization of Logic\u003c/a\u003e. \u003ca href=\"https://bitbucket.org/jasmin_blanchette/isafol\"\u003ehttps://bitbucket.org/jasmin_blanchette/isafol\u003c/a\u003e.",
	"authors": [
	"Anders Schlichtkrull"
	],
	"date": "2016-06-30",
	- "id": 465,
	+ "id": 468,
	"link": "/entries/Resolution_FOL.html",
	"permalink": "/entries/Resolution_FOL.html",
	"shortname": "Resolution_FOL",
	"title": "The Resolution Calculus for First-Order Logic",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the Z property introduced by Dehornoy and van Oostrom. First we show that for any abstract rewrite system, Z implies confluence. Then we give two examples of proofs using Z: confluence of lambda-calculus with respect to beta-reduction and confluence of combinatory logic.",
	"authors": [
	"Bertram Felgenhauer",
	"Julian Nagele",
	"Vincent van Oostrom",
	"Christian Sternagel"
	],
	"date": "2016-06-30",
	- "id": 466,
	+ "id": 469,
	"link": "/entries/Rewriting_Z.html",
	"permalink": "/entries/Rewriting_Z.html",
	"shortname": "Rewriting_Z",
	"title": "The Z Property",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "The paper \"Compositional Verification and Refinement of Concurrent Value-Dependent Noninterference\" by Murray et. al. (CSF 2016) presents a compositional theory of refinement for a value-dependent noninterference property, defined in (Murray, PLAS 2015), for concurrent programs. This development formalises that refinement theory, and demonstrates its application on some small examples.",
	"authors": [
	"Toby Murray",
	"Robert Sison",
	"Edward Pierzchalski",
	"Christine Rizkallah"
	],
	"date": "2016-06-28",
	- "id": 467,
	+ "id": 470,
	"link": "/entries/Dependent_SIFUM_Refinement.html",
	"permalink": "/entries/Dependent_SIFUM_Refinement.html",
	"shortname": "Dependent_SIFUM_Refinement",
	"title": "Compositional Security-Preserving Refinement for Concurrent Imperative Programs",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains a definition of IP addresses and a library to work with them. Generic IP addresses are modeled as machine words of arbitrary length. Derived from this generic definition, IPv4 addresses are 32bit machine words, IPv6 addresses are 128bit words. Additionally, IPv4 addresses can be represented in dot-decimal notation and IPv6 addresses in (compressed) colon-separated notation. We support toString functions and parsers for both notations. Sets of IP addresses can be represented with a netmask (e.g. 192.168.0.0/255.255.0.0) or in CIDR notation (e.g. 192.168.0.0/16). To provide executable code for set operations on IP address ranges, the library includes a datatype to work on arbitrary intervals of machine words.",
	"authors": [
	"Cornelius Diekmann",
	"Julius Michaelis",
	"Lars Hupel"
	],
	"date": "2016-06-28",
	- "id": 468,
	+ "id": 471,
	"link": "/entries/IP_Addresses.html",
	"permalink": "/entries/IP_Addresses.html",
	"shortname": "IP_Addresses",
	"title": "IP Addresses",
	"topic_links": [
	"computer-science/networks"
	],
	"topics": [
	"Computer science/Networks"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eThis entry provides three lemmas to count the number of multisets of a given size and finite carrier set. The first lemma provides a cardinality formula assuming that the multiset's elements are chosen from the given carrier set. The latter two lemmas provide formulas assuming that the multiset's elements also cover the given carrier set, i.e., each element of the carrier set occurs in the multiset at least once.\u003c/p\u003e \u003cp\u003eThe proof of the first lemma uses the argument of the recurrence relation for counting multisets. The proof of the second lemma is straightforward, and the proof of the third lemma is easily obtained using the first cardinality lemma. A challenge for the formalization is the derivation of the required induction rule, which is a special combination of the induction rules for finite sets and natural numbers. The induction rule is derived by defining a suitable inductive predicate and transforming the predicate's induction rule.\u003c/p\u003e",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-06-26",
	- "id": 469,
	+ "id": 472,
	"link": "/entries/Card_Multisets.html",
	"permalink": "/entries/Card_Multisets.html",
	"shortname": "Card_Multisets",
	"title": "Cardinality of Multisets",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e This article attempts to develop a usable framework for doing category theory in Isabelle/HOL. Our point of view, which to some extent differs from that of the previous AFP articles on the subject, is to try to explore how category theory can be done efficaciously within HOL, rather than trying to match exactly the way things are done using a traditional approach. To this end, we define the notion of category in an \"object-free\" style, in which a category is represented by a single partial composition operation on arrows. This way of defining categories provides some advantages in the context of HOL, including the ability to avoid the use of records and the possibility of defining functors and natural transformations simply as certain functions on arrows, rather than as composite objects. We define various constructions associated with the basic notions, including: dual category, product category, functor category, discrete category, free category, functor composition, and horizontal and vertical composite of natural transformations. A \"set category\" locale is defined that axiomatizes the notion \"category of all sets at a type and all functions between them,\" and a fairly extensive set of properties of set categories is derived from the locale assumptions. The notion of a set category is used to prove the Yoneda Lemma in a general setting of a category equipped with a \"hom embedding,\" which maps arrows of the category to the \"universe\" of the set category. We also give a treatment of adjunctions, defining adjunctions via left and right adjoint functors, natural bijections between hom-sets, and unit and counit natural transformations, and showing the equivalence of these definitions. We also develop the theory of limits, including representations of functors, diagrams and cones, and diagonal functors. We show that right adjoint functors preserve limits, and that limits can be constructed via products and equalizers. We characterize the conditions under which limits exist in a set category. We also examine the case of limits in a functor category, ultimately culminating in a proof that the Yoneda embedding preserves limits. \u003c/p\u003e\u003cp\u003e Revisions made subsequent to the first version of this article added material on equivalence of categories, cartesian categories, categories with pullbacks, categories with finite limits, and cartesian closed categories. A construction was given of the category of hereditarily finite sets and functions between them, and it was shown that this category is cartesian closed. Using \"ZFC_in_HOL\", a construction was also given of the (large) category of small sets and functions between them, and it was shown that this category is small-complete. \u003c/p\u003e",
	"authors": [
	"Eugene W. Stark"
	],
	"date": "2016-06-26",
	- "id": 470,
	+ "id": 473,
	"link": "/entries/Category3.html",
	"permalink": "/entries/Category3.html",
	"shortname": "Category3",
	"title": "Category Theory with Adjunctions and Limits",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 1
	},
	{
	"abstract": "The paper \"Compositional Verification and Refinement of Concurrent Value-Dependent Noninterference\" by Murray et. al. (CSF 2016) presents a dependent security type system for compositionally verifying a value-dependent noninterference property, defined in (Murray, PLAS 2015), for concurrent programs. This development formalises that security definition, the type system and its soundness proof, and demonstrates its application on some small examples. It was derived from the SIFUM_Type_Systems AFP entry, by Sylvia Grewe, Heiko Mantel and Daniel Schoepe, and whose structure it inherits.",
	"authors": [
	"Toby Murray",
	"Robert Sison",
	"Edward Pierzchalski",
	"Christine Rizkallah"
	],
	"date": "2016-06-25",
	- "id": 471,
	+ "id": 474,
	"link": "/entries/Dependent_SIFUM_Type_Systems.html",
	"permalink": "/entries/Dependent_SIFUM_Type_Systems.html",
	"shortname": "Dependent_SIFUM_Type_Systems",
	"title": "A Dependent Security Type System for Concurrent Imperative Programs",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eIn this work, we define the Catalan numbers \u003cem\u003eC\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and prove several equivalent definitions (including some closed-form formulae). We also show one of their applications (counting the number of binary trees of size \u003cem\u003en\u003c/em\u003e), prove the asymptotic growth approximation \u003cem\u003eC\u003csub\u003en\u003c/sub\u003e \u0026sim; 4\u003csup\u003en\u003c/sup\u003e / (\u0026radic;\u003cspan style=\"text-decoration: overline\"\u003e\u0026pi;\u003c/span\u003e \u0026middot; n\u003csup\u003e1.5\u003c/sup\u003e)\u003c/em\u003e, and provide reasonably efficient executable code to compute them.\u003c/p\u003e \u003cp\u003eThe derivation of the closed-form formulae uses algebraic manipulations of the ordinary generating function of the Catalan numbers, and the asymptotic approximation is then done using generalised binomial coefficients and the Gamma function. Thanks to these highly non-elementary mathematical tools, the proofs are very short and simple.\u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2016-06-21",
	- "id": 472,
	+ "id": 475,
	"link": "/entries/Catalan_Numbers.html",
	"permalink": "/entries/Catalan_Numbers.html",
	"shortname": "Catalan_Numbers",
	"title": "Catalan Numbers",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "Variants of Kleene algebra support program construction and verification by algebraic reasoning. This entry provides a verification component for Hoare logic based on Kleene algebra with tests, verification components for weakest preconditions and strongest postconditions based on Kleene algebra with domain and a component for step-wise refinement based on refinement Kleene algebra with tests. In addition to these components for the partial correctness of while programs, a verification component for total correctness based on divergence Kleene algebras and one for (partial correctness) of recursive programs based on domain quantales are provided. Finally we have integrated memory models for programs with pointers and a program trace semantics into the weakest precondition component.",
	"authors": [
	"Victor B. F. Gomes",
	"Georg Struth"
	],
	"date": "2016-06-18",
	- "id": 473,
	+ "id": 476,
	"link": "/entries/Algebraic_VCs.html",
	"permalink": "/entries/Algebraic_VCs.html",
	"shortname": "Algebraic_VCs",
	"title": "Program Construction and Verification Components Based on Kleene Algebra",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eIn his outstanding work on Communicating Sequential Processes, Hoare has defined two fundamental binary operations allowing to compose the input processes into another, typically more complex, process: sequential composition and concurrent composition. Particularly, the output of the latter operation is a process in which any event not shared by both operands can occur whenever the operand that admits the event can engage in it, whereas any event shared by both operands can occur just in case both can engage in it.\u003c/p\u003e \u003cp\u003eThis paper formalizes Hoare's definition of concurrent composition and proves, in the general case of a possibly intransitive policy, that CSP noninterference security is conserved under this operation. This result, along with the previous analogous one concerning sequential composition, enables the construction of more and more complex processes enforcing noninterference security by composing, sequentially or concurrently, simpler secure processes, whose security can in turn be proven using either the definition of security, or unwinding theorems.\u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2016-06-13",
	- "id": 474,
	+ "id": 477,
	"link": "/entries/Noninterference_Concurrent_Composition.html",
	"permalink": "/entries/Noninterference_Concurrent_Composition.html",
	"shortname": "Noninterference_Concurrent_Composition",
	"title": "Conservation of CSP Noninterference Security under Concurrent Composition",
	"topic_links": [
	"computer-science/security",
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry contains an extension to the Isabelle library for fixed-width machine words. In particular, the entry adds quickcheck setup for words, printing as hexadecimals, additional operations, reasoning about alignment, signed words, enumerations of words, normalisation of word numerals, and an extensive library of properties about generic fixed-width words, as well as an instantiation of many of these to the commonly used 32 and 64-bit bases.",
	"authors": [
	"Joel Beeren",
	"Matthew Fernandez",
	"Xin Gao",
	"Gerwin Klein",
	"Rafal Kolanski",
	"Japheth Lim",
	"Corey Lewis",
	"Daniel Matichuk",
	"Thomas Sewell"
	],
	"date": "2016-06-09",
	- "id": 475,
	+ "id": 478,
	"link": "/entries/Word_Lib.html",
	"permalink": "/entries/Word_Lib.html",
	"shortname": "Word_Lib",
	"title": "Finite Machine Word Library",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 9
	},
	{
	"abstract": "We formalize tree decompositions and tree width in Isabelle/HOL, proving that trees have treewidth 1. We also show that every edge of a tree decomposition is a separation of the underlying graph. As an application of this theorem we prove that complete graphs of size n have treewidth n-1.",
	"authors": [
	"Christoph Dittmann"
	],
	"date": "2016-05-31",
	- "id": 476,
	+ "id": 479,
	"link": "/entries/Tree_Decomposition.html",
	"permalink": "/entries/Tree_Decomposition.html",
	"shortname": "Tree_Decomposition",
	"title": "Tree Decomposition",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides formulae for counting the number of equivalence relations and partial equivalence relations over a finite carrier set with given cardinality. To count the number of equivalence relations, we provide bijections between equivalence relations and set partitions, and then transfer the main results of the two AFP entries, Cardinality of Set Partitions and Spivey's Generalized Recurrence for Bell Numbers, to theorems on equivalence relations. To count the number of partial equivalence relations, we observe that counting partial equivalence relations over a set A is equivalent to counting all equivalence relations over all subsets of the set A. From this observation and the results on equivalence relations, we show that the cardinality of partial equivalence relations over a finite set of cardinality n is equal to the n+1-th Bell number.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-05-24",
	- "id": 477,
	+ "id": 480,
	"link": "/entries/Card_Equiv_Relations.html",
	"permalink": "/entries/Card_Equiv_Relations.html",
	"shortname": "Card_Equiv_Relations",
	"title": "Cardinality of Equivalence Relations",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eBrzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Their algorithm generates POSIX values which encode the information of how a regular expression matches a string--—that is, which part of the string is matched by which part of the regular expression. In this paper we give our inductive definition of what a POSIX value is and show (i) that such a value is unique (for given regular expression and string being matched) and (ii) that Sulzmann and Lu’s algorithm always generates such a value (provided that the regular expression matches the string). This holds also when optimisations are included. Finally we show that (iii) our inductive definition of a POSIX value is equivalent to an alternative definition by Okui and Suzuki which identifies POSIX values as least elements according to an ordering of values.\u003c/p\u003e",
	"authors": [
	"Fahad Ausaf",
	"Roy Dyckhoff",
	"Christian Urban"
	],
	"date": "2016-05-24",
	- "id": 478,
	+ "id": 481,
	"link": "/entries/Posix-Lexing.html",
	"permalink": "/entries/Posix-Lexing.html",
	"shortname": "Posix-Lexing",
	"title": "POSIX Lexing with Derivatives of Regular Expressions",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eThe spectral radius of a matrix A is the maximum norm of all eigenvalues of A. In previous work we already formalized that for a complex matrix A, the values in A\u003csup\u003en\u003c/sup\u003e grow polynomially in n if and only if the spectral radius is at most one. One problem with the above characterization is the determination of all \u003cem\u003ecomplex\u003c/em\u003e eigenvalues. In case A contains only non-negative real values, a simplification is possible with the help of the Perron\u0026ndash;Frobenius theorem, which tells us that it suffices to consider only the \u003cem\u003ereal\u003c/em\u003e eigenvalues of A, i.e., applying Sturm's method can decide the polynomial growth of A\u003csup\u003en\u003c/sup\u003e. \u003c/p\u003e\u003cp\u003e We formalize the Perron\u0026ndash;Frobenius theorem based on a proof via Brouwer's fixpoint theorem, which is available in the HOL multivariate analysis (HMA) library. Since the results on the spectral radius is based on matrices in the Jordan normal form (JNF) library, we further develop a connection which allows us to easily transfer theorems between HMA and JNF. With this connection we derive the combined result: if A is a non-negative real matrix, and no real eigenvalue of A is strictly larger than one, then A\u003csup\u003en\u003c/sup\u003e is polynomially bounded in n. \u003c/p\u003e",
	"authors": [
	"Jose Divasón",
	"Ondřej Kunčar",
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2016-05-20",
	- "id": 479,
	+ "id": 482,
	"link": "/entries/Perron_Frobenius.html",
	"permalink": "/entries/Perron_Frobenius.html",
	"shortname": "Perron_Frobenius",
	"title": "Perron-Frobenius Theorem for Spectral Radius Analysis",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 5
	},
	{
	"abstract": "The \u003ca href=\"http://incredible.pm\"\u003eIncredible Proof Machine\u003c/a\u003e is an interactive visual theorem prover which represents proofs as port graphs. We model this proof representation in Isabelle, and prove that it is just as powerful as natural deduction.",
	"authors": [
	"Joachim Breitner",
	"Denis Lohner"
	],
	"date": "2016-05-20",
	- "id": 480,
	+ "id": 483,
	"link": "/entries/Incredible_Proof_Machine.html",
	"permalink": "/entries/Incredible_Proof_Machine.html",
	"shortname": "Incredible_Proof_Machine",
	"title": "The meta theory of the Incredible Proof Machine",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The impossibility of distributed consensus with one faulty process is a result with important consequences for real world distributed systems e.g., commits in replicated databases. Since proofs are not immune to faults and even plausible proofs with a profound formalism can conclude wrong results, we validate the fundamental result named FLP after Fischer, Lynch and Paterson. We present a formalization of distributed systems and the aforementioned consensus problem. Our proof is based on Hagen Völzer's paper \"A constructive proof for FLP\". In addition to the enhanced confidence in the validity of Völzer's proof, we contribute the missing gaps to show the correctness in Isabelle/HOL. We clarify the proof details and even prove fairness of the infinite execution that contradicts consensus. Our Isabelle formalization can also be reused for further proofs of properties of distributed systems.",
	"authors": [
	"Benjamin Bisping",
	"Paul-David Brodmann",
	"Tim Jungnickel",
	"Christina Rickmann",
	"Henning Seidler",
	"Anke Stüber",
	"Arno Wilhelm-Weidner",
	"Kirstin Peters",
	"Uwe Nestmann"
	],
	"date": "2016-05-18",
	- "id": 481,
	+ "id": 484,
	"link": "/entries/FLP.html",
	"permalink": "/entries/FLP.html",
	"shortname": "FLP",
	"title": "A Constructive Proof for FLP",
	"topic_links": [
	"computer-science/concurrency"
	],
	"topics": [
	"Computer science/Concurrency"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalises a proof of the maximum-flow minimal-cut theorem for networks with countably many edges. A network is a directed graph with non-negative real-valued edge labels and two dedicated vertices, the source and the sink. A flow in a network assigns non-negative real numbers to the edges such that for all vertices except for the source and the sink, the sum of values on incoming edges equals the sum of values on outgoing edges. A cut is a subset of the vertices which contains the source, but not the sink. Our theorem states that in every network, there is a flow and a cut such that the flow saturates all the edges going out of the cut and is zero on all the incoming edges. The proof is based on the paper \u003cemph\u003eThe Max-Flow Min-Cut theorem for countable networks\u003c/emph\u003e by Aharoni et al. Additionally, we prove a characterisation of the lifting operation for relations on discrete probability distributions, which leads to a concise proof of its distributivity over relation composition.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2016-05-09",
	- "id": 482,
	+ "id": 485,
	"link": "/entries/MFMC_Countable.html",
	"permalink": "/entries/MFMC_Countable.html",
	"shortname": "MFMC_Countable",
	"title": "A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 1
	},
	{
	"abstract": "This work contains a formalisation of basic Randomised Social Choice, including Stochastic Dominance and Social Decision Schemes (SDSs) along with some of their most important properties (Anonymity, Neutrality, ex-post- and SD-Efficiency, SD-Strategy-Proofness) and two particular SDSs – Random Dictatorship and Random Serial Dictatorship (with proofs of the properties that they satisfy). Many important properties of these concepts are also proven – such as the two equivalent characterisations of Stochastic Dominance and the fact that SD-efficiency of a lottery only depends on the support. The entry also provides convenient commands to define Preference Profiles, prove their well-formedness, and automatically derive restrictions that sufficiently nice SDSs need to satisfy on the defined profiles. Currently, the formalisation focuses on weak preferences and Stochastic Dominance, but it should be easy to extend it to other domains – such as strict preferences – or other lottery extensions – such as Bilinear Dominance or Pairwise Comparison.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2016-05-05",
	- "id": 483,
	+ "id": 486,
	"link": "/entries/Randomised_Social_Choice.html",
	"permalink": "/entries/Randomised_Social_Choice.html",
	"shortname": "Randomised_Social_Choice",
	"title": "Randomised Social Choice Theory",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 3
	},
	{
	"abstract": "This entry defines the Bell numbers as the cardinality of set partitions for a carrier set of given size, and derives Spivey's generalized recurrence relation for Bell numbers following his elegant and intuitive combinatorial proof. \u003cp\u003e As the set construction for the combinatorial proof requires construction of three intermediate structures, the main difficulty of the formalization is handling the overall combinatorial argument in a structured way. The introduced proof structure allows us to compose the combinatorial argument from its subparts, and supports to keep track how the detailed proof steps are related to the overall argument. To obtain this structure, this entry uses set monad notation for the set construction's definition, introduces suitable predicates and rules, and follows a repeating structure in its Isar proof.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-05-04",
	- "id": 484,
	+ "id": 487,
	"link": "/entries/Bell_Numbers_Spivey.html",
	"permalink": "/entries/Bell_Numbers_Spivey.html",
	"shortname": "Bell_Numbers_Spivey",
	"title": "Spivey's Generalized Recurrence for Bell Numbers",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 2
	},
	{
	"abstract": "This formalisation contains the proof that there is no anonymous and neutral Social Decision Scheme for at least four voters and alternatives that fulfils both SD-Efficiency and SD-Strategy- Proofness. The proof is a fully structured and quasi-human-redable one. It was derived from the (unstructured) SMT proof of the case for exactly four voters and alternatives by Brandl et al. Their proof relies on an unverified translation of the original problem to SMT, and the proof that lifts the argument for exactly four voters and alternatives to the general case is also not machine-checked. In this Isabelle proof, on the other hand, all of these steps are fully proven and machine-checked. This is particularly important seeing as a previously published informal proof of a weaker statement contained a mistake in precisely this lifting step.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2016-05-04",
	- "id": 485,
	+ "id": 488,
	"link": "/entries/SDS_Impossibility.html",
	"permalink": "/entries/SDS_Impossibility.html",
	"shortname": "SDS_Impossibility",
	"title": "The Incompatibility of SD-Efficiency and SD-Strategy-Proofness",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalization is concerned with the theory of Gröbner bases in (commutative) multivariate polynomial rings over fields, originally developed by Buchberger in his 1965 PhD thesis. Apart from the statement and proof of the main theorem of the theory, the formalization also implements Buchberger's algorithm for actually computing Gröbner bases as a tail-recursive function, thus allowing to effectively decide ideal membership in finitely generated polynomial ideals. Furthermore, all functions can be executed on a concrete representation of multivariate polynomials as association lists.",
	"authors": [
	"Fabian Immler",
	"Alexander Maletzky"
	],
	"date": "2016-05-02",
	- "id": 486,
	+ "id": 489,
	"link": "/entries/Groebner_Bases.html",
	"permalink": "/entries/Groebner_Bases.html",
	"shortname": "Groebner_Bases",
	"title": "Gröbner Bases Theory",
	"topic_links": [
	"mathematics/algebra",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Algebra",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 4
	},
	{
	"abstract": "We provide a formal proof within First Order Relativity Theory that no observer can travel faster than the speed of light. Originally reported in Stannett \u0026 Németi (2014) \"Using Isabelle/HOL to verify first-order relativity theory\", Journal of Automated Reasoning 52(4), pp. 361-378.",
	"authors": [
	"Mike Stannett",
	"István Németi"
	],
	"date": "2016-04-28",
	- "id": 487,
	+ "id": 490,
	"link": "/entries/No_FTL_observers.html",
	"permalink": "/entries/No_FTL_observers.html",
	"shortname": "No_FTL_observers",
	"title": "No Faster-Than-Light Observers",
	"topic_links": [
	"mathematics/physics"
	],
	"topics": [
	"Mathematics/Physics"
	],
	"used_by": 0
	},
	{
	"abstract": "The theory provides a formalisation of the Cocke-Younger-Kasami algorithm (CYK for short), an approach to solving the word problem for context-free languages. CYK decides if a word is in the languages generated by a context-free grammar in Chomsky normal form. The formalized algorithm is executable.",
	"authors": [
	"Maksym Bortin"
	],
	"date": "2016-04-27",
	- "id": 488,
	+ "id": 491,
	"link": "/entries/CYK.html",
	"permalink": "/entries/CYK.html",
	"shortname": "CYK",
	"title": "A formalisation of the Cocke-Younger-Kasami algorithm",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a verified and executable implementation of ROBDDs in Isabelle/HOL. Our implementation relates pointer-based computation in the Heap monad to operations on an abstract definition of boolean functions. Internally, we implemented the if-then-else combinator in a recursive fashion, following the Shannon decomposition of the argument functions. The implementation mixes and adapts known techniques and is built with efficiency in mind.",
	"authors": [
	"Julius Michaelis",
	"Max W. Haslbeck",
	"Peter Lammich",
	"Lars Hupel"
	],
	"date": "2016-04-27",
	- "id": 489,
	+ "id": 492,
	"link": "/entries/ROBDD.html",
	"permalink": "/entries/ROBDD.html",
	"shortname": "ROBDD",
	"title": "Algorithms for Reduced Ordered Binary Decision Diagrams",
	"topic_links": [
	"computer-science/algorithms",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Algorithms",
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003eIn his outstanding work on Communicating Sequential Processes, Hoare has defined two fundamental binary operations allowing to compose the input processes into another, typically more complex, process: sequential composition and concurrent composition. Particularly, the output of the former operation is a process that initially behaves like the first operand, and then like the second operand once the execution of the first one has terminated successfully, as long as it does.\u003c/p\u003e \u003cp\u003eThis paper formalizes Hoare's definition of sequential composition and proves, in the general case of a possibly intransitive policy, that CSP noninterference security is conserved under this operation, provided that successful termination cannot be affected by confidential events and cannot occur as an alternative to other events in the traces of the first operand. Both of these assumptions are shown, by means of counterexamples, to be necessary for the theorem to hold.\u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2016-04-26",
	- "id": 490,
	+ "id": 493,
	"link": "/entries/Noninterference_Sequential_Composition.html",
	"permalink": "/entries/Noninterference_Sequential_Composition.html",
	"shortname": "Noninterference_Sequential_Composition",
	"title": "Conservation of CSP Noninterference Security under Sequential Composition",
	"topic_links": [
	"computer-science/security",
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 1
	},
	{
	"abstract": "Kleene algebras with domain are Kleene algebras endowed with an operation that maps each element of the algebra to its domain of definition (or its complement) in abstract fashion. They form a simple algebraic basis for Hoare logics, dynamic logics or predicate transformer semantics. We formalise a modular hierarchy of algebras with domain and antidomain (domain complement) operations in Isabelle/HOL that ranges from domain and antidomain semigroups to modal Kleene algebras and divergence Kleene algebras. We link these algebras with models of binary relations and program traces. We include some examples from modal logics, termination and program analysis.",
	"authors": [
	"Victor B. F. Gomes",
	"Walter Guttmann",
	"Peter Höfner",
	"Georg Struth",
	"Tjark Weber"
	],
	"date": "2016-04-12",
	- "id": 491,
	+ "id": 494,
	"link": "/entries/KAD.html",
	"permalink": "/entries/KAD.html",
	"shortname": "KAD",
	"title": "Kleene Algebras with Domain",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/automata-and-formal-languages",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Automata and formal languages",
	"Mathematics/Algebra"
	],
	- "used_by": 2
	+ "used_by": 3
	},
	{
	"abstract": "We provide formal proofs in Isabelle-HOL (using mostly structured Isar proofs) of the soundness and completeness of the Resolution rule in propositional logic. The completeness proofs take into account the usual redundancy elimination rules (tautology elimination and subsumption), and several refinements of the Resolution rule are considered: ordered resolution (with selection functions), positive and negative resolution, semantic resolution and unit resolution (the latter refinement is complete only for clause sets that are Horn- renamable). We also define a concrete procedure for computing saturated sets and establish its soundness and completeness. The clause sets are not assumed to be finite, so that the results can be applied to formulas obtained by grounding sets of first-order clauses (however, a total ordering among atoms is assumed to be given). Next, we show that the unrestricted Resolution rule is deductive- complete, in the sense that it is able to generate all (prime) implicates of any set of propositional clauses (i.e., all entailment- minimal, non-valid, clausal consequences of the considered set). The generation of prime implicates is an important problem, with many applications in artificial intelligence and verification (for abductive reasoning, knowledge compilation, diagnosis, debugging etc.). We also show that implicates can be computed in an incremental way, by fixing an ordering among all the atoms in the considered sets and resolving upon these atoms one by one in the considered order (with no backtracking). This feature is critical for the efficient computation of prime implicates. Building on these results, we provide a procedure for computing such implicates and establish its soundness and completeness.",
	"authors": [
	"Nicolas Peltier"
	],
	"date": "2016-03-11",
	- "id": 492,
	+ "id": 495,
	"link": "/entries/PropResPI.html",
	"permalink": "/entries/PropResPI.html",
	"shortname": "PropResPI",
	"title": "Propositional Resolution and Prime Implicates Generation",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "The Cartan fixed point theorems concern the group of holomorphic automorphisms on a connected open set of C\u003csup\u003en\u003c/sup\u003e. Ciolli et al. have formalised the one-dimensional case of these theorems in HOL Light. This entry contains their proofs, ported to Isabelle/HOL. Thus it addresses the authors' remark that \"it would be important to write a formal proof in a language that can be read by both humans and machines\".",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2016-03-08",
	- "id": 493,
	+ "id": 496,
	"link": "/entries/Cartan_FP.html",
	"permalink": "/entries/Cartan_FP.html",
	"shortname": "Cartan_FP",
	"title": "The Cartan Fixed Point Theorems",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Timed automata are a widely used formalism for modeling real-time systems, which is employed in a class of successful model checkers such as UPPAAL [LPY97], HyTech [HHWt97] or Kronos [Yov97]. This work formalizes the theory for the subclass of diagonal-free timed automata, which is sufficient to model many interesting problems. We first define the basic concepts and semantics of diagonal-free timed automata. Based on this, we prove two types of decidability results for the language emptiness problem. The first is the classic result of Alur and Dill [AD90, AD94], which uses a finite partitioning of the state space into so-called `regions`. Our second result focuses on an approach based on `Difference Bound Matrices (DBMs)`, which is practically used by model checkers. We prove the correctness of the basic forward analysis operations on DBMs. One of these operations is the Floyd-Warshall algorithm for the all-pairs shortest paths problem. To obtain a finite search space, a widening operation has to be used for this kind of analysis. We use Patricia Bouyer's [Bou04] approach to prove that this widening operation is correct in the sense that DBM-based forward analysis in combination with the widening operation also decides language emptiness. The interesting property of this proof is that the first decidability result is reused to obtain the second one.",
	"authors": [
	"Simon Wimmer"
	],
	"date": "2016-03-08",
	- "id": 494,
	+ "id": 497,
	"link": "/entries/Timed_Automata.html",
	"permalink": "/entries/Timed_Automata.html",
	"shortname": "Timed_Automata",
	"title": "Timed Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "This theory provides a formalisation of linear temporal logic (LTL) and unifies previous formalisations within the AFP. This entry establishes syntax and semantics for this logic and decouples it from existing entries, yielding a common environment for theories reasoning about LTL. Furthermore a parser written in SML and an executable simplifier are provided.",
	"authors": [
	"Salomon Sickert"
	],
	"date": "2016-03-01",
	- "id": 495,
	+ "id": 498,
	"link": "/entries/LTL.html",
	"permalink": "/entries/LTL.html",
	"shortname": "LTL",
	"title": "Linear Temporal Logic",
	"topic_links": [
	"logic/general-logic/temporal-logic",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Logic/General logic/Temporal logic",
	"Computer science/Automata and formal languages"
	],
	"used_by": 6
	},
	{
	"abstract": "\u003cp\u003e These theories formalize the quantitative analysis of a number of classical algorithms for the list update problem: 2-competitiveness of move-to-front, the lower bound of 2 for the competitiveness of deterministic list update algorithms and 1.6-competitiveness of the randomized COMB algorithm, the best randomized list update algorithm known to date. The material is based on the first two chapters of \u003ci\u003eOnline Computation and Competitive Analysis\u003c/i\u003e by Borodin and El-Yaniv. \u003c/p\u003e \u003cp\u003e For an informal description see the FSTTCS 2016 publication \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs/fsttcs16.html\"\u003eVerified Analysis of List Update Algorithms\u003c/a\u003e by Haslbeck and Nipkow. \u003c/p\u003e",
	"authors": [
	"Maximilian P. L. Haslbeck",
	"Tobias Nipkow"
	],
	"date": "2016-02-17",
	- "id": 496,
	+ "id": 499,
	"link": "/entries/List_Update.html",
	"permalink": "/entries/List_Update.html",
	"shortname": "List_Update",
	"title": "Analysis of List Update Algorithms",
	"topic_links": [
	"computer-science/algorithms/online",
	"computer-science/algorithms/randomized"
	],
	"topics": [
	"Computer science/Algorithms/Online",
	"Computer science/Algorithms/Randomized"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e We define a functional variant of the static single assignment (SSA) form construction algorithm described by \u003ca href=\"https://doi.org/10.1007/978-3-642-37051-9_6\"\u003eBraun et al.\u003c/a\u003e, which combines simplicity and efficiency. The definition is based on a general, abstract control flow graph representation using Isabelle locales. \u003c/p\u003e \u003cp\u003e We prove that the algorithm's output is semantically equivalent to the input according to a small-step semantics, and that it is in minimal SSA form for the common special case of reducible inputs. We then show the satisfiability of the locale assumptions by giving instantiations for a simple While language. \u003c/p\u003e \u003cp\u003e Furthermore, we use a generic instantiation based on typedefs in order to extract OCaml code and replace the unverified SSA construction algorithm of the \u003ca href=\"https://doi.org/10.1145/2579080\"\u003eCompCertSSA project\u003c/a\u003e with it. \u003c/p\u003e \u003cp\u003e A more detailed description of the verified SSA construction can be found in the paper \u003ca href=\"https://doi.org/10.1145/2892208.2892211\"\u003eVerified Construction of Static Single Assignment Form\u003c/a\u003e, CC 2016. \u003c/p\u003e",
	"authors": [
	"Sebastian Ullrich",
	"Denis Lohner"
	],
	"date": "2016-02-05",
	- "id": 497,
	+ "id": 500,
	"link": "/entries/Formal_SSA.html",
	"permalink": "/entries/Formal_SSA.html",
	"shortname": "Formal_SSA",
	"title": "Verified Construction of Static Single Assignment Form",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 1
	},
	{
	"abstract": "Based on existing libraries for polynomial interpolation and matrices, we formalized several factorization algorithms for polynomials, including Kronecker's algorithm for integer polynomials, Yun's square-free factorization algorithm for field polynomials, and Berlekamp's algorithm for polynomials over finite fields. By combining the last one with Hensel's lifting, we derive an efficient factorization algorithm for the integer polynomials, which is then lifted for rational polynomials by mechanizing Gauss' lemma. Finally, we assembled a combined factorization algorithm for rational polynomials, which combines all the mentioned algorithms and additionally uses the explicit formula for roots of quadratic polynomials and a rational root test. \u003cp\u003e As side products, we developed division algorithms for polynomials over integral domains, as well as primality-testing and prime-factorization algorithms for integers.",
	"authors": [
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2016-01-29",
	- "id": 498,
	+ "id": 501,
	"link": "/entries/Polynomial_Factorization.html",
	"permalink": "/entries/Polynomial_Factorization.html",
	"shortname": "Polynomial_Factorization",
	"title": "Polynomial Factorization",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 12
	},
	{
	"abstract": "We formalized three algorithms for polynomial interpolation over arbitrary fields: Lagrange's explicit expression, the recursive algorithm of Neville and Aitken, and the Newton interpolation in combination with an efficient implementation of divided differences. Variants of these algorithms for integer polynomials are also available, where sometimes the interpolation can fail; e.g., there is no linear integer polynomial \u003ci\u003ep\u003c/i\u003e such that \u003ci\u003ep(0) = 0\u003c/i\u003e and \u003ci\u003ep(2) = 1\u003c/i\u003e. Moreover, for the Newton interpolation for integer polynomials, we proved that all intermediate results that are computed during the algorithm must be integers. This admits an early failure detection in the implementation. Finally, we proved the uniqueness of polynomial interpolation. \u003cp\u003e The development also contains improved code equations to speed up the division of integers in target languages.",
	"authors": [
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2016-01-29",
	- "id": 499,
	+ "id": 502,
	"link": "/entries/Polynomial_Interpolation.html",
	"permalink": "/entries/Polynomial_Interpolation.html",
	"shortname": "Polynomial_Interpolation",
	"title": "Polynomial Interpolation",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 9
	},
	{
	"abstract": "This work contains a formalization of some topics in knot theory. The concepts that were formalized include definitions of tangles, links, framed links and link/tangle equivalence. The formalization is based on a formulation of links in terms of tangles. We further construct and prove the invariance of the Bracket polynomial. Bracket polynomial is an invariant of framed links closely linked to the Jones polynomial. This is perhaps the first attempt to formalize any aspect of knot theory in an interactive proof assistant.",
	"authors": [
	"T.V.H. Prathamesh"
	],
	"date": "2016-01-20",
	- "id": 500,
	+ "id": 503,
	"link": "/entries/Knot_Theory.html",
	"permalink": "/entries/Knot_Theory.html",
	"shortname": "Knot_Theory",
	"title": "Knot Theory",
	"topic_links": [
	"mathematics/topology"
	],
	"topics": [
	"Mathematics/Topology"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work, the Kronecker tensor product of matrices and the proofs of some of its properties are formalized. Properties which have been formalized include associativity of the tensor product and the mixed-product property.",
	"authors": [
	"T.V.H. Prathamesh"
	],
	"date": "2016-01-18",
	- "id": 501,
	+ "id": 504,
	"link": "/entries/Matrix_Tensor.html",
	"permalink": "/entries/Matrix_Tensor.html",
	"shortname": "Matrix_Tensor",
	"title": "Tensor Product of Matrices",
	"topic_links": [
	"computer-science/data-structures",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Data structures",
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "This entry provides a basic library for number partitions, defines the two-argument partition function through its recurrence relation and relates this partition function to the cardinality of number partitions. The main proof shows that the recursively-defined partition function with arguments n and k equals the cardinality of number partitions of n with exactly k parts. The combinatorial proof follows the proof sketch of Theorem 2.4.1 in Mazur's textbook `Combinatorics: A Guided Tour`. This entry can serve as starting point for various more intrinsic properties about number partitions, the partition function and related recurrence relations.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2016-01-14",
	- "id": 502,
	+ "id": 505,
	"link": "/entries/Card_Number_Partitions.html",
	"permalink": "/entries/Card_Number_Partitions.html",
	"shortname": "Card_Number_Partitions",
	"title": "Cardinality of Number Partitions",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 3
	},
	{
	"abstract": "\u003cp\u003e This entry contains a definition of angles between vectors and between three points. Building on this, we prove basic geometric properties of triangles, such as the Isosceles Triangle Theorem, the Law of Sines and the Law of Cosines, that the sum of the angles of a triangle is π, and the congruence theorems for triangles. \u003c/p\u003e\u003cp\u003e The definitions and proofs were developed following those by John Harrison in HOL Light. However, due to Isabelle's type class system, all definitions and theorems in the Isabelle formalisation hold for all real inner product spaces. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-12-28",
	- "id": 503,
	+ "id": 506,
	"link": "/entries/Triangle.html",
	"permalink": "/entries/Triangle.html",
	"shortname": "Triangle",
	"title": "Basic Geometric Properties of Triangles",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 3
	},
	{
	"abstract": "\u003cp\u003e Descartes' Rule of Signs relates the number of positive real roots of a polynomial with the number of sign changes in its coefficient sequence. \u003c/p\u003e\u003cp\u003e Our proof follows the simple inductive proof given by Rob Arthan, which was also used by John Harrison in his HOL Light formalisation. We proved most of the lemmas for arbitrary linearly-ordered integrity domains (e.g. integers, rationals, reals); the main result, however, requires the intermediate value theorem and was therefore only proven for real polynomials. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-12-28",
	- "id": 504,
	+ "id": 507,
	"link": "/entries/Descartes_Sign_Rule.html",
	"permalink": "/entries/Descartes_Sign_Rule.html",
	"shortname": "Descartes_Sign_Rule",
	"title": "Descartes' Rule of Signs",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e Liouville numbers are a class of transcendental numbers that can be approximated particularly well with rational numbers. Historically, they were the first numbers whose transcendence was proven. \u003c/p\u003e\u003cp\u003e In this entry, we define the concept of Liouville numbers as well as the standard construction to obtain Liouville numbers (including Liouville's constant) and we prove their most important properties: irrationality and transcendence. \u003c/p\u003e\u003cp\u003e The proof is very elementary and requires only standard arithmetic, the Mean Value Theorem for polynomials, and the boundedness of polynomials on compact intervals. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-12-28",
	- "id": 505,
	+ "id": 508,
	"link": "/entries/Liouville_Numbers.html",
	"permalink": "/entries/Liouville_Numbers.html",
	"shortname": "Liouville_Numbers",
	"title": "Liouville numbers",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e In this work, we prove the lower bound \u003cspan class=\"nobr\"\u003eln(H_n) - ln(5/3)\u003c/span\u003e for the partial sum of the Prime Harmonic series and, based on this, the divergence of the Prime Harmonic Series \u003cspan class=\"nobr\"\u003e∑[p\u0026thinsp;prime]\u0026thinsp;·\u0026thinsp;1/p.\u003c/span\u003e \u003c/p\u003e\u003cp\u003e The proof relies on the unique squarefree decomposition of natural numbers. This is similar to Euler's original proof (which was highly informal and morally questionable). Its advantage over proofs by contradiction, like the famous one by Paul Erdős, is that it provides a relatively good lower bound for the partial sums. \u003c/p\u003e",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-12-28",
	- "id": 506,
	+ "id": 509,
	"link": "/entries/Prime_Harmonic_Series.html",
	"permalink": "/entries/Prime_Harmonic_Series.html",
	"shortname": "Prime_Harmonic_Series",
	"title": "The Divergence of the Prime Harmonic Series",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Based on existing libraries for matrices, factorization of rational polynomials, and Sturm's theorem, we formalized algebraic numbers in Isabelle/HOL. Our development serves as an implementation for real and complex numbers, and it admits to compute roots and completely factorize real and complex polynomials, provided that all coefficients are rational numbers. Moreover, we provide two implementations to display algebraic numbers, an injective and expensive one, or a faster but approximative version. \u003c/p\u003e\u003cp\u003e To this end, we mechanized several results on resultants, which also required us to prove that polynomials over a unique factorization domain form again a unique factorization domain. \u003c/p\u003e",
	"authors": [
	"René Thiemann",
	"Akihisa Yamada",
	"Sebastiaan J. C. Joosten"
	],
	"date": "2015-12-22",
	- "id": 507,
	+ "id": 510,
	"link": "/entries/Algebraic_Numbers.html",
	"permalink": "/entries/Algebraic_Numbers.html",
	"shortname": "Algebraic_Numbers",
	"title": "Algebraic Numbers in Isabelle/HOL",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 7
	},
	{
	"abstract": "Applicative functors augment computations with effects by lifting function application to types which model the effects. As the structure of the computation cannot depend on the effects, applicative expressions can be analysed statically. This allows us to lift universally quantified equations to the effectful types, as observed by Hinze. Thus, equational reasoning over effectful computations can be reduced to pure types. \u003c/p\u003e\u003cp\u003e This entry provides a package for registering applicative functors and two proof methods for lifting of equations over applicative functors. The first method normalises applicative expressions according to the laws of applicative functors. This way, equations whose two sides contain the same list of variables can be lifted to every applicative functor. \u003c/p\u003e\u003cp\u003e To lift larger classes of equations, the second method exploits a number of additional properties (e.g., commutativity of effects) provided the properties have been declared for the concrete applicative functor at hand upon registration. \u003c/p\u003e\u003cp\u003e We declare several types from the Isabelle library as applicative functors and illustrate the use of the methods with two examples: the lifting of the arithmetic type class hierarchy to streams and the verification of a relabelling function on binary trees. We also formalise and verify the normalisation algorithm used by the first proof method. \u003c/p\u003e",
	"authors": [
	"Andreas Lochbihler",
	"Joshua Schneider"
	],
	"date": "2015-12-22",
	- "id": 508,
	+ "id": 511,
	"link": "/entries/Applicative_Lifting.html",
	"permalink": "/entries/Applicative_Lifting.html",
	"shortname": "Applicative_Lifting",
	"title": "Applicative Lifting",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 4
	},
	{
	"abstract": "The Stern-Brocot tree contains all rational numbers exactly once and in their lowest terms. We formalise the Stern-Brocot tree as a coinductive tree using recursive and iterative specifications, which we have proven equivalent, and show that it indeed contains all the numbers as stated. Following Hinze, we prove that the Stern-Brocot tree can be linearised looplessly into Stern's diatonic sequence (also known as Dijkstra's fusc function) and that it is a permutation of the Bird tree. \u003c/p\u003e\u003cp\u003e The reasoning stays at an abstract level by appealing to the uniqueness of solutions of guarded recursive equations and lifting algebraic laws point-wise to trees and streams using applicative functors. \u003c/p\u003e",
	"authors": [
	"Peter Gammie",
	"Andreas Lochbihler"
	],
	"date": "2015-12-22",
	- "id": 509,
	+ "id": 512,
	"link": "/entries/Stern_Brocot.html",
	"permalink": "/entries/Stern_Brocot.html",
	"shortname": "Stern_Brocot",
	"title": "The Stern-Brocot Tree",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The theory's main theorem states that the cardinality of set partitions of size k on a carrier set of size n is expressed by Stirling numbers of the second kind. In Isabelle, Stirling numbers of the second kind are defined in the AFP entry `Discrete Summation` through their well-known recurrence relation. The main theorem relates them to the alternative definition as cardinality of set partitions. The proof follows the simple and short explanation in Richard P. Stanley's `Enumerative Combinatorics: Volume 1` and Wikipedia, and unravels the full details and implicit reasoning steps of these explanations.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2015-12-12",
	- "id": 510,
	+ "id": 513,
	"link": "/entries/Card_Partitions.html",
	"permalink": "/entries/Card_Partitions.html",
	"shortname": "Card_Partitions",
	"title": "Cardinality of Set Partitions",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 4
	},
	{
	"abstract": "A Latin Square is a n x n table filled with integers from 1 to n where each number appears exactly once in each row and each column. A Latin Rectangle is a partially filled n x n table with r filled rows and n-r empty rows, such that each number appears at most once in each row and each column. The main result of this theory is that any Latin Rectangle can be completed to a Latin Square.",
	"authors": [
	"Alexander Bentkamp"
	],
	"date": "2015-12-02",
	- "id": 511,
	+ "id": 514,
	"link": "/entries/Latin_Square.html",
	"permalink": "/entries/Latin_Square.html",
	"shortname": "Latin_Square",
	"title": "Latin Square",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "Ergodic theory is the branch of mathematics that studies the behaviour of measure preserving transformations, in finite or infinite measure. It interacts both with probability theory (mainly through measure theory) and with geometry as a lot of interesting examples are from geometric origin. We implement the first definitions and theorems of ergodic theory, including notably Poicaré recurrence theorem for finite measure preserving systems (together with the notion of conservativity in general), induced maps, Kac's theorem, Birkhoff theorem (arguably the most important theorem in ergodic theory), and variations around it such as conservativity of the corresponding skew product, or Atkinson lemma.",
	"authors": [
	"Sebastien Gouezel"
	],
	"date": "2015-12-01",
	- "id": 512,
	+ "id": 515,
	"link": "/entries/Ergodic_Theory.html",
	"permalink": "/entries/Ergodic_Theory.html",
	"shortname": "Ergodic_Theory",
	"title": "Ergodic Theory",
	"topic_links": [
	"mathematics/probability-theory"
	],
	"topics": [
	"Mathematics/Probability theory"
	],
	"used_by": 4
	},
	{
	"abstract": "Euler's Partition Theorem states that the number of partitions with only distinct parts is equal to the number of partitions with only odd parts. The combinatorial proof follows John Harrison's HOL Light formalization. This theorem is the 45th theorem of the Top 100 Theorems list.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2015-11-19",
	- "id": 513,
	+ "id": 516,
	"link": "/entries/Euler_Partition.html",
	"permalink": "/entries/Euler_Partition.html",
	"shortname": "Euler_Partition",
	"title": "Euler's Partition Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the Tortoise and Hare cycle-finding algorithm ascribed to Floyd by Knuth, and an improved version due to Brent.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2015-11-18",
	- "id": 514,
	+ "id": 517,
	"link": "/entries/TortoiseHare.html",
	"permalink": "/entries/TortoiseHare.html",
	"shortname": "TortoiseHare",
	"title": "The Tortoise and Hare Algorithm",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This development provides a formalization of planarity based on combinatorial maps and proves that Kuratowski's theorem implies combinatorial planarity. Moreover, it contains verified implementations of programs checking certificates for planarity (i.e., a combinatorial map) or non-planarity (i.e., a Kuratowski subgraph).",
	"authors": [
	"Lars Noschinski"
	],
	"date": "2015-11-11",
	- "id": 515,
	+ "id": 518,
	"link": "/entries/Planarity_Certificates.html",
	"permalink": "/entries/Planarity_Certificates.html",
	"shortname": "Planarity_Certificates",
	"title": "Planarity Certificates",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of parity games (a two-player game on directed graphs) and a proof of their positional determinacy in Isabelle/HOL. This proof works for both finite and infinite games.",
	"authors": [
	"Christoph Dittmann"
	],
	"date": "2015-11-02",
	- "id": 516,
	+ "id": 519,
	"link": "/entries/Parity_Game.html",
	"permalink": "/entries/Parity_Game.html",
	"shortname": "Parity_Game",
	"title": "Positional Determinacy of Parity Games",
	"topic_links": [
	"mathematics/games-and-economics",
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Games and economics",
	"Mathematics/Graph theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We represent a theory \u003ci\u003eof\u003c/i\u003e (a fragment of) Isabelle/HOL \u003ci\u003ein\u003c/i\u003e Isabelle/HOL. The purpose of this exercise is to write packages for domain-specific specifications such as class models, B-machines, ..., and generally speaking, any domain-specific languages whose abstract syntax can be defined by a HOL \"datatype\". On this basis, the Isabelle code-generator can then be used to generate code for global context transformations as well as tactic code. \u003cp\u003e Consequently the package is geared towards parsing, printing and code-generation to the Isabelle API. It is at the moment not sufficiently rich for doing meta theory on Isabelle itself. Extensions in this direction are possible though. \u003cp\u003e Moreover, the chosen fragment is fairly rudimentary. However it should be easily adapted to one's needs if a package is written on top of it. The supported API contains types, terms, transformation of global context like definitions and data-type declarations as well as infrastructure for Isar-setups. \u003cp\u003e This theory is drawn from the \u003ca href=\"http://isa-afp.org/entries/Featherweight_OCL.html\"\u003eFeatherweight OCL\u003c/a\u003e project where it is used to construct a package for object-oriented data-type theories generated from UML class diagrams. The Featherweight OCL, for example, allows for both the direct execution of compiled tactic code by the Isabelle API as well as the generation of \".thy\"-files for debugging purposes. \u003cp\u003e Gained experience from this project shows that the compiled code is sufficiently efficient for practical purposes while being based on a formal \u003ci\u003emodel\u003c/i\u003e on which properties of the package can be proven such as termination of certain transformations, correctness, etc.",
	"authors": [
	"Frédéric Tuong",
	"Burkhart Wolff"
	],
	"date": "2015-09-16",
	- "id": 517,
	+ "id": 520,
	"link": "/entries/Isabelle_Meta_Model.html",
	"permalink": "/entries/Isabelle_Meta_Model.html",
	"shortname": "Isabelle_Meta_Model",
	"title": "A Meta-Model for the Isabelle API",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "Recently, Javier Esparza and Jan Kretinsky proposed a new method directly translating linear temporal logic (LTL) formulas to deterministic (generalized) Rabin automata. Compared to the existing approaches of constructing a non-deterministic Buechi-automaton in the first step and then applying a determinization procedure (e.g. some variant of Safra's construction) in a second step, this new approach preservers a relation between the formula and the states of the resulting automaton. While the old approach produced a monolithic structure, the new method is compositional. Furthermore, in some cases the resulting automata are much smaller than the automata generated by existing approaches. In order to ensure the correctness of the construction, this entry contains a complete formalisation and verification of the translation. Furthermore from this basis executable code is generated.",
	"authors": [
	"Salomon Sickert"
	],
	"date": "2015-09-04",
	- "id": 518,
	+ "id": 521,
	"link": "/entries/LTL_to_DRA.html",
	"permalink": "/entries/LTL_to_DRA.html",
	"shortname": "LTL_to_DRA",
	"title": "Converting Linear Temporal Logic to Deterministic (Generalized) Rabin Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e Matrix interpretations are useful as measure functions in termination proving. In order to use these interpretations also for complexity analysis, the growth rate of matrix powers has to examined. Here, we formalized a central result of spectral radius theory, namely that the growth rate is polynomially bounded if and only if the spectral radius of a matrix is at most one. \u003c/p\u003e\u003cp\u003e To formally prove this result we first studied the growth rates of matrices in Jordan normal form, and prove the result that every complex matrix has a Jordan normal form using a constructive prove via Schur decomposition. \u003c/p\u003e\u003cp\u003e The whole development is based on a new abstract type for matrices, which is also executable by a suitable setup of the code generator. It completely subsumes our former AFP-entry on executable matrices, and its main advantage is its close connection to the HMA-representation which allowed us to easily adapt existing proofs on determinants. \u003c/p\u003e\u003cp\u003e All the results have been applied to improve CeTA, our certifier to validate termination and complexity proof certificates. \u003c/p\u003e",
	"authors": [
	"René Thiemann",
	"Akihisa Yamada"
	],
	"date": "2015-08-21",
	- "id": 519,
	+ "id": 522,
	"link": "/entries/Jordan_Normal_Form.html",
	"permalink": "/entries/Jordan_Normal_Form.html",
	"shortname": "Jordan_Normal_Form",
	"title": "Matrices, Jordan Normal Forms, and Spectral Radius Theory",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 15
	},
	{
	"abstract": "This theory formalizes the commutation version of decreasing diagrams for Church-Rosser modulo. The proof follows Felgenhauer and van Oostrom (RTA 2013). The theory also provides important specializations, in particular van Oostrom’s conversion version (TCS 2008) of decreasing diagrams.",
	"authors": [
	"Bertram Felgenhauer"
	],
	"date": "2015-08-20",
	- "id": 520,
	+ "id": 523,
	"link": "/entries/Decreasing-Diagrams-II.html",
	"permalink": "/entries/Decreasing-Diagrams-II.html",
	"shortname": "Decreasing-Diagrams-II",
	"title": "Decreasing Diagrams II",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e The necessary and sufficient condition for CSP noninterference security stated by the Ipurge Unwinding Theorem is expressed in terms of a pair of event lists varying over the set of process traces. This does not render it suitable for the subsequent application of rule induction in the case of a process defined inductively, since rule induction may rather be applied to a single variable ranging over an inductively defined set. \u003c/p\u003e\u003cp\u003e Starting from the Ipurge Unwinding Theorem, this paper derives a necessary and sufficient condition for CSP noninterference security that involves a single event list varying over the set of process traces, and is thus suitable for rule induction; hence its name, Inductive Unwinding Theorem. Similarly to the Ipurge Unwinding Theorem, the new theorem only requires to consider individual accepted and refused events for each process trace, and applies to the general case of a possibly intransitive noninterference policy. Specific variants of this theorem are additionally proven for deterministic processes and trace set processes. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2015-08-18",
	- "id": 521,
	+ "id": 524,
	"link": "/entries/Noninterference_Inductive_Unwinding.html",
	"permalink": "/entries/Noninterference_Inductive_Unwinding.html",
	"shortname": "Noninterference_Inductive_Unwinding",
	"title": "The Inductive Unwinding Theorem for CSP Noninterference Security",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group_add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, including Maschke's theorem, Schur's lemma, and Frobenius reciprocity. We also prove that every irreducible representation is isomorphic to a submodule of the group ring, leading to the fact that for a finite group there are only finitely many isomorphism classes of irreducible representations. In all of this, no restriction is made on the characteristic of the ring or field of scalars until the definition of a group representation, and then the only restriction made is that the characteristic must not divide the order of the group.",
	"authors": [
	"Jeremy Sylvestre"
	],
	"date": "2015-08-12",
	- "id": 522,
	+ "id": 525,
	"link": "/entries/Rep_Fin_Groups.html",
	"permalink": "/entries/Rep_Fin_Groups.html",
	"shortname": "Rep_Fin_Groups",
	"title": "Representations of Finite Groups",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "Encodings or the proof of their absence are the main way to compare process calculi. To analyse the quality of encodings and to rule out trivial or meaningless encodings, they are augmented with quality criteria. There exists a bunch of different criteria and different variants of criteria in order to reason in different settings. This leads to incomparable results. Moreover it is not always clear whether the criteria used to obtain a result in a particular setting do indeed fit to this setting. We show how to formally reason about and compare encodability criteria by mapping them on requirements on a relation between source and target terms that is induced by the encoding function. In particular we analyse the common criteria full abstraction, operational correspondence, divergence reflection, success sensitiveness, and respect of barbs; e.g. we analyse the exact nature of the simulation relation (coupled simulation versus bisimulation) that is induced by different variants of operational correspondence. This way we reduce the problem of analysing or comparing encodability criteria to the better understood problem of comparing relations on processes.",
	"authors": [
	"Kirstin Peters",
	"Rob van Glabbeek"
	],
	"date": "2015-08-10",
	- "id": 523,
	+ "id": 526,
	"link": "/entries/Encodability_Process_Calculi.html",
	"permalink": "/entries/Encodability_Process_Calculi.html",
	"shortname": "Encodability_Process_Calculi",
	"title": "Analysing and Comparing Encodability Criteria for Process Calculi",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "Isabelle/Isar provides named cases to structure proofs. This article contains an implementation of a proof method \u003ctt\u003ecasify\u003c/tt\u003e, which can be used to easily extend proof tools with support for named cases. Such a proof tool must produce labeled subgoals, which are then interpreted by \u003ctt\u003ecasify\u003c/tt\u003e. \u003cp\u003e As examples, this work contains verification condition generators producing named cases for three languages: The Hoare language from \u003ctt\u003eHOL/Library\u003c/tt\u003e, a monadic language for computations with failure (inspired by the AutoCorres tool), and a language of conditional expressions. These VCGs are demonstrated by a number of example programs.",
	"authors": [
	"Lars Noschinski"
	],
	"date": "2015-07-21",
	- "id": 524,
	+ "id": 527,
	"link": "/entries/Case_Labeling.html",
	"permalink": "/entries/Case_Labeling.html",
	"shortname": "Case_Labeling",
	"title": "Generating Cases from Labeled Subgoals",
	"topic_links": [
	"tools",
	"computer-science/programming-languages/misc"
	],
	"topics": [
	"Tools",
	"Computer science/Programming languages/Misc"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry provides Landau symbols to describe and reason about the asymptotic growth of functions for sufficiently large inputs. A number of simplification procedures are provided for additional convenience: cancelling of dominated terms in sums under a Landau symbol, cancelling of common factors in products, and a decision procedure for Landau expressions containing products of powers of functions like x, ln(x), ln(ln(x)) etc.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-07-14",
	- "id": 525,
	+ "id": 528,
	"link": "/entries/Landau_Symbols.html",
	"permalink": "/entries/Landau_Symbols.html",
	"shortname": "Landau_Symbols",
	"title": "Landau Symbols",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 11
	},
	{
	"abstract": "This article contains a formalisation of the Akra-Bazzi method based on a proof by Leighton. It is a generalisation of the well-known Master Theorem for analysing the complexity of Divide \u0026 Conquer algorithms. We also include a generalised version of the Master theorem based on the Akra-Bazzi theorem, which is easier to apply than the Akra-Bazzi theorem itself. \u003cp\u003e Some proof methods that facilitate applying the Master theorem are also included. For a more detailed explanation of the formalisation and the proof methods, see the accompanying paper (publication forthcoming).",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2015-07-14",
	- "id": 526,
	+ "id": 529,
	"link": "/entries/Akra_Bazzi.html",
	"permalink": "/entries/Akra_Bazzi.html",
	"shortname": "Akra_Bazzi",
	"title": "The Akra-Bazzi theorem and the Master theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "Hermite Normal Form is a canonical matrix analogue of Reduced Echelon Form, but involving matrices over more general rings. In this work we formalise an algorithm to compute the Hermite Normal Form of a matrix by means of elementary row operations, taking advantage of the Echelon Form AFP entry. We have proven the correctness of such an algorithm and refined it to immutable arrays. Furthermore, we have also formalised the uniqueness of the Hermite Normal Form of a matrix. Code can be exported and some examples of execution involving integer matrices and polynomial matrices are presented as well.",
	"authors": [
	"Jose Divasón",
	"Jesús Aransay"
	],
	"date": "2015-07-07",
	- "id": 527,
	+ "id": 530,
	"link": "/entries/Hermite.html",
	"permalink": "/entries/Hermite.html",
	"shortname": "Hermite",
	"title": "Hermite Normal Form",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "The Derangements Formula describes the number of fixpoint-free permutations as a closed formula. This theorem is the 88th theorem in a list of the “\u003ca href=\"http://www.cs.ru.nl/~freek/100/\"\u003eTop 100 Mathematical Theorems\u003c/a\u003e”.",
	"authors": [
	"Lukas Bulwahn"
	],
	"date": "2015-06-27",
	- "id": 528,
	+ "id": 531,
	"link": "/entries/Derangements.html",
	"permalink": "/entries/Derangements.html",
	"shortname": "Derangements",
	"title": "Derangements Formula",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "Binary multirelations associate elements of a set with its subsets; hence they are binary relations from a set to its power set. Applications include alternating automata, models and logics for games, program semantics with dual demonic and angelic nondeterministic choices and concurrent dynamic logics. This proof document supports an arXiv article that formalises the basic algebra of multirelations and proposes axiom systems for them, ranging from weak bi-monoids to weak bi-quantales.",
	"authors": [
	"Hitoshi Furusawa",
	"Georg Struth"
	],
	"date": "2015-06-11",
	- "id": 529,
	+ "id": 532,
	"link": "/entries/Multirelations.html",
	"permalink": "/entries/Multirelations.html",
	"shortname": "Multirelations",
	"title": "Binary Multirelations",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e Among the various mathematical tools introduced in his outstanding work on Communicating Sequential Processes, Hoare has defined \"interleaves\" as the predicate satisfied by any three lists such that the first list may be split into sublists alternately extracted from the other two ones, whatever is the criterion for extracting an item from either one list or the other in each step. \u003c/p\u003e\u003cp\u003e This paper enriches Hoare's definition by identifying such criterion with the truth value of a predicate taking as inputs the head and the tail of the first list. This enhanced \"interleaves\" predicate turns out to permit the proof of equalities between lists without the need of an induction. Some rules that allow to infer \"interleaves\" statements without induction, particularly applying to the addition or removal of a prefix to the input lists, are also proven. Finally, a stronger version of the predicate, named \"Interleaves\", is shown to fulfil further rules applying to the addition or removal of a suffix to the input lists. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2015-06-11",
	- "id": 530,
	+ "id": 533,
	"link": "/entries/List_Interleaving.html",
	"permalink": "/entries/List_Interleaving.html",
	"shortname": "List_Interleaving",
	"title": "Reasoning about Lists via List Interleaving",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e The classical definition of noninterference security for a deterministic state machine with outputs requires to consider the outputs produced by machine actions after any trace, i.e. any indefinitely long sequence of actions, of the machine. In order to render the verification of the security of such a machine more straightforward, there is a need of some sufficient condition for security such that just individual actions, rather than unbounded sequences of actions, have to be considered. \u003c/p\u003e\u003cp\u003e By extending previous results applying to transitive noninterference policies, Rushby has proven an unwinding theorem that provides a sufficient condition of this kind in the general case of a possibly intransitive policy. This condition has to be satisfied by a generic function mapping security domains into equivalence relations over machine states. \u003c/p\u003e\u003cp\u003e An analogous problem arises for CSP noninterference security, whose definition requires to consider any possible future, i.e. any indefinitely long sequence of subsequent events and any indefinitely large set of refused events associated to that sequence, for each process trace. \u003c/p\u003e\u003cp\u003e This paper provides a sufficient condition for CSP noninterference security, which indeed requires to just consider individual accepted and refused events and applies to the general case of a possibly intransitive policy. This condition follows Rushby's one for classical noninterference security, and has to be satisfied by a generic function mapping security domains into equivalence relations over process traces; hence its name, Generic Unwinding Theorem. Variants of this theorem applying to deterministic processes and trace set processes are also proven. Finally, the sufficient condition for security expressed by the theorem is shown not to be a necessary condition as well, viz. there exists a secure process such that no domain-relation map satisfying the condition exists. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2015-06-11",
	- "id": 531,
	+ "id": 534,
	"link": "/entries/Noninterference_Generic_Unwinding.html",
	"permalink": "/entries/Noninterference_Generic_Unwinding.html",
	"shortname": "Noninterference_Generic_Unwinding",
	"title": "The Generic Unwinding Theorem for CSP Noninterference Security",
	"topic_links": [
	"computer-science/security",
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e The definition of noninterference security for Communicating Sequential Processes requires to consider any possible future, i.e. any indefinitely long sequence of subsequent events and any indefinitely large set of refused events associated to that sequence, for each process trace. In order to render the verification of the security of a process more straightforward, there is a need of some sufficient condition for security such that just individual accepted and refused events, rather than unbounded sequences and sets of events, have to be considered. \u003c/p\u003e\u003cp\u003e Of course, if such a sufficient condition were necessary as well, it would be even more valuable, since it would permit to prove not only that a process is secure by verifying that the condition holds, but also that a process is not secure by verifying that the condition fails to hold. \u003c/p\u003e\u003cp\u003e This paper provides a necessary and sufficient condition for CSP noninterference security, which indeed requires to just consider individual accepted and refused events and applies to the general case of a possibly intransitive policy. This condition follows Rushby's output consistency for deterministic state machines with outputs, and has to be satisfied by a specific function mapping security domains into equivalence relations over process traces. The definition of this function makes use of an intransitive purge function following Rushby's one; hence the name given to the condition, Ipurge Unwinding Theorem. \u003c/p\u003e\u003cp\u003e Furthermore, in accordance with Hoare's formal definition of deterministic processes, it is shown that a process is deterministic just in case it is a trace set process, i.e. it may be identified by means of a trace set alone, matching the set of its traces, in place of a failures-divergences pair. Then, variants of the Ipurge Unwinding Theorem are proven for deterministic processes and trace set processes. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2015-06-11",
	- "id": 532,
	+ "id": 535,
	"link": "/entries/Noninterference_Ipurge_Unwinding.html",
	"permalink": "/entries/Noninterference_Ipurge_Unwinding.html",
	"shortname": "Noninterference_Ipurge_Unwinding",
	"title": "The Ipurge Unwinding Theorem for CSP Noninterference Security",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 3
	},
	{
	"abstract": "This article formalizes the amortized analysis of dynamic tables parameterized with their minimal and maximal load factors and the expansion and contraction factors. \u003cP\u003e A full description is found in a \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs\"\u003ecompanion paper\u003c/a\u003e.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2015-06-07",
	- "id": 533,
	+ "id": 536,
	"link": "/entries/Dynamic_Tables.html",
	"permalink": "/entries/Dynamic_Tables.html",
	"shortname": "Dynamic_Tables",
	"title": "Parameterized Dynamic Tables",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize new decision procedures for WS1S, M2L(Str), and Presburger Arithmetics. Formulas of these logics denote regular languages. Unlike traditional decision procedures, we do \u003cem\u003enot\u003c/em\u003e translate formulas into automata (nor into regular expressions), at least not explicitly. Instead we devise notions of derivatives (inspired by Brzozowski derivatives for regular expressions) that operate on formulas directly and compute a syntactic bisimulation using these derivatives. The treatment of Boolean connectives and quantifiers is uniform for all mentioned logics and is abstracted into a locale. This locale is then instantiated by different atomic formulas and their derivatives (which may differ even for the same logic under different encodings of interpretations as formal words). \u003cp\u003e The WS1S instance is described in the draft paper \u003ca href=\"https://people.inf.ethz.ch/trayteld/papers/csl15-ws1s_derivatives/index.html\"\u003eA Coalgebraic Decision Procedure for WS1S\u003c/a\u003e by the author.",
	"authors": [
	"Dmitriy Traytel"
	],
	"date": "2015-05-28",
	- "id": 534,
	+ "id": 537,
	"link": "/entries/Formula_Derivatives.html",
	"permalink": "/entries/Formula_Derivatives.html",
	"shortname": "Formula_Derivatives",
	"title": "Derivatives of Logical Formulas",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/general-logic/decidability-of-theories"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/General logic/Decidability of theories"
	],
	"used_by": 1
	},
	{
	"abstract": "Numerous models of probabilistic systems are studied in the literature. Coalgebra has been used to classify them into system types and compare their expressiveness. We formalize the resulting hierarchy of probabilistic system types by modeling the semantics of the different systems as codatatypes. This approach yields simple and concise proofs, as bisimilarity coincides with equality for codatatypes. \u003cp\u003e This work is described in detail in the ITP 2015 publication by the authors.",
	"authors": [
	"Johannes Hölzl",
	"Andreas Lochbihler",
	"Dmitriy Traytel"
	],
	"date": "2015-05-27",
	- "id": 535,
	+ "id": 538,
	"link": "/entries/Probabilistic_System_Zoo.html",
	"permalink": "/entries/Probabilistic_System_Zoo.html",
	"shortname": "Probabilistic_System_Zoo",
	"title": "A Zoo of Probabilistic Systems",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "A VCG auction (named after their inventors Vickrey, Clarke, and Groves) is a generalization of the single-good, second price Vickrey auction to the case of a combinatorial auction (multiple goods, from which any participant can bid on each possible combination). We formalize in this entry VCG auctions, including tie-breaking and prove that the functions for the allocation and the price determination are well-defined. Furthermore we show that the allocation function allocates goods only to participants, only goods in the auction are allocated, and no good is allocated twice. We also show that the price function is non-negative. These properties also hold for the automatically extracted Scala code.",
	"authors": [
	"Marco B. Caminati",
	"Manfred Kerber",
	"Christoph Lange",
	"Colin Rowat"
	],
	"date": "2015-04-30",
	- "id": 536,
	+ "id": 539,
	"link": "/entries/Vickrey_Clarke_Groves.html",
	"permalink": "/entries/Vickrey_Clarke_Groves.html",
	"shortname": "Vickrey_Clarke_Groves",
	"title": "VCG - Combinatorial Vickrey-Clarke-Groves Auctions",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "The theory of residuated lattices, first proposed by Ward and Dilworth, is formalised in Isabelle/HOL. This includes concepts of residuated functions; their adjoints and conjugates. It also contains necessary and sufficient conditions for the existence of these operations in an arbitrary lattice. The mathematical components for residuated lattices are linked to the AFP entry for relation algebra. In particular, we prove Jonsson and Tsinakis conditions for a residuated boolean algebra to form a relation algebra.",
	"authors": [
	"Victor B. F. Gomes",
	"Georg Struth"
	],
	"date": "2015-04-15",
	- "id": 537,
	+ "id": 540,
	"link": "/entries/Residuated_Lattices.html",
	"permalink": "/entries/Residuated_Lattices.html",
	"shortname": "Residuated_Lattices",
	"title": "Residuated Lattices",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "ConcurrentIMP extends the small imperative language IMP with control non-determinism and constructs for synchronous message passing.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2015-04-13",
	- "id": 538,
	+ "id": 541,
	"link": "/entries/ConcurrentIMP.html",
	"permalink": "/entries/ConcurrentIMP.html",
	"shortname": "ConcurrentIMP",
	"title": "Concurrent IMP",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e We use ConcurrentIMP to model Schism, a state-of-the-art real-time garbage collection scheme for weak memory, and show that it is safe on x86-TSO.\u003c/p\u003e \u003cp\u003e This development accompanies the PLDI 2015 paper of the same name. \u003c/p\u003e",
	"authors": [
	"Peter Gammie",
	"Tony Hosking",
	"Kai Engelhardt"
	],
	"date": "2015-04-13",
	- "id": 539,
	+ "id": 542,
	"link": "/entries/ConcurrentGC.html",
	"permalink": "/entries/ConcurrentGC.html",
	"shortname": "ConcurrentGC",
	"title": "Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO",
	"topic_links": [
	"computer-science/algorithms/concurrent"
	],
	"topics": [
	"Computer science/Algorithms/Concurrent"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes the “trie” data structure invented by Fredkin [CACM 1960]. It also provides a specialization where the entries in the trie are lists.",
	"authors": [
	"Andreas Lochbihler",
	"Tobias Nipkow"
	],
	"date": "2015-03-30",
	- "id": 540,
	+ "id": 543,
	"link": "/entries/Trie.html",
	"permalink": "/entries/Trie.html",
	"shortname": "Trie",
	"title": "Trie",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 5
	},
	{
	"abstract": "Algorithms for solving the consensus problem are fundamental to distributed computing. Despite their brevity, their ability to operate in concurrent, asynchronous and failure-prone environments comes at the cost of complex and subtle behaviors. Accordingly, understanding how they work and proving their correctness is a non-trivial endeavor where abstraction is immensely helpful. Moreover, research on consensus has yielded a large number of algorithms, many of which appear to share common algorithmic ideas. A natural question is whether and how these similarities can be distilled and described in a precise, unified way. In this work, we combine stepwise refinement and lockstep models to provide an abstract and unified view of a sizeable family of consensus algorithms. Our models provide insights into the design choices underlying the different algorithms, and classify them based on those choices.",
	"authors": [
	"Ognjen Marić",
	"Christoph Sprenger"
	],
	"date": "2015-03-18",
	- "id": 541,
	+ "id": 544,
	"link": "/entries/Consensus_Refined.html",
	"permalink": "/entries/Consensus_Refined.html",
	"shortname": "Consensus_Refined",
	"title": "Consensus Refined",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eWe provide a framework for registering automatic methods to derive class instances of datatypes, as it is possible using Haskell's “\u003ctt\u003ederiving Ord, Show, ...\u003c/tt\u003e” feature.\u003c/p\u003e \u003cp\u003eWe further implemented such automatic methods to derive comparators, linear orders, parametrizable equality functions, and hash-functions which are required in the Isabelle Collection Framework and the Container Framework. Moreover, for the tactic of Blanchette to show that a datatype is countable, we implemented a wrapper so that this tactic becomes accessible in our framework. All of the generators are based on the infrastructure that is provided by the BNF-based datatype package.\u003c/p\u003e \u003cp\u003eOur formalization was performed as part of the \u003ca href=\"http://cl-informatik.uibk.ac.at/software/ceta\"\u003eIsaFoR/CeTA\u003c/a\u003e project. With our new tactics we could remove several tedious proofs for (conditional) linear orders, and conditional equality operators within IsaFoR and the Container Framework.\u003c/p\u003e",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2015-03-11",
	- "id": 542,
	+ "id": 545,
	"link": "/entries/Deriving.html",
	"permalink": "/entries/Deriving.html",
	"shortname": "Deriving",
	"title": "Deriving class instances for datatypes",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 13
	},
	{
	"abstract": "We formalize the Call Arity analysis, as implemented in GHC, and prove both functional correctness and, more interestingly, safety (i.e. the transformation does not increase allocation). \u003cp\u003e We use syntax and the denotational semantics from the entry \"Launchbury\", where we formalized Launchbury's natural semantics for lazy evaluation. \u003cp\u003e The functional correctness of Call Arity is proved with regard to that denotational semantics. The operational properties are shown with regard to a small-step semantics akin to Sestoft's mark 1 machine, which we prove to be equivalent to Launchbury's semantics. \u003cp\u003e We use Christian Urban's Nominal2 package to define our terms and make use of Brian Huffman's HOLCF package for the domain-theoretical aspects of the development.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2015-02-20",
	- "id": 543,
	+ "id": 546,
	"link": "/entries/Call_Arity.html",
	"permalink": "/entries/Call_Arity.html",
	"shortname": "Call_Arity",
	"title": "The Safety of Call Arity",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize an algorithm to compute the Echelon Form of a matrix. We have proved its existence over Bézout domains and made it executable over Euclidean domains, such as the integer ring and the univariate polynomials over a field. This allows us to compute determinants, inverses and characteristic polynomials of matrices. The work is based on the HOL-Multivariate Analysis library, and on both the Gauss-Jordan and Cayley-Hamilton AFP entries. As a by-product, some algebraic structures have been implemented (principal ideal domains, Bézout domains...). The algorithm has been refined to immutable arrays and code can be generated to functional languages as well.",
	"authors": [
	"Jose Divasón",
	"Jesús Aransay"
	],
	"date": "2015-02-12",
	- "id": 544,
	+ "id": 547,
	"link": "/entries/Echelon_Form.html",
	"permalink": "/entries/Echelon_Form.html",
	"shortname": "Echelon_Form",
	"title": "Echelon Form",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "QR decomposition is an algorithm to decompose a real matrix A into the product of two other matrices Q and R, where Q is orthogonal and R is invertible and upper triangular. The algorithm is useful for the least squares problem; i.e., the computation of the best approximation of an unsolvable system of linear equations. As a side-product, the Gram-Schmidt process has also been formalized. A refinement using immutable arrays is presented as well. The development relies, among others, on the AFP entry \"Implementing field extensions of the form Q[sqrt(b)]\" by René Thiemann, which allows execution of the algorithm using symbolic computations. Verified code can be generated and executed using floats as well.",
	"authors": [
	"Jose Divasón",
	"Jesús Aransay"
	],
	"date": "2015-02-12",
	- "id": 545,
	+ "id": 548,
	"link": "/entries/QR_Decomposition.html",
	"permalink": "/entries/QR_Decomposition.html",
	"shortname": "QR_Decomposition",
	"title": "QR Decomposition",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "Finite Automata, both deterministic and non-deterministic, for regular languages. The Myhill-Nerode Theorem. Closure under intersection, concatenation, etc. Regular expressions define regular languages. Closure under reversal; the powerset construction mapping NFAs to DFAs. Left and right languages; minimal DFAs. Brzozowski's minimization algorithm. Uniqueness up to isomorphism of minimal DFAs.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2015-02-05",
	- "id": 546,
	+ "id": 549,
	"link": "/entries/Finite_Automata_HF.html",
	"permalink": "/entries/Finite_Automata_HF.html",
	"shortname": "Finite_Automata_HF",
	"title": "Finite Automata in Hereditarily Finite Set Theory",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "The UpDown scheme is a recursive scheme used to compute the stiffness matrix on a special form of sparse grids. Usually, when discretizing a Euclidean space of dimension d we need O(n^d) points, for n points along each dimension. Sparse grids are a hierarchical representation where the number of points is reduced to O(n * log(n)^d). One disadvantage of such sparse grids is that the algorithm now operate recursively in the dimensions and levels of the sparse grid. \u003cp\u003e The UpDown scheme allows us to compute the stiffness matrix on such a sparse grid. The stiffness matrix represents the influence of each representation function on the L^2 scalar product. For a detailed description see Dirk Pflüger's PhD thesis. This formalization was developed as an interdisciplinary project (IDP) at the Technische Universität München.",
	"authors": [
	"Johannes Hölzl"
	],
	"date": "2015-01-28",
	- "id": 547,
	+ "id": 550,
	"link": "/entries/UpDown_Scheme.html",
	"permalink": "/entries/UpDown_Scheme.html",
	"shortname": "UpDown_Scheme",
	"title": "Verification of the UpDown Scheme",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "We present the Unified Policy Framework (UPF), a generic framework for modelling security (access-control) policies. UPF emphasizes the view that a policy is a policy decision function that grants or denies access to resources, permissions, etc. In other words, instead of modelling the relations of permitted or prohibited requests directly, we model the concrete function that implements the policy decision point in a system. In more detail, UPF is based on the following four principles: 1) Functional representation of policies, 2) No conflicts are possible, 3) Three-valued decision type (allow, deny, undefined), 4) Output type not containing the decision only.",
	"authors": [
	"Achim D. Brucker",
	"Lukas Brügger",
	"Burkhart Wolff"
	],
	"date": "2014-11-28",
	- "id": 548,
	+ "id": 551,
	"link": "/entries/UPF.html",
	"permalink": "/entries/UPF.html",
	"shortname": "UPF",
	"title": "The Unified Policy Framework (UPF)",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e The Ad hoc On-demand Distance Vector (AODV) routing protocol allows the nodes in a Mobile Ad hoc Network (MANET) or a Wireless Mesh Network (WMN) to know where to forward data packets. Such a protocol is ‘loop free’ if it never leads to routing decisions that forward packets in circles. \u003cp\u003e This development mechanises an existing pen-and-paper proof of loop freedom of AODV. The protocol is modelled in the Algebra of Wireless Networks (AWN), which is the subject of an earlier paper and AFP mechanization. The proof relies on a novel compositional approach for lifting invariants to networks of nodes. \u003c/p\u003e\u003cp\u003e We exploit the mechanization to analyse several variants of AODV and show that Isabelle/HOL can re-establish most proof obligations automatically and identify exactly the steps that are no longer valid. \u003c/p\u003e",
	"authors": [
	"Timothy Bourke",
	"Peter Höfner"
	],
	"date": "2014-10-23",
	- "id": 549,
	+ "id": 552,
	"link": "/entries/AODV.html",
	"permalink": "/entries/AODV.html",
	"shortname": "AODV",
	"title": "Loop freedom of the (untimed) AODV routing protocol",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "We implemented a command that can be used to easily generate elements of a restricted type \u003ctt\u003e{x :: 'a. P x}\u003c/tt\u003e, provided the definition is of the form \u003ctt\u003ef ys = (if check ys then Some(generate ys :: 'a) else None)\u003c/tt\u003e where \u003ctt\u003eys\u003c/tt\u003e is a list of variables \u003ctt\u003ey1 ... yn\u003c/tt\u003e and \u003ctt\u003echeck ys ==\u003e P(generate ys)\u003c/tt\u003e can be proved. \u003cp\u003e In principle, such a definition is also directly possible using the \u003ctt\u003elift_definition\u003c/tt\u003e command. However, then this definition will not be suitable for code-generation. To this end, we automated a more complex construction of Joachim Breitner which is amenable for code-generation, and where the test \u003ctt\u003echeck ys\u003c/tt\u003e will only be performed once. In the automation, one auxiliary type is created, and Isabelle's lifting- and transfer-package is invoked several times.",
	"authors": [
	"René Thiemann"
	],
	"date": "2014-10-13",
	- "id": 550,
	+ "id": 553,
	"link": "/entries/Lifting_Definition_Option.html",
	"permalink": "/entries/Lifting_Definition_Option.html",
	"shortname": "Lifting_Definition_Option",
	"title": "Lifting Definition Option",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "Stream Fusion is a system for removing intermediate list data structures from functional programs, in particular Haskell. This entry adapts stream fusion to Isabelle/HOL and its code generator. We define stream types for finite and possibly infinite lists and stream versions for most of the fusible list functions in the theories List and Coinductive_List, and prove them correct with respect to the conversion functions between lists and streams. The Stream Fusion transformation itself is implemented as a simproc in the preprocessor of the code generator. [Brian Huffman's \u003ca href=\"http://isa-afp.org/entries/Stream-Fusion.html\"\u003eAFP entry\u003c/a\u003e formalises stream fusion in HOLCF for the domain of lazy lists to prove the GHC compiler rewrite rules correct. In contrast, this work enables Isabelle's code generator to perform stream fusion itself. To that end, it covers both finite and coinductive lists from the HOL library and the Coinductive entry. The fusible list functions require specification and proof principles different from Huffman's.]",
	"authors": [
	"Andreas Lochbihler",
	"Alexandra Maximova"
	],
	"date": "2014-10-10",
	- "id": 551,
	+ "id": 554,
	"link": "/entries/Stream_Fusion_Code.html",
	"permalink": "/entries/Stream_Fusion_Code.html",
	"shortname": "Stream_Fusion_Code",
	"title": "Stream Fusion in HOL with Code Generation",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003ca href=\"https://doi.org/10.1007/978-3-642-36742-7_35\"\u003eBhat et al. [TACAS 2013]\u003c/a\u003e developed an inductive compiler that computes density functions for probability spaces described by programs in a probabilistic functional language. In this work, we implement such a compiler for a modified version of this language within the theorem prover Isabelle and give a formal proof of its soundness w.r.t. the semantics of the source and target language. Together with Isabelle's code generation for inductive predicates, this yields a fully verified, executable density compiler. The proof is done in two steps: First, an abstract compiler working with abstract functions modelled directly in the theorem prover's logic is defined and proved sound. Then, this compiler is refined to a concrete version that returns a target-language expression. \u003cp\u003e An article with the same title and authors is published in the proceedings of ESOP 2015. A detailed presentation of this work can be found in the first author's master's thesis with the same title.",
	"authors": [
	"Manuel Eberl",
	"Johannes Hölzl",
	"Tobias Nipkow"
	],
	"date": "2014-10-09",
	- "id": 552,
	+ "id": 555,
	"link": "/entries/Density_Compiler.html",
	"permalink": "/entries/Density_Compiler.html",
	"shortname": "Density_Compiler",
	"title": "A Verified Compiler for Probability Density Functions",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of refinement calculus for reactive systems. Refinement calculus is based on monotonic predicate transformers (monotonic functions from sets of post-states to sets of pre-states), and it is a powerful formalism for reasoning about imperative programs. We model reactive systems as monotonic property transformers that transform sets of output infinite sequences into sets of input infinite sequences. Within this semantics we can model refinement of reactive systems, (unbounded) angelic and demonic nondeterminism, sequential composition, and other semantic properties. We can model systems that may fail for some inputs, and we can model compatibility of systems. We can specify systems that have liveness properties using linear temporal logic, and we can refine system specifications into systems based on symbolic transitions systems, suitable for implementations.",
	"authors": [
	"Viorel Preoteasa"
	],
	"date": "2014-10-08",
	- "id": 553,
	+ "id": 556,
	"link": "/entries/RefinementReactive.html",
	"permalink": "/entries/RefinementReactive.html",
	"shortname": "RefinementReactive",
	"title": "Formalization of Refinement Calculus for Reactive Systems",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides several monads intended for the development of stand-alone certifiers via code generation from Isabelle/HOL. More specifically, there are three flavors of error monads (the sum type, for the case where all monadic functions are total; an instance of the former, the so called check monad, yielding either success without any further information or an error message; as well as a variant of the sum type that accommodates partial functions by providing an explicit bottom element) and a parser monad built on top. All of this monads are heavily used in the IsaFoR/CeTA project which thus provides many examples of their usage.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2014-10-03",
	- "id": 554,
	+ "id": 557,
	"link": "/entries/Certification_Monads.html",
	"permalink": "/entries/Certification_Monads.html",
	"shortname": "Certification_Monads",
	"title": "Certification Monads",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 3
	},
	{
	"abstract": "This entry provides an XML library for Isabelle/HOL. This includes parsing and pretty printing of XML trees as well as combinators for transforming XML trees into arbitrary user-defined data. The main contribution of this entry is an interface (fit for code generation) that allows for communication between verified programs formalized in Isabelle/HOL and the outside world via XML. This library was developed as part of the IsaFoR/CeTA project to which we refer for examples of its usage.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2014-10-03",
	- "id": 555,
	+ "id": 558,
	"link": "/entries/XML.html",
	"permalink": "/entries/XML.html",
	"shortname": "XML",
	"title": "XML",
	"topic_links": [
	"computer-science/functional-programming",
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Functional programming",
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "The insertion sort algorithm of Cormen et al. (Introduction to Algorithms) is expressed in Imperative HOL and proved to be correct and terminating. For this purpose we also provide a theory about imperative loop constructs with accompanying induction/invariant rules for proving partial and total correctness. Furthermore, the formalized algorithm is fit for code generation.",
	"authors": [
	"Christian Sternagel"
	],
	"date": "2014-09-25",
	- "id": 556,
	+ "id": 559,
	"link": "/entries/Imperative_Insertion_Sort.html",
	"permalink": "/entries/Imperative_Insertion_Sort.html",
	"shortname": "Imperative_Insertion_Sort",
	"title": "Imperative Insertion Sort",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We have formalized the Sturm–Tarski theorem (also referred as the Tarski theorem), which generalizes Sturm's theorem. Sturm's theorem is usually used as a way to count distinct real roots, while the Sturm-Tarksi theorem forms the basis for Tarski's classic quantifier elimination for real closed field.",
	"authors": [
	"Wenda Li"
	],
	"date": "2014-09-19",
	- "id": 557,
	+ "id": 560,
	"link": "/entries/Sturm_Tarski.html",
	"permalink": "/entries/Sturm_Tarski.html",
	"shortname": "Sturm_Tarski",
	"title": "The Sturm–Tarski Theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 4
	},
	{
	"abstract": "This document contains a proof of the Cayley-Hamilton theorem based on the development of matrices in HOL/Multivariate Analysis.",
	"authors": [
	"Stephan Adelsberger",
	"Stefan Hetzl",
	"Florian Pollak"
	],
	"date": "2014-09-15",
	- "id": 558,
	+ "id": 561,
	"link": "/entries/Cayley_Hamilton.html",
	"permalink": "/entries/Cayley_Hamilton.html",
	"shortname": "Cayley_Hamilton",
	"title": "The Cayley-Hamilton Theorem",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theorem for maximal normal subgroups as well as many useful lemmas about normal subgroups and factor groups. The proof is inspired by course notes of Stuart Rankin.",
	"authors": [
	"Jakob von Raumer"
	],
	"date": "2014-09-09",
	- "id": 559,
	+ "id": 562,
	"link": "/entries/Jordan_Hoelder.html",
	"permalink": "/entries/Jordan_Hoelder.html",
	"shortname": "Jordan_Hoelder",
	"title": "The Jordan-Hölder Theorem",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry verifies priority queues based on Braun trees. Insertion and deletion take logarithmic time and preserve the balanced nature of Braun trees. Two implementations of deletion are provided.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2014-09-04",
	- "id": 560,
	+ "id": 563,
	"link": "/entries/Priority_Queue_Braun.html",
	"permalink": "/entries/Priority_Queue_Braun.html",
	"shortname": "Priority_Queue_Braun",
	"title": "Priority Queues Based on Braun Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "The Gauss-Jordan algorithm states that any matrix over a field can be transformed by means of elementary row operations to a matrix in reduced row echelon form. The formalization is based on the Rank Nullity Theorem entry of the AFP and on the HOL-Multivariate-Analysis session of Isabelle, where matrices are represented as functions over finite types. We have set up the code generator to make this representation executable. In order to improve the performance, a refinement to immutable arrays has been carried out. We have formalized some of the applications of the Gauss-Jordan algorithm. Thanks to this development, the following facts can be computed over matrices whose elements belong to a field: Ranks, Determinants, Inverses, Bases and dimensions and Solutions of systems of linear equations. Code can be exported to SML and Haskell.",
	"authors": [
	"Jose Divasón",
	"Jesús Aransay"
	],
	"date": "2014-09-03",
	- "id": 561,
	+ "id": 564,
	"link": "/entries/Gauss_Jordan.html",
	"permalink": "/entries/Gauss_Jordan.html",
	"shortname": "Gauss_Jordan",
	"title": "Gauss-Jordan Algorithm and Its Applications",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 4
	},
	{
	"abstract": "This development proves upper and lower bounds for several familiar real-valued functions. For sin, cos, exp and sqrt, it defines and verifies infinite families of upper and lower bounds, mostly based on Taylor series expansions. For arctan, ln and exp, it verifies a finite collection of upper and lower bounds, originally obtained from the functions' continued fraction expansions using the computer algebra system Maple. A common theme in these proofs is to take the difference between a function and its approximation, which should be zero at one point, and then consider the sign of the derivative. The immediate purpose of this development is to verify axioms used by MetiTarski, an automatic theorem prover for real-valued special functions. Crucial to MetiTarski's operation is the provision of upper and lower bounds for each function of interest.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2014-08-29",
	- "id": 562,
	+ "id": 565,
	"link": "/entries/Special_Function_Bounds.html",
	"permalink": "/entries/Special_Function_Bounds.html",
	"shortname": "Special_Function_Bounds",
	"title": "Real-Valued Special Functions: Upper and Lower Bounds",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalisation of basic linear algebra is based completely on locales, building off HOL-Algebra. It includes basic definitions: linear combinations, span, linear independence; linear transformations; interpretation of function spaces as vector spaces; the direct sum of vector spaces, sum of subspaces; the replacement theorem; existence of bases in finite-dimensional; vector spaces, definition of dimension; the rank-nullity theorem. Some concepts are actually defined and proved for modules as they also apply there. Infinite-dimensional vector spaces are supported, but dimension is only supported for finite-dimensional vector spaces. The proofs are standard; the proofs of the replacement theorem and rank-nullity theorem roughly follow the presentation in Linear Algebra by Friedberg, Insel, and Spence. The rank-nullity theorem generalises the existing development in the Archive of Formal Proof (originally using type classes, now using a mix of type classes and locales).",
	"authors": [
	"Holden Lee"
	],
	"date": "2014-08-29",
	- "id": 563,
	+ "id": 566,
	"link": "/entries/VectorSpace.html",
	"permalink": "/entries/VectorSpace.html",
	"shortname": "VectorSpace",
	"title": "Vector Spaces",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 3
	},
	{
	"abstract": "Skew heaps are an amazingly simple and lightweight implementation of priority queues. They were invented by Sleator and Tarjan [SIAM 1986] and have logarithmic amortized complexity. This entry provides executable and verified functional skew heaps. \u003cp\u003e The amortized complexity of skew heaps is analyzed in the AFP entry \u003ca href=\"http://isa-afp.org/entries/Amortized_Complexity.html\"\u003eAmortized Complexity\u003c/a\u003e.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2014-08-13",
	- "id": 564,
	+ "id": 567,
	"link": "/entries/Skew_Heap.html",
	"permalink": "/entries/Skew_Heap.html",
	"shortname": "Skew_Heap",
	"title": "Skew Heap",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "Splay trees are self-adjusting binary search trees which were invented by Sleator and Tarjan [JACM 1985]. This entry provides executable and verified functional splay trees as well as the related splay heaps (due to Okasaki). \u003cp\u003e The amortized complexity of splay trees and heaps is analyzed in the AFP entry \u003ca href=\"http://isa-afp.org/entries/Amortized_Complexity.html\"\u003eAmortized Complexity\u003c/a\u003e.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2014-08-12",
	- "id": 565,
	+ "id": 568,
	"link": "/entries/Splay_Tree.html",
	"permalink": "/entries/Splay_Tree.html",
	"shortname": "Splay_Tree",
	"title": "Splay Tree",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "We implemented a type class for \"to-string\" functions, similar to Haskell's Show class. Moreover, we provide instantiations for Isabelle/HOL's standard types like bool, prod, sum, nats, ints, and rats. It is further possible, to automatically derive show functions for arbitrary user defined datatypes similar to Haskell's \"deriving Show\".",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2014-07-29",
	- "id": 566,
	+ "id": 569,
	"link": "/entries/Show.html",
	"permalink": "/entries/Show.html",
	"shortname": "Show",
	"title": "Haskell's Show Class in Isabelle/HOL",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 16
	},
	{
	"abstract": "\u003cp\u003eIntransitive noninterference has been a widely studied topic in the last few decades. Several well-established methodologies apply interactive theorem proving to formulate a noninterference theorem over abstract academic models. In joint work with several industrial and academic partners throughout Europe, we are helping in the certification process of PikeOS, an industrial separation kernel developed at SYSGO. In this process, established theories could not be applied. We present a new generic model of separation kernels and a new theory of intransitive noninterference. The model is rich in detail, making it suitable for formal verification of realistic and industrial systems such as PikeOS. Using a refinement-based theorem proving approach, we ensure that proofs remain manageable.\u003c/p\u003e \u003cp\u003e This document corresponds to the deliverable D31.1 of the EURO-MILS Project \u003ca href=\"http://www.euromils.eu\"\u003ehttp://www.euromils.eu\u003c/a\u003e.\u003c/p\u003e",
	"authors": [
	"Freek Verbeek",
	"Sergey Tverdyshev",
	"Oto Havle",
	"Holger Blasum",
	"Bruno Langenstein",
	"Werner Stephan",
	"Yakoub Nemouchi",
	"Abderrahmane Feliachi",
	"Burkhart Wolff",
	"Julien Schmaltz"
	],
	"date": "2014-07-18",
	- "id": 567,
	+ "id": 570,
	"link": "/entries/CISC-Kernel.html",
	"permalink": "/entries/CISC-Kernel.html",
	"shortname": "CISC-Kernel",
	"title": "Formal Specification of a Generic Separation Kernel",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003epGCL is both a programming language and a specification language that incorporates both probabilistic and nondeterministic choice, in a unified manner. Program verification is by refinement or annotation (or both), using either Hoare triples, or weakest-precondition entailment, in the style of GCL.\u003c/p\u003e \u003cp\u003e This package provides both a shallow embedding of the language primitives, and an annotation and refinement framework. The generated document includes a brief tutorial.\u003c/p\u003e",
	"authors": [
	"David Cock"
	],
	"date": "2014-07-13",
	- "id": 568,
	+ "id": 571,
	"link": "/entries/pGCL.html",
	"permalink": "/entries/pGCL.html",
	"shortname": "pGCL",
	"title": "pGCL for Isabelle",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "A framework for the analysis of the amortized complexity of functional data structures is formalized in Isabelle/HOL and applied to a number of standard examples and to the folowing non-trivial ones: skew heaps, splay trees, splay heaps and pairing heaps.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2014-07-07",
	- "id": 569,
	+ "id": 572,
	"link": "/entries/Amortized_Complexity.html",
	"permalink": "/entries/Amortized_Complexity.html",
	"shortname": "Amortized_Complexity",
	"title": "Amortized Complexity Verified",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 2
	},
	{
	"abstract": "We present a unified theory for verifying network security policies. A security policy is represented as directed graph. To check high-level security goals, security invariants over the policy are expressed. We cover monotonic security invariants, i.e. prohibiting more does not harm security. We provide the following contributions for the security invariant theory. \u003cul\u003e \u003cli\u003eSecure auto-completion of scenario-specific knowledge, which eases usability.\u003c/li\u003e \u003cli\u003eSecurity violations can be repaired by tightening the policy iff the security invariants hold for the deny-all policy.\u003c/li\u003e \u003cli\u003eAn algorithm to compute a security policy.\u003c/li\u003e \u003cli\u003eA formalization of stateful connection semantics in network security mechanisms.\u003c/li\u003e \u003cli\u003eAn algorithm to compute a secure stateful implementation of a policy.\u003c/li\u003e \u003cli\u003eAn executable implementation of all the theory.\u003c/li\u003e \u003cli\u003eExamples, ranging from an aircraft cabin data network to the analysis of a large real-world firewall.\u003c/li\u003e \u003cli\u003eMore examples: A fully automated translation of high-level security goals to both firewall and SDN configurations (see Examples/Distributed_WebApp.thy).\u003c/li\u003e \u003c/ul\u003e For a detailed description, see \u003cul\u003e \u003cli\u003eC. Diekmann, A. Korsten, and G. Carle. \u003ca href=\"http://www.net.in.tum.de/fileadmin/bibtex/publications/papers/diekmann2015mansdnnfv.pdf\"\u003eDemonstrating topoS: Theorem-prover-based synthesis of secure network configurations.\u003c/a\u003e In 2nd International Workshop on Management of SDN and NFV Systems, manSDN/NFV, Barcelona, Spain, November 2015.\u003c/li\u003e \u003cli\u003eC. Diekmann, S.-A. Posselt, H. Niedermayer, H. Kinkelin, O. Hanka, and G. Carle. \u003ca href=\"http://www.net.in.tum.de/pub/diekmann/forte14.pdf\"\u003eVerifying Security Policies using Host Attributes.\u003c/a\u003e In FORTE, 34th IFIP International Conference on Formal Techniques for Distributed Objects, Components and Systems, Berlin, Germany, June 2014.\u003c/li\u003e \u003cli\u003eC. Diekmann, L. Hupel, and G. Carle. Directed Security Policies: \u003ca href=\"http://rvg.web.cse.unsw.edu.au/eptcs/paper.cgi?ESSS2014.3\"\u003eA Stateful Network Implementation.\u003c/a\u003e In J. Pang and Y. Liu, editors, Engineering Safety and Security Systems, volume 150 of Electronic Proceedings in Theoretical Computer Science, pages 20-34, Singapore, May 2014. Open Publishing Association.\u003c/li\u003e \u003c/ul\u003e",
	"authors": [
	"Cornelius Diekmann"
	],
	"date": "2014-07-04",
	- "id": 570,
	+ "id": 573,
	"link": "/entries/Network_Security_Policy_Verification.html",
	"permalink": "/entries/Network_Security_Policy_Verification.html",
	"shortname": "Network_Security_Policy_Verification",
	"title": "Network Security Policy Verification",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Pop-refinement is an approach to stepwise refinement, carried out inside an interactive theorem prover by constructing a monotonically decreasing sequence of predicates over deeply embedded target programs. The sequence starts with a predicate that characterizes the possible implementations, and ends with a predicate that characterizes a unique program in explicit syntactic form. Pop-refinement enables more requirements (e.g. program-level and non-functional) to be captured in the initial specification and preserved through refinement. Security requirements expressed as hyperproperties (i.e. predicates over sets of traces) are always preserved by pop-refinement, unlike the popular notion of refinement as trace set inclusion. Two simple examples in Isabelle/HOL are presented, featuring program-level requirements, non-functional requirements, and hyperproperties.",
	"authors": [
	"Alessandro Coglio"
	],
	"date": "2014-07-03",
	- "id": 571,
	+ "id": 574,
	"link": "/entries/Pop_Refinement.html",
	"permalink": "/entries/Pop_Refinement.html",
	"shortname": "Pop_Refinement",
	"title": "Pop-Refinement",
	"topic_links": [
	"computer-science/programming-languages/misc"
	],
	"topics": [
	"Computer science/Programming languages/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "Monadic second-order logic on finite words (MSO) is a decidable yet expressive logic into which many decision problems can be encoded. Since MSO formulas correspond to regular languages, equivalence of MSO formulas can be reduced to the equivalence of some regular structures (e.g. automata). We verify an executable decision procedure for MSO formulas that is not based on automata but on regular expressions. \u003cp\u003e Decision procedures for regular expression equivalence have been formalized before, usually based on Brzozowski derivatives. Yet, for a straightforward embedding of MSO formulas into regular expressions an extension of regular expressions with a projection operation is required. We prove total correctness and completeness of an equivalence checker for regular expressions extended in that way. We also define a language-preserving translation of formulas into regular expressions with respect to two different semantics of MSO. ",
	"authors": [
	"Dmitriy Traytel",
	"Tobias Nipkow"
	],
	"date": "2014-06-12",
	- "id": 572,
	+ "id": 575,
	"link": "/entries/MSO_Regex_Equivalence.html",
	"permalink": "/entries/MSO_Regex_Equivalence.html",
	"shortname": "MSO_Regex_Equivalence",
	"title": "Decision Procedures for MSO on Words Based on Derivatives of Regular Expressions",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/general-logic/decidability-of-theories"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/General logic/Decidability of theories"
	],
	"used_by": 0
	},
	{
	"abstract": "This entry provides executable checkers for the following properties of boolean expressions: satisfiability, tautology and equivalence. Internally, the checkers operate on binary decision trees and are reasonably efficient (for purely functional algorithms).",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2014-06-08",
	- "id": 573,
	+ "id": 576,
	"link": "/entries/Boolean_Expression_Checkers.html",
	"permalink": "/entries/Boolean_Expression_Checkers.html",
	"shortname": "Boolean_Expression_Checkers",
	"title": "Boolean Expression Checkers",
	"topic_links": [
	"computer-science/algorithms",
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Computer science/Algorithms",
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 2
	},
	{
	"abstract": "We present an LTL model checker whose code has been completely verified using the Isabelle theorem prover. The checker consists of over 4000 lines of ML code. The code is produced using the Isabelle Refinement Framework, which allows us to split its correctness proof into (1) the proof of an abstract version of the checker, consisting of a few hundred lines of “formalized pseudocode”, and (2) a verified refinement step in which mathematical sets and other abstract structures are replaced by implementations of efficient structures like red-black trees and functional arrays. This leads to a checker that, while still slower than unverified checkers, can already be used as a trusted reference implementation against which advanced implementations can be tested.",
	"authors": [
	"Javier Esparza",
	"Peter Lammich",
	"René Neumann",
	"Tobias Nipkow",
	"Alexander Schimpf",
	"Jan-Georg Smaus"
	],
	"date": "2014-05-28",
	- "id": 574,
	+ "id": 577,
	"link": "/entries/CAVA_LTL_Modelchecker.html",
	"permalink": "/entries/CAVA_LTL_Modelchecker.html",
	"shortname": "CAVA_LTL_Modelchecker",
	"title": "A Fully Verified Executable LTL Model Checker",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize linear-time temporal logic (LTL) and the algorithm by Gerth et al. to convert LTL formulas to generalized Büchi automata. We also formalize some syntactic rewrite rules that can be applied to optimize the LTL formula before conversion. Moreover, we integrate the Stuttering Equivalence AFP-Entry by Stefan Merz, adapting the lemma that next-free LTL formula cannot distinguish between stuttering equivalent runs to our setting. \u003cp\u003e We use the Isabelle Refinement and Collection framework, as well as the Autoref tool, to obtain a refined version of our algorithm, from which efficiently executable code can be extracted.",
	"authors": [
	"Alexander Schimpf",
	"Peter Lammich"
	],
	"date": "2014-05-28",
	- "id": 575,
	+ "id": 578,
	"link": "/entries/LTL_to_GBA.html",
	"permalink": "/entries/LTL_to_GBA.html",
	"shortname": "LTL_to_GBA",
	"title": "Converting Linear-Time Temporal Logic to Generalized Büchi Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "We present an executable formalization of the language Promela, the description language for models of the model checker SPIN. This formalization is part of the work for a completely verified model checker (CAVA), but also serves as a useful (and executable!) description of the semantics of the language itself, something that is currently missing. The formalization uses three steps: It takes an abstract syntax tree generated from an SML parser, removes syntactic sugar and enriches it with type information. This further gets translated into a transition system, on which the semantic engine (read: successor function) operates.",
	"authors": [
	"René Neumann"
	],
	"date": "2014-05-28",
	- "id": 576,
	+ "id": 579,
	"link": "/entries/Promela.html",
	"permalink": "/entries/Promela.html",
	"shortname": "Promela",
	"title": "Promela Formalization",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 1
	},
	{
	"abstract": "We report on the graph and automata library that is used in the fully verified LTL model checker CAVA. As most components of CAVA use some type of graphs or automata, a common automata library simplifies assembly of the components and reduces redundancy. \u003cp\u003e The CAVA Automata Library provides a hierarchy of graph and automata classes, together with some standard algorithms. Its object oriented design allows for sharing of algorithms, theorems, and implementations between its classes, and also simplifies extensions of the library. Moreover, it is integrated into the Automatic Refinement Framework, supporting automatic refinement of the abstract automata types to efficient data structures. \u003cp\u003e Note that the CAVA Automata Library is work in progress. Currently, it is very specifically tailored towards the requirements of the CAVA model checker. Nevertheless, the formalization techniques presented here allow an extension of the library to a wider scope. Moreover, they are not limited to graph libraries, but apply to class hierarchies in general. \u003cp\u003e The CAVA Automata Library is described in the paper: Peter Lammich, The CAVA Automata Library, Isabelle Workshop 2014.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2014-05-28",
	- "id": 577,
	+ "id": 580,
	"link": "/entries/CAVA_Automata.html",
	"permalink": "/entries/CAVA_Automata.html",
	"shortname": "CAVA_Automata",
	"title": "The CAVA Automata Library",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 8
	},
	{
	"abstract": "We present an Isabelle/HOL formalization of Gabow's algorithm for finding the strongly connected components of a directed graph. Using data refinement techniques, we extract efficient code that performs comparable to a reference implementation in Java. Our style of formalization allows for re-using large parts of the proofs when defining variants of the algorithm. We demonstrate this by verifying an algorithm for the emptiness check of generalized Büchi automata, re-using most of the existing proofs.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2014-05-28",
	- "id": 578,
	+ "id": 581,
	"link": "/entries/Gabow_SCC.html",
	"permalink": "/entries/Gabow_SCC.html",
	"shortname": "Gabow_SCC",
	"title": "Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm",
	"topic_links": [
	"computer-science/algorithms/graph",
	"mathematics/graph-theory"
	],
	"topics": [
	"Computer science/Algorithms/Graph",
	"Mathematics/Graph theory"
	],
	"used_by": 2
	},
	{
	"abstract": "\u003cp\u003e An extension of classical noninterference security for deterministic state machines, as introduced by Goguen and Meseguer and elegantly formalized by Rushby, to nondeterministic systems should satisfy two fundamental requirements: it should be based on a mathematically precise theory of nondeterminism, and should be equivalent to (or at least not weaker than) the classical notion in the degenerate deterministic case. \u003c/p\u003e \u003cp\u003e This paper proposes a definition of noninterference security applying to Hoare's Communicating Sequential Processes (CSP) in the general case of a possibly intransitive noninterference policy, and proves the equivalence of this security property to classical noninterference security for processes representing deterministic state machines. \u003c/p\u003e \u003cp\u003e Furthermore, McCullough's generalized noninterference security is shown to be weaker than both the proposed notion of CSP noninterference security for a generic process, and classical noninterference security for processes representing deterministic state machines. This renders CSP noninterference security preferable as an extension of classical noninterference security to nondeterministic systems. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2014-05-23",
	- "id": 579,
	+ "id": 582,
	"link": "/entries/Noninterference_CSP.html",
	"permalink": "/entries/Noninterference_CSP.html",
	"shortname": "Noninterference_CSP",
	"title": "Noninterference Security in Communicating Sequential Processes",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 1
	},
	{
	"abstract": "This formulation of the Roy-Floyd-Warshall algorithm for the transitive closure bypasses matrices and arrays, but uses a more direct mathematical model with adjacency functions for immediate predecessors and successors. This can be implemented efficiently in functional programming languages and is particularly adequate for sparse relations.",
	"authors": [
	"Makarius Wenzel"
	],
	"date": "2014-05-23",
	- "id": 580,
	+ "id": 583,
	"link": "/entries/Roy_Floyd_Warshall.html",
	"permalink": "/entries/Roy_Floyd_Warshall.html",
	"shortname": "Roy_Floyd_Warshall",
	"title": "Transitive closure according to Roy-Floyd-Warshall",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "Regular algebras axiomatise the equational theory of regular expressions as induced by regular language identity. We use Isabelle/HOL for a detailed systematic study of regular algebras given by Boffa, Conway, Kozen and Salomaa. We investigate the relationships between these classes, formalise a soundness proof for the smallest class (Salomaa's) and obtain completeness of the largest one (Boffa's) relative to a deep result by Krob. In addition we provide a large collection of regular identities in the general setting of Boffa's axiom. Our regular algebra hierarchy is orthogonal to the Kleene algebra hierarchy in the Archive of Formal Proofs; we have not aimed at an integration for pragmatic reasons.",
	"authors": [
	"Simon Foster",
	"Georg Struth"
	],
	"date": "2014-05-21",
	- "id": 581,
	+ "id": 584,
	"link": "/entries/Regular_Algebras.html",
	"permalink": "/entries/Regular_Algebras.html",
	"shortname": "Regular_Algebras",
	"title": "Regular Algebras",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This set of theories presents a formalisation in Isabelle/HOL of data dependencies between components. The approach allows to analyse system structure oriented towards efficient checking of system: it aims at elaborating for a concrete system, which parts of the system are necessary to check a given property.",
	"authors": [
	"Maria Spichkova"
	],
	"date": "2014-04-28",
	- "id": 582,
	+ "id": 585,
	"link": "/entries/ComponentDependencies.html",
	"permalink": "/entries/ComponentDependencies.html",
	"shortname": "ComponentDependencies",
	"title": "Formalisation and Analysis of Component Dependencies",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Research in information-flow security aims at developing methods to identify undesired information leaks within programs from private (high) sources to public (low) sinks. For a concurrent system, it is desirable to have compositional analysis methods that allow for analyzing each thread independently and that nevertheless guarantee that the parallel composition of successfully analyzed threads satisfies a global security guarantee. However, such a compositional analysis should not be overly pessimistic about what an environment might do with shared resources. Otherwise, the analysis will reject many intuitively secure programs. \u003cp\u003e The paper \"Assumptions and Guarantees for Compositional Noninterference\" by Mantel et. al. presents one solution for this problem: an approach for compositionally reasoning about non-interference in concurrent programs via rely-guarantee-style reasoning. We present an Isabelle/HOL formalization of the concepts and proofs of this approach.",
	"authors": [
	"Sylvia Grewe",
	"Heiko Mantel",
	"Daniel Schoepe"
	],
	"date": "2014-04-23",
	- "id": 583,
	+ "id": 586,
	"link": "/entries/SIFUM_Type_Systems.html",
	"permalink": "/entries/SIFUM_Type_Systems.html",
	"shortname": "SIFUM_Type_Systems",
	"title": "A Formalization of Assumptions and Guarantees for Compositional Noninterference",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "Research in information-flow security aims at developing methods to identify undesired information leaks within programs from private sources to public sinks. Noninterference captures this intuition by requiring that no information whatsoever flows from private sources to public sinks. However, in practice this definition is often too strict: Depending on the intuitive desired security policy, the controlled declassification of certain private information (WHAT) at certain points in the program (WHERE) might not result in an undesired information leak. \u003cp\u003e We present an Isabelle/HOL formalization of such a security property for controlled declassification, namely WHAT\u0026WHERE-security from \"Scheduler-Independent Declassification\" by Lux, Mantel, and Perner. The formalization includes compositionality proofs for and a soundness proof for a security type system that checks for programs in a simple while language with dynamic thread creation. \u003cp\u003e Our formalization of the security type system is abstract in the language for expressions and in the semantic side conditions for expressions. It can easily be instantiated with different syntactic approximations for these side conditions. The soundness proof of such an instantiation boils down to showing that these syntactic approximations imply the semantic side conditions. \u003cp\u003e This Isabelle/HOL formalization uses theories from the entry Strong Security.",
	"authors": [
	"Sylvia Grewe",
	"Alexander Lux",
	"Heiko Mantel",
	"Jens Sauer"
	],
	"date": "2014-04-23",
	- "id": 584,
	+ "id": 587,
	"link": "/entries/WHATandWHERE_Security.html",
	"permalink": "/entries/WHATandWHERE_Security.html",
	"shortname": "WHATandWHERE_Security",
	"title": "A Formalization of Declassification with WHAT-and-WHERE-Security",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "Research in information-flow security aims at developing methods to identify undesired information leaks within programs from private sources to public sinks. Noninterference captures this intuition. Strong security from Sabelfeld and Sands formalizes noninterference for concurrent systems. \u003cp\u003e We present an Isabelle/HOL formalization of strong security for arbitrary security lattices (Sabelfeld and Sands use a two-element security lattice in the original publication). The formalization includes compositionality proofs for strong security and a soundness proof for a security type system that checks strong security for programs in a simple while language with dynamic thread creation. \u003cp\u003e Our formalization of the security type system is abstract in the language for expressions and in the semantic side conditions for expressions. It can easily be instantiated with different syntactic approximations for these side conditions. The soundness proof of such an instantiation boils down to showing that these syntactic approximations imply the semantic side conditions.",
	"authors": [
	"Sylvia Grewe",
	"Alexander Lux",
	"Heiko Mantel",
	"Jens Sauer"
	],
	"date": "2014-04-23",
	- "id": 585,
	+ "id": 588,
	"link": "/entries/Strong_Security.html",
	"permalink": "/entries/Strong_Security.html",
	"shortname": "Strong_Security",
	"title": "A Formalization of Strong Security",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 1
	},
	{
	"abstract": "This is a formalization of bounded-deducibility security (BD security), a flexible notion of information-flow security applicable to arbitrary transition systems. It generalizes Sutherland's classic notion of nondeducibility by factoring in declassification bounds and trigger, whereas nondeducibility states that, in a system, information cannot flow between specified sources and sinks, BD security indicates upper bounds for the flow and triggers under which these upper bounds are no longer guaranteed.",
	"authors": [
	"Andrei Popescu",
	"Peter Lammich",
	"Thomas Bauereiss"
	],
	"date": "2014-04-22",
	- "id": 586,
	+ "id": 589,
	"link": "/entries/Bounded_Deducibility_Security.html",
	"permalink": "/entries/Bounded_Deducibility_Security.html",
	"shortname": "Bounded_Deducibility_Security",
	"title": "Bounded-Deducibility Security",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 3
	},
	{
	"abstract": "We formalize HyperCTL, a temporal logic for expressing security properties. We first define a shallow embedding of HyperCTL, within which we prove inductive and coinductive rules for the operators. Then we show that a HyperCTL* formula captures Goguen-Meseguer noninterference, a landmark information flow property. We also define a deep embedding and connect it to the shallow embedding by a denotational semantics, for which we prove sanity w.r.t. dependence on the free variables. Finally, we show that under some finiteness assumptions about the model, noninterference is given by a (finitary) syntactic formula.",
	"authors": [
	"Markus N. Rabe",
	"Peter Lammich",
	"Andrei Popescu"
	],
	"date": "2014-04-16",
	- "id": 587,
	+ "id": 590,
	"link": "/entries/HyperCTL.html",
	"permalink": "/entries/HyperCTL.html",
	"shortname": "HyperCTL",
	"title": "A shallow embedding of HyperCTL*",
	"topic_links": [
	"computer-science/security",
	"logic/general-logic/temporal-logic"
	],
	"topics": [
	"Computer science/Security",
	"Logic/General logic/Temporal logic"
	],
	"used_by": 0
	},
	{
	"abstract": "A formalization of an abstract property of possibly infinite derivation trees (modeled by a codatatype), representing the core of a proof (in Beth/Hintikka style) of the first-order logic completeness theorem, independent of the concrete syntax or inference rules. This work is described in detail in the IJCAR 2014 publication by the authors. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of FOL---e.g., with or without predicates, equality, or sorts. Here, we give only a toy example instantiation with classical propositional logic. A more serious instance---many-sorted FOL with equality---is described elsewhere [Blanchette and Popescu, FroCoS 2013].",
	"authors": [
	"Jasmin Christian Blanchette",
	"Andrei Popescu",
	"Dmitriy Traytel"
	],
	"date": "2014-04-16",
	- "id": 588,
	+ "id": 591,
	"link": "/entries/Abstract_Completeness.html",
	"permalink": "/entries/Abstract_Completeness.html",
	"shortname": "Abstract_Completeness",
	"title": "Abstract Completeness",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 4
	},
	{
	"abstract": "These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided.",
	"authors": [
	"Florian Haftmann"
	],
	"date": "2014-04-13",
	- "id": 589,
	+ "id": 592,
	"link": "/entries/Discrete_Summation.html",
	"permalink": "/entries/Discrete_Summation.html",
	"shortname": "Discrete_Summation",
	"title": "Discrete Summation",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 3
	},
	{
	"abstract": "This document accompanies the article \"The Design and Implementation of a Verification Technique for GPU Kernels\" by Adam Betts, Nathan Chong, Alastair F. Donaldson, Jeroen Ketema, Shaz Qadeer, Paul Thomson and John Wickerson. It formalises all of the definitions provided in Sections 3 and 4 of the article.",
	"authors": [
	"John Wickerson"
	],
	"date": "2014-04-03",
	- "id": 590,
	+ "id": 593,
	"link": "/entries/GPU_Kernel_PL.html",
	"permalink": "/entries/GPU_Kernel_PL.html",
	"shortname": "GPU_Kernel_PL",
	"title": "Syntax and semantics of a GPU kernel programming language",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize a probabilistic noninterference for a multi-threaded language with uniform scheduling, where probabilistic behaviour comes from both the scheduler and the individual threads. We define notions probabilistic noninterference in two variants: resumption-based and trace-based. For the resumption-based notions, we prove compositionality w.r.t. the language constructs and establish sound type-system-like syntactic criteria. This is a formalization of the mathematical development presented at CPP 2013 and CALCO 2013. It is the probabilistic variant of the Possibilistic Noninterference AFP entry.",
	"authors": [
	"Andrei Popescu",
	"Johannes Hölzl"
	],
	"date": "2014-03-11",
	- "id": 591,
	+ "id": 594,
	"link": "/entries/Probabilistic_Noninterference.html",
	"permalink": "/entries/Probabilistic_Noninterference.html",
	"shortname": "Probabilistic_Noninterference",
	"title": "Probabilistic Noninterference",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e AWN is a process algebra developed for modelling and analysing protocols for Mobile Ad hoc Networks (MANETs) and Wireless Mesh Networks (WMNs). AWN models comprise five distinct layers: sequential processes, local parallel compositions, nodes, partial networks, and complete networks.\u003c/p\u003e \u003cp\u003e This development mechanises the original operational semantics of AWN and introduces a variant 'open' operational semantics that enables the compositional statement and proof of invariants across distinct network nodes. It supports labels (for weakening invariants) and (abstract) data state manipulations. A framework for compositional invariant proofs is developed, including a tactic (inv_cterms) for inductive invariant proofs of sequential processes, lifting rules for the open versions of the higher layers, and a rule for transferring lifted properties back to the standard semantics. A notion of 'control terms' reduces proof obligations to the subset of subterms that act directly (in contrast to operators for combining terms and joining processes).\u003c/p\u003e",
	"authors": [
	"Timothy Bourke"
	],
	"date": "2014-03-08",
	- "id": 592,
	+ "id": 595,
	"link": "/entries/AWN.html",
	"permalink": "/entries/AWN.html",
	"shortname": "AWN",
	"title": "Mechanization of the Algebra for Wireless Networks (AWN)",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 1
	},
	{
	"abstract": "We provide a wrapper around the partial-function command that supports mutual recursion.",
	"authors": [
	"René Thiemann"
	],
	"date": "2014-02-18",
	- "id": 593,
	+ "id": 596,
	"link": "/entries/Partial_Function_MR.html",
	"permalink": "/entries/Partial_Function_MR.html",
	"shortname": "Partial_Function_MR",
	"title": "Mutually Recursive Partial Functions",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 2
	},
	{
	"abstract": "Random graphs are graphs with a fixed number of vertices, where each edge is present with a fixed probability. We are interested in the probability that a random graph contains a certain pattern, for example a cycle or a clique. A very high edge probability gives rise to perhaps too many edges (which degrades performance for many algorithms), whereas a low edge probability might result in a disconnected graph. We prove a theorem about a threshold probability such that a higher edge probability will asymptotically almost surely produce a random graph with the desired subgraph.",
	"authors": [
	"Lars Hupel"
	],
	"date": "2014-02-13",
	- "id": 594,
	+ "id": 597,
	"link": "/entries/Random_Graph_Subgraph_Threshold.html",
	"permalink": "/entries/Random_Graph_Subgraph_Threshold.html",
	"shortname": "Random_Graph_Subgraph_Threshold",
	"title": "Properties of Random Graphs -- Subgraph Containment",
	"topic_links": [
	"mathematics/graph-theory",
	"mathematics/probability-theory"
	],
	"topics": [
	"Mathematics/Graph theory",
	"Mathematics/Probability theory"
	],
	"used_by": 3
	},
	{
	"abstract": "Stepwise program refinement techniques can be used to simplify program verification. Programs are better understood since their main properties are clearly stated, and verification of rather complex algorithms is reduced to proving simple statements connecting successive program specifications. Additionally, it is easy to analyze similar algorithms and to compare their properties within a single formalization. Usually, formal analysis is not done in educational setting due to complexity of verification and a lack of tools and procedures to make comparison easy. Verification of an algorithm should not only give correctness proof, but also better understanding of an algorithm. If the verification is based on small step program refinement, it can become simple enough to be demonstrated within the university-level computer science curriculum. In this paper we demonstrate this and give a formal analysis of two well known algorithms (Selection Sort and Heap Sort) using proof assistant Isabelle/HOL and program refinement techniques.",
	"authors": [
	"Danijela Petrovic"
	],
	"date": "2014-02-11",
	- "id": 595,
	+ "id": 598,
	"link": "/entries/Selection_Heap_Sort.html",
	"permalink": "/entries/Selection_Heap_Sort.html",
	"shortname": "Selection_Heap_Sort",
	"title": "Verification of Selection and Heap Sort Using Locales",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "We give a formalization of affine forms as abstract representations of zonotopes. We provide affine operations as well as overapproximations of some non-affine operations like multiplication and division. Expressions involving those operations can automatically be turned into (executable) functions approximating the original expression in affine arithmetic.",
	"authors": [
	"Fabian Immler"
	],
	"date": "2014-02-07",
	- "id": 596,
	+ "id": 599,
	"link": "/entries/Affine_Arithmetic.html",
	"permalink": "/entries/Affine_Arithmetic.html",
	"shortname": "Affine_Arithmetic",
	"title": "Affine Arithmetic",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "We apply data refinement to implement the real numbers, where we support all numbers in the field extension Q[sqrt(b)], i.e., all numbers of the form p + q * sqrt(b) for rational numbers p and q and some fixed natural number b. To this end, we also developed algorithms to precisely compute roots of a rational number, and to perform a factorization of natural numbers which eliminates duplicate prime factors. \u003cp\u003e Our results have been used to certify termination proofs which involve polynomial interpretations over the reals.",
	"authors": [
	"René Thiemann"
	],
	"date": "2014-02-06",
	- "id": 597,
	+ "id": 600,
	"link": "/entries/Real_Impl.html",
	"permalink": "/entries/Real_Impl.html",
	"shortname": "Real_Impl",
	"title": "Implementing field extensions of the form Q[sqrt(b)]",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "We formalize a unified framework for verified decision procedures for regular expression equivalence. Five recently published formalizations of such decision procedures (three based on derivatives, two on marked regular expressions) can be obtained as instances of the framework. We discover that the two approaches based on marked regular expressions, which were previously thought to be the same, are different, and one seems to produce uniformly smaller automata. The common framework makes it possible to compare the performance of the different decision procedures in a meaningful way. ",
	"authors": [
	"Tobias Nipkow",
	"Dmitriy Traytel"
	],
	"date": "2014-01-30",
	- "id": 598,
	+ "id": 601,
	"link": "/entries/Regex_Equivalence.html",
	"permalink": "/entries/Regex_Equivalence.html",
	"shortname": "Regex_Equivalence",
	"title": "Unified Decision Procedures for Regular Expression Equivalence",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "These theories extend the existing proof of the first Sylow theorem (written by Florian Kammueller and L. C. Paulson) by what are often called the second, third and fourth Sylow theorems. These theorems state propositions about the number of Sylow p-subgroups of a group and the fact that they are conjugate to each other. The proofs make use of an implementation of group actions and their properties.",
	"authors": [
	"Jakob von Raumer"
	],
	"date": "2014-01-28",
	- "id": 599,
	+ "id": 602,
	"link": "/entries/Secondary_Sylow.html",
	"permalink": "/entries/Secondary_Sylow.html",
	"shortname": "Secondary_Sylow",
	"title": "Secondary Sylow Theorems",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "Tarski's algebra of binary relations is formalised along the lines of the standard textbooks of Maddux and Schmidt and Ströhlein. This includes relation-algebraic concepts such as subidentities, vectors and a domain operation as well as various notions associated to functions. Relation algebras are also expanded by a reflexive transitive closure operation, and they are linked with Kleene algebras and models of binary relations and Boolean matrices.",
	"authors": [
	"Alasdair Armstrong",
	"Simon Foster",
	"Georg Struth",
	"Tjark Weber"
	],
	"date": "2014-01-25",
	- "id": 600,
	+ "id": 603,
	"link": "/entries/Relation_Algebra.html",
	"permalink": "/entries/Relation_Algebra.html",
	"shortname": "Relation_Algebra",
	"title": "Relation Algebra",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "We formalise Kleene algebra with tests (KAT) and demonic refinement algebra (DRA) in Isabelle/HOL. KAT is relevant for program verification and correctness proofs in the partial correctness setting. While DRA targets similar applications in the context of total correctness. Our formalisation contains the two most important models of these algebras: binary relations in the case of KAT and predicate transformers in the case of DRA. In addition, we derive the inference rules for Hoare logic in KAT and its relational model and present a simple formally verified program verification tool prototype based on the algebraic approach.",
	"authors": [
	"Alasdair Armstrong",
	"Victor B. F. Gomes",
	"Georg Struth"
	],
	"date": "2014-01-23",
	- "id": 601,
	+ "id": 604,
	"link": "/entries/KAT_and_DRA.html",
	"permalink": "/entries/KAT_and_DRA.html",
	"shortname": "KAT_and_DRA",
	"title": "Kleene Algebra with Tests and Demonic Refinement Algebras",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/automata-and-formal-languages",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Automata and formal languages",
	"Mathematics/Algebra"
	],
	"used_by": 2
	},
	{
	"abstract": "The Unified Modeling Language (UML) is one of the few modeling languages that is widely used in industry. While UML is mostly known as diagrammatic modeling language (e.g., visualizing class models), it is complemented by a textual language, called Object Constraint Language (OCL). The current version of OCL is based on a four-valued logic that turns UML into a formal language. Any type comprises the elements \"invalid\" and \"null\" which are propagated as strict and non-strict, respectively. Unfortunately, the former semi-formal semantics of this specification language, captured in the \"Annex A\" of the OCL standard, leads to different interpretations of corner cases. We formalize the core of OCL: denotational definitions, a logical calculus and operational rules that allow for the execution of OCL expressions by a mixture of term rewriting and code compilation. Our formalization reveals several inconsistencies and contradictions in the current version of the OCL standard. Overall, this document is intended to provide the basis for a machine-checked text \"Annex A\" of the OCL standard targeting at tool implementors.",
	"authors": [
	"Achim D. Brucker",
	"Frédéric Tuong",
	"Burkhart Wolff"
	],
	"date": "2014-01-16",
	- "id": 602,
	+ "id": 605,
	"link": "/entries/Featherweight_OCL.html",
	"permalink": "/entries/Featherweight_OCL.html",
	"shortname": "Featherweight_OCL",
	"title": "Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5",
	"topic_links": [
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/System description languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This paper presents an Isabelle/HOL set of theories which allows the specification of crypto-based components and the verification of their composition properties wrt. cryptographic aspects. We introduce a formalisation of the security property of data secrecy, the corresponding definitions and proofs. Please note that here we import the Isabelle/HOL theory ListExtras.thy, presented in the AFP entry FocusStreamsCaseStudies-AFP.",
	"authors": [
	"Maria Spichkova"
	],
	"date": "2014-01-11",
	- "id": 603,
	+ "id": 606,
	"link": "/entries/CryptoBasedCompositionalProperties.html",
	"permalink": "/entries/CryptoBasedCompositionalProperties.html",
	"shortname": "CryptoBasedCompositionalProperties",
	"title": "Compositional Properties of Crypto-Based Components",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Sturm's Theorem states that polynomial sequences with certain properties, so-called Sturm sequences, can be used to count the number of real roots of a real polynomial. This work contains a proof of Sturm's Theorem and code for constructing Sturm sequences efficiently. It also provides the “sturm” proof method, which can decide certain statements about the roots of real polynomials, such as “the polynomial P has exactly n roots in the interval I” or “P(x) \u003e Q(x) for all x \u0026#8712; \u0026#8477;”.",
	"authors": [
	"Manuel Eberl"
	],
	"date": "2014-01-11",
	- "id": 604,
	+ "id": 607,
	"link": "/entries/Sturm_Sequences.html",
	"permalink": "/entries/Sturm_Sequences.html",
	"shortname": "Sturm_Sequences",
	"title": "Sturm's Theorem",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 4
	},
	{
	"abstract": "\u003cp\u003e Tail-recursive function definitions are sometimes more straightforward than alternatives, but proving theorems on them may be roundabout because of the peculiar form of the resulting recursion induction rules. \u003c/p\u003e\u003cp\u003e This paper describes a proof method that provides a general solution to this problem by means of suitable invariants over inductive sets, and illustrates the application of such method by examining two case studies. \u003c/p\u003e",
	"authors": [
	"Pasquale Noce"
	],
	"date": "2013-12-01",
	- "id": 605,
	+ "id": 608,
	"link": "/entries/Tail_Recursive_Functions.html",
	"permalink": "/entries/Tail_Recursive_Functions.html",
	"shortname": "Tail_Recursive_Functions",
	"title": "A General Method for the Proof of Theorems on Tail-recursive Functions",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "Gödel's two incompleteness theorems are formalised, following a careful \u003ca href=\"http://journals.impan.gov.pl/dm/Inf/422-0-1.html\"\u003epresentation\u003c/a\u003e by Swierczkowski, in the theory of \u003ca href=\"HereditarilyFinite.html\"\u003ehereditarily finite sets\u003c/a\u003e. This represents the first ever machine-assisted proof of the second incompleteness theorem. Compared with traditional formalisations using Peano arithmetic (see e.g. Boolos), coding is simpler, with no need to formalise the notion of multiplication (let alone that of a prime number) in the formalised calculus upon which the theorem is based. However, other technical problems had to be solved in order to complete the argument.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2013-11-17",
	- "id": 606,
	+ "id": 609,
	"link": "/entries/Incompleteness.html",
	"permalink": "/entries/Incompleteness.html",
	"shortname": "Incompleteness",
	"title": "Gödel's Incompleteness Theorems",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 2
	},
	{
	"abstract": "The theory of hereditarily finite sets is formalised, following the \u003ca href=\"http://journals.impan.gov.pl/dm/Inf/422-0-1.html\"\u003edevelopment\u003c/a\u003e of Swierczkowski. An HF set is a finite collection of other HF sets; they enjoy an induction principle and satisfy all the axioms of ZF set theory apart from the axiom of infinity, which is negated. All constructions that are possible in ZF set theory (Cartesian products, disjoint sums, natural numbers, functions) without using infinite sets are possible here. The definition of addition for the HF sets follows Kirby. This development forms the foundation for the Isabelle proof of Gödel's incompleteness theorems, which has been \u003ca href=\"Incompleteness.html\"\u003eformalised separately\u003c/a\u003e.",
	"authors": [
	"Lawrence C. Paulson"
	],
	"date": "2013-11-17",
	- "id": 607,
	+ "id": 610,
	"link": "/entries/HereditarilyFinite.html",
	"permalink": "/entries/HereditarilyFinite.html",
	"shortname": "HereditarilyFinite",
	"title": "The Hereditarily Finite Sets",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 5
	},
	{
	"abstract": "\u003cp\u003eWe define formal languages as a codataype of infinite trees branching over the alphabet. Each node in such a tree indicates whether the path to this node constitutes a word inside or outside of the language. This codatatype is isormorphic to the set of lists representation of languages, but caters for definitions by corecursion and proofs by coinduction.\u003c/p\u003e \u003cp\u003eRegular operations on languages are then defined by primitive corecursion. A difficulty arises here, since the standard definitions of concatenation and iteration from the coalgebraic literature are not primitively corecursive-they require guardedness up-to union/concatenation. Without support for up-to corecursion, these operation must be defined as a composition of primitive ones (and proved being equal to the standard definitions). As an exercise in coinduction we also prove the axioms of Kleene algebra for the defined regular operations.\u003c/p\u003e \u003cp\u003eFurthermore, a language for context-free grammars given by productions in Greibach normal form and an initial nonterminal is constructed by primitive corecursion, yielding an executable decision procedure for the word problem without further ado.\u003c/p\u003e",
	"authors": [
	"Dmitriy Traytel"
	],
	"date": "2013-11-15",
	- "id": 608,
	+ "id": 611,
	"link": "/entries/Coinductive_Languages.html",
	"permalink": "/entries/Coinductive_Languages.html",
	"shortname": "Coinductive_Languages",
	"title": "A Codatatype of Formal Languages",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "This set of theories presents an Isabelle/HOL formalisation of stream processing components introduced in Focus, a framework for formal specification and development of interactive systems. This is an extended and updated version of the formalisation, which was elaborated within the methodology \"Focus on Isabelle\". In addition, we also applied the formalisation on three case studies that cover different application areas: process control (Steam Boiler System), data transmission (FlexRay communication protocol), memory and processing components (Automotive-Gateway System).",
	"authors": [
	"Maria Spichkova"
	],
	"date": "2013-11-14",
	- "id": 609,
	+ "id": 612,
	"link": "/entries/FocusStreamsCaseStudies.html",
	"permalink": "/entries/FocusStreamsCaseStudies.html",
	"shortname": "FocusStreamsCaseStudies",
	"title": "Stream Processing Components: Isabelle/HOL Formalisation and Case Studies",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "Dana Scott's version of Gödel's proof of God's existence is formalized in quantified modal logic KB (QML KB). QML KB is modeled as a fragment of classical higher-order logic (HOL); thus, the formalization is essentially a formalization in HOL.",
	"authors": [
	"Christoph Benzmüller",
	"Bruno Woltzenlogel Paleo"
	],
	"date": "2013-11-12",
	- "id": 610,
	+ "id": 613,
	"link": "/entries/GoedelGod.html",
	"permalink": "/entries/GoedelGod.html",
	"shortname": "GoedelGod",
	"title": "Gödel's God in Isabelle/HOL",
	"topic_links": [
	"logic/philosophical-aspects"
	],
	"topics": [
	"Logic/Philosophical aspects"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory contains a formalization of decreasing diagrams showing that any locally decreasing abstract rewrite system is confluent. We consider the valley (van Oostrom, TCS 1994) and the conversion version (van Oostrom, RTA 2008) and closely follow the original proofs. As an application we prove Newman's lemma.",
	"authors": [
	"Harald Zankl"
	],
	"date": "2013-11-01",
	- "id": 611,
	+ "id": 614,
	"link": "/entries/Decreasing-Diagrams.html",
	"permalink": "/entries/Decreasing-Diagrams.html",
	"shortname": "Decreasing-Diagrams",
	"title": "Decreasing Diagrams",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 0
	},
	{
	"abstract": "We present the Autoref tool for Isabelle/HOL, which automatically refines algorithms specified over abstract concepts like maps and sets to algorithms over concrete implementations like red-black-trees, and produces a refinement theorem. It is based on ideas borrowed from relational parametricity due to Reynolds and Wadler. The tool allows for rapid prototyping of verified, executable algorithms. Moreover, it can be configured to fine-tune the result to the user~s needs. Our tool is able to automatically instantiate generic algorithms, which greatly simplifies the implementation of executable data structures. \u003cp\u003e This AFP-entry provides the basic tool, which is then used by the Refinement and Collection Framework to provide automatic data refinement for the nondeterminism monad and various collection datastructures.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2013-10-02",
	- "id": 612,
	+ "id": 615,
	"link": "/entries/Automatic_Refinement.html",
	"permalink": "/entries/Automatic_Refinement.html",
	"shortname": "Automatic_Refinement",
	"title": "Automatic Data Refinement",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 10
	},
	{
	"abstract": "This entry makes machine words and machine arithmetic available for code generation from Isabelle/HOL. It provides a common abstraction that hides the differences between the different target languages. The code generator maps these operations to the APIs of the target languages. Apart from that, we extend the available bit operations on types int and integer, and map them to the operations in the target languages.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2013-09-17",
	- "id": 613,
	+ "id": 616,
	"link": "/entries/Native_Word.html",
	"permalink": "/entries/Native_Word.html",
	"shortname": "Native_Word",
	"title": "Native Word",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 10
	},
	{
	"abstract": "This development provides a formal model of IEEE-754 floating-point arithmetic. This formalization, including formal specification of the standard and proofs of important properties of floating-point arithmetic, forms the foundation for verifying programs with floating-point computation. There is also a code generation setup for floats so that we can execute programs using this formalization in functional programming languages.",
	"authors": [
	"Lei Yu"
	],
	"date": "2013-07-27",
	- "id": 614,
	+ "id": 617,
	"link": "/entries/IEEE_Floating_Point.html",
	"permalink": "/entries/IEEE_Floating_Point.html",
	"shortname": "IEEE_Floating_Point",
	"title": "A Formal Model of IEEE Floating Point Arithmetic",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 2
	},
	{
	"abstract": "In 1927, Lehmer presented criterions for primality, based on the converse of Fermat's litte theorem. This work formalizes the second criterion from Lehmer's paper, a necessary and sufficient condition for primality. \u003cp\u003e As a side product we formalize some properties of Euler's phi-function, the notion of the order of an element of a group, and the cyclicity of the multiplicative group of a finite field.",
	"authors": [
	"Simon Wimmer",
	"Lars Noschinski"
	],
	"date": "2013-07-22",
	- "id": 615,
	+ "id": 618,
	"link": "/entries/Lehmer.html",
	"permalink": "/entries/Lehmer.html",
	"shortname": "Lehmer",
	"title": "Lehmer's Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 1
	},
	{
	"abstract": "In 1975, Pratt introduced a proof system for certifying primes. He showed that a number \u003ci\u003ep\u003c/i\u003e is prime iff a primality certificate for \u003ci\u003ep\u003c/i\u003e exists. By showing a logarithmic upper bound on the length of the certificates in size of the prime number, he concluded that the decision problem for prime numbers is in NP. This work formalizes soundness and completeness of Pratt's proof system as well as an upper bound for the size of the certificate.",
	"authors": [
	"Simon Wimmer",
	"Lars Noschinski"
	],
	"date": "2013-07-22",
	- "id": 616,
	+ "id": 619,
	"link": "/entries/Pratt_Certificate.html",
	"permalink": "/entries/Pratt_Certificate.html",
	"shortname": "Pratt_Certificate",
	"title": "Pratt's Primality Certificates",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 2
	},
	{
	"abstract": "This development provides a formalization of undirected graphs and simple graphs, which are based on Benedikt Nordhoff and Peter Lammich's simple formalization of labelled directed graphs in the archive. Then, with our formalization of graphs, we show both necessary and sufficient conditions for Eulerian trails and circuits as well as the fact that the Königsberg Bridge Problem does not have a solution. In addition, we show the Friendship Theorem in simple graphs.",
	"authors": [
	"Wenda Li"
	],
	"date": "2013-07-19",
	- "id": 617,
	+ "id": 620,
	"link": "/entries/Koenigsberg_Friendship.html",
	"permalink": "/entries/Koenigsberg_Friendship.html",
	"shortname": "Koenigsberg_Friendship",
	"title": "The Königsberg Bridge Problem and the Friendship Theorem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a formalization of the soundness and completeness properties for various efficient encodings of sorts in unsorted first-order logic used by Isabelle's Sledgehammer tool. \u003cp\u003e Essentially, the encodings proceed as follows: a many-sorted problem is decorated with (as few as possible) tags or guards that make the problem monotonic; then sorts can be soundly erased. \u003cp\u003e The development employs a formalization of many-sorted first-order logic in clausal form (clauses, structures and the basic properties of the satisfaction relation), which could be of interest as the starting point for other formalizations of first-order logic metatheory.",
	"authors": [
	"Jasmin Christian Blanchette",
	"Andrei Popescu"
	],
	"date": "2013-06-27",
	- "id": 618,
	+ "id": 621,
	"link": "/entries/Sort_Encodings.html",
	"permalink": "/entries/Sort_Encodings.html",
	"shortname": "Sort_Encodings",
	"title": "Sound and Complete Sort Encodings for First-Order Logic",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory is split into two sections. In the first section, we give a formal proof that a well-known axiomatic characterization of the single-source shortest path problem is correct. Namely, we prove that in a directed graph with a non-negative cost function on the edges the single-source shortest path function is the only function that satisfies a set of four axioms. In the second section, we give a formal proof of the correctness of an axiomatic characterization of the single-source shortest path problem for directed graphs with general cost functions. The axioms here are more involved because we have to account for potential negative cycles in the graph. The axioms are summarized in three Isabelle locales.",
	"authors": [
	"Christine Rizkallah"
	],
	"date": "2013-05-22",
	- "id": 619,
	+ "id": 622,
	"link": "/entries/ShortestPath.html",
	"permalink": "/entries/ShortestPath.html",
	"shortname": "ShortestPath",
	"title": "An Axiomatic Characterization of the Single-Source Shortest Path Problem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This development provides a formalization of directed graphs, supporting (labelled) multi-edges and infinite graphs. A polymorphic edge type allows edges to be treated as pairs of vertices, if multi-edges are not required. Formalized properties are i.a. walks (and related concepts), connectedness and subgraphs and basic properties of isomorphisms. \u003cp\u003e This formalization is used to prove characterizations of Euler Trails, Shortest Paths and Kuratowski subgraphs.",
	"authors": [
	"Lars Noschinski"
	],
	"date": "2013-04-28",
	- "id": 620,
	+ "id": 623,
	"link": "/entries/Graph_Theory.html",
	"permalink": "/entries/Graph_Theory.html",
	"shortname": "Graph_Theory",
	"title": "Graph Theory",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 7
	},
	{
	"abstract": "This development provides a framework for container types like sets and maps such that generated code implements these containers with different (efficient) data structures. Thanks to type classes and refinement during code generation, this light-weight approach can seamlessly replace Isabelle's default setup for code generation. Heuristics automatically pick one of the available data structures depending on the type of elements to be stored, but users can also choose on their own. The extensible design permits to add more implementations at any time. \u003cp\u003e To support arbitrary nesting of sets, we define a linear order on sets based on a linear order of the elements and provide efficient implementations. It even allows to compare complements with non-complements.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2013-04-15",
	- "id": 621,
	+ "id": 624,
	"link": "/entries/Containers.html",
	"permalink": "/entries/Containers.html",
	"shortname": "Containers",
	"title": "Light-weight Containers",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 9
	},
	{
	"abstract": "\u003cp\u003eDealing with binders, renaming of bound variables, capture-avoiding substitution, etc., is very often a major problem in formal proofs, especially in proofs by structural and rule induction. Nominal Isabelle is designed to make such proofs easy to formalise: it provides an infrastructure for declaring nominal datatypes (that is alpha-equivalence classes) and for defining functions over them by structural recursion. It also provides induction principles that have Barendregt’s variable convention already built in. \u003c/p\u003e\u003cp\u003e This entry can be used as a more advanced replacement for HOL/Nominal in the Isabelle distribution. \u003c/p\u003e",
	"authors": [
	"Christian Urban",
	"Stefan Berghofer",
	"Cezary Kaliszyk"
	],
	"date": "2013-02-21",
	- "id": 622,
	+ "id": 625,
	"link": "/entries/Nominal2.html",
	"permalink": "/entries/Nominal2.html",
	"shortname": "Nominal2",
	"title": "Nominal 2",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 8
	},
	{
	"abstract": "In his seminal paper \"Natural Semantics for Lazy Evaluation\", John Launchbury proves his semantics correct with respect to a denotational semantics, and outlines an adequacy proof. We have formalized both semantics and machine-checked the correctness proof, clarifying some details. Furthermore, we provide a new and more direct adequacy proof that does not require intermediate operational semantics.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2013-01-31",
	- "id": 623,
	+ "id": 626,
	"link": "/entries/Launchbury.html",
	"permalink": "/entries/Launchbury.html",
	"shortname": "Launchbury",
	"title": "The Correctness of Launchbury's Natural Semantics for Lazy Evaluation",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi",
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi",
	"Computer science/Semantics and reasoning"
	],
	"used_by": 1
	},
	{
	"abstract": "This document concerns the theory of ribbon proofs: a diagrammatic proof system, based on separation logic, for verifying program correctness. We include the syntax, proof rules, and soundness results for two alternative formalisations of ribbon proofs. \u003cp\u003e Compared to traditional proof outlines, ribbon proofs emphasise the structure of a proof, so are intelligible and pedagogical. Because they contain less redundancy than proof outlines, and allow each proof step to be checked locally, they may be more scalable. Where proof outlines are cumbersome to modify, ribbon proofs can be visually manoeuvred to yield proofs of variant programs.",
	"authors": [
	"John Wickerson"
	],
	"date": "2013-01-19",
	- "id": 624,
	+ "id": 627,
	"link": "/entries/Ribbon_Proofs.html",
	"permalink": "/entries/Ribbon_Proofs.html",
	"shortname": "Ribbon_Proofs",
	"title": "Ribbon Proofs",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "In this contribution, we present some formalizations based on the HOL-Multivariate-Analysis session of Isabelle. Firstly, a generalization of several theorems of such library are presented. Secondly, some definitions and proofs involving Linear Algebra and the four fundamental subspaces of a matrix are shown. Finally, we present a proof of the result known in Linear Algebra as the “Rank-Nullity Theorem”, which states that, given any linear map f from a finite dimensional vector space V to a vector space W, then the dimension of V is equal to the dimension of the kernel of f (which is a subspace of V) and the dimension of the range of f (which is a subspace of W). The proof presented here is based on the one given by Sheldon Axler in his book \u003ci\u003eLinear Algebra Done Right\u003c/i\u003e. As a corollary of the previous theorem, and taking advantage of the relationship between linear maps and matrices, we prove that, for every matrix A (which has associated a linear map between finite dimensional vector spaces), the sum of its null space and its column space (which is equal to the range of the linear map) is equal to the number of columns of A.",
	"authors": [
	"Jose Divasón",
	"Jesús Aransay"
	],
	"date": "2013-01-16",
	- "id": 625,
	+ "id": 628,
	"link": "/entries/Rank_Nullity_Theorem.html",
	"permalink": "/entries/Rank_Nullity_Theorem.html",
	"shortname": "Rank_Nullity_Theorem",
	"title": "Rank-Nullity Theorem in Linear Algebra",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 4
	},
	{
	"abstract": "These files contain a formalisation of variants of Kleene algebras and their most important models as axiomatic type classes in Isabelle/HOL. Kleene algebras are foundational structures in computing with applications ranging from automata and language theory to computational modeling, program construction and verification. \u003cp\u003e We start with formalising dioids, which are additively idempotent semirings, and expand them by axiomatisations of the Kleene star for finite iteration and an omega operation for infinite iteration. We show that powersets over a given monoid, (regular) languages, sets of paths in a graph, sets of computation traces, binary relations and formal power series form Kleene algebras, and consider further models based on lattices, max-plus semirings and min-plus semirings. We also demonstrate that dioids are closed under the formation of matrices (proofs for Kleene algebras remain to be completed). \u003cp\u003e On the one hand we have aimed at a reference formalisation of variants of Kleene algebras that covers a wide range of variants and the core theorems in a structured and modular way and provides readable proofs at text book level. On the other hand, we intend to use this algebraic hierarchy and its models as a generic algebraic middle-layer from which programming applications can quickly be explored, implemented and verified.",
	"authors": [
	"Alasdair Armstrong",
	"Georg Struth",
	"Tjark Weber"
	],
	"date": "2013-01-15",
	- "id": 626,
	+ "id": 629,
	"link": "/entries/Kleene_Algebra.html",
	"permalink": "/entries/Kleene_Algebra.html",
	"shortname": "Kleene_Algebra",
	"title": "Kleene Algebra",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/automata-and-formal-languages",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Automata and formal languages",
	"Mathematics/Algebra"
	],
	- "used_by": 6
	+ "used_by": 7
	},
	{
	"abstract": "We implement the Babylonian method to compute n-th roots of numbers. We provide precise algorithms for naturals, integers and rationals, and offer an approximation algorithm for square roots over linear ordered fields. Moreover, there are precise algorithms to compute the floor and the ceiling of n-th roots.",
	"authors": [
	"René Thiemann"
	],
	"date": "2013-01-03",
	- "id": 627,
	+ "id": 630,
	"link": "/entries/Sqrt_Babylonian.html",
	"permalink": "/entries/Sqrt_Babylonian.html",
	"shortname": "Sqrt_Babylonian",
	"title": "Computing N-th Roots using the Babylonian Method",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 4
	},
	{
	"abstract": "We provide a framework for separation-logic based correctness proofs of Imperative HOL programs. Our framework comes with a set of proof methods to automate canonical tasks such as verification condition generation and frame inference. Moreover, we provide a set of examples that show the applicability of our framework. The examples include algorithms on lists, hash-tables, and union-find trees. We also provide abstract interfaces for lists, maps, and sets, that allow to develop generic imperative algorithms and use data-refinement techniques. \u003cbr\u003e As we target Imperative HOL, our programs can be translated to efficiently executable code in various target languages, including ML, OCaml, Haskell, and Scala.",
	"authors": [
	"Peter Lammich",
	"Rene Meis"
	],
	"date": "2012-11-14",
	- "id": 628,
	+ "id": 631,
	"link": "/entries/Separation_Logic_Imperative_HOL.html",
	"permalink": "/entries/Separation_Logic_Imperative_HOL.html",
	"shortname": "Separation_Logic_Imperative_HOL",
	"title": "A Separation Logic Framework for Imperative HOL",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 2
	},
	{
	"abstract": "A proof of the open induction schema based on J.-C. Raoult, Proving open properties by induction, \u003ci\u003eInformation Processing Letters\u003c/i\u003e 29, 1988, pp.19-23. \u003cp\u003eThis research was supported by the Austrian Science Fund (FWF): J3202.\u003c/p\u003e",
	"authors": [
	"Mizuhito Ogawa",
	"Christian Sternagel"
	],
	"date": "2012-11-02",
	- "id": 629,
	+ "id": 632,
	"link": "/entries/Open_Induction.html",
	"permalink": "/entries/Open_Induction.html",
	"shortname": "Open_Induction",
	"title": "Open Induction",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 4
	},
	{
	"abstract": "Tarski's axioms of plane geometry are formalized and, using the standard real Cartesian model, shown to be consistent. A substantial theory of the projective plane is developed. Building on this theory, the Klein-Beltrami model of the hyperbolic plane is defined and shown to satisfy all of Tarski's axioms except his Euclidean axiom; thus Tarski's Euclidean axiom is shown to be independent of his other axioms of plane geometry. \u003cp\u003e An earlier version of this work was the subject of the author's \u003ca href=\"http://researcharchive.vuw.ac.nz/handle/10063/2315\"\u003eMSc thesis\u003c/a\u003e, which contains natural-language explanations of some of the more interesting proofs.",
	"authors": [
	"T. J. M. Makarios"
	],
	"date": "2012-10-30",
	- "id": 630,
	+ "id": 633,
	"link": "/entries/Tarskis_Geometry.html",
	"permalink": "/entries/Tarskis_Geometry.html",
	"shortname": "Tarskis_Geometry",
	"title": "The independence of Tarski's Euclidean axiom",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "A proof of Bondy's theorem following B. Bollabas, Combinatorics, 1986, Cambridge University Press.",
	"authors": [
	"Jeremy Avigad",
	"Stefan Hetzl"
	],
	"date": "2012-10-27",
	- "id": 631,
	+ "id": 634,
	"link": "/entries/Bondy.html",
	"permalink": "/entries/Bondy.html",
	"shortname": "Bondy",
	"title": "Bondy's Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize a wide variety of Volpano/Smith-style noninterference notions for a while language with parallel composition. We systematize and classify these notions according to compositionality w.r.t. the language constructs. Compositionality yields sound syntactic criteria (a.k.a. type systems) in a uniform way. \u003cp\u003e An \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs/cpp12.html\"\u003earticle\u003c/a\u003e about these proofs is published in the proceedings of the conference Certified Programs and Proofs 2012.",
	"authors": [
	"Andrei Popescu",
	"Johannes Hölzl"
	],
	"date": "2012-09-10",
	- "id": 632,
	+ "id": 635,
	"link": "/entries/Possibilistic_Noninterference.html",
	"permalink": "/entries/Possibilistic_Noninterference.html",
	"shortname": "Possibilistic_Noninterference",
	"title": "Possibilistic Noninterference",
	"topic_links": [
	"computer-science/security",
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Security",
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a framework for registering automatic methods to derive class instances of datatypes, as it is possible using Haskell's “\u003ctt\u003ederiving Ord, Show, ...\u003c/tt\u003e” feature. \u003cp\u003e We further implemented such automatic methods to derive (linear) orders or hash-functions which are required in the Isabelle Collection Framework. Moreover, for the tactic of Huffman and Krauss to show that a datatype is countable, we implemented a wrapper so that this tactic becomes accessible in our framework. \u003cp\u003e Our formalization was performed as part of the \u003ca href=\"http://cl-informatik.uibk.ac.at/software/ceta\"\u003eIsaFoR/CeTA\u003c/a\u003e project. With our new tactic we could completely remove tedious proofs for linear orders of two datatypes. \u003cp\u003e This development is aimed at datatypes generated by the \"\u003ctt\u003eold_datatype\u003c/tt\u003e\" command.",
	"authors": [
	"René Thiemann"
	],
	"date": "2012-08-07",
	- "id": 633,
	+ "id": 636,
	"link": "/entries/Datatype_Order_Generator.html",
	"permalink": "/entries/Datatype_Order_Generator.html",
	"shortname": "Datatype_Order_Generator",
	"title": "Generating linear orders for datatypes",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 5
	},
	{
	"abstract": "Squaring the circle, doubling the cube and trisecting an angle, using a compass and straightedge alone, are classic unsolved problems first posed by the ancient Greeks. All three problems were proved to be impossible in the 19th century. The following document presents the proof of the impossibility of solving the latter two problems using Isabelle/HOL, following a proof by Carrega. The proof uses elementary methods: no Galois theory or field extensions. The set of points constructible using a compass and straightedge is defined inductively. Radical expressions, which involve only square roots and arithmetic of rational numbers, are defined, and we find that all constructive points have radical coordinates. Finally, doubling the cube and trisecting certain angles requires solving certain cubic equations that can be proved to have no rational roots. The Isabelle proofs require a great many detailed calculations.",
	"authors": [
	"Ralph Romanos",
	"Lawrence C. Paulson"
	],
	"date": "2012-08-05",
	- "id": 634,
	+ "id": 637,
	"link": "/entries/Impossible_Geometry.html",
	"permalink": "/entries/Impossible_Geometry.html",
	"shortname": "Impossible_Geometry",
	"title": "Proving the Impossibility of Trisecting an Angle and Doubling the Cube",
	"topic_links": [
	"mathematics/algebra",
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Algebra",
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "Distributed computing is inherently based on replication, promising increased tolerance to failures of individual computing nodes or communication channels. Realizing this promise, however, involves quite subtle algorithmic mechanisms, and requires precise statements about the kinds and numbers of faults that an algorithm tolerates (such as process crashes, communication faults or corrupted values). The landmark theorem due to Fischer, Lynch, and Paterson shows that it is impossible to achieve Consensus among N asynchronously communicating nodes in the presence of even a single permanent failure. Existing solutions must rely on assumptions of \"partial synchrony\". \u003cp\u003e Indeed, there have been numerous misunderstandings on what exactly a given algorithm is supposed to realize in what kinds of environments. Moreover, the abundance of subtly different computational models complicates comparisons between different algorithms. Charron-Bost and Schiper introduced the Heard-Of model for representing algorithms and failure assumptions in a uniform framework, simplifying comparisons between algorithms. \u003cp\u003e In this contribution, we represent the Heard-Of model in Isabelle/HOL. We define two semantics of runs of algorithms with different unit of atomicity and relate these through a reduction theorem that allows us to verify algorithms in the coarse-grained semantics (where proofs are easier) and infer their correctness for the fine-grained one (which corresponds to actual executions). We instantiate the framework by verifying six Consensus algorithms that differ in the underlying algorithmic mechanisms and the kinds of faults they tolerate.",
	"authors": [
	"Henri Debrat",
	"Stephan Merz"
	],
	"date": "2012-07-27",
	- "id": 635,
	+ "id": 638,
	"link": "/entries/Heard_Of.html",
	"permalink": "/entries/Heard_Of.html",
	"shortname": "Heard_Of",
	"title": "Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 1
	},
	{
	"abstract": "We apply Andy Pitts's methods of defining relations over domains to several classical results in the literature. We show that the Y combinator coincides with the domain-theoretic fixpoint operator, that parallel-or and the Plotkin existential are not definable in PCF, that the continuation semantics for PCF coincides with the direct semantics, and that our domain-theoretic semantics for PCF is adequate for reasoning about contextual equivalence in an operational semantics. Our version of PCF is untyped and has both strict and non-strict function abstractions. The development is carried out in HOLCF.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2012-07-01",
	- "id": 636,
	+ "id": 639,
	"link": "/entries/PCF.html",
	"permalink": "/entries/PCF.html",
	"shortname": "PCF",
	"title": "Logical Relations for PCF",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "These theories contain a formalization of first class type constructors and axiomatic constructor classes for HOLCF. This work is described in detail in the ICFP 2012 paper \u003ci\u003eFormal Verification of Monad Transformers\u003c/i\u003e by the author. The formalization is a revised and updated version of earlier joint work with Matthews and White. \u003cP\u003e Based on the hierarchy of type classes in Haskell, we define classes for functors, monads, monad-plus, etc. Each one includes all the standard laws as axioms. We also provide a new user command, tycondef, for defining new type constructors in HOLCF. Using tycondef, we instantiate the type class hierarchy with various monads and monad transformers.",
	"authors": [
	"Brian Huffman"
	],
	"date": "2012-06-26",
	- "id": 637,
	+ "id": 640,
	"link": "/entries/Tycon.html",
	"permalink": "/entries/Tycon.html",
	"shortname": "Tycon",
	"title": "Type Constructor Classes and Monad Transformers",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise a large portion of CCS as described in Milner's book 'Communication and Concurrency' using the nominal datatype package in Isabelle. Our results include many of the standard theorems of bisimulation equivalence and congruence, for both weak and strong versions. One main goal of this formalisation is to keep the machine-checked proofs as close to their pen-and-paper counterpart as possible. \u003cp\u003e This entry is described in detail in \u003ca href=\"http://www.itu.dk/people/jebe/files/thesis.pdf\"\u003eBengtson's thesis\u003c/a\u003e.",
	"authors": [
	"Jesper Bengtson"
	],
	"date": "2012-05-29",
	- "id": 638,
	+ "id": 641,
	"link": "/entries/CCS.html",
	"permalink": "/entries/CCS.html",
	"shortname": "CCS",
	"title": "CCS in nominal logic",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "Psi-calculi are extensions of the pi-calculus, accommodating arbitrary nominal datatypes to represent not only data but also communication channels, assertions and conditions, giving it an expressive power beyond the applied pi-calculus and the concurrent constraint pi-calculus. \u003cp\u003e We have formalised psi-calculi in the interactive theorem prover Isabelle using its nominal datatype package. One distinctive feature is that the framework needs to treat binding sequences, as opposed to single binders, in an efficient way. While different methods for formalising single binder calculi have been proposed over the last decades, representations for such binding sequences are not very well explored. \u003cp\u003e The main effort in the formalisation is to keep the machine checked proofs as close to their pen-and-paper counterparts as possible. This includes treating all binding sequences as atomic elements, and creating custom induction and inversion rules that to remove the bulk of manual alpha-conversions. \u003cp\u003e This entry is described in detail in \u003ca href=\"http://www.itu.dk/people/jebe/files/thesis.pdf\"\u003eBengtson's thesis\u003c/a\u003e.",
	"authors": [
	"Jesper Bengtson"
	],
	"date": "2012-05-29",
	- "id": 639,
	+ "id": 642,
	"link": "/entries/Psi_Calculi.html",
	"permalink": "/entries/Psi_Calculi.html",
	"shortname": "Psi_Calculi",
	"title": "Psi-calculi in Isabelle",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalise the pi-calculus using the nominal datatype package, based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a uniform manner. We thus provide one of the most extensive formalisations of a the pi-calculus ever done inside a theorem prover. \u003cp\u003e A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic. \u003cp\u003e This entry is described in detail in \u003ca href=\"http://www.itu.dk/people/jebe/files/thesis.pdf\"\u003eBengtson's thesis\u003c/a\u003e.",
	"authors": [
	"Jesper Bengtson"
	],
	"date": "2012-05-29",
	- "id": 640,
	+ "id": 643,
	"link": "/entries/Pi_Calculus.html",
	"permalink": "/entries/Pi_Calculus.html",
	"shortname": "Pi_Calculus",
	"title": "The pi-calculus in nominal logic",
	"topic_links": [
	"computer-science/concurrency/process-calculi"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "The Circus specification language combines elements for complex data and behavior specifications, using an integration of Z and CSP with a refinement calculus. Its semantics is based on Hoare and He's Unifying Theories of Programming (UTP). Isabelle/Circus is a formalization of the UTP and the Circus language in Isabelle/HOL. It contains proof rules and tactic support that allows for proofs of refinement for Circus processes (involving both data and behavioral aspects). \u003cp\u003e The Isabelle/Circus environment supports a syntax for the semantic definitions which is close to textbook presentations of Circus. This article contains an extended version of corresponding VSTTE Paper together with the complete formal development of its underlying commented theories.",
	"authors": [
	"Abderrahmane Feliachi",
	"Burkhart Wolff",
	"Marie-Claude Gaudel"
	],
	"date": "2012-05-27",
	- "id": 641,
	+ "id": 644,
	"link": "/entries/Circus.html",
	"permalink": "/entries/Circus.html",
	"shortname": "Circus",
	"title": "Isabelle/Circus",
	"topic_links": [
	"computer-science/concurrency/process-calculi",
	"computer-science/system-description-languages"
	],
	"topics": [
	"Computer science/Concurrency/Process calculi",
	"Computer science/System description languages"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a generic type class implementation of separation algebra for Isabelle/HOL as well as lemmas and generic tactics which can be used directly for any instantiation of the type class. \u003cP\u003e The ex directory contains example instantiations that include structures such as a heap or virtual memory. \u003cP\u003e The abstract separation algebra is based upon \"Abstract Separation Logic\" by Calcagno et al. These theories are also the basis of the ITP 2012 rough diamond \"Mechanised Separation Algebra\" by the authors. \u003cP\u003e The aim of this work is to support and significantly reduce the effort for future separation logic developments in Isabelle/HOL by factoring out the part of separation logic that can be treated abstractly once and for all. This includes developing typical default rule sets for reasoning as well as automated tactic support for separation logic.",
	"authors": [
	"Gerwin Klein",
	"Rafal Kolanski",
	"Andrew Boyton"
	],
	"date": "2012-05-11",
	- "id": 642,
	+ "id": 645,
	"link": "/entries/Separation_Algebra.html",
	"permalink": "/entries/Separation_Algebra.html",
	"shortname": "Separation_Algebra",
	"title": "Separation Algebra",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 3
	},
	{
	"abstract": "\u003cp\u003eTwo omega-sequences are stuttering equivalent if they differ only by finite repetitions of elements. Stuttering equivalence is a fundamental concept in the theory of concurrent and distributed systems. Notably, Lamport argues that refinement notions for such systems should be insensitive to finite stuttering. Peled and Wilke showed that all PLTL (propositional linear-time temporal logic) properties that are insensitive to stuttering equivalence can be expressed without the next-time operator. Stuttering equivalence is also important for certain verification techniques such as partial-order reduction for model checking.\u003c/p\u003e \u003cp\u003eWe formalize stuttering equivalence in Isabelle/HOL. Our development relies on the notion of stuttering sampling functions that may skip blocks of identical sequence elements. We also encode PLTL and prove the theorem due to Peled and Wilke.\u003c/p\u003e",
	"authors": [
	"Stephan Merz"
	],
	"date": "2012-05-07",
	- "id": 643,
	+ "id": 646,
	"link": "/entries/Stuttering_Equivalence.html",
	"permalink": "/entries/Stuttering_Equivalence.html",
	"shortname": "Stuttering_Equivalence",
	"title": "Stuttering Equivalence",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 5
	},
	{
	"abstract": "This document contains the full theory files accompanying article \u003ci\u003eInductive Study of Confidentiality --- for Everyone\u003c/i\u003e in \u003ci\u003eFormal Aspects of Computing\u003c/i\u003e. They aim at an illustrative and didactic presentation of the Inductive Method of protocol analysis, focusing on the treatment of one of the main goals of security protocols: confidentiality against a threat model. The treatment of confidentiality, which in fact forms a key aspect of all protocol analysis tools, has been found cryptic by many learners of the Inductive Method, hence the motivation for this work. The theory files in this document guide the reader step by step towards design and proof of significant confidentiality theorems. These are developed against two threat models, the standard Dolev-Yao and a more audacious one, the General Attacker, which turns out to be particularly useful also for teaching purposes.",
	"authors": [
	"Giampaolo Bella"
	],
	"date": "2012-05-02",
	- "id": 644,
	+ "id": 647,
	"link": "/entries/Inductive_Confidentiality.html",
	"permalink": "/entries/Inductive_Confidentiality.html",
	"shortname": "Inductive_Confidentiality",
	"title": "Inductive Study of Confidentiality",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003eSession Ordinary-Differential-Equations formalizes ordinary differential equations (ODEs) and initial value problems. This work comprises proofs for local and global existence of unique solutions (Picard-Lindelöf theorem). Moreover, it contains a formalization of the (continuous or even differentiable) dependency of the flow on initial conditions as the \u003ci\u003eflow\u003c/i\u003e of ODEs.\u003c/p\u003e \u003cp\u003e Not in the generated document are the following sessions: \u003cul\u003e \u003cli\u003e HOL-ODE-Numerics: Rigorous numerical algorithms for computing enclosures of solutions based on Runge-Kutta methods and affine arithmetic. Reachability analysis with splitting and reduction at hyperplanes.\u003c/li\u003e \u003cli\u003e HOL-ODE-Examples: Applications of the numerical algorithms to concrete systems of ODEs.\u003c/li\u003e \u003cli\u003e Lorenz_C0, Lorenz_C1: Verified algorithms for checking C1-information according to Tucker's proof, computation of C0-information.\u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e",
	"authors": [
	"Fabian Immler",
	"Johannes Hölzl"
	],
	"date": "2012-04-26",
	- "id": 645,
	+ "id": 648,
	"link": "/entries/Ordinary_Differential_Equations.html",
	"permalink": "/entries/Ordinary_Differential_Equations.html",
	"shortname": "Ordinary_Differential_Equations",
	"title": "Ordinary Differential Equations",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 3
	},
	{
	"abstract": "Based on Isabelle/HOL's type class for preorders, we introduce a type class for well-quasi-orders (wqo) which is characterized by the absence of \"bad\" sequences (our proofs are along the lines of the proof of Nash-Williams, from which we also borrow terminology). Our main results are instantiations for the product type, the list type, and a type of finite trees, which (almost) directly follow from our proofs of (1) Dickson's Lemma, (2) Higman's Lemma, and (3) Kruskal's Tree Theorem. More concretely: \u003cul\u003e \u003cli\u003eIf the sets A and B are wqo then their Cartesian product is wqo.\u003c/li\u003e \u003cli\u003eIf the set A is wqo then the set of finite lists over A is wqo.\u003c/li\u003e \u003cli\u003eIf the set A is wqo then the set of finite trees over A is wqo.\u003c/li\u003e \u003c/ul\u003e The research was funded by the Austrian Science Fund (FWF): J3202.",
	"authors": [
	"Christian Sternagel"
	],
	"date": "2012-04-13",
	- "id": 646,
	+ "id": 649,
	"link": "/entries/Well_Quasi_Orders.html",
	"permalink": "/entries/Well_Quasi_Orders.html",
	"shortname": "Well_Quasi_Orders",
	"title": "Well-Quasi-Orders",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 5
	},
	{
	"abstract": "We define the Abortable Linearizable Module automaton (ALM for short) and prove its key composition property using the IOA theory of HOLCF. The ALM is at the heart of the Speculative Linearizability framework. This framework simplifies devising correct speculative algorithms by enabling their decomposition into independent modules that can be analyzed and proved correct in isolation. It is particularly useful when working in a distributed environment, where the need to tolerate faults and asynchrony has made current monolithic protocols so intricate that it is no longer tractable to check their correctness. Our theory contains a typical example of a refinement proof in the I/O-automata framework of Lynch and Tuttle.",
	"authors": [
	"Rachid Guerraoui",
	"Viktor Kuncak",
	"Giuliano Losa"
	],
	"date": "2012-03-01",
	- "id": 647,
	+ "id": 650,
	"link": "/entries/Abortable_Linearizable_Modules.html",
	"permalink": "/entries/Abortable_Linearizable_Modules.html",
	"shortname": "Abortable_Linearizable_Modules",
	"title": "Abortable Linearizable Modules",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e We provide a generic work-list algorithm to compute the (reflexive-)transitive closure of relations where only successors of newly detected states are generated. In contrast to our previous work, the relations do not have to be finite, but each element must only have finitely many (indirect) successors. Moreover, a subsumption relation can be used instead of pure equality. An executable variant of the algorithm is available where the generic operations are instantiated with list operations. \u003c/p\u003e\u003cp\u003e This formalization was performed as part of the IsaFoR/CeTA project, and it has been used to certify size-change termination proofs where large transitive closures have to be computed. \u003c/p\u003e",
	"authors": [
	"René Thiemann"
	],
	"date": "2012-02-29",
	- "id": 648,
	+ "id": 651,
	"link": "/entries/Transitive-Closure-II.html",
	"permalink": "/entries/Transitive-Closure-II.html",
	"shortname": "Transitive-Closure-II",
	"title": "Executable Transitive Closures",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "This works presents a formalization of the Girth-Chromatic number theorem in graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. The proof uses the theory of Random Graphs to prove the existence with probabilistic arguments.",
	"authors": [
	"Lars Noschinski"
	],
	"date": "2012-02-06",
	- "id": 649,
	+ "id": 652,
	"link": "/entries/Girth_Chromatic.html",
	"permalink": "/entries/Girth_Chromatic.html",
	"shortname": "Girth_Chromatic",
	"title": "A Probabilistic Proof of the Girth-Chromatic Number Theorem",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 5
	},
	{
	"abstract": "We implement and prove correct Dijkstra's algorithm for the single source shortest path problem, conceived in 1956 by E. Dijkstra. The algorithm is implemented using the data refinement framework for monadic, nondeterministic programs. An efficient implementation is derived using data structures from the Isabelle Collection Framework.",
	"authors": [
	"Benedikt Nordhoff",
	"Peter Lammich"
	],
	"date": "2012-01-30",
	- "id": 650,
	+ "id": 653,
	"link": "/entries/Dijkstra_Shortest_Path.html",
	"permalink": "/entries/Dijkstra_Shortest_Path.html",
	"shortname": "Dijkstra_Shortest_Path",
	"title": "Dijkstra's Shortest Path Algorithm",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 3
	},
	{
	"abstract": "We provide a framework for program and data refinement in Isabelle/HOL. The framework is based on a nondeterminism-monad with assertions, i.e., the monad carries a set of results or an assertion failure. Recursion is expressed by fixed points. For convenience, we also provide while and foreach combinators. \u003cp\u003e The framework provides tools to automatize canonical tasks, such as verification condition generation, finding appropriate data refinement relations, and refine an executable program to a form that is accepted by the Isabelle/HOL code generator. \u003cp\u003e This submission comes with a collection of examples and a user-guide, illustrating the usage of the framework.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2012-01-30",
	- "id": 651,
	+ "id": 654,
	"link": "/entries/Refine_Monadic.html",
	"permalink": "/entries/Refine_Monadic.html",
	"shortname": "Refine_Monadic",
	"title": "Refinement for Monadic Programs",
	"topic_links": [
	"computer-science/semantics-and-reasoning"
	],
	"topics": [
	"Computer science/Semantics and reasoning"
	],
	"used_by": 3
	},
	{
	"abstract": "This is a formalization of Markov models in Isabelle/HOL. It builds on Isabelle's probability theory. The available models are currently Discrete-Time Markov Chains and a extensions of them with rewards. \u003cp\u003e As application of these models we formalize probabilistic model checking of pCTL formulas, analysis of IPv4 address allocation in ZeroConf and an analysis of the anonymity of the Crowds protocol. \u003ca href=\"http://arxiv.org/abs/1212.3870\"\u003eSee here for the corresponding paper.\u003c/a\u003e",
	"authors": [
	"Johannes Hölzl",
	"Tobias Nipkow"
	],
	"date": "2012-01-03",
	- "id": 652,
	+ "id": 655,
	"link": "/entries/Markov_Models.html",
	"permalink": "/entries/Markov_Models.html",
	"shortname": "Markov_Models",
	"title": "Markov Models",
	"topic_links": [
	"mathematics/probability-theory",
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Mathematics/Probability theory",
	"Computer science/Automata and formal languages"
	],
	"used_by": 4
	},
	{
	"abstract": "We mechanise the logic TLA* \u003ca href=\"http://www.springerlink.com/content/ax3qk557qkdyt7n6/\"\u003e[Merz 1999]\u003c/a\u003e, an extension of Lamport's Temporal Logic of Actions (TLA) \u003ca href=\"http://dl.acm.org/citation.cfm?doid=177492.177726\"\u003e[Lamport 1994]\u003c/a\u003e for specifying and reasoning about concurrent and reactive systems. Aiming at a framework for mechanising] the verification of TLA (or TLA) specifications, this contribution reuses some elements from a previous axiomatic encoding of TLA in Isabelle/HOL by the second author [Merz 1998], which has been part of the Isabelle distribution. In contrast to that previous work, we give here a shallow, definitional embedding, with the following highlights: \u003cul\u003e \u003cli\u003ea theory of infinite sequences, including a formalisation of the concepts of stuttering invariance central to TLA and TLA; \u003cli\u003ea definition of the semantics of TLA, which extends TLA by a mutually-recursive definition of formulas and pre-formulas, generalising TLA action formulas; \u003cli\u003ea substantial set of derived proof rules, including the TLA axioms and Lamport's proof rules for system verification; \u003cli\u003ea set of examples illustrating the usage of Isabelle/TLA* for reasoning about systems. \u003c/ul\u003e Note that this work is unrelated to the ongoing development of a proof system for the specification language TLA+, which includes an encoding of TLA+ as a new Isabelle object logic \u003ca href=\"http://www.springerlink.com/content/354026160p14j175/\"\u003e[Chaudhuri et al 2010]\u003c/a\u003e.",
	"authors": [
	"Gudmund Grov",
	"Stephan Merz"
	],
	"date": "2011-11-19",
	- "id": 653,
	+ "id": 656,
	"link": "/entries/TLA.html",
	"permalink": "/entries/TLA.html",
	"shortname": "TLA",
	"title": "A Definitional Encoding of TLA* in Isabelle/HOL",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a formalization of the mergesort algorithm as used in GHC's Data.List module, proving correctness and stability. Furthermore, experimental data suggests that generated (Haskell-)code for this algorithm is much faster than for previous algorithms available in the Isabelle distribution.",
	"authors": [
	"Christian Sternagel"
	],
	"date": "2011-11-09",
	- "id": 654,
	+ "id": 657,
	"link": "/entries/Efficient-Mergesort.html",
	"permalink": "/entries/Efficient-Mergesort.html",
	"shortname": "Efficient-Mergesort",
	"title": "Efficient Mergesort",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 2
	},
	{
	"abstract": "Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We formalize here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of this algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).",
	"authors": [
	"Viorel Preoteasa"
	],
	"date": "2011-09-22",
	- "id": 655,
	+ "id": 658,
	"link": "/entries/MonoBoolTranAlgebra.html",
	"permalink": "/entries/MonoBoolTranAlgebra.html",
	"shortname": "MonoBoolTranAlgebra",
	"title": "Algebra of Monotonic Boolean Transformers",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 1
	},
	{
	"abstract": "This formalization introduces and collects some algebraic structures based on lattices and complete lattices for use in other developments. The structures introduced are modular, and lattice ordered groups. In addition to the results proved for the new lattices, this formalization also introduces theorems about latices and complete lattices in general.",
	"authors": [
	"Viorel Preoteasa"
	],
	"date": "2011-09-22",
	- "id": 656,
	+ "id": 659,
	"link": "/entries/LatticeProperties.html",
	"permalink": "/entries/LatticeProperties.html",
	"shortname": "LatticeProperties",
	"title": "Lattice Properties",
	"topic_links": [
	"mathematics/order"
	],
	"topics": [
	"Mathematics/Order"
	],
	"used_by": 3
	},
	{
	"abstract": "Pseudo-hoops are algebraic structures introduced by B. Bosbach under the name of complementary semigroups. In this formalization we prove some properties of pseudo-hoops and we define the basic concepts of filter and normal filter. The lattice of normal filters is isomorphic with the lattice of congruences of a pseudo-hoop. We also study some important classes of pseudo-hoops. Bounded Wajsberg pseudo-hoops are equivalent to pseudo-Wajsberg algebras and bounded basic pseudo-hoops are equivalent to pseudo-BL algebras. Some examples of pseudo-hoops are given in the last section of the formalization.",
	"authors": [
	"George Georgescu",
	"Laurentiu Leustean",
	"Viorel Preoteasa"
	],
	"date": "2011-09-22",
	- "id": 657,
	+ "id": 660,
	"link": "/entries/PseudoHoops.html",
	"permalink": "/entries/PseudoHoops.html",
	"shortname": "PseudoHoops",
	"title": "Pseudo Hoops",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "There are many proofs of the Myhill-Nerode theorem using automata. In this library we give a proof entirely based on regular expressions, since regularity of languages can be conveniently defined using regular expressions (it is more painful in HOL to define regularity in terms of automata). We prove the first direction of the Myhill-Nerode theorem by solving equational systems that involve regular expressions. For the second direction we give two proofs: one using tagging-functions and another using partial derivatives. We also establish various closure properties of regular languages. Most details of the theories are described in our ITP 2011 paper.",
	"authors": [
	"Chunhan Wu",
	"Xingyuan Zhang",
	"Christian Urban"
	],
	"date": "2011-08-26",
	- "id": 658,
	+ "id": 661,
	"link": "/entries/Myhill-Nerode.html",
	"permalink": "/entries/Myhill-Nerode.html",
	"shortname": "Myhill-Nerode",
	"title": "The Myhill-Nerode Theorem Based on Regular Expressions",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory provides a compact formulation of Gauss-Jordan elimination for matrices represented as functions. Its distinctive feature is succinctness. It is not meant for large computations.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2011-08-19",
	- "id": 659,
	+ "id": 662,
	"link": "/entries/Gauss-Jordan-Elim-Fun.html",
	"permalink": "/entries/Gauss-Jordan-Elim-Fun.html",
	"shortname": "Gauss-Jordan-Elim-Fun",
	"title": "Gauss-Jordan Elimination for Matrices Represented as Functions",
	"topic_links": [
	"computer-science/algorithms/mathematical",
	"mathematics/algebra"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical",
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "\u003cp\u003e A \u003cem\u003ematching\u003c/em\u003e in a graph \u003ci\u003eG\u003c/i\u003e is a subset \u003ci\u003eM\u003c/i\u003e of the edges of \u003ci\u003eG\u003c/i\u003e such that no two share an endpoint. A matching has maximum cardinality if its cardinality is at least as large as that of any other matching. An \u003cem\u003eodd-set cover\u003c/em\u003e \u003ci\u003eOSC\u003c/i\u003e of a graph \u003ci\u003eG\u003c/i\u003e is a labeling of the nodes of \u003ci\u003eG\u003c/i\u003e with integers such that every edge of \u003ci\u003eG\u003c/i\u003e is either incident to a node labeled 1 or connects two nodes labeled with the same number \u003ci\u003ei \u0026ge; 2\u003c/i\u003e. \u003c/p\u003e\u003cp\u003e This article proves Edmonds theorem:\u003cbr\u003e Let \u003ci\u003eM\u003c/i\u003e be a matching in a graph \u003ci\u003eG\u003c/i\u003e and let \u003ci\u003eOSC\u003c/i\u003e be an odd-set cover of \u003ci\u003eG\u003c/i\u003e. For any \u003ci\u003ei \u0026ge; 0\u003c/i\u003e, let \u003cvar\u003en(i)\u003c/var\u003e be the number of nodes labeled \u003ci\u003ei\u003c/i\u003e. If \u003ci\u003e\|M\| = n(1) + \u0026sum;\u003csub\u003ei \u0026ge; 2\u003c/sub\u003e(n(i) div 2)\u003c/i\u003e, then \u003ci\u003eM\u003c/i\u003e is a maximum cardinality matching. \u003c/p\u003e",
	"authors": [
	"Christine Rizkallah"
	],
	"date": "2011-07-21",
	- "id": 660,
	+ "id": 663,
	"link": "/entries/Max-Card-Matching.html",
	"permalink": "/entries/Max-Card-Matching.html",
	"shortname": "Max-Card-Matching",
	"title": "Maximum Cardinality Matching",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Knowledge-based programs (KBPs) are a formalism for directly relating agents' knowledge and behaviour. Here we present a general scheme for compiling KBPs to executable automata with a proof of correctness in Isabelle/HOL. We develop the algorithm top-down, using Isabelle's locale mechanism to structure these proofs, and show that two classic examples can be synthesised using Isabelle's code generator.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2011-05-17",
	- "id": 661,
	+ "id": 664,
	"link": "/entries/KBPs.html",
	"permalink": "/entries/KBPs.html",
	"shortname": "KBPs",
	"title": "Knowledge-based programs",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 1
	},
	{
	"abstract": "Some acute-angled triangles are special, e.g. right-angled or isoscele triangles. Some are not of this kind, but, without measuring angles, look as if they were. In that sense, there is exactly one general triangle. This well-known fact is proven here formally.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2011-04-01",
	- "id": 662,
	+ "id": 665,
	"link": "/entries/General-Triangle.html",
	"permalink": "/entries/General-Triangle.html",
	"shortname": "General-Triangle",
	"title": "The General Triangle Is Unique",
	"topic_links": [
	"mathematics/geometry"
	],
	"topics": [
	"Mathematics/Geometry"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide a generic work-list algorithm to compute the transitive closure of finite relations where only successors of newly detected states are generated. This algorithm is then instantiated for lists over arbitrary carriers and red black trees (which are faster but require a linear order on the carrier), respectively. Our formalization was performed as part of the IsaFoR/CeTA project where reflexive transitive closures of large tree automata have to be computed.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2011-03-14",
	- "id": 663,
	+ "id": 666,
	"link": "/entries/Transitive-Closure.html",
	"permalink": "/entries/Transitive-Closure.html",
	"shortname": "Transitive-Closure",
	"title": "Executable Transitive Closures of Finite Relations",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 3
	},
	{
	"abstract": "We formalize the AutoFocus Semantics (a time-synchronous subset of the Focus formalism) as stream processing functions on finite and infinite message streams represented as finite/infinite lists. The formalization comprises both the conventional single-clocking semantics (uniform global clock for all components and communications channels) and its extension to multi-clocking semantics (internal execution clocking of a component may be a multiple of the external communication clocking). The semantics is defined by generic stream processing functions making it suitable for simulation/code generation in Isabelle/HOL. Furthermore, a number of AutoFocus semantics properties are formalized using definitions from the IntervalLogic theories.",
	"authors": [
	"David Trachtenherz"
	],
	"date": "2011-02-23",
	- "id": 664,
	+ "id": 667,
	"link": "/entries/AutoFocus-Stream.html",
	"permalink": "/entries/AutoFocus-Stream.html",
	"shortname": "AutoFocus-Stream",
	"title": "AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "We introduce a theory of infinite lists in HOL formalized as functions over naturals (folder ListInf, theories ListInf and ListInf_Prefix). It also provides additional results for finite lists (theory ListInf/List2), natural numbers (folder CommonArith, esp. division/modulo, naturals with infinity), sets (folder CommonSet, esp. cutting/truncating sets, traversing sets of naturals).",
	"authors": [
	"David Trachtenherz"
	],
	"date": "2011-02-23",
	- "id": 665,
	+ "id": 668,
	"link": "/entries/List-Infinite.html",
	"permalink": "/entries/List-Infinite.html",
	"shortname": "List-Infinite",
	"title": "Infinite Lists",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "We introduce a theory of temporal logic operators using sets of natural numbers as time domain, formalized in a shallow embedding manner. The theory comprises special natural intervals (theory IL_Interval: open and closed intervals, continuous and modulo intervals, interval traversing results), operators for shifting intervals to left/right on the number axis as well as expanding/contracting intervals by constant factors (theory IL_IntervalOperators.thy), and ultimately definitions and results for unary and binary temporal operators on arbitrary natural sets (theory IL_TemporalOperators).",
	"authors": [
	"David Trachtenherz"
	],
	"date": "2011-02-23",
	- "id": 666,
	+ "id": 669,
	"link": "/entries/Nat-Interval-Logic.html",
	"permalink": "/entries/Nat-Interval-Logic.html",
	"shortname": "Nat-Interval-Logic",
	"title": "Interval Temporal Logic on Natural Numbers",
	"topic_links": [
	"logic/general-logic/temporal-logic"
	],
	"topics": [
	"Logic/General logic/Temporal logic"
	],
	"used_by": 1
	},
	{
	"abstract": "A fully-formalized and extensible minimal imperative fragment of Java.",
	"authors": [
	"Rok Strniša",
	"Matthew Parkinson"
	],
	"date": "2011-02-07",
	- "id": 667,
	+ "id": 670,
	"link": "/entries/LightweightJava.html",
	"permalink": "/entries/LightweightJava.html",
	"shortname": "LightweightJava",
	"title": "Lightweight Java",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "This work presents a verification of an implementation in SPARK/ADA of the cryptographic hash-function RIPEMD-160. A functional specification of RIPEMD-160 is given in Isabelle/HOL. Proofs for the verification conditions generated by the static-analysis toolset of SPARK certify the functional correctness of the implementation.",
	"authors": [
	"Fabian Immler"
	],
	"date": "2011-01-10",
	- "id": 668,
	+ "id": 671,
	"link": "/entries/RIPEMD-160-SPARK.html",
	"permalink": "/entries/RIPEMD-160-SPARK.html",
	"shortname": "RIPEMD-160-SPARK",
	"title": "RIPEMD-160",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. We also give several equivalent characterizations of lower semicontinuity. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set. Also, we introduce the notion of the lower semicontinuous hull of an arbitrary function and prove its basic properties.",
	"authors": [
	"Bogdan Grechuk"
	],
	"date": "2011-01-08",
	- "id": 669,
	+ "id": 672,
	"link": "/entries/Lower_Semicontinuous.html",
	"permalink": "/entries/Lower_Semicontinuous.html",
	"shortname": "Lower_Semicontinuous",
	"title": "Lower Semicontinuous Functions",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Two proofs of Hall's Marriage Theorem: one due to Halmos and Vaughan, one due to Rado.",
	"authors": [
	"Dongchen Jiang",
	"Tobias Nipkow"
	],
	"date": "2010-12-17",
	- "id": 670,
	+ "id": 673,
	"link": "/entries/Marriage.html",
	"permalink": "/entries/Marriage.html",
	"shortname": "Marriage",
	"title": "Hall's Marriage Theorem",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 1
	},
	{
	"abstract": "In his dissertation, Olin Shivers introduces a concept of control flow graphs for functional languages, provides an algorithm to statically derive a safe approximation of the control flow graph and proves this algorithm correct. In this research project, Shivers' algorithms and proofs are formalized in the HOLCF extension of HOL.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2010-11-16",
	- "id": 671,
	+ "id": 674,
	"link": "/entries/Shivers-CFA.html",
	"permalink": "/entries/Shivers-CFA.html",
	"shortname": "Shivers-CFA",
	"title": "Shivers' Control Flow Analysis",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "We implement and prove correct binomial heaps and skew binomial heaps. Both are data-structures for priority queues. While binomial heaps have logarithmic \u003cem\u003efindMin\u003c/em\u003e, \u003cem\u003edeleteMin\u003c/em\u003e, \u003cem\u003einsert\u003c/em\u003e, and \u003cem\u003emeld\u003c/em\u003e operations, skew binomial heaps have constant time \u003cem\u003efindMin\u003c/em\u003e, \u003cem\u003einsert\u003c/em\u003e, and \u003cem\u003emeld\u003c/em\u003e operations, and only the \u003cem\u003edeleteMin\u003c/em\u003e-operation is logarithmic. This is achieved by using \u003cem\u003eskew links\u003c/em\u003e to avoid cascading linking on \u003cem\u003einsert\u003c/em\u003e-operations, and \u003cem\u003edata-structural bootstrapping\u003c/em\u003e to get constant-time \u003cem\u003efindMin\u003c/em\u003e and \u003cem\u003emeld\u003c/em\u003e operations. Our implementation follows the paper by Brodal and Okasaki.",
	"authors": [
	"Rene Meis",
	"Finn Nielsen",
	"Peter Lammich"
	],
	"date": "2010-10-28",
	- "id": 672,
	+ "id": 675,
	"link": "/entries/Binomial-Heaps.html",
	"permalink": "/entries/Binomial-Heaps.html",
	"shortname": "Binomial-Heaps",
	"title": "Binomial Heaps and Skew Binomial Heaps",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 2
	},
	{
	"abstract": "We implement and prove correct 2-3 finger trees. Finger trees are a general purpose data structure, that can be used to efficiently implement other data structures, such as priority queues. Intuitively, a finger tree is an annotated sequence, where the annotations are elements of a monoid. Apart from operations to access the ends of the sequence, the main operation is to split the sequence at the point where a \u003cem\u003emonotone predicate\u003c/em\u003e over the sum of the left part of the sequence becomes true for the first time. The implementation follows the paper of Hinze and Paterson. The code generator can be used to get efficient, verified code.",
	"authors": [
	"Benedikt Nordhoff",
	"Stefan Körner",
	"Peter Lammich"
	],
	"date": "2010-10-28",
	- "id": 673,
	+ "id": 676,
	"link": "/entries/Finger-Trees.html",
	"permalink": "/entries/Finger-Trees.html",
	"shortname": "Finger-Trees",
	"title": "Finger Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 3
	},
	{
	"abstract": "Priority queues are an important data structure and efficient implementations of them are crucial. We implement a functional variant of binomial queues in Isabelle/HOL and show its functional correctness. A verification against an abstract reference specification of priority queues has also been attempted, but could not be achieved to the full extent.",
	"authors": [
	"René Neumann"
	],
	"date": "2010-10-28",
	- "id": 674,
	+ "id": 677,
	"link": "/entries/Binomial-Queues.html",
	"permalink": "/entries/Binomial-Queues.html",
	"shortname": "Binomial-Queues",
	"title": "Functional Binomial Queues",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Handling variable binding is one of the main difficulties in formal proofs. In this context, Moggi's computational metalanguage serves as an interesting case study. It features monadic types and a commuting conversion rule that rearranges the binding structure. Lindley and Stark have given an elegant proof of strong normalization for this calculus. The key construction in their proof is a notion of relational TT-lifting, using stacks of elimination contexts to obtain a Girard-Tait style logical relation. I give a formalization of their proof in Isabelle/HOL-Nominal with a particular emphasis on the treatment of bound variables.",
	"authors": [
	"Christian Doczkal"
	],
	"date": "2010-08-29",
	- "id": 675,
	+ "id": 678,
	"link": "/entries/Lam-ml-Normalization.html",
	"permalink": "/entries/Lam-ml-Normalization.html",
	"shortname": "Lam-ml-Normalization",
	"title": "Strong Normalization of Moggis's Computational Metalanguage",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "We define multivariate polynomials over arbitrary (ordered) semirings in combination with (executable) operations like addition, multiplication, and substitution. We also define (weak) monotonicity of polynomials and comparison of polynomials where we provide standard estimations like absolute positiveness or the more recent approach of Neurauter, Zankl, and Middeldorp. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over polynomials. Our formalization was performed as part of the \u003ca href=\"http://cl-informatik.uibk.ac.at/software/ceta\"\u003eIsaFoR/CeTA-system\u003c/a\u003e which contains several termination techniques. The provided theories have been essential to formalize polynomial interpretations. \u003cp\u003e This formalization also contains an abstract representation as coefficient functions with finite support and a type of power-products. If this type is ordered by a linear (term) ordering, various additional notions, such as leading power-product, leading coefficient etc., are introduced as well. Furthermore, a lot of generic properties of, and functions on, multivariate polynomials are formalized, including the substitution and evaluation homomorphisms, embeddings of polynomial rings into larger rings (i.e. with one additional indeterminate), homogenization and dehomogenization of polynomials, and the canonical isomorphism between R[X,Y] and R[X][Y].",
	"authors": [
	"Christian Sternagel",
	"René Thiemann",
	"Alexander Maletzky",
	"Fabian Immler",
	"Florian Haftmann",
	"Andreas Lochbihler",
	"Alexander Bentkamp"
	],
	"date": "2010-08-10",
	- "id": 676,
	+ "id": 679,
	"link": "/entries/Polynomials.html",
	"permalink": "/entries/Polynomials.html",
	"shortname": "Polynomials",
	"title": "Executable Multivariate Polynomials",
	"topic_links": [
	"mathematics/analysis",
	"mathematics/algebra",
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Mathematics/Analysis",
	"Mathematics/Algebra",
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 7
	},
	{
	"abstract": "We formalize in Isabelle/HOL the abtract syntax and a synchronous step semantics for the specification language Statecharts. The formalization is based on Hierarchical Automata which allow a structural decomposition of Statecharts into Sequential Automata. To support the composition of Statecharts, we introduce calculating operators to construct a Hierarchical Automaton in a stepwise manner. Furthermore, we present a complete semantics of Statecharts including a theory of data spaces, which enables the modelling of racing effects. We also adapt CTL for Statecharts to build a bridge for future combinations with model checking. However the main motivation of this work is to provide a sound and complete basis for reasoning on Statecharts. As a central meta theorem we prove that the well-formedness of a Statechart is preserved by the semantics.",
	"authors": [
	"Steffen Helke",
	"Florian Kammüller"
	],
	"date": "2010-08-08",
	- "id": 677,
	+ "id": 680,
	"link": "/entries/Statecharts.html",
	"permalink": "/entries/Statecharts.html",
	"shortname": "Statecharts",
	"title": "Formalizing Statecharts using Hierarchical Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Free Groups are, in a sense, the most generic kind of group. They are defined over a set of generators with no additional relations in between them. They play an important role in the definition of group presentations and in other fields. This theory provides the definition of Free Group as the set of fully canceled words in the generators. The universal property is proven, as well as some isomorphisms results about Free Groups.",
	"authors": [
	"Joachim Breitner"
	],
	"date": "2010-06-24",
	- "id": 678,
	+ "id": 681,
	"link": "/entries/Free-Groups.html",
	"permalink": "/entries/Free-Groups.html",
	"shortname": "Free-Groups",
	"title": "Free Groups",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This article presents a development of Category Theory in Isabelle/HOL. A Category is defined using records and locales. Functors and Natural Transformations are also defined. The main result that has been formalized is that the Yoneda functor is a full and faithful embedding. We also formalize the completeness of many sorted monadic equational logic. Extensive use is made of the HOLZF theory in both cases. For an informal description see \u003ca href=\"http://www.srcf.ucam.org/~apk32/Isabelle/Category/Cat.pdf\"\u003ehere [pdf]\u003c/a\u003e.",
	"authors": [
	"Alexander Katovsky"
	],
	"date": "2010-06-20",
	- "id": 679,
	+ "id": 682,
	"link": "/entries/Category2.html",
	"permalink": "/entries/Category2.html",
	"shortname": "Category2",
	"title": "Category Theory",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We provide the operations of matrix addition, multiplication, transposition, and matrix comparisons as executable functions over ordered semirings. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over matrices. We further show that the standard semirings over the naturals, integers, and rationals, as well as the arctic semirings satisfy the axioms that are required by our matrix theory. Our formalization is part of the \u003ca href=\"http://cl-informatik.uibk.ac.at/software/ceta\"\u003eCeTA\u003c/a\u003e system which contains several termination techniques. The provided theories have been essential to formalize matrix-interpretations and arctic interpretations.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2010-06-17",
	- "id": 680,
	+ "id": 683,
	"link": "/entries/Matrix.html",
	"permalink": "/entries/Matrix.html",
	"shortname": "Matrix",
	"title": "Executable Matrix Operations on Matrices of Arbitrary Dimensions",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 5
	},
	{
	"abstract": "We present an Isabelle formalization of abstract rewriting (see, e.g., the book by Baader and Nipkow). First, we define standard relations like \u003ci\u003ejoinability\u003c/i\u003e, \u003ci\u003emeetability\u003c/i\u003e, \u003ci\u003econversion\u003c/i\u003e, etc. Then, we formalize important properties of abstract rewrite systems, e.g., confluence and strong normalization. Our main concern is on strong normalization, since this formalization is the basis of \u003ca href=\"http://cl-informatik.uibk.ac.at/software/ceta\"\u003eCeTA\u003c/a\u003e (which is mainly about strong normalization of term rewrite systems). Hence lemmas involving strong normalization constitute by far the biggest part of this theory. One of those is Newman's lemma.",
	"authors": [
	"Christian Sternagel",
	"René Thiemann"
	],
	"date": "2010-06-14",
	- "id": 681,
	+ "id": 684,
	"link": "/entries/Abstract-Rewriting.html",
	"permalink": "/entries/Abstract-Rewriting.html",
	"shortname": "Abstract-Rewriting",
	"title": "Abstract Rewriting",
	"topic_links": [
	"logic/rewriting"
	],
	"topics": [
	"Logic/Rewriting"
	],
	"used_by": 10
	},
	{
	"abstract": "The invariant based programming is a technique of constructing correct programs by first identifying the basic situations (pre- and post-conditions and invariants) that can occur during the execution of the program, and then defining the transitions and proving that they preserve the invariants. Data refinement is a technique of building correct programs working on concrete datatypes as refinements of more abstract programs. In the theories presented here we formalize the predicate transformer semantics for invariant based programs and their data refinement.",
	"authors": [
	"Viorel Preoteasa",
	"Ralph-Johan Back"
	],
	"date": "2010-05-28",
	- "id": 682,
	+ "id": 685,
	"link": "/entries/DataRefinementIBP.html",
	"permalink": "/entries/DataRefinementIBP.html",
	"shortname": "DataRefinementIBP",
	"title": "Semantics and Data Refinement of Invariant Based Programs",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 1
	},
	{
	"abstract": "The verification of the Deutsch-Schorr-Waite graph marking algorithm is used as a benchmark in many formalizations of pointer programs. The main purpose of this mechanization is to show how data refinement of invariant based programs can be used in verifying practical algorithms. The verification starts with an abstract algorithm working on a graph given by a relation \u003ci\u003enext\u003c/i\u003e on nodes. Gradually the abstract program is refined into Deutsch-Schorr-Waite graph marking algorithm where only one bit per graph node of additional memory is used for marking.",
	"authors": [
	"Viorel Preoteasa",
	"Ralph-Johan Back"
	],
	"date": "2010-05-28",
	- "id": 683,
	+ "id": 686,
	"link": "/entries/GraphMarkingIBP.html",
	"permalink": "/entries/GraphMarkingIBP.html",
	"shortname": "GraphMarkingIBP",
	"title": "Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "This document gives a formalization of the proof of the Robbins conjecture, following A. Mann, \u003ci\u003eA Complete Proof of the Robbins Conjecture\u003c/i\u003e, 2003.",
	"authors": [
	"Matthew Doty"
	],
	"date": "2010-05-22",
	- "id": 684,
	+ "id": 687,
	"link": "/entries/Robbins-Conjecture.html",
	"permalink": "/entries/Robbins-Conjecture.html",
	"shortname": "Robbins-Conjecture",
	"title": "A Complete Proof of the Robbins Conjecture",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a library of constructions on regular expressions and languages. It provides the operations of concatenation, Kleene star and derivative on languages. Regular expressions and their meaning are defined. An executable equivalence checker for regular expressions is verified; it does not need automata but works directly on regular expressions. \u003ci\u003eBy mapping regular expressions to binary relations, an automatic and complete proof method for (in)equalities of binary relations over union, concatenation and (reflexive) transitive closure is obtained.\u003c/i\u003e \u003cP\u003e Extended regular expressions with complement and intersection are also defined and an equivalence checker is provided.",
	"authors": [
	"Alexander Krauss",
	"Tobias Nipkow"
	],
	"date": "2010-05-12",
	- "id": 685,
	+ "id": 688,
	"link": "/entries/Regular-Sets.html",
	"permalink": "/entries/Regular-Sets.html",
	"shortname": "Regular-Sets",
	"title": "Regular Sets and Expressions",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 12
	},
	{
	"abstract": "We present a Theory of Objects based on the original functional sigma-calculus by Abadi and Cardelli but with an additional parameter to methods. We prove confluence of the operational semantics following the outline of Nipkow's proof of confluence for the lambda-calculus reusing his theory Commutation, a generic diamond lemma reduction. We furthermore formalize a simple type system for our sigma-calculus including a proof of type safety. The entire development uses the concept of Locally Nameless representation for binders. We reuse an earlier proof of confluence for a simpler sigma-calculus based on de Bruijn indices and lists to represent objects.",
	"authors": [
	"Ludovic Henrio",
	"Florian Kammüller",
	"Bianca Lutz",
	"Henry Sudhof"
	],
	"date": "2010-04-30",
	- "id": 686,
	+ "id": 689,
	"link": "/entries/Locally-Nameless-Sigma.html",
	"permalink": "/entries/Locally-Nameless-Sigma.html",
	"shortname": "Locally-Nameless-Sigma",
	"title": "Locally Nameless Sigma Calculus",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory defines a type constructor representing the free Boolean algebra over a set of generators. Values of type (α)\u003ci\u003eformula\u003c/i\u003e represent propositional formulas with uninterpreted variables from type α, ordered by implication. In addition to all the standard Boolean algebra operations, the library also provides a function for building homomorphisms to any other Boolean algebra type.",
	"authors": [
	"Brian Huffman"
	],
	"date": "2010-03-29",
	- "id": 687,
	+ "id": 690,
	"link": "/entries/Free-Boolean-Algebra.html",
	"permalink": "/entries/Free-Boolean-Algebra.html",
	"shortname": "Free-Boolean-Algebra",
	"title": "Free Boolean Algebra",
	"topic_links": [
	"logic/general-logic/classical-propositional-logic"
	],
	"topics": [
	"Logic/General logic/Classical propositional logic"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e In this contribution, we show how correctness proofs for \u003ca href=\"Slicing.html\"\u003eintra-\u003c/a\u003e and \u003ca href=\"HRB-Slicing.html\"\u003einterprocedural slicing\u003c/a\u003e can be used to prove that slicing is able to guarantee information flow noninterference. Moreover, we also illustrate how to lift the control flow graphs of the respective frameworks such that they fulfil the additional assumptions needed in the noninterference proofs. A detailed description of the intraprocedural proof and its interplay with the slicing framework can be found in the PLAS'09 paper by Wasserrab et al. \u003c/p\u003e \u003cp\u003e This entry contains the part for intra-procedural slicing. See entry \u003ca href=\"InformationFlowSlicing_Inter.html\"\u003eInformationFlowSlicing_Inter\u003c/a\u003e for the inter-procedural part. \u003c/p\u003e",
	"authors": [
	"Daniel Wasserrab"
	],
	"date": "2010-03-23",
	- "id": 688,
	+ "id": 691,
	"link": "/entries/InformationFlowSlicing.html",
	"permalink": "/entries/InformationFlowSlicing.html",
	"shortname": "InformationFlowSlicing",
	"title": "Information Flow Noninterference via Slicing",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "\u003cp\u003e In this contribution, we show how correctness proofs for \u003ca href=\"Slicing.html\"\u003eintra-\u003c/a\u003e and \u003ca href=\"HRB-Slicing.html\"\u003einterprocedural slicing\u003c/a\u003e can be used to prove that slicing is able to guarantee information flow noninterference. Moreover, we also illustrate how to lift the control flow graphs of the respective frameworks such that they fulfil the additional assumptions needed in the noninterference proofs. A detailed description of the intraprocedural proof and its interplay with the slicing framework can be found in the PLAS'09 paper by Wasserrab et al. \u003c/p\u003e \u003cp\u003e This entry contains the part for inter-procedural slicing. See entry \u003ca href=\"InformationFlowSlicing.html\"\u003eInformationFlowSlicing\u003c/a\u003e for the intra-procedural part. \u003c/p\u003e",
	"authors": [
	"Daniel Wasserrab"
	],
	"date": "2010-03-23",
	- "id": 689,
	+ "id": 692,
	"link": "/entries/InformationFlowSlicing_Inter.html",
	"permalink": "/entries/InformationFlowSlicing_Inter.html",
	"shortname": "InformationFlowSlicing_Inter",
	"title": "Inter-Procedural Information Flow Noninterference via Slicing",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory provides functions for finding the index of an element in a list, by predicate and by value.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2010-02-20",
	- "id": 690,
	+ "id": 693,
	"link": "/entries/List-Index.html",
	"permalink": "/entries/List-Index.html",
	"shortname": "List-Index",
	"title": "List Index",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 23
	},
	{
	"abstract": "This article collects formalisations of general-purpose coinductive data types and sets. Currently, it contains coinductive natural numbers, coinductive lists, i.e. lazy lists or streams, infinite streams, coinductive terminated lists, coinductive resumptions, a library of operations on coinductive lists, and a version of König's lemma as an application for coinductive lists.\u003cbr\u003eThe initial theory was contributed by Paulson and Wenzel. Extensions and other coinductive formalisations of general interest are welcome.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2010-02-12",
	- "id": 691,
	+ "id": 694,
	"link": "/entries/Coinductive.html",
	"permalink": "/entries/Coinductive.html",
	"shortname": "Coinductive",
	"title": "Coinductive",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 13
	},
	{
	"abstract": "This contribution contains a fast SAT solver for Isabelle written in Standard ML. By loading the theory \u003ctt\u003eDPT_SAT_Solver\u003c/tt\u003e, the SAT solver installs itself (under the name “\u003ctt\u003edptsat\u003c/tt\u003e”) and certain Isabelle tools like Refute will start using it automatically. This is a port of the DPT (Decision Procedure Toolkit) SAT Solver written in OCaml.",
	"authors": [
	"Armin Heller"
	],
	"date": "2009-12-09",
	- "id": 692,
	+ "id": 695,
	"link": "/entries/DPT-SAT-Solver.html",
	"permalink": "/entries/DPT-SAT-Solver.html",
	"shortname": "DPT-SAT-Solver",
	"title": "A Fast SAT Solver for Isabelle in Standard ML",
	"topic_links": [
	"tools"
	],
	"topics": [
	"Tools"
	],
	"used_by": 0
	},
	{
	"abstract": "This work presents a formalization of a library for automata on bit strings. It forms the basis of a reflection-based decision procedure for Presburger arithmetic, which is efficiently executable thanks to Isabelle's code generator. With this work, we therefore provide a mechanized proof of a well-known connection between logic and automata theory. The formalization is also described in a publication [TPHOLs 2009].",
	"authors": [
	"Stefan Berghofer",
	"Markus Reiter"
	],
	"date": "2009-12-03",
	- "id": 693,
	+ "id": 696,
	"link": "/entries/Presburger-Automata.html",
	"permalink": "/entries/Presburger-Automata.html",
	"shortname": "Presburger-Automata",
	"title": "Formalizing the Logic-Automaton Connection",
	"topic_links": [
	"computer-science/automata-and-formal-languages",
	"logic/general-logic/decidability-of-theories"
	],
	"topics": [
	"Computer science/Automata and formal languages",
	"Logic/General logic/Decidability of theories"
	],
	"used_by": 0
	},
	{
	"abstract": "This development provides an efficient, extensible, machine checked collections framework. The library adopts the concepts of interface, implementation and generic algorithm from object-oriented programming and implements them in Isabelle/HOL. The framework features the use of data refinement techniques to refine an abstract specification (using high-level concepts like sets) to a more concrete implementation (using collection datastructures, like red-black-trees). The code-generator of Isabelle/HOL can be used to generate efficient code.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2009-11-25",
	- "id": 694,
	+ "id": 697,
	"link": "/entries/Collections.html",
	"permalink": "/entries/Collections.html",
	"shortname": "Collections",
	"title": "Collections Framework",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 19
	},
	{
	"abstract": "This work presents a machine-checked tree automata library for Standard-ML, OCaml and Haskell. The algorithms are efficient by using appropriate data structures like RB-trees. The available algorithms for non-deterministic automata include membership query, reduction, intersection, union, and emptiness check with computation of a witness for non-emptiness. The executable algorithms are derived from less-concrete, non-executable algorithms using data-refinement techniques. The concrete data structures are from the Isabelle Collections Framework. Moreover, this work contains a formalization of the class of tree-regular languages and its closure properties under set operations.",
	"authors": [
	"Peter Lammich"
	],
	"date": "2009-11-25",
	- "id": 695,
	+ "id": 698,
	"link": "/entries/Tree-Automata.html",
	"permalink": "/entries/Tree-Automata.html",
	"shortname": "Tree-Automata",
	"title": "Tree Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "These theories present the mechanised proof of the Perfect Number Theorem.",
	"authors": [
	"Mark Ijbema"
	],
	"date": "2009-11-22",
	- "id": 696,
	+ "id": 699,
	"link": "/entries/Perfect-Number-Thm.html",
	"permalink": "/entries/Perfect-Number-Thm.html",
	"shortname": "Perfect-Number-Thm",
	"title": "Perfect Number Theorem",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "After verifying \u003ca href=\"Slicing.html\"\u003edynamic and static interprocedural slicing\u003c/a\u003e, we present a modular framework for static interprocedural slicing. To this end, we formalized the standard two-phase slicer from Horwitz, Reps and Binkley (see their TOPLAS 12(1) 1990 paper) together with summary edges as presented by Reps et al. (see FSE 1994). The framework is again modular in the programming language by using an abstract CFG, defined via structural and well-formedness properties. Using a weak simulation between the original and sliced graph, we were able to prove the correctness of static interprocedural slicing. We also instantiate our framework with a simple While language with procedures. This shows that the chosen abstractions are indeed valid.",
	"authors": [
	"Daniel Wasserrab"
	],
	"date": "2009-11-13",
	- "id": 697,
	+ "id": 700,
	"link": "/entries/HRB-Slicing.html",
	"permalink": "/entries/HRB-Slicing.html",
	"shortname": "HRB-Slicing",
	"title": "Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "Gill and Hutton formalise the worker/wrapper transformation, building on the work of Launchbury and Peyton-Jones who developed it as a way of changing the type at which a recursive function operates. This development establishes the soundness of the technique and several examples of its use.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2009-10-30",
	- "id": 698,
	+ "id": 701,
	"link": "/entries/WorkerWrapper.html",
	"permalink": "/entries/WorkerWrapper.html",
	"shortname": "WorkerWrapper",
	"title": "The Worker/Wrapper Transformation",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "We develop a basic theory of ordinals and cardinals in Isabelle/HOL, up to the point where some cardinality facts relevant for the “working mathematician” become available. Unlike in set theory, here we do not have at hand canonical notions of ordinal and cardinal. Therefore, here an ordinal is merely a well-order relation and a cardinal is an ordinal minim w.r.t. order embedding on its field.",
	"authors": [
	"Andrei Popescu"
	],
	"date": "2009-09-01",
	- "id": 699,
	+ "id": 702,
	"link": "/entries/Ordinals_and_Cardinals.html",
	"permalink": "/entries/Ordinals_and_Cardinals.html",
	"shortname": "Ordinals_and_Cardinals",
	"title": "Ordinals and Cardinals",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 0
	},
	{
	"abstract": "The invertibility of the rules of a sequent calculus is important for guiding proof search and can be used in some formalised proofs of Cut admissibility. We present sufficient conditions for when a rule is invertible with respect to a calculus. We illustrate the conditions with examples. It must be noted we give purely syntactic criteria; no guarantees are given as to the suitability of the rules.",
	"authors": [
	"Peter Chapman"
	],
	"date": "2009-08-28",
	- "id": 700,
	+ "id": 703,
	"link": "/entries/SequentInvertibility.html",
	"permalink": "/entries/SequentInvertibility.html",
	"shortname": "SequentInvertibility",
	"title": "Invertibility in Sequent Calculi",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the usual proof that the group generated by the function k -\u003e k + 1 on the integers gives rise to a cofinitary group.",
	"authors": [
	"Bart Kastermans"
	],
	"date": "2009-08-04",
	- "id": 701,
	+ "id": 704,
	"link": "/entries/CofGroups.html",
	"permalink": "/entries/CofGroups.html",
	"shortname": "CofGroups",
	"title": "An Example of a Cofinitary Group in Isabelle/HOL",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "FinFuns are total functions that are constant except for a finite set of points, i.e. a generalisation of finite maps. They are formalised as a new type in Isabelle/HOL such that the code generator can handle equality tests and quantification on FinFuns. On the code output level, FinFuns are explicitly represented by constant functions and pointwise updates, similarly to associative lists. Inside the logic, they behave like ordinary functions with extensionality. Via the update/constant pattern, a recursion combinator and an induction rule for FinFuns allow for defining and reasoning about operators on FinFun that are also executable.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2009-05-06",
	- "id": 702,
	+ "id": 705,
	"link": "/entries/FinFun.html",
	"permalink": "/entries/FinFun.html",
	"shortname": "FinFun",
	"title": "Code Generation for Functions as Data",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 4
	},
	{
	"abstract": "Stream Fusion is a system for removing intermediate list structures from Haskell programs; it consists of a Haskell library along with several compiler rewrite rules. (The library is available \u003ca href=\"http://hackage.haskell.org/package/stream-fusion\"\u003eonline\u003c/a\u003e.)\u003cbr\u003e\u003cbr\u003eThese theories contain a formalization of much of the Stream Fusion library in HOLCF. Lazy list and stream types are defined, along with coercions between the two types, as well as an equivalence relation for streams that generate the same list. List and stream versions of map, filter, foldr, enumFromTo, append, zipWith, and concatMap are defined, and the stream versions are shown to respect stream equivalence.",
	"authors": [
	"Brian Huffman"
	],
	"date": "2009-04-29",
	- "id": 703,
	+ "id": 706,
	"link": "/entries/Stream-Fusion.html",
	"permalink": "/entries/Stream-Fusion.html",
	"shortname": "Stream-Fusion",
	"title": "Stream Fusion",
	"topic_links": [
	"computer-science/functional-programming"
	],
	"topics": [
	"Computer science/Functional programming"
	],
	"used_by": 0
	},
	{
	"abstract": "This document contains the Isabelle/HOL sources underlying the paper \u003ci\u003eA bytecode logic for JML and types\u003c/i\u003e by Beringer and Hofmann, updated to Isabelle 2008. We present a program logic for a subset of sequential Java bytecode that is suitable for representing both, features found in high-level specification language JML as well as interpretations of high-level type systems. To this end, we introduce a fine-grained collection of assertions, including strong invariants, local annotations and VDM-reminiscent partial-correctness specifications. Thanks to a goal-oriented structure and interpretation of judgements, verification may proceed without recourse to an additional control flow analysis. The suitability for interpreting intensional type systems is illustrated by the proof-carrying-code style encoding of a type system for a first-order functional language which guarantees a constant upper bound on the number of objects allocated throughout an execution, be the execution terminating or non-terminating. Like the published paper, the formal development is restricted to a comparatively small subset of the JVML, lacking (among other features) exceptions, arrays, virtual methods, and static fields. This shortcoming has been overcome meanwhile, as our paper has formed the basis of the Mobius base logic, a program logic for the full sequential fragment of the JVML. Indeed, the present formalisation formed the basis of a subsequent formalisation of the Mobius base logic in the proof assistant Coq, which includes a proof of soundness with respect to the Bicolano operational semantics by Pichardie.",
	"authors": [
	"Lennart Beringer",
	"Martin Hofmann"
	],
	"date": "2008-12-12",
	- "id": 704,
	+ "id": 707,
	"link": "/entries/BytecodeLogicJmlTypes.html",
	"permalink": "/entries/BytecodeLogicJmlTypes.html",
	"shortname": "BytecodeLogicJmlTypes",
	"title": "A Bytecode Logic for JML and Types",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "We present interpretations of type systems for secure information flow in Hoare logic, complementing previous encodings in relational program logics. We first treat the imperative language IMP, extended by a simple procedure call mechanism. For this language we consider base-line non-interference in the style of Volpano et al. and the flow-sensitive type system by Hunt and Sands. In both cases, we show how typing derivations may be used to automatically generate proofs in the program logic that certify the absence of illicit flows. We then add instructions for object creation and manipulation, and derive appropriate proof rules for base-line non-interference. As a consequence of our work, standard verification technology may be used for verifying that a concrete program satisfies the non-interference property.\u003cbr\u003e\u003cbr\u003eThe present proof development represents an update of the formalisation underlying our paper [CSF 2007] and is intended to resolve any ambiguities that may be present in the paper.",
	"authors": [
	"Lennart Beringer",
	"Martin Hofmann"
	],
	"date": "2008-11-10",
	- "id": 705,
	+ "id": 708,
	"link": "/entries/SIFPL.html",
	"permalink": "/entries/SIFPL.html",
	"shortname": "SIFPL",
	"title": "Secure information flow and program logics",
	"topic_links": [
	"computer-science/programming-languages/logics",
	"computer-science/security"
	],
	"topics": [
	"Computer science/Programming languages/Logics",
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "Drawing on Sen's landmark work \"Collective Choice and Social Welfare\" (1970), this development proves Arrow's General Possibility Theorem, Sen's Liberal Paradox and May's Theorem in a general setting. The goal was to make precise the classical statements and proofs of these results, and to provide a foundation for more recent results such as the Gibbard-Satterthwaite and Duggan-Schwartz theorems.",
	"authors": [
	"Peter Gammie"
	],
	"date": "2008-11-09",
	- "id": 706,
	+ "id": 709,
	"link": "/entries/SenSocialChoice.html",
	"permalink": "/entries/SenSocialChoice.html",
	"shortname": "SenSocialChoice",
	"title": "Some classical results in Social Choice Theory",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "Tilings are defined inductively. It is shown that one form of mutilated chess board cannot be tiled with dominoes, while another one can be tiled with L-shaped tiles. Please add further fun examples of this kind!",
	"authors": [
	"Tobias Nipkow",
	"Lawrence C. Paulson"
	],
	"date": "2008-11-07",
	- "id": 707,
	+ "id": 710,
	"link": "/entries/FunWithTilings.html",
	"permalink": "/entries/FunWithTilings.html",
	"shortname": "FunWithTilings",
	"title": "Fun With Tilings",
	"topic_links": [
	"mathematics/misc"
	],
	"topics": [
	"Mathematics/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "Huffman's algorithm is a procedure for constructing a binary tree with minimum weighted path length. This report presents a formal proof of the correctness of Huffman's algorithm written using Isabelle/HOL. Our proof closely follows the sketches found in standard algorithms textbooks, uncovering a few snags in the process. Another distinguishing feature of our formalization is the use of custom induction rules to help Isabelle's automatic tactics, leading to very short proofs for most of the lemmas.",
	"authors": [
	"Jasmin Christian Blanchette"
	],
	"date": "2008-10-15",
	- "id": 708,
	+ "id": 711,
	"link": "/entries/Huffman.html",
	"permalink": "/entries/Huffman.html",
	"shortname": "Huffman",
	"title": "The Textbook Proof of Huffman's Algorithm",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "Slicing is a widely-used technique with applications in e.g. compiler technology and software security. Thus verification of algorithms in these areas is often based on the correctness of slicing, which should ideally be proven independent of concrete programming languages and with the help of well-known verifying techniques such as proof assistants. As a first step in this direction, this contribution presents a framework for dynamic and static intraprocedural slicing based on control flow and program dependence graphs. Abstracting from concrete syntax we base the framework on a graph representation of the program fulfilling certain structural and well-formedness properties.\u003cbr\u003e\u003cbr\u003eThe formalization consists of the basic framework (in subdirectory Basic/), the correctness proof for dynamic slicing (in subdirectory Dynamic/), the correctness proof for static intraprocedural slicing (in subdirectory StaticIntra/) and instantiations of the framework with a simple While language (in subdirectory While/) and the sophisticated object-oriented bytecode language of Jinja (in subdirectory JinjaVM/). For more information on the framework, see the TPHOLS 2008 paper by Wasserrab and Lochbihler and the PLAS 2009 paper by Wasserrab et al.",
	"authors": [
	"Daniel Wasserrab"
	],
	"date": "2008-09-16",
	- "id": 709,
	+ "id": 712,
	"link": "/entries/Slicing.html",
	"permalink": "/entries/Slicing.html",
	"shortname": "Slicing",
	"title": "Towards Certified Slicing",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 2
	},
	{
	"abstract": "The Volpano/Smith/Irvine security type systems requires that variables are annotated as high (secret) or low (public), and provides typing rules which guarantee that secret values cannot leak to public output ports. This property of a program is called confidentiality. For a simple while-language without threads, our proof shows that typeability in the Volpano/Smith system guarantees noninterference. Noninterference means that if two initial states for program execution are low-equivalent, then the final states are low-equivalent as well. This indeed implies that secret values cannot leak to public ports. The proof defines an abstract syntax and operational semantics for programs, formalizes noninterference, and then proceeds by rule induction on the operational semantics. The mathematically most intricate part is the treatment of implicit flows. Note that the Volpano/Smith system is not flow-sensitive and thus quite unprecise, resulting in false alarms. However, due to the correctness property, all potential breaks of confidentiality are discovered.",
	"authors": [
	"Gregor Snelting",
	"Daniel Wasserrab"
	],
	"date": "2008-09-02",
	- "id": 710,
	+ "id": 713,
	"link": "/entries/VolpanoSmith.html",
	"permalink": "/entries/VolpanoSmith.html",
	"shortname": "VolpanoSmith",
	"title": "A Correctness Proof for the Volpano/Smith Security Typing System",
	"topic_links": [
	"computer-science/programming-languages/type-systems",
	"computer-science/security"
	],
	"topics": [
	"Computer science/Programming languages/Type systems",
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes two proofs of Arrow's impossibility theorem due to Geanakoplos and derives the Gibbard-Satterthwaite theorem as a corollary. One formalization is based on utility functions, the other one on strict partial orders.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2008-09-01",
	- "id": 711,
	+ "id": 714,
	"link": "/entries/ArrowImpossibilityGS.html",
	"permalink": "/entries/ArrowImpossibilityGS.html",
	"shortname": "ArrowImpossibilityGS",
	"title": "Arrow and Gibbard-Satterthwaite",
	"topic_links": [
	"mathematics/games-and-economics"
	],
	"topics": [
	"Mathematics/Games and economics"
	],
	"used_by": 0
	},
	{
	"abstract": "This is a collection of cute puzzles of the form ”Show that if a function satisfies the following constraints, it must be ...µ Please add further examples to this collection!",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2008-08-26",
	- "id": 712,
	+ "id": 715,
	"link": "/entries/FunWithFunctions.html",
	"permalink": "/entries/FunWithFunctions.html",
	"shortname": "FunWithFunctions",
	"title": "Fun With Functions",
	"topic_links": [
	"mathematics/misc"
	],
	"topics": [
	"Mathematics/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "This document contains formal correctness proofs of modern SAT solvers. Following (Krstic et al, 2007) and (Nieuwenhuis et al., 2006), solvers are described using state-transition systems. Several different SAT solver descriptions are given and their partial correctness and termination is proved. These include: \u003cul\u003e \u003cli\u003e a solver based on classical DPLL procedure (using only a backtrack-search with unit propagation),\u003c/li\u003e \u003cli\u003e a very general solver with backjumping and learning (similar to the description given in (Nieuwenhuis et al., 2006)), and\u003c/li\u003e \u003cli\u003e a solver with a specific conflict analysis algorithm (similar to the description given in (Krstic et al., 2007)).\u003c/li\u003e \u003c/ul\u003e Within the SAT solver correctness proofs, a large number of lemmas about propositional logic and CNF formulae are proved. This theory is self-contained and could be used for further exploring of properties of CNF based SAT algorithms.",
	"authors": [
	"Filip Marić"
	],
	"date": "2008-07-23",
	- "id": 713,
	+ "id": 716,
	"link": "/entries/SATSolverVerification.html",
	"permalink": "/entries/SATSolverVerification.html",
	"shortname": "SATSolverVerification",
	"title": "Formal Verification of Modern SAT Solvers",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This document presents the formalization of introductory material from recursion theory --- definitions and basic properties of primitive recursive functions, Cantor pairing function and computably enumerable sets (including a proof of existence of a one-complete computably enumerable set and a proof of the Rice's theorem).",
	"authors": [
	"Michael Nedzelsky"
	],
	"date": "2008-04-05",
	- "id": 714,
	+ "id": 717,
	"link": "/entries/Recursion-Theory-I.html",
	"permalink": "/entries/Recursion-Theory-I.html",
	"shortname": "Recursion-Theory-I",
	"title": "Recursion Theory I",
	"topic_links": [
	"logic/computability"
	],
	"topics": [
	"Logic/Computability"
	],
	"used_by": 1
	},
	{
	"abstract": "We present the theory of Simpl, a sequential imperative programming language. We introduce its syntax, its semantics (big and small-step operational semantics) and Hoare logics for both partial as well as total correctness. We prove soundness and completeness of the Hoare logic. We integrate and automate the Hoare logic in Isabelle/HOL to obtain a practically usable verification environment for imperative programs. Simpl is independent of a concrete programming language but expressive enough to cover all common language features: mutually recursive procedures, abrupt termination and exceptions, runtime faults, local and global variables, pointers and heap, expressions with side effects, pointers to procedures, partial application and closures, dynamic method invocation and also unbounded nondeterminism.",
	"authors": [
	"Norbert Schirmer"
	],
	"date": "2008-02-29",
	- "id": 715,
	+ "id": 718,
	"link": "/entries/Simpl.html",
	"permalink": "/entries/Simpl.html",
	"shortname": "Simpl",
	"title": "A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment",
	"topic_links": [
	"computer-science/programming-languages/language-definitions",
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions",
	"Computer science/Programming languages/Logics"
	],
	"used_by": 2
	},
	{
	"abstract": "We present the verification of the normalisation of a binary decision diagram (BDD). The normalisation follows the original algorithm presented by Bryant in 1986 and transforms an ordered BDD in a reduced, ordered and shared BDD. The verification is based on Hoare logics.",
	"authors": [
	"Veronika Ortner",
	"Norbert Schirmer"
	],
	"date": "2008-02-29",
	- "id": 716,
	+ "id": 719,
	"link": "/entries/BDD.html",
	"permalink": "/entries/BDD.html",
	"shortname": "BDD",
	"title": "BDD Normalisation",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes normalization by evaluation as implemented in Isabelle. Lambda calculus plus term rewriting is compiled into a functional program with pattern matching. It is proved that the result of a successful evaluation is a) correct, i.e. equivalent to the input, and b) in normal form.",
	"authors": [
	"Klaus Aehlig",
	"Tobias Nipkow"
	],
	"date": "2008-02-18",
	- "id": 717,
	+ "id": 720,
	"link": "/entries/NormByEval.html",
	"permalink": "/entries/NormByEval.html",
	"shortname": "NormByEval",
	"title": "Normalization by Evaluation",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "This article formalizes quantifier elimination procedures for dense linear orders, linear real arithmetic and Presburger arithmetic. In each case both a DNF-based non-elementary algorithm and one or more (doubly) exponential NNF-based algorithms are formalized, including the well-known algorithms by Ferrante and Rackoff and by Cooper. The NNF-based algorithms for dense linear orders are new but based on Ferrante and Rackoff and on an algorithm by Loos and Weisspfenning which simulates infenitesimals. All algorithms are directly executable. In particular, they yield reflective quantifier elimination procedures for HOL itself. The formalization makes heavy use of locales and is therefore highly modular.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2008-01-11",
	- "id": 718,
	+ "id": 721,
	"link": "/entries/LinearQuantifierElim.html",
	"permalink": "/entries/LinearQuantifierElim.html",
	"shortname": "LinearQuantifierElim",
	"title": "Quantifier Elimination for Linear Arithmetic",
	"topic_links": [
	"logic/general-logic/decidability-of-theories"
	],
	"topics": [
	"Logic/General logic/Decidability of theories"
	],
	"used_by": 0
	},
	{
	"abstract": "In this work we formally verify the soundness and precision of a static program analysis that detects conflicts (e. g. data races) in programs with procedures, thread creation and monitors with the Isabelle theorem prover. As common in static program analysis, our program model abstracts guarded branching by nondeterministic branching, but completely interprets the call-/return behavior of procedures, synchronization by monitors, and thread creation. The analysis is based on the observation that all conflicts already occur in a class of particularly restricted schedules. These restricted schedules are suited to constraint-system-based program analysis. The formalization is based upon a flowgraph-based program model with an operational semantics as reference point.",
	"authors": [
	"Peter Lammich",
	"Markus Müller-Olm"
	],
	"date": "2007-12-14",
	- "id": 719,
	+ "id": 722,
	"link": "/entries/Program-Conflict-Analysis.html",
	"permalink": "/entries/Program-Conflict-Analysis.html",
	"shortname": "Program-Conflict-Analysis",
	"title": "Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors",
	"topic_links": [
	"computer-science/programming-languages/static-analysis"
	],
	"topics": [
	"Computer science/Programming languages/Static analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "We extend the Jinja source code semantics by Klein and Nipkow with Java-style arrays and threads. Concurrency is captured in a generic framework semantics for adding concurrency through interleaving to a sequential semantics, which features dynamic thread creation, inter-thread communication via shared memory, lock synchronisation and joins. Also, threads can suspend themselves and be notified by others. We instantiate the framework with the adapted versions of both Jinja source and byte code and show type safety for the multithreaded case. Equally, the compiler from source to byte code is extended, for which we prove weak bisimilarity between the source code small step semantics and the defensive Jinja virtual machine. On top of this, we formalise the JMM and show the DRF guarantee and consistency. For description of the different parts, see Lochbihler's papers at FOOL 2008, ESOP 2010, ITP 2011, and ESOP 2012.",
	"authors": [
	"Andreas Lochbihler"
	],
	"date": "2007-12-03",
	- "id": 720,
	+ "id": 723,
	"link": "/entries/JinjaThreads.html",
	"permalink": "/entries/JinjaThreads.html",
	"shortname": "JinjaThreads",
	"title": "Jinja with Threads",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "This article is an Isabelle formalisation of a paper with the same title. In a similar way as Knuth's 0-1-principle for sorting algorithms, that paper develops a 0-1-2-principle for parallel prefix computations.",
	"authors": [
	"Sascha Böhme"
	],
	"date": "2007-11-06",
	- "id": 721,
	+ "id": 724,
	"link": "/entries/MuchAdoAboutTwo.html",
	"permalink": "/entries/MuchAdoAboutTwo.html",
	"shortname": "MuchAdoAboutTwo",
	"title": "Much Ado About Two",
	"topic_links": [
	"computer-science/algorithms"
	],
	"topics": [
	"Computer science/Algorithms"
	],
	"used_by": 0
	},
	{
	"abstract": "This document presents the mechanised proofs of\u003cul\u003e\u003cli\u003eFermat's Last Theorem for exponents 3 and 4 and\u003c/li\u003e\u003cli\u003ethe parametrisation of Pythagorean Triples.\u003c/li\u003e\u003c/ul\u003e",
	"authors": [
	"Roelof Oosterhuis"
	],
	"date": "2007-08-12",
	- "id": 722,
	+ "id": 725,
	"link": "/entries/Fermat3_4.html",
	"permalink": "/entries/Fermat3_4.html",
	"shortname": "Fermat3_4",
	"title": "Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This document presents the mechanised proofs of the following results:\u003cul\u003e\u003cli\u003eany prime number of the form 4m+1 can be written as the sum of two squares;\u003c/li\u003e\u003cli\u003eany natural number can be written as the sum of four squares\u003c/li\u003e\u003c/ul\u003e",
	"authors": [
	"Roelof Oosterhuis"
	],
	"date": "2007-08-12",
	- "id": 723,
	+ "id": 726,
	"link": "/entries/SumSquares.html",
	"permalink": "/entries/SumSquares.html",
	"shortname": "SumSquares",
	"title": "Sums of Two and Four Squares",
	"topic_links": [
	"mathematics/number-theory"
	],
	"topics": [
	"Mathematics/Number theory"
	],
	"used_by": 0
	},
	{
	"abstract": "Convergence with respect to a valuation is discussed as convergence of a Cauchy sequence. Cauchy sequences of polynomials are defined. They are used to formalize Hensel's lemma.",
	"authors": [
	"Hidetsune Kobayashi"
	],
	"date": "2007-08-08",
	- "id": 724,
	+ "id": 727,
	"link": "/entries/Valuation.html",
	"permalink": "/entries/Valuation.html",
	"shortname": "Valuation",
	"title": "Fundamental Properties of Valuation Theory and Hensel's Lemma",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 0
	},
	{
	"abstract": "We present a formalization of parts of Melvin Fitting's book \"First-Order Logic and Automated Theorem Proving\". The formalization covers the syntax of first-order logic, its semantics, the model existence theorem, a natural deduction proof calculus together with a proof of correctness and completeness, as well as the Löwenheim-Skolem theorem.",
	"authors": [
	"Stefan Berghofer"
	],
	"date": "2007-08-02",
	- "id": 725,
	+ "id": 728,
	"link": "/entries/FOL-Fitting.html",
	"permalink": "/entries/FOL-Fitting.html",
	"shortname": "FOL-Fitting",
	"title": "First-Order Logic According to Fitting",
	"topic_links": [
	"logic/general-logic/classical-first-order-logic"
	],
	"topics": [
	"Logic/General logic/Classical first-order logic"
	],
	"used_by": 2
	},
	{
	"abstract": "We present a solution to the POPLmark challenge designed by Aydemir et al., which has as a goal the formalization of the meta-theory of System F\u003csub\u003e\u0026lt;:\u003c/sub\u003e. The formalization is carried out in the theorem prover Isabelle/HOL using an encoding based on de Bruijn indices. We start with a relatively simple formalization covering only the basic features of System F\u003csub\u003e\u0026lt;:\u003c/sub\u003e, and explain how it can be extended to also cover records and more advanced binding constructs.",
	"authors": [
	"Stefan Berghofer"
	],
	"date": "2007-08-02",
	- "id": 726,
	+ "id": 729,
	"link": "/entries/POPLmark-deBruijn.html",
	"permalink": "/entries/POPLmark-deBruijn.html",
	"shortname": "POPLmark-deBruijn",
	"title": "POPLmark Challenge Via de Bruijn Indices",
	"topic_links": [
	"computer-science/programming-languages/lambda-calculi"
	],
	"topics": [
	"Computer science/Programming languages/Lambda calculi"
	],
	"used_by": 0
	},
	{
	"abstract": "Two models of an electronic hotel key card system are contrasted: a state based and a trace based one. Both are defined, verified, and proved equivalent in the theorem prover Isabelle/HOL. It is shown that if a guest follows a certain safety policy regarding her key cards, she can be sure that nobody but her can enter her room.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2006-09-09",
	- "id": 727,
	+ "id": 730,
	"link": "/entries/HotelKeyCards.html",
	"permalink": "/entries/HotelKeyCards.html",
	"shortname": "HotelKeyCards",
	"title": "Hotel Key Card System",
	"topic_links": [
	"computer-science/security"
	],
	"topics": [
	"Computer science/Security"
	],
	"used_by": 0
	},
	{
	"abstract": "These therories describe Hoare logics for a number of imperative language constructs, from while-loops to mutually recursive procedures. Both partial and total correctness are treated. In particular a proof system for total correctness of recursive procedures in the presence of unbounded nondeterminism is presented.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2006-08-08",
	- "id": 728,
	+ "id": 731,
	"link": "/entries/Abstract-Hoare-Logics.html",
	"permalink": "/entries/Abstract-Hoare-Logics.html",
	"shortname": "Abstract-Hoare-Logics",
	"title": "Abstract Hoare Logics",
	"topic_links": [
	"computer-science/programming-languages/logics"
	],
	"topics": [
	"Computer science/Programming languages/Logics"
	],
	"used_by": 0
	},
	{
	"abstract": "These theories present the verified enumeration of \u003ci\u003etame\u003c/i\u003e plane graphs as defined by Thomas C. Hales in his proof of the Kepler Conjecture in his book \u003ci\u003eDense Sphere Packings. A Blueprint for Formal Proofs.\u003c/i\u003e [CUP 2012]. The values of the constants in the definition of tameness are identical to those in the \u003ca href=\"https://code.google.com/p/flyspeck/\"\u003eFlyspeck project\u003c/a\u003e. The \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs/Flyspeck/\"\u003eIJCAR 2006 paper by Nipkow, Bauer and Schultz\u003c/a\u003e refers to the original version of Hales' proof, the \u003ca href=\"http://www21.in.tum.de/~nipkow/pubs/itp11.html\"\u003eITP 2011 paper by Nipkow\u003c/a\u003e refers to the Blueprint version of the proof.",
	"authors": [
	"Gertrud Bauer",
	"Tobias Nipkow"
	],
	"date": "2006-05-22",
	- "id": 729,
	+ "id": 732,
	"link": "/entries/Flyspeck-Tame.html",
	"permalink": "/entries/Flyspeck-Tame.html",
	"shortname": "Flyspeck-Tame",
	"title": "Flyspeck I: Tame Graphs",
	"topic_links": [
	"mathematics/graph-theory"
	],
	"topics": [
	"Mathematics/Graph theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We present an operational semantics and type safety proof for multiple inheritance in C++. The semantics models the behavior of method calls, field accesses, and two forms of casts in C++ class hierarchies. For explanations see the OOPSLA 2006 paper by Wasserrab, Nipkow, Snelting and Tip.",
	"authors": [
	"Daniel Wasserrab"
	],
	"date": "2006-05-15",
	- "id": 730,
	+ "id": 733,
	"link": "/entries/CoreC++.html",
	"permalink": "/entries/CoreC++.html",
	"shortname": "CoreC++",
	"title": "CoreC++",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the type system, small-step operational semantics, and type soundness proof for Featherweight Java, a simple object calculus, in Isabelle/HOL.",
	"authors": [
	"J. Nathan Foster",
	"Dimitrios Vytiniotis"
	],
	"date": "2006-03-31",
	- "id": 731,
	+ "id": 734,
	"link": "/entries/FeatherweightJava.html",
	"permalink": "/entries/FeatherweightJava.html",
	"shortname": "FeatherweightJava",
	"title": "A Theory of Featherweight Java in Isabelle/HOL",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 0
	},
	{
	"abstract": "F. B. Schneider (\"Understanding protocols for Byzantine clock synchronization\") generalizes a number of protocols for Byzantine fault-tolerant clock synchronization and presents a uniform proof for their correctness. In Schneider's schema, each processor maintains a local clock by periodically adjusting each value to one computed by a convergence function applied to the readings of all the clocks. Then, correctness of an algorithm, i.e. that the readings of two clocks at any time are within a fixed bound of each other, is based upon some conditions on the convergence function. To prove that a particular clock synchronization algorithm is correct it suffices to show that the convergence function used by the algorithm meets Schneider's conditions. Using the theorem prover Isabelle, we formalize the proofs that the convergence functions of two algorithms, namely, the Interactive Convergence Algorithm (ICA) of Lamport and Melliar-Smith and the Fault-tolerant Midpoint algorithm of Lundelius-Lynch, meet Schneider's conditions. Furthermore, we experiment on handling some parts of the proofs with fully automatic tools like ICS and CVC-lite. These theories are part of a joint work with Alwen Tiu and Leonor P. Nieto \u003ca href=\"http://users.rsise.anu.edu.au/~tiu/clocksync.pdf\"\u003e\"Verification of Clock Synchronization Algorithms: Experiments on a combination of deductive tools\"\u003c/a\u003e in proceedings of AVOCS 2005. In this work the correctness of Schneider schema was also verified using Isabelle (entry \u003ca href=\"GenClock.html\"\u003eGenClock\u003c/a\u003e in AFP).",
	"authors": [
	"Damián Barsotti"
	],
	"date": "2006-03-15",
	- "id": 732,
	+ "id": 735,
	"link": "/entries/ClockSynchInst.html",
	"permalink": "/entries/ClockSynchInst.html",
	"shortname": "ClockSynchInst",
	"title": "Instances of Schneider's generalized protocol of clock synchronization",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "This document presents the mechanised proofs of two popular theorems attributed to Augustin Louis Cauchy - Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality.",
	"authors": [
	"Benjamin Porter"
	],
	"date": "2006-03-14",
	- "id": 733,
	+ "id": 736,
	"link": "/entries/Cauchy.html",
	"permalink": "/entries/Cauchy.html",
	"shortname": "Cauchy",
	"title": "Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 1
	},
	{
	"abstract": "This development defines a well-ordered type of countable ordinals. It includes notions of continuous and normal functions, recursively defined functions over ordinals, least fixed-points, and derivatives. Much of ordinal arithmetic is formalized, including exponentials and logarithms. The development concludes with formalizations of Cantor Normal Form and Veblen hierarchies over normal functions.",
	"authors": [
	"Brian Huffman"
	],
	"date": "2005-11-11",
	- "id": 734,
	+ "id": 737,
	"link": "/entries/Ordinal.html",
	"permalink": "/entries/Ordinal.html",
	"shortname": "Ordinal",
	"title": "Countable Ordinals",
	"topic_links": [
	"logic/set-theory"
	],
	"topics": [
	"Logic/Set theory"
	],
	"used_by": 1
	},
	{
	"abstract": "We formalise a functional implementation of the FFT algorithm over the complex numbers, and its inverse. Both are shown equivalent to the usual definitions of these operations through Vandermonde matrices. They are also shown to be inverse to each other, more precisely, that composition of the inverse and the transformation yield the identity up to a scalar.",
	"authors": [
	"Clemens Ballarin"
	],
	"date": "2005-10-12",
	- "id": 735,
	+ "id": 738,
	"link": "/entries/FFT.html",
	"permalink": "/entries/FFT.html",
	"shortname": "FFT",
	"title": "Fast Fourier Transform",
	"topic_links": [
	"computer-science/algorithms/mathematical"
	],
	"topics": [
	"Computer science/Algorithms/Mathematical"
	],
	"used_by": 0
	},
	{
	"abstract": "We formalize the generalized Byzantine fault-tolerant clock synchronization protocol of Schneider. This protocol abstracts from particular algorithms or implementations for clock synchronization. This abstraction includes several assumptions on the behaviors of physical clocks and on general properties of concrete algorithms/implementations. Based on these assumptions the correctness of the protocol is proved by Schneider. His proof was later verified by Shankar using the theorem prover EHDM (precursor to PVS). Our formalization in Isabelle/HOL is based on Shankar's formalization.",
	"authors": [
	"Alwen Tiu"
	],
	"date": "2005-06-24",
	- "id": 736,
	+ "id": 739,
	"link": "/entries/GenClock.html",
	"permalink": "/entries/GenClock.html",
	"shortname": "GenClock",
	"title": "Formalization of a Generalized Protocol for Clock Synchronization",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "Disk Paxos is an algorithm for building arbitrary fault-tolerant distributed systems. The specification of Disk Paxos has been proved correct informally and tested using the TLC model checker, but up to now, it has never been fully formally verified. In this work we have formally verified its correctness using the Isabelle theorem prover and the HOL logic system, showing that Isabelle is a practical tool for verifying properties of TLA+ specifications.",
	"authors": [
	"Mauro Jaskelioff",
	"Stephan Merz"
	],
	"date": "2005-06-22",
	- "id": 737,
	+ "id": 740,
	"link": "/entries/DiskPaxos.html",
	"permalink": "/entries/DiskPaxos.html",
	"shortname": "DiskPaxos",
	"title": "Proving the Correctness of Disk Paxos",
	"topic_links": [
	"computer-science/algorithms/distributed"
	],
	"topics": [
	"Computer science/Algorithms/Distributed"
	],
	"used_by": 0
	},
	{
	"abstract": "This document presents the formalization of an object-oriented data and store model in Isabelle/HOL. This model is being used in the Java Interactive Verification Environment, Jive.",
	"authors": [
	"Nicole Rauch",
	"Norbert Schirmer"
	],
	"date": "2005-06-20",
	- "id": 738,
	+ "id": 741,
	"link": "/entries/JiveDataStoreModel.html",
	"permalink": "/entries/JiveDataStoreModel.html",
	"shortname": "JiveDataStoreModel",
	"title": "Jive Data and Store Model",
	"topic_links": [
	"computer-science/programming-languages/misc"
	],
	"topics": [
	"Computer science/Programming languages/Misc"
	],
	"used_by": 0
	},
	{
	"abstract": "We introduce Jinja, a Java-like programming language with a formal semantics designed to exhibit core features of the Java language architecture. Jinja is a compromise between realism of the language and tractability and clarity of the formal semantics. The following aspects are formalised: a big and a small step operational semantics for Jinja and a proof of their equivalence; a type system and a definite initialisation analysis; a type safety proof of the small step semantics; a virtual machine (JVM), its operational semantics and its type system; a type safety proof for the JVM; a bytecode verifier, i.e. data flow analyser for the JVM; a correctness proof of the bytecode verifier w.r.t. the type system; a compiler and a proof that it preserves semantics and well-typedness. The emphasis of this work is not on particular language features but on providing a unified model of the source language, the virtual machine and the compiler. The whole development has been carried out in the theorem prover Isabelle/HOL.",
	"authors": [
	"Gerwin Klein",
	"Tobias Nipkow"
	],
	"date": "2005-06-01",
	- "id": 739,
	+ "id": 742,
	"link": "/entries/Jinja.html",
	"permalink": "/entries/Jinja.html",
	"shortname": "Jinja",
	"title": "Jinja is not Java",
	"topic_links": [
	"computer-science/programming-languages/language-definitions"
	],
	"topics": [
	"Computer science/Programming languages/Language definitions"
	],
	"used_by": 4
	},
	{
	"abstract": "Formal verification is getting more and more important in computer science. However the state of the art formal verification methods in cryptography are very rudimentary. These theories are one step to provide a tool box allowing the use of formal methods in every aspect of cryptography. Moreover we present a proof of concept for the feasibility of verification techniques to a standard signature algorithm.",
	"authors": [
	"Christina Lindenberg",
	"Kai Wirt"
	],
	"date": "2005-05-02",
	- "id": 740,
	+ "id": 743,
	"link": "/entries/RSAPSS.html",
	"permalink": "/entries/RSAPSS.html",
	"shortname": "RSAPSS",
	"title": "SHA1, RSA, PSS and more",
	"topic_links": [
	"computer-science/security/cryptography"
	],
	"topics": [
	"Computer science/Security/Cryptography"
	],
	"used_by": 0
	},
	{
	"abstract": "This development proves Yoneda's lemma and aims to be readable by humans. It only defines what is needed for the lemma: categories, functors and natural transformations. Limits, adjunctions and other important concepts are not included.",
	"authors": [
	"Greg O'Keefe"
	],
	"date": "2005-04-21",
	- "id": 741,
	+ "id": 744,
	"link": "/entries/Category.html",
	"permalink": "/entries/Category.html",
	"shortname": "Category",
	"title": "Category Theory to Yoneda's Lemma",
	"topic_links": [
	"mathematics/category-theory"
	],
	"topics": [
	"Mathematics/Category theory"
	],
	"used_by": 0
	},
	{
	"abstract": "These theories illustrates the verification of basic file operations (file creation, file read and file write) in the Isabelle theorem prover. We describe a file at two levels of abstraction: an abstract file represented as a resizable array, and a concrete file represented using data blocks.",
	"authors": [
	"Karen Zee",
	"Viktor Kuncak"
	],
	"date": "2004-12-09",
	- "id": 742,
	+ "id": 745,
	"link": "/entries/FileRefinement.html",
	"permalink": "/entries/FileRefinement.html",
	"shortname": "FileRefinement",
	"title": "File Refinement",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "Lebesgue-style integration plays a major role in advanced probability. We formalize concepts of elementary measure theory, real-valued random variables as Borel-measurable functions, and a stepwise inductive definition of the integral itself. All proofs are carried out in human readable style using the Isar language.",
	"authors": [
	"Stefan Richter"
	],
	"date": "2004-11-19",
	- "id": 743,
	+ "id": 746,
	"link": "/entries/Integration.html",
	"permalink": "/entries/Integration.html",
	"shortname": "Integration",
	"title": "Integration theory and random variables",
	"topic_links": [
	"mathematics/analysis"
	],
	"topics": [
	"Mathematics/Analysis"
	],
	"used_by": 0
	},
	{
	"abstract": "Soundness and completeness for a system of first order logic are formally proved, building on James Margetson's formalization of work by Wainer and Wallen. The completeness proofs naturally suggest an algorithm to derive proofs. This algorithm, which can be implemented tail recursively, is formalized in Isabelle/HOL. The algorithm can be executed via the rewriting tactics of Isabelle. Alternatively, the definitions can be exported to OCaml, yielding a directly executable program.",
	"authors": [
	"Tom Ridge"
	],
	"date": "2004-09-28",
	- "id": 744,
	+ "id": 747,
	"link": "/entries/Verified-Prover.html",
	"permalink": "/entries/Verified-Prover.html",
	"shortname": "Verified-Prover",
	"title": "A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic",
	"topic_links": [
	"logic/general-logic/mechanization-of-proofs"
	],
	"topics": [
	"Logic/General logic/Mechanization of proofs"
	],
	"used_by": 0
	},
	{
	"abstract": "The completeness of first-order logic is proved, following the first five pages of Wainer and Wallen's chapter of the book \u003ci\u003eProof Theory\u003c/i\u003e by Aczel et al., CUP, 1992. Their presentation of formulas allows the proofs to use symmetry arguments. Margetson formalized this theorem by early 2000. The Isar conversion is thanks to Tom Ridge. A paper describing the formalization is available \u003ca href=\"Completeness-paper.pdf\"\u003e[pdf]\u003c/a\u003e.",
	"authors": [
	"James Margetson",
	"Tom Ridge"
	],
	"date": "2004-09-20",
	- "id": 745,
	+ "id": 748,
	"link": "/entries/Completeness.html",
	"permalink": "/entries/Completeness.html",
	"shortname": "Completeness",
	"title": "Completeness theorem",
	"topic_links": [
	"logic/proof-theory"
	],
	"topics": [
	"Logic/Proof theory"
	],
	"used_by": 0
	},
	{
	"abstract": "This formalization of Ramsey's theorem (infinitary version) is taken from Boolos and Jeffrey, \u003ci\u003eComputability and Logic\u003c/i\u003e, 3rd edition, Chapter 26. It differs slightly from the text by assuming a slightly stronger hypothesis. In particular, the induction hypothesis is stronger, holding for any infinite subset of the naturals. This avoids the rather peculiar mapping argument between kj and aikj on p.263, which is unnecessary and slightly mars this really beautiful result.",
	"authors": [
	"Tom Ridge"
	],
	"date": "2004-09-20",
	- "id": 746,
	+ "id": 749,
	"link": "/entries/Ramsey-Infinite.html",
	"permalink": "/entries/Ramsey-Infinite.html",
	"shortname": "Ramsey-Infinite",
	"title": "Ramsey's theorem, infinitary version",
	"topic_links": [
	"mathematics/combinatorics"
	],
	"topics": [
	"Mathematics/Combinatorics"
	],
	"used_by": 0
	},
	{
	"abstract": "An exception compilation scheme that dynamically creates and removes exception handler entries on the stack. A formalization of an article of the same name by \u003ca href=\"http://www.cs.nott.ac.uk/~gmh/\"\u003eHutton\u003c/a\u003e and Wright.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2004-07-09",
	- "id": 747,
	+ "id": 750,
	"link": "/entries/Compiling-Exceptions-Correctly.html",
	"permalink": "/entries/Compiling-Exceptions-Correctly.html",
	"shortname": "Compiling-Exceptions-Correctly",
	"title": "Compiling Exceptions Correctly",
	"topic_links": [
	"computer-science/programming-languages/compiling"
	],
	"topics": [
	"Computer science/Programming languages/Compiling"
	],
	"used_by": 0
	},
	{
	"abstract": "Depth-first search of a graph is formalized with recdef. It is shown that it visits all of the reachable nodes from a given list of nodes. Executable ML code of depth-first search is obtained using the code generation feature of Isabelle/HOL.",
	"authors": [
	"Toshiaki Nishihara",
	"Yasuhiko Minamide"
	],
	"date": "2004-06-24",
	- "id": 748,
	+ "id": 751,
	"link": "/entries/Depth-First-Search.html",
	"permalink": "/entries/Depth-First-Search.html",
	"shortname": "Depth-First-Search",
	"title": "Depth First Search",
	"topic_links": [
	"computer-science/algorithms/graph"
	],
	"topics": [
	"Computer science/Algorithms/Graph"
	],
	"used_by": 0
	},
	{
	"abstract": "The theory of groups, rings and modules is developed to a great depth. Group theory results include Zassenhaus's theorem and the Jordan-Hoelder theorem. The ring theory development includes ideals, quotient rings and the Chinese remainder theorem. The module development includes the Nakayama lemma, exact sequences and Tensor products.",
	"authors": [
	"Hidetsune Kobayashi",
	"L. Chen",
	"H. Murao"
	],
	"date": "2004-05-18",
	- "id": 749,
	+ "id": 752,
	"link": "/entries/Group-Ring-Module.html",
	"permalink": "/entries/Group-Ring-Module.html",
	"shortname": "Group-Ring-Module",
	"title": "Groups, Rings and Modules",
	"topic_links": [
	"mathematics/algebra"
	],
	"topics": [
	"Mathematics/Algebra"
	],
	"used_by": 1
	},
	{
	"abstract": "This theory contains some useful extensions to the LList (lazy list) theory by \u003ca href=\"http://www.cl.cam.ac.uk/~lp15/\"\u003eLarry Paulson\u003c/a\u003e, including finite, infinite, and positive llists over an alphabet, as well as the new constants take and drop and the prefix order of llists. Finally, the notions of safety and liveness in the sense of Alpern and Schneider (1985) are defined.",
	"authors": [
	"Stefan Friedrich"
	],
	"date": "2004-04-26",
	- "id": 750,
	+ "id": 753,
	"link": "/entries/Lazy-Lists-II.html",
	"permalink": "/entries/Lazy-Lists-II.html",
	"shortname": "Lazy-Lists-II",
	"title": "Lazy Lists II",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 1
	},
	{
	"abstract": "This entry contains two theories. The first, \u003ctt\u003eTopology\u003c/tt\u003e, develops the basic notions of general topology. The second, which can be viewed as a demonstration of the first, is called \u003ctt\u003eLList_Topology\u003c/tt\u003e. It develops the topology of lazy lists.",
	"authors": [
	"Stefan Friedrich"
	],
	"date": "2004-04-26",
	- "id": 751,
	+ "id": 754,
	"link": "/entries/Topology.html",
	"permalink": "/entries/Topology.html",
	"shortname": "Topology",
	"title": "Topology",
	"topic_links": [
	"mathematics/topology"
	],
	"topics": [
	"Mathematics/Topology"
	],
	"used_by": 0
	},
	{
	"abstract": "The correctness is shown of binary search tree operations (lookup, insert and remove) implementing a set. Two versions are given, for both structured and linear (tactic-style) proofs. An implementation of integer-indexed maps is also verified.",
	"authors": [
	"Viktor Kuncak"
	],
	"date": "2004-04-05",
	- "id": 752,
	+ "id": 755,
	"link": "/entries/BinarySearchTree.html",
	"permalink": "/entries/BinarySearchTree.html",
	"shortname": "BinarySearchTree",
	"title": "Binary Search Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory defines deterministic and nondeterministic automata in a functional representation: the transition function/relation and the finality predicate are just functions. Hence the state space may be infinite. It is shown how to convert regular expressions into such automata. A scanner (generator) is implemented with the help of functional automata: the scanner chops the input up into longest recognized substrings. Finally we also show how to convert a certain subclass of functional automata (essentially the finite deterministic ones) into regular sets.",
	"authors": [
	"Tobias Nipkow"
	],
	"date": "2004-03-30",
	- "id": 753,
	+ "id": 756,
	"link": "/entries/Functional-Automata.html",
	"permalink": "/entries/Functional-Automata.html",
	"shortname": "Functional-Automata",
	"title": "Functional Automata",
	"topic_links": [
	"computer-science/automata-and-formal-languages"
	],
	"topics": [
	"Computer science/Automata and formal languages"
	],
	"used_by": 0
	},
	{
	"abstract": "Two formalizations of AVL trees with room for extensions. The first formalization is monolithic and shorter, the second one in two stages, longer and a bit simpler. The final implementation is the same. If you are interested in developing this further, please contact \u003ctt\u003egerwin.klein@nicta.com.au\u003c/tt\u003e.",
	"authors": [
	"Tobias Nipkow",
	"Cornelia Pusch"
	],
	"date": "2004-03-19",
	- "id": 754,
	+ "id": 757,
	"link": "/entries/AVL-Trees.html",
	"permalink": "/entries/AVL-Trees.html",
	"shortname": "AVL-Trees",
	"title": "AVL Trees",
	"topic_links": [
	"computer-science/data-structures"
	],
	"topics": [
	"Computer science/Data structures"
	],
	"used_by": 0
	},
	{
	"abstract": "This theory defines the type inference rules and the type inference algorithm \u003ci\u003eW\u003c/i\u003e for MiniML (simply-typed lambda terms with \u003ctt\u003elet\u003c/tt\u003e) due to Milner. It proves the soundness and completeness of \u003ci\u003eW\u003c/i\u003e w.r.t. the rules.",
	"authors": [
	"Wolfgang Naraschewski",
	"Tobias Nipkow"
	],
	"date": "2004-03-19",
	- "id": 755,
	+ "id": 758,
	"link": "/entries/MiniML.html",
	"permalink": "/entries/MiniML.html",
	"shortname": "MiniML",
	"title": "Mini ML",
	"topic_links": [
	"computer-science/programming-languages/type-systems"
	],
	"topics": [
	"Computer science/Programming languages/Type systems"
	],
	"used_by": 0
	}
	]
	\ No newline at end of file
	diff --git a/web/index.xml b/web/index.xml
	--- a/web/index.xml
	+++ b/web/index.xml
	@@ -1,13844 +1,13898 @@
	<?xml version="1.0" encoding="utf-8" standalone="yes"?>
	<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
	<channel>
	<title>Archive of Formal Proofs</title>
	<link>/</link>
	<description>Recent content on Archive of Formal Proofs</description>
	<generator>Hugo -- gohugo.io</generator>
	<language>en-gb</language>
	- <lastBuildDate>Tue, 11 Jul 2023 00:00:00 +0000</lastBuildDate><atom:link href="/index.xml" rel="self" type="application/rss+xml" />
	+ <lastBuildDate>Thu, 10 Aug 2023 00:00:00 +0000</lastBuildDate><atom:link href="/index.xml" rel="self" type="application/rss+xml" />
	+ <item>
	+ <title>Polygonal Number Theorem</title>
	+ <link>/entries/Polygonal_Number_Theorem.html</link>
	+ <pubDate>Thu, 10 Aug 2023 00:00:00 +0000</pubDate>
	+
	+ <guid>/entries/Polygonal_Number_Theorem.html</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	+ <title>Modal quantales, involutive quantales, Dedekind Quantales</title>
	+ <link>/entries/Quantales_Converse.html</link>
	+ <pubDate>Tue, 25 Jul 2023 00:00:00 +0000</pubDate>
	+
	+ <guid>/entries/Quantales_Converse.html</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	+ <title>Earley Parser</title>
	+ <link>/entries/Earley_Parser.html</link>
	+ <pubDate>Sun, 16 Jul 2023 00:00:00 +0000</pubDate>
	+
	+ <guid>/entries/Earley_Parser.html</guid>
	+ <description></description>
	+ </item>
	+
	<item>
	<title>Gray Codes for Arbitrary Numeral Systems</title>
	<link>/entries/Gray_Codes.html</link>
	<pubDate>Tue, 11 Jul 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Gray_Codes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Randomized Algorithms</title>
	<link>/entries/Executable_Randomized_Algorithms.html</link>
	<pubDate>Mon, 19 Jun 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Executable_Randomized_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>DCR Syntax and Execution Equivalent Markings</title>
	<link>/entries/DCR-ExecutionEquivalence.html</link>
	<pubDate>Fri, 16 Jun 2023 00:00:00 +0000</pubDate>

	<guid>/entries/DCR-ExecutionEquivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Zeckendorf’s Theorem</title>
	<link>/entries/Zeckendorf.html</link>
	<pubDate>Mon, 12 Jun 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Zeckendorf.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cryptographic Standards</title>
	<link>/entries/Crypto_Standards.html</link>
	<pubDate>Tue, 06 Jun 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Crypto_Standards.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Efficient Implementation of the Weighted Path Order</title>
	<link>/entries/Efficient_Weighted_Path_Order.html</link>
	<pubDate>Thu, 01 Jun 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Efficient_Weighted_Path_Order.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Results on Directed Sets</title>
	<link>/entries/Directed_Sets.html</link>
	<pubDate>Wed, 24 May 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Directed_Sets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inner Structure, Determinism and Modal Algebra of Multirelations</title>
	<link>/entries/Multirelations_Heterogeneous.html</link>
	<pubDate>Mon, 22 May 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Multirelations_Heterogeneous.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Enumeration</title>
	<link>/entries/Tree_Enumeration.html</link>
	<pubDate>Tue, 09 May 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Tree_Enumeration.html</guid>
	<description></description>
	</item>

	<item>
	<title>MLSS Decision Procedure</title>
	<link>/entries/MLSS_Decision_Proc.html</link>
	<pubDate>Fri, 05 May 2023 00:00:00 +0000</pubDate>

	<guid>/entries/MLSS_Decision_Proc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Three Squares Theorem</title>
	<link>/entries/Three_Squares.html</link>
	<pubDate>Wed, 03 May 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Three_Squares.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Halting Problem is Soluble in Malament-Hogarth Spacetimes</title>
	<link>/entries/MHComputation.html</link>
	<pubDate>Sat, 29 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/MHComputation.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Schwartz-Zippel Lemma</title>
	<link>/entries/Schwartz_Zippel.html</link>
	<pubDate>Thu, 27 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Schwartz_Zippel.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of the SCL(FOL) Calculus: Simple Clause Learning for First-Order Logic</title>
	<link>/entries/Simple_Clause_Learning.html</link>
	<pubDate>Thu, 20 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Simple_Clause_Learning.html</guid>
	<description></description>
	</item>

	<item>
	<title>The CHSH inequality: Tsirelson's upper-bound and other results</title>
	<link>/entries/TsirelsonBound.html</link>
	<pubDate>Tue, 18 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/TsirelsonBound.html</guid>
	<description></description>
	</item>

	<item>
	<title>Distributed Distinct Elements</title>
	<link>/entries/Distributed_Distinct_Elements.html</link>
	<pubDate>Mon, 03 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Distributed_Distinct_Elements.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Hyper Hoare Logic: A Logic to (Dis-)Prove Program Hyperproperties</title>
	<link>/entries/HyperHoareLogic.html</link>
	<pubDate>Mon, 03 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/HyperHoareLogic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Positional Notation for Natural Numbers in an Arbitrary Base</title>
	<link>/entries/DigitsInBase.html</link>
	<pubDate>Mon, 03 Apr 2023 00:00:00 +0000</pubDate>

	<guid>/entries/DigitsInBase.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of CommCSL: A Relational Concurrent Separation Logic for Proving Information Flow Security in Concurrent Programs</title>
	<link>/entries/CommCSL.html</link>
	<pubDate>Wed, 15 Mar 2023 00:00:00 +0000</pubDate>

	<guid>/entries/CommCSL.html</guid>
	<description></description>
	</item>

	<item>
	<title>No Faster-Than-Light Observers (GenRel)</title>
	<link>/entries/No_FTL_observers_Gen_Rel.html</link>
	<pubDate>Sun, 05 Mar 2023 00:00:00 +0000</pubDate>

	<guid>/entries/No_FTL_observers_Gen_Rel.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expander Graphs</title>
	<link>/entries/Expander_Graphs.html</link>
	<pubDate>Fri, 03 Mar 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Expander_Graphs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Renaming-Enriched Sets (Rensets) and Renaming-Based Recursion</title>
	<link>/entries/Rensets.html</link>
	<pubDate>Tue, 28 Feb 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Rensets.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sound and Complete Calculus for Probability Inequalities</title>
	<link>/entries/Probability_Inequality_Completeness.html</link>
	<pubDate>Mon, 20 Feb 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Probability_Inequality_Completeness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Group Law of Edwards Elliptic Curves</title>
	<link>/entries/Edwards_Elliptic_Curves_Group.html</link>
	<pubDate>Thu, 16 Feb 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Edwards_Elliptic_Curves_Group.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hardness of Lattice Problems</title>
	<link>/entries/CVP_Hardness.html</link>
	<pubDate>Thu, 02 Feb 2023 00:00:00 +0000</pubDate>

	<guid>/entries/CVP_Hardness.html</guid>
	<description></description>
	</item>

	<item>
	<title>ABY3 Multiplication and Array Shuffling</title>
	<link>/entries/ABY3_Protocols.html</link>
	<pubDate>Fri, 27 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/ABY3_Protocols.html</guid>
	<description></description>
	</item>

	<item>
	<title>Given Clause Loops</title>
	<link>/entries/Given_Clause_Loops.html</link>
	<pubDate>Wed, 25 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Given_Clause_Loops.html</guid>
	<description></description>
	</item>

	<item>
	<title>Suppes' Theorem For Probability Logic</title>
	<link>/entries/Suppes_Theorem.html</link>
	<pubDate>Sun, 22 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Suppes_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Hoare Logic for Diverging Programs</title>
	<link>/entries/HoareForDivergence.html</link>
	<pubDate>Fri, 20 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/HoareForDivergence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strict Omega Categories</title>
	<link>/entries/StrictOmegaCategories.html</link>
	<pubDate>Sat, 14 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/StrictOmegaCategories.html</guid>
	<description></description>
	</item>

	<item>
	<title>Synthetic Completeness</title>
	<link>/entries/Synthetic_Completeness.html</link>
	<pubDate>Mon, 09 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Synthetic_Completeness.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cook-Levin theorem</title>
	<link>/entries/Cook_Levin.html</link>
	<pubDate>Sun, 08 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Cook_Levin.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary codes that do not preserve primitivity</title>
	<link>/entries/Binary_Code_Imprimitive.html</link>
	<pubDate>Tue, 03 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Binary_Code_Imprimitive.html</guid>
	<description></description>
	</item>

	<item>
	<title>Intersection of two monoids generated by two element codes</title>
	<link>/entries/Two_Generated_Word_Monoids_Intersection.html</link>
	<pubDate>Tue, 03 Jan 2023 00:00:00 +0000</pubDate>

	<guid>/entries/Two_Generated_Word_Monoids_Intersection.html</guid>
	<description></description>
	</item>

	<item>
	<title>A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL</title>
	<link>/entries/Quantifier_Elimination_Hybrid.html</link>
	<pubDate>Thu, 15 Dec 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Quantifier_Elimination_Hybrid.html</guid>
	<description></description>
	</item>

	<item>
	<title>Class-based Classical Propositional Logic</title>
	<link>/entries/Propositional_Logic_Class.html</link>
	<pubDate>Thu, 15 Dec 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Propositional_Logic_Class.html</guid>
	<description></description>
	</item>

	<item>
	<title>Birkhoff's Representation Theorem For Finite Distributive Lattices</title>
	<link>/entries/Birkhoff_Finite_Distributive_Lattices.html</link>
	<pubDate>Tue, 06 Dec 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Birkhoff_Finite_Distributive_Lattices.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automation of Boolos' Curious Inference in Isabelle/HOL</title>
	<link>/entries/Boolos_Curious_Inference_Automated.html</link>
	<pubDate>Mon, 05 Dec 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Boolos_Curious_Inference_Automated.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Translation of Multitape Turing Machines into Singletape Turing Machines</title>
	<link>/entries/Multitape_To_Singletape_TM.html</link>
	<pubDate>Wed, 30 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Multitape_To_Singletape_TM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Object Theory</title>
	<link>/entries/AOT.html</link>
	<pubDate>Mon, 28 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/AOT.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal CHERI-C Memory Model</title>
	<link>/entries/CHERI-C_Memory_Model.html</link>
	<pubDate>Fri, 25 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/CHERI-C_Memory_Model.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sauer-Shelah Lemma</title>
	<link>/entries/Sauer_Shelah_Lemma.html</link>
	<pubDate>Thu, 24 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Sauer_Shelah_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kneser's Theorem and the Cauchy–Davenport Theorem</title>
	<link>/entries/Kneser_Cauchy_Davenport.html</link>
	<pubDate>Mon, 21 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Kneser_Cauchy_Davenport.html</guid>
	<description></description>
	</item>

	<item>
	<title>Turán's Graph Theorem</title>
	<link>/entries/Turans_Graph_Theorem.html</link>
	<pubDate>Mon, 14 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Turans_Graph_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial Enumeration Algorithms</title>
	<link>/entries/Combinatorial_Enumeration_Algorithms.html</link>
	<pubDate>Fri, 11 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorial_Enumeration_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Balog–Szemerédi–Gowers Theorem</title>
	<link>/entries/Balog_Szemeredi_Gowers.html</link>
	<pubDate>Fri, 11 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Balog_Szemeredi_Gowers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of Strategy-Proofness and Representation in Party-Approval Multi-Winner Elections</title>
	<link>/entries/PAPP_Impossibility.html</link>
	<pubDate>Thu, 10 Nov 2022 00:00:00 +0000</pubDate>

	<guid>/entries/PAPP_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of Query Optimization Algorithms</title>
	<link>/entries/Query_Optimization.html</link>
	<pubDate>Tue, 04 Oct 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Query_Optimization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum Segment Sum</title>
	<link>/entries/Maximum_Segment_Sum.html</link>
	<pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Maximum_Segment_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Undirected Graph Theory</title>
	<link>/entries/Undirected_Graph_Theory.html</link>
	<pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Undirected_Graph_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Making Arbitrary Relational Calculus Queries Safe-Range</title>
	<link>/entries/Safe_Range_RC.html</link>
	<pubDate>Wed, 28 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Range_RC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stalnaker's Epistemic Logic</title>
	<link>/entries/Stalnaker_Logic.html</link>
	<pubDate>Fri, 23 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Stalnaker_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>p-adic Fields and p-adic Semialgebraic Sets</title>
	<link>/entries/Padic_Field.html</link>
	<pubDate>Thu, 22 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Padic_Field.html</guid>
	<description></description>
	</item>

	<item>
	<title>Risk-Free Lending</title>
	<link>/entries/Risk_Free_Lending.html</link>
	<pubDate>Sun, 18 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Risk_Free_Lending.html</guid>
	<description></description>
	</item>

	<item>
	<title>Soundness and Completeness of Implicational Logic</title>
	<link>/entries/Implicational_Logic.html</link>
	<pubDate>Tue, 13 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Implicational_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>CRYSTALS-Kyber</title>
	<link>/entries/CRYSTALS-Kyber.html</link>
	<pubDate>Thu, 08 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/CRYSTALS-Kyber.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unbounded Separation Logic</title>
	<link>/entries/Separation_Logic_Unbounded.html</link>
	<pubDate>Mon, 05 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Unbounded.html</guid>
	<description></description>
	</item>

	<item>
	<title>Khovanskii&#x27;s Theorem</title>
	<link>/entries/Khovanskii_Theorem.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Khovanskii_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hales–Jewett Theorem</title>
	<link>/entries/Hales_Jewett.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Hales_Jewett.html</guid>
	<description></description>
	</item>

	<item>
	<title>Number Theoretic Transform</title>
	<link>/entries/Number_Theoretic_Transform.html</link>
	<pubDate>Thu, 18 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Number_Theoretic_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Correctness of a Set-based Algorithm for Computing Strongly Connected Components of a Graph</title>
	<link>/entries/SCC_Bloemen_Sequential.html</link>
	<pubDate>Wed, 17 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/SCC_Bloemen_Sequential.html</guid>
	<description></description>
	</item>

	<item>
	<title>From THE BOOK: Two Squares via Involutions</title>
	<link>/entries/Involutions2Squares.html</link>
	<pubDate>Mon, 15 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Involutions2Squares.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Complete Test Strategies for Finite State Machines</title>
	<link>/entries/FSM_Tests.html</link>
	<pubDate>Tue, 09 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FSM_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Nano JSON: Working with JSON formatted data in Isabelle/HOL and Isabelle/ML</title>
	<link>/entries/Nano_JSON.html</link>
	<pubDate>Fri, 29 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Nano_JSON.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL</title>
	<link>/entries/Solidity.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Solidity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simultaneous diagonalization of pairwise commuting Hermitian matrices</title>
	<link>/entries/Commuting_Hermitian.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Commuting_Hermitian.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pólya’s Proof of the Weighted Arithmetic–Geometric Mean Inequality</title>
	<link>/entries/Weighted_Arithmetic_Geometric_Mean.html</link>
	<pubDate>Mon, 11 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Weighted_Arithmetic_Geometric_Mean.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reuse-Based Multi-Stage Compiler Verification for Language IMP</title>
	<link>/entries/IMP_Compiler_Reuse.html</link>
	<pubDate>Sun, 10 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler_Reuse.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Time Double-Ended Queue</title>
	<link>/entries/Real_Time_Deque.html</link>
	<pubDate>Thu, 23 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Time_Deque.html</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos's Curious Inference in Isabelle/HOL</title>
	<link>/entries/Boolos_Curious_Inference.html</link>
	<pubDate>Mon, 20 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Boolos_Curious_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Fields</title>
	<link>/entries/Finite_Fields.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Fields.html</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet: Formalization of a Verification Framework for Secure Data Plane Protocols</title>
	<link>/entries/IsaNet.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IsaNet.html</guid>
	<description></description>
	</item>

	<item>
	<title>Diophantine Equations and the DPRM Theorem</title>
	<link>/entries/DPRM_Theorem.html</link>
	<pubDate>Mon, 06 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/DPRM_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Reducing Rewrite Properties to Properties on Ground Terms</title>
	<link>/entries/Rewrite_Properties_Reduction.html</link>
	<pubDate>Thu, 02 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Rewrite_Properties_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Restricted Definition of the Magic Wand to Soundly Combine Fractions of a Wand</title>
	<link>/entries/Combinable_Wands.html</link>
	<pubDate>Mon, 30 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Combinable_Wands.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Plünnecke-Ruzsa Inequality</title>
	<link>/entries/Pluennecke_Ruzsa_Inequality.html</link>
	<pubDate>Thu, 26 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Pluennecke_Ruzsa_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Framework for the Sound Automation of Magic Wands</title>
	<link>/entries/Package_logic.html</link>
	<pubDate>Wed, 18 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Package_logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clique is not solvable by monotone circuits of polynomial size</title>
	<link>/entries/Clique_and_Monotone_Circuits.html</link>
	<pubDate>Sun, 08 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Clique_and_Monotone_Circuits.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</title>
	<link>/entries/Fishers_Inequality.html</link>
	<pubDate>Thu, 21 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Fishers_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Digit Expansions</title>
	<link>/entries/Digit_Expansions.html</link>
	<pubDate>Wed, 20 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Digit_Expansions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generalized Multiset Ordering is NP-Complete</title>
	<link>/entries/Multiset_Ordering_NPC.html</link>
	<pubDate>Wed, 20 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Multiset_Ordering_NPC.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sophomore's Dream</title>
	<link>/entries/Sophomores_Dream.html</link>
	<pubDate>Sun, 10 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Sophomores_Dream.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Combinator Library for Prefix-Free Codes</title>
	<link>/entries/Prefix_Free_Code_Combinators.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Prefix_Free_Code_Combinators.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Randomized Approximation Algorithms for Frequency Moments</title>
	<link>/entries/Frequency_Moments.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Frequency_Moments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructing the Reals as Dedekind Cuts of Rationals</title>
	<link>/entries/Dedekind_Real.html</link>
	<pubDate>Thu, 24 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Dedekind_Real.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ackermann's Function Is Not Primitive Recursive</title>
	<link>/entries/Ackermanns_not_PR.html</link>
	<pubDate>Wed, 23 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Ackermanns_not_PR.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Naive Prover for First-Order Logic</title>
	<link>/entries/FOL_Seq_Calc3.html</link>
	<pubDate>Tue, 22 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc3.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent</title>
	<link>/entries/Cotangent_PFD_Formula.html</link>
	<pubDate>Tue, 15 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Cotangent_PFD_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Independence of the Continuum Hypothesis in Isabelle/ZF</title>
	<link>/entries/Independence_CH.html</link>
	<pubDate>Sun, 06 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Independence_CH.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive Models of Fragments of ZFC</title>
	<link>/entries/Transitive_Models.html</link>
	<pubDate>Thu, 03 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Transition Systems</title>
	<link>/entries/ResiduatedTransitionSystem.html</link>
	<pubDate>Mon, 28 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/ResiduatedTransitionSystem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Hash Families</title>
	<link>/entries/Universal_Hash_Families.html</link>
	<pubDate>Sun, 20 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Hash_Families.html</guid>
	<description></description>
	</item>

	<item>
	<title>Wetzel's Problem and the Continuum Hypothesis</title>
	<link>/entries/Wetzels_Problem.html</link>
	<pubDate>Fri, 18 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Wetzels_Problem.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Query Evaluation</title>
	<link>/entries/Eval_FO.html</link>
	<pubDate>Tue, 15 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Eval_FO.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Head Monitoring of Metric Dynamic Logic</title>
	<link>/entries/VYDRA_MDL.html</link>
	<pubDate>Sun, 13 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/VYDRA_MDL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Enumeration of Equivalence Relations</title>
	<link>/entries/Equivalence_Relation_Enumeration.html</link>
	<pubDate>Fri, 04 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Equivalence_Relation_Enumeration.html</guid>
	<description></description>
	</item>

	<item>
	<title>Duality of Linear Programming</title>
	<link>/entries/LP_Duality.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/LP_Duality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi-Borel Spaces</title>
	<link>/entries/Quasi_Borel_Spaces.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Quasi_Borel_Spaces.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Theory of Rewriting</title>
	<link>/entries/FO_Theory_Rewriting.html</link>
	<pubDate>Wed, 02 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FO_Theory_Rewriting.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sequent Calculus Prover for First-Order Logic with Functions</title>
	<link>/entries/FOL_Seq_Calc2.html</link>
	<pubDate>Mon, 31 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc2.html</guid>
	<description></description>
	</item>

	<item>
	<title>Young's Inequality for Increasing Functions</title>
	<link>/entries/Youngs_Inequality.html</link>
	<pubDate>Mon, 31 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Youngs_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interpolation Polynomials (in HOL-Algebra)</title>
	<link>/entries/Interpolation_Polynomials_HOL_Algebra.html</link>
	<pubDate>Sat, 29 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Interpolation_Polynomials_HOL_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Median Method</title>
	<link>/entries/Median_Method.html</link>
	<pubDate>Tue, 25 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Actuarial Mathematics</title>
	<link>/entries/Actuarial_Mathematics.html</link>
	<pubDate>Sun, 23 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Actuarial_Mathematics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational numbers from THE BOOK</title>
	<link>/entries/Irrationals_From_THEBOOK.html</link>
	<pubDate>Sat, 08 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationals_From_THEBOOK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knight's Tour Revisited Revisited</title>
	<link>/entries/Knights_Tour.html</link>
	<pubDate>Tue, 04 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Knights_Tour.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual Numbers and Forward Differentiation</title>
	<link>/entries/Hyperdual.html</link>
	<pubDate>Fri, 31 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hyperdual.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Shapley Algorithm</title>
	<link>/entries/Gale_Shapley.html</link>
	<pubDate>Wed, 29 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Gale_Shapley.html</guid>
	<description></description>
	</item>

	<item>
	<title>Roth's Theorem on Arithmetic Progressions</title>
	<link>/entries/Roth_Arithmetic_Progressions.html</link>
	<pubDate>Tue, 28 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Roth_Arithmetic_Progressions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Decision Processes with Rewards</title>
	<link>/entries/MDP-Rewards.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Rewards.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Algorithms for Solving Markov Decision Processes</title>
	<link>/entries/MDP-Algorithms.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Tree Relations</title>
	<link>/entries/Regular_Tree_Relations.html</link>
	<pubDate>Wed, 15 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Tree_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simplicial Complexes and Boolean functions</title>
	<link>/entries/Simplicial_complexes_and_boolean_functions.html</link>
	<pubDate>Mon, 29 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Simplicial_complexes_and_boolean_functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>van Emde Boas Trees</title>
	<link>/entries/Van_Emde_Boas_Trees.html</link>
	<pubDate>Tue, 23 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_Emde_Boas_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Foundation of geometry in planes, and some complements: Excluding the parallel axioms</title>
	<link>/entries/Foundation_of_geometry.html</link>
	<pubDate>Mon, 22 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Foundation_of_geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hahn and Jordan Decomposition Theorems</title>
	<link>/entries/Hahn_Jordan_Decomposition.html</link>
	<pubDate>Fri, 19 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hahn_Jordan_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automating Public Announcement Logic and the Wise Men Puzzle in Isabelle/HOL</title>
	<link>/entries/PAL.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/PAL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Exploring Simplified Variants of Gödel’s Ontological Argument in Isabelle/HOL</title>
	<link>/entries/SimplifiedOntologicalArgument.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/SimplifiedOntologicalArgument.html</guid>
	<description></description>
	</item>

	<item>
	<title>Factorization of Polynomials with Algebraic Coefficients</title>
	<link>/entries/Factor_Algebraic_Polynomial.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Factor_Algebraic_Polynomial.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real Exponents as the Limits of Sequences of Rational Exponents</title>
	<link>/entries/Real_Power.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Power.html</guid>
	<description></description>
	</item>

	<item>
	<title>Szemerédi's Regularity Lemma</title>
	<link>/entries/Szemeredi_Regularity.html</link>
	<pubDate>Fri, 05 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Szemeredi_Regularity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum and Classical Registers</title>
	<link>/entries/Registers.html</link>
	<pubDate>Thu, 28 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Registers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Belief Revision Theory</title>
	<link>/entries/Belief_Revision.html</link>
	<pubDate>Tue, 19 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Belief_Revision.html</guid>
	<description></description>
	</item>

	<item>
	<title>X86 instruction semantics and basic block symbolic execution</title>
	<link>/entries/X86_Semantics.html</link>
	<pubDate>Wed, 13 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/X86_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebras for Iteration, Infinite Executions and Correctness of Sequential Computations</title>
	<link>/entries/Correctness_Algebras.html</link>
	<pubDate>Tue, 12 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Correctness_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Quadratic Virtual Substitution for Real Arithmetic</title>
	<link>/entries/Virtual_Substitution.html</link>
	<pubDate>Sat, 02 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Virtual_Substitution.html</guid>
	<description></description>
	</item>

	<item>
	<title>Soundness and Completeness of an Axiomatic System for First-Order Logic</title>
	<link>/entries/FOL_Axiomatic.html</link>
	<pubDate>Fri, 24 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Axiomatic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Bounded Operators</title>
	<link>/entries/Complex_Bounded_Operators.html</link>
	<pubDate>Sat, 18 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Bounded_Operators.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Weighted Path Orders and Recursive Path Orders</title>
	<link>/entries/Weighted_Path_Order.html</link>
	<pubDate>Thu, 16 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Weighted_Path_Order.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL I: Foundations: Design Patterns, Set Theory, Digraphs, Semicategories</title>
	<link>/entries/CZH_Foundations.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Foundations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL II: Elementary Theory of 1-Categories</title>
	<link>/entries/CZH_Elementary_Categories.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Elementary_Categories.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL III: Universal Constructions</title>
	<link>/entries/CZH_Universal_Constructions.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Universal_Constructions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional Simplification</title>
	<link>/entries/Conditional_Simplification.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Conditional_Simplification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional Transfer Rule</title>
	<link>/entries/Conditional_Transfer_Rule.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Conditional_Transfer_Rule.html</guid>
	<description></description>
	</item>

	<item>
	<title>Extension of Types-To-Sets</title>
	<link>/entries/Types_To_Sets_Extension.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Types_To_Sets_Extension.html</guid>
	<description></description>
	</item>

	<item>
	<title>IDE: Introduction, Destruction, Elimination</title>
	<link>/entries/Intro_Dest_Elim.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Intro_Dest_Elim.html</guid>
	<description></description>
	</item>

	<item>
	<title>A data flow analysis algorithm for computing dominators</title>
	<link>/entries/Dominance_CHK.html</link>
	<pubDate>Sun, 05 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Dominance_CHK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Solving Cubic and Quartic Equations</title>
	<link>/entries/Cubic_Quartic_Equations.html</link>
	<pubDate>Fri, 03 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Cubic_Quartic_Equations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logging-independent Message Anonymity in the Relational Method</title>
	<link>/entries/Logging_Independent_Anonymity.html</link>
	<pubDate>Thu, 26 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Logging_Independent_Anonymity.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Theorem of Three Circles</title>
	<link>/entries/Three_Circles.html</link>
	<pubDate>Sat, 21 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Three_Circles.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoCon: A Confidentiality-Verified Conference Management System</title>
	<link>/entries/CoCon.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoCon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional BD Security</title>
	<link>/entries/BD_Security_Compositional.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BD_Security_Compositional.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMed: A confidentiality-verified social media platform</title>
	<link>/entries/CoSMed.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMed.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMeDis: A confidentiality-verified distributed social media platform</title>
	<link>/entries/CoSMeDis.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMeDis.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fresh identifiers</title>
	<link>/entries/Fresh_Identifiers.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Fresh_Identifiers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial Design Theory</title>
	<link>/entries/Design_Theory.html</link>
	<pubDate>Fri, 13 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Design_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Forests</title>
	<link>/entries/Relational_Forests.html</link>
	<pubDate>Tue, 03 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz' Independent Axioms for Minkowski Spacetime</title>
	<link>/entries/Schutz_Spacetime.html</link>
	<pubDate>Tue, 27 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Schutz_Spacetime.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finitely Generated Abelian Groups</title>
	<link>/entries/Finitely_Generated_Abelian_Groups.html</link>
	<pubDate>Wed, 07 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Finitely_Generated_Abelian_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>SpecCheck - Specification-Based Testing for Isabelle/ML</title>
	<link>/entries/SpecCheck.html</link>
	<pubDate>Thu, 01 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/SpecCheck.html</guid>
	<description></description>
	</item>

	<item>
	<title>Van der Waerden's Theorem</title>
	<link>/entries/Van_der_Waerden.html</link>
	<pubDate>Tue, 22 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_der_Waerden.html</guid>
	<description></description>
	</item>

	<item>
	<title>MiniSail - A kernel language for the ISA specification language SAIL</title>
	<link>/entries/MiniSail.html</link>
	<pubDate>Fri, 18 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MiniSail.html</guid>
	<description></description>
	</item>

	<item>
	<title>Public Announcement Logic</title>
	<link>/entries/Public_Announcement_Logic.html</link>
	<pubDate>Thu, 17 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Public_Announcement_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Shorter Compiler Correctness Proof for Language IMP</title>
	<link>/entries/IMP_Compiler.html</link>
	<pubDate>Fri, 04 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics on Words Basics</title>
	<link>/entries/Combinatorics_Words.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Lemma</title>
	<link>/entries/Combinatorics_Words_Graph_Lemma.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Graph_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lyndon words</title>
	<link>/entries/Combinatorics_Words_Lyndon.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Lyndon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regression Test Selection</title>
	<link>/entries/Regression_Test_Selection.html</link>
	<pubDate>Fri, 30 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regression_Test_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle's Metalogic: Formalization and Proof Checker</title>
	<link>/entries/Metalogic_ProofChecker.html</link>
	<pubDate>Tue, 27 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Metalogic_ProofChecker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting the Exponent</title>
	<link>/entries/Lifting_the_Exponent.html</link>
	<pubDate>Tue, 27 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_the_Exponent.html</guid>
	<description></description>
	</item>

	<item>
	<title>The BKR Decision Procedure for Univariate Real Arithmetic</title>
	<link>/entries/BenOr_Kozen_Reif.html</link>
	<pubDate>Sat, 24 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BenOr_Kozen_Reif.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Stewart Games</title>
	<link>/entries/GaleStewart_Games.html</link>
	<pubDate>Fri, 23 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/GaleStewart_Games.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Timely Dataflow's Progress Tracking Protocol</title>
	<link>/entries/Progress_Tracking.html</link>
	<pubDate>Tue, 13 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Progress_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Control via Dependency Tracking</title>
	<link>/entries/IFC_Tracking.html</link>
	<pubDate>Thu, 01 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IFC_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck's Schemes in Algebraic Geometry</title>
	<link>/entries/Grothendieck_Schemes.html</link>
	<pubDate>Mon, 29 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Grothendieck_Schemes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hensel's Lemma for the p-adic Integers</title>
	<link>/entries/Padic_Ints.html</link>
	<pubDate>Tue, 23 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Padic_Ints.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL: the Communication Modeling Aspect</title>
	<link>/entries/Constructive_Cryptography_CM.html</link>
	<pubDate>Wed, 17 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography_CM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation</title>
	<link>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</link>
	<pubDate>Fri, 12 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum projective measurements and the CHSH inequality</title>
	<link>/entries/Projective_Measurements.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Measurements.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hermite–Lindemann–Weierstraß Transcendence Theorem</title>
	<link>/entries/Hermite_Lindemann.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite_Lindemann.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mereology</title>
	<link>/entries/Mereology.html</link>
	<pubDate>Mon, 01 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Mereology.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sunflower Lemma of Erdős and Rado</title>
	<link>/entries/Sunflowers.html</link>
	<pubDate>Thu, 25 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Sunflowers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Imperative Implementation of B-Trees</title>
	<link>/entries/BTree.html</link>
	<pubDate>Wed, 24 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BTree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Puiseux Series</title>
	<link>/entries/Formal_Puiseux_Series.html</link>
	<pubDate>Wed, 17 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_Puiseux_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Laws of Large Numbers</title>
	<link>/entries/Laws_of_Large_Numbers.html</link>
	<pubDate>Wed, 10 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Laws_of_Large_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid</title>
	<link>/entries/IsaGeoCoq.html</link>
	<pubDate>Sun, 31 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IsaGeoCoq.html</guid>
	<description></description>
	</item>

	<item>
	<title>Solution to the xkcd Blue Eyes puzzle</title>
	<link>/entries/Blue_Eyes.html</link>
	<pubDate>Sat, 30 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Blue_Eyes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hood-Melville Queue</title>
	<link>/entries/Hood_Melville_Queue.html</link>
	<pubDate>Mon, 18 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hood_Melville_Queue.html</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaDCI: a Java semantics with dynamic class initialization</title>
	<link>/entries/JinjaDCI.html</link>
	<pubDate>Mon, 11 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/JinjaDCI.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cofinality and the Delta System Lemma</title>
	<link>/entries/Delta_System_Lemma.html</link>
	<pubDate>Sun, 27 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Delta_System_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Topological semantics for paraconsistent and paracomplete logics</title>
	<link>/entries/Topological_Semantics.html</link>
	<pubDate>Thu, 17 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Topological_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Minimum Spanning Tree Algorithms</title>
	<link>/entries/Relational_Minimum_Spanning_Trees.html</link>
	<pubDate>Tue, 08 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Minimum_Spanning_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inline Caching and Unboxing Optimization for Interpreters</title>
	<link>/entries/Interpreter_Optimizations.html</link>
	<pubDate>Mon, 07 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Interpreter_Optimizations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Relational Method with Message Anonymity for the Verification of Cryptographic Protocols</title>
	<link>/entries/Relational_Method.html</link>
	<pubDate>Sat, 05 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information</title>
	<link>/entries/Isabelle_Marries_Dirac.html</link>
	<pubDate>Sun, 22 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Marries_Dirac.html</guid>
	<description></description>
	</item>

	<item>
	<title>The HOL-CSP Refinement Toolkit</title>
	<link>/entries/CSP_RefTK.html</link>
	<pubDate>Thu, 19 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/CSP_RefTK.html</guid>
	<description></description>
	</item>

	<item>
	<title>AI Planning Languages Semantics</title>
	<link>/entries/AI_Planning_Languages_Semantics.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/AI_Planning_Languages_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified SAT-Based AI Planning</title>
	<link>/entries/Verified_SAT_Based_AI_Planning.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Verified_SAT_Based_AI_Planning.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sound Type System for Physical Quantities, Units, and Measurements</title>
	<link>/entries/Physical_Quantities.html</link>
	<pubDate>Tue, 20 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Physical_Quantities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Map Extras</title>
	<link>/entries/Finite-Map-Extras.html</link>
	<pubDate>Mon, 12 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Finite-Map-Extras.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Safely Composable Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Safely Composable Web Components</title>
	<link>/entries/SC_DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/SC_DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Web Components</title>
	<link>/entries/DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safely Composable DOM</title>
	<link>/entries/Core_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Core_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Abstract Formalization of G&ouml;del's Incompleteness Theorems</title>
	<link>/entries/Goedel_Incompleteness.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_Incompleteness.html</guid>
	<description></description>
	</item>

	<item>
	<title>From Abstract to Concrete G&ouml;del's Incompleteness Theorems&mdash;Part I</title>
	<link>/entries/Goedel_HFSet_Semantic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_HFSet_Semantic.html</guid>
	<description></description>
	</item>

	<item>
	<title>From Abstract to Concrete G&ouml;del's Incompleteness Theorems&mdash;Part II</title>
	<link>/entries/Goedel_HFSet_Semanticless.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goedel_HFSet_Semanticless.html</guid>
	<description></description>
	</item>

	<item>
	<title>Robinson Arithmetic</title>
	<link>/entries/Robinson_Arithmetic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Robinson_Arithmetic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax-Independent Logic Infrastructure</title>
	<link>/entries/Syntax_Independent_Logic.html</link>
	<pubDate>Wed, 16 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Syntax_Independent_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machines.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machines.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inference of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machine_Inference.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machine_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Practical Algebraic Calculus Checker</title>
	<link>/entries/PAC_Checker.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/PAC_Checker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Some classical results in inductive inference of recursive functions</title>
	<link>/entries/Inductive_Inference.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Disjoint-Set Forests</title>
	<link>/entries/Relational_Disjoint_Set_Forests.html</link>
	<pubDate>Wed, 26 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Disjoint_Set_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Extensions to the Comprehensive Framework for Saturation Theorem Proving</title>
	<link>/entries/Saturation_Framework_Extensions.html</link>
	<pubDate>Tue, 25 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Saturation_Framework_Extensions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Putting the `K' into Bird's derivation of Knuth-Morris-Pratt string matching</title>
	<link>/entries/BirdKMP.html</link>
	<pubDate>Tue, 25 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/BirdKMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amicable Numbers</title>
	<link>/entries/Amicable_Numbers.html</link>
	<pubDate>Tue, 04 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Amicable_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal Partitions</title>
	<link>/entries/Ordinal_Partitions.html</link>
	<pubDate>Mon, 03 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinal_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of The Chandy--Lamport Distributed Snapshot Algorithm</title>
	<link>/entries/Chandy_Lamport.html</link>
	<pubDate>Tue, 21 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Chandy_Lamport.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Characterisations of Paths</title>
	<link>/entries/Relational_Paths.html</link>
	<pubDate>Mon, 13 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Paths.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles</title>
	<link>/entries/Safe_Distance.html</link>
	<pubDate>Mon, 01 Jun 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Distance.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified algorithm for computing the Smith normal form of a matrix</title>
	<link>/entries/Smith_Normal_Form.html</link>
	<pubDate>Sat, 23 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Smith_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Nash-Williams Partition Theorem</title>
	<link>/entries/Nash_Williams.html</link>
	<pubDate>Sat, 16 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Nash_Williams.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Knuth–Bendix Orders</title>
	<link>/entries/Knuth_Bendix_Order.html</link>
	<pubDate>Wed, 13 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Bendix_Order.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality Criteria for Series by Erdős and Straus</title>
	<link>/entries/Irrational_Series_Erdos_Straus.html</link>
	<pubDate>Tue, 12 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Irrational_Series_Erdos_Straus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion Theorem in ZF</title>
	<link>/entries/Recursion-Addition.html</link>
	<pubDate>Mon, 11 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Recursion-Addition.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Normalisation Procedure for Linear Temporal Logic: Isabelle/HOL Formalisation</title>
	<link>/entries/LTL_Normal_Form.html</link>
	<pubDate>Fri, 08 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Forcing in Isabelle/ZF</title>
	<link>/entries/Forcing.html</link>
	<pubDate>Wed, 06 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Forcing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Banach-Steinhaus Theorem</title>
	<link>/entries/Banach_Steinhaus.html</link>
	<pubDate>Sat, 02 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Banach_Steinhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Attack Trees in Isabelle for GDPR compliance of IoT healthcare systems</title>
	<link>/entries/Attack_Trees.html</link>
	<pubDate>Mon, 27 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Attack_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gaussian Integers</title>
	<link>/entries/Gaussian_Integers.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Gaussian_Integers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Power Sum Polynomials</title>
	<link>/entries/Power_Sum_Polynomials.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Power_Sum_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Lambert W Function on the Reals</title>
	<link>/entries/Lambert_W.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lambert_W.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices for ODEs</title>
	<link>/entries/Matrices_for_ODEs.html</link>
	<pubDate>Sun, 19 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Matrices_for_ODEs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Authenticated Data Structures As Functors</title>
	<link>/entries/ADS_Functor.html</link>
	<pubDate>Thu, 16 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/ADS_Functor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Algorithm for Greedily Computing Associative Aggregations on Sliding Windows</title>
	<link>/entries/Sliding_Window_Algorithm.html</link>
	<pubDate>Fri, 10 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Sliding_Window_Algorithm.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Comprehensive Framework for Saturation Theorem Proving</title>
	<link>/entries/Saturation_Framework.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Saturation_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations</title>
	<link>/entries/MFODL_Monitor_Optimized.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/MFODL_Monitor_Optimized.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automated Stateful Protocol Verification</title>
	<link>/entries/Automated_Stateful_Protocol_Verification.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Automated_Stateful_Protocol_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stateful Protocol Composition and Typing</title>
	<link>/entries/Stateful_Protocol_Composition_and_Typing.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Stateful_Protocol_Composition_and_Typing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lucas's Theorem</title>
	<link>/entries/Lucas_Theorem.html</link>
	<pubDate>Tue, 07 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lucas_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Eventual Consistency of the Collaborative Editing Framework WOOT</title>
	<link>/entries/WOOT_Strong_Eventual_Consistency.html</link>
	<pubDate>Wed, 25 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/WOOT_Strong_Eventual_Consistency.html</guid>
	<description></description>
	</item>

	<item>
	<title>Furstenberg's topology and his proof of the infinitude of primes</title>
	<link>/entries/Furstenberg_Topology.html</link>
	<pubDate>Sun, 22 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Furstenberg_Topology.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Under-Approximate Relational Logic</title>
	<link>/entries/Relational-Incorrectness-Logic.html</link>
	<pubDate>Thu, 12 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational-Incorrectness-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hello World</title>
	<link>/entries/Hello_World.html</link>
	<pubDate>Sat, 07 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Hello_World.html</guid>
	<description></description>
	</item>

	<item>
	<title>Implementing the Goodstein Function in &lambda;-Calculus</title>
	<link>/entries/Goodstein_Lambda.html</link>
	<pubDate>Fri, 21 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Goodstein_Lambda.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Generic Framework for Verified Compilers</title>
	<link>/entries/VeriComp.html</link>
	<pubDate>Mon, 10 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/VeriComp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Arithmetic progressions and relative primes</title>
	<link>/entries/Arith_Prog_Rel_Primes.html</link>
	<pubDate>Sat, 01 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Arith_Prog_Rel_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Hierarchy of Algebras for Boolean Subsets</title>
	<link>/entries/Subset_Boolean_Algebras.html</link>
	<pubDate>Fri, 31 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Subset_Boolean_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mersenne primes and the Lucas–Lehmer test</title>
	<link>/entries/Mersenne_Primes.html</link>
	<pubDate>Fri, 17 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Mersenne_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Approximation Algorithms</title>
	<link>/entries/Approximation_Algorithms.html</link>
	<pubDate>Thu, 16 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Approximation_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Closest Pair of Points Algorithms</title>
	<link>/entries/Closest_Pair_Points.html</link>
	<pubDate>Mon, 13 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Closest_Pair_Points.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skip Lists</title>
	<link>/entries/Skip_Lists.html</link>
	<pubDate>Thu, 09 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Skip_Lists.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bicategories</title>
	<link>/entries/Bicategory.html</link>
	<pubDate>Mon, 06 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Bicategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Irrationality of ζ(3)</title>
	<link>/entries/Zeta_3_Irrational.html</link>
	<pubDate>Fri, 27 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_3_Irrational.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing a Seligman-Style Tableau System for Hybrid Logic</title>
	<link>/entries/Hybrid_Logic.html</link>
	<pubDate>Fri, 20 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Poincaré-Bendixson Theorem</title>
	<link>/entries/Poincare_Bendixson.html</link>
	<pubDate>Wed, 18 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Bendixson.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Geometry</title>
	<link>/entries/Complex_Geometry.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Poincaré Disc Model</title>
	<link>/entries/Poincare_Disc.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Disc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss Sums and the Pólya–Vinogradov Inequality</title>
	<link>/entries/Gauss_Sums.html</link>
	<pubDate>Tue, 10 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Sums.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</title>
	<link>/entries/Generalized_Counting_Sort.html</link>
	<pubDate>Wed, 04 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generalized_Counting_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interval Arithmetic on 32-bit Words</title>
	<link>/entries/Interval_Arithmetic_Word32.html</link>
	<pubDate>Wed, 27 Nov 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Interval_Arithmetic_Word32.html</guid>
	<description></description>
	</item>

	<item>
	<title>Zermelo Fraenkel Set Theory in Higher-Order Logic</title>
	<link>/entries/ZFC_in_HOL.html</link>
	<pubDate>Thu, 24 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/ZFC_in_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/C</title>
	<link>/entries/Isabelle_C.html</link>
	<pubDate>Tue, 22 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_C.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2019 -- Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2019.html</link>
	<pubDate>Wed, 16 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Aristotle's Assertoric Syllogistic</title>
	<link>/entries/Aristotles_Assertoric_Syllogistic.html</link>
	<pubDate>Tue, 08 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Aristotles_Assertoric_Syllogistic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sigma Protocols and Commitment Schemes</title>
	<link>/entries/Sigma_Commit_Crypto.html</link>
	<pubDate>Mon, 07 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Sigma_Commit_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clean - An Abstract Imperative Programming Language and its Theory</title>
	<link>/entries/Clean.html</link>
	<pubDate>Fri, 04 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Clean.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Multiway-Join Algorithms</title>
	<link>/entries/Generic_Join.html</link>
	<pubDate>Mon, 16 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Join.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification Components for Hybrid Systems</title>
	<link>/entries/Hybrid_Systems_VCs.html</link>
	<pubDate>Tue, 10 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Systems_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fourier Series</title>
	<link>/entries/Fourier.html</link>
	<pubDate>Fri, 06 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Case Study in Basic Algebra</title>
	<link>/entries/Jacobson_Basic_Algebra.html</link>
	<pubDate>Fri, 30 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Jacobson_Basic_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation of an Adaptive State Counting Algorithm</title>
	<link>/entries/Adaptive_State_Counting.html</link>
	<pubDate>Fri, 16 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Adaptive_State_Counting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Laplace Transform</title>
	<link>/entries/Laplace_Transform.html</link>
	<pubDate>Wed, 14 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Laplace_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Communicating Concurrent Kleene Algebra for Distributed Systems Specification</title>
	<link>/entries/C2KA_DistributedSystems.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/C2KA_DistributedSystems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Programming</title>
	<link>/entries/Linear_Programming.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Programming.html</guid>
	<description></description>
	</item>

	<item>
	<title>Selected Problems from the International Mathematical Olympiad 2019</title>
	<link>/entries/IMO2019.html</link>
	<pubDate>Mon, 05 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMO2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar Quorum Systems</title>
	<link>/entries/Stellar_Quorums.html</link>
	<pubDate>Thu, 01 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Stellar_Quorums.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Development of a Polychronous Polytimed Coordination Language</title>
	<link>/entries/TESL_Language.html</link>
	<pubDate>Tue, 30 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/TESL_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>Order Extension and Szpilrajn's Extension Theorem</title>
	<link>/entries/Szpilrajn.html</link>
	<pubDate>Sat, 27 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Szpilrajn.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sequent Calculus for First-Order Logic</title>
	<link>/entries/FOL_Seq_Calc1.html</link>
	<pubDate>Thu, 18 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Seq_Calc1.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Code Generator from Isabelle/HOL to CakeML</title>
	<link>/entries/CakeML_Codegen.html</link>
	<pubDate>Mon, 08 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML_Codegen.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</title>
	<link>/entries/MFOTL_Monitor.html</link>
	<pubDate>Thu, 04 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/MFOTL_Monitor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complete Non-Orders and Fixed Points</title>
	<link>/entries/Complete_Non_Orders.html</link>
	<pubDate>Thu, 27 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complete_Non_Orders.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Search Trees</title>
	<link>/entries/Priority_Search_Trees.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Search_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra</title>
	<link>/entries/Prim_Dijkstra_Simple.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prim_Dijkstra_Simple.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Inequalities</title>
	<link>/entries/Linear_Inequalities.html</link>
	<pubDate>Fri, 21 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Inequalities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hilbert's Nullstellensatz</title>
	<link>/entries/Nullstellensatz.html</link>
	<pubDate>Sun, 16 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Nullstellensatz.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds</title>
	<link>/entries/Groebner_Macaulay.html</link>
	<pubDate>Sat, 15 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Macaulay.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Heaps for IMP2</title>
	<link>/entries/IMP2_Binary_Heap.html</link>
	<pubDate>Thu, 13 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2_Binary_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Game Logic</title>
	<link>/entries/Differential_Game_Logic.html</link>
	<pubDate>Mon, 03 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Game_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multidimensional Binary Search Trees</title>
	<link>/entries/KD_Tree.html</link>
	<pubDate>Thu, 30 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/KD_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Generic Authenticated Data Structures</title>
	<link>/entries/LambdaAuth.html</link>
	<pubDate>Tue, 14 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaAuth.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Party Computation</title>
	<link>/entries/Multi_Party_Computation.html</link>
	<pubDate>Thu, 09 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Multi_Party_Computation.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP Version 2.0</title>
	<link>/entries/HOL-CSP.html</link>
	<pubDate>Fri, 26 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/HOL-CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Compositional and Unified Translation of LTL into ω-Automata</title>
	<link>/entries/LTL_Master_Theorem.html</link>
	<pubDate>Tue, 16 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Master_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Theory of Syntax with Bindings</title>
	<link>/entries/Binding_Syntax_Theory.html</link>
	<pubDate>Sat, 06 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Binding_Syntax_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of Certain Infinite Series</title>
	<link>/entries/Transcendence_Series_Hancl_Rucki.html</link>
	<pubDate>Wed, 27 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Transcendence_Series_Hancl_Rucki.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum Hoare Logic</title>
	<link>/entries/QHLProver.html</link>
	<pubDate>Sun, 24 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/QHLProver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Safe OCL</title>
	<link>/entries/Safe_OCL.html</link>
	<pubDate>Sat, 09 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Elementary Facts About the Distribution of Primes</title>
	<link>/entries/Prime_Distribution_Elementary.html</link>
	<pubDate>Thu, 21 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Distribution_Elementary.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal's Algorithm for Minimum Spanning Forest</title>
	<link>/entries/Kruskal.html</link>
	<pubDate>Thu, 14 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Kruskal.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Primality Testing</title>
	<link>/entries/Probabilistic_Prime_Tests.html</link>
	<pubDate>Mon, 11 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Prime_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Turing Machine</title>
	<link>/entries/Universal_Turing_Machine.html</link>
	<pubDate>Fri, 08 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Turing_Machine.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/UTP: Mechanised Theory Engineering for Unifying Theories of Programming</title>
	<link>/entries/UTP.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/UTP.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inversions of a List</title>
	<link>/entries/List_Inversions.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/List_Inversions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Farkas' Lemma and Motzkin's Transposition Theorem</title>
	<link>/entries/Farkas.html</link>
	<pubDate>Thu, 17 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Farkas.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Algebra for Higher-Order Terms</title>
	<link>/entries/Higher_Order_Terms.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Higher_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2 – Simple Program Verification in Isabelle/HOL</title>
	<link>/entries/IMP2.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reduction Theorem for Store Buffers</title>
	<link>/entries/Store_Buffer_Reduction.html</link>
	<pubDate>Mon, 07 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Store_Buffer_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model</title>
	<link>/entries/Core_DOM.html</link>
	<pubDate>Wed, 26 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Core_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Concurrent Revisions</title>
	<link>/entries/Concurrent_Revisions.html</link>
	<pubDate>Tue, 25 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Revisions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Imperative Programs using Auto2</title>
	<link>/entries/Auto2_Imperative_HOL.html</link>
	<pubDate>Fri, 21 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL</title>
	<link>/entries/Constructive_Cryptography.html</link>
	<pubDate>Mon, 17 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Orderings and Lattices</title>
	<link>/entries/Order_Lattice_Props.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Order_Lattice_Props.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantales</title>
	<link>/entries/Quantales.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quantales.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer Semantics</title>
	<link>/entries/Transformer_Semantics.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Transformer_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Functional Implementation of Bachmair and Ganzinger's Ordered Resolution Prover</title>
	<link>/entries/Functional_Ordered_Resolution_Prover.html</link>
	<pubDate>Fri, 23 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Functional_Ordered_Resolution_Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Saturation</title>
	<link>/entries/Graph_Saturation.html</link>
	<pubDate>Fri, 23 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Graph_Saturation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2 Prover</title>
	<link>/entries/Auto2_HOL.html</link>
	<pubDate>Tue, 20 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matroids</title>
	<link>/entries/Matroids.html</link>
	<pubDate>Fri, 16 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Matroids.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving generic class instances for datatypes</title>
	<link>/entries/Generic_Deriving.html</link>
	<pubDate>Tue, 06 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation and Evaluation of Alan Gewirth's Proof for the Principle of Generic Consistency in Isabelle/HOL</title>
	<link>/entries/GewirthPGCProof.html</link>
	<pubDate>Tue, 30 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/GewirthPGCProof.html</guid>
	<description></description>
	</item>

	<item>
	<title>Epistemic Logic: Completeness of Modal Logics</title>
	<link>/entries/Epistemic_Logic.html</link>
	<pubDate>Mon, 29 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Epistemic_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth Manifolds</title>
	<link>/entries/Smooth_Manifolds.html</link>
	<pubDate>Mon, 22 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Smooth_Manifolds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_EPO.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_EPO.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Binary Search Trees</title>
	<link>/entries/Randomised_BSTs.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Upper Bounding Diameters of State Spaces of Factored Transition Systems</title>
	<link>/entries/Factored_Transition_System_Bounding.html</link>
	<pubDate>Fri, 12 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Factored_Transition_System_Bounding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of π</title>
	<link>/entries/Pi_Transcendental.html</link>
	<pubDate>Fri, 28 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>Symmetric Polynomials</title>
	<link>/entries/Symmetric_Polynomials.html</link>
	<pubDate>Tue, 25 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Symmetric_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
	<link>/entries/Signature_Groebner.html</link>
	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Signature_Groebner.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Prime Number Theorem</title>
	<link>/entries/Prime_Number_Theorem.html</link>
	<pubDate>Wed, 19 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Number_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Aggregation Algebras</title>
	<link>/entries/Aggregation_Algebras.html</link>
	<pubDate>Sat, 15 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Aggregation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/entries/Octonions.html</link>
	<pubDate>Fri, 14 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Octonions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/entries/Quaternions.html</link>
	<pubDate>Wed, 05 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quaternions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Budan–Fourier Theorem and Counting Real Roots with Multiplicity</title>
	<link>/entries/Budan_Fourier.html</link>
	<pubDate>Sun, 02 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Budan_Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Incremental Simplex Algorithm with Unsatisfiable Core Generation</title>
	<link>/entries/Simplex.html</link>
	<pubDate>Fri, 24 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Simplex.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minsky Machines</title>
	<link>/entries/Minsky_Machines.html</link>
	<pubDate>Tue, 14 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Minsky_Machines.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pricing in discrete financial models</title>
	<link>/entries/DiscretePricing.html</link>
	<pubDate>Mon, 16 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/DiscretePricing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Von-Neumann-Morgenstern Utility Theorem</title>
	<link>/entries/Neumann_Morgenstern_Utility.html</link>
	<pubDate>Wed, 04 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Neumann_Morgenstern_Utility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pell's Equation</title>
	<link>/entries/Pell.html</link>
	<pubDate>Sat, 23 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pell.html</guid>
	<description></description>
	</item>

	<item>
	<title>Projective Geometry</title>
	<link>/entries/Projective_Geometry.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Localization of a Commutative Ring</title>
	<link>/entries/Localization_Ring.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Localization_Ring.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Order Reduction</title>
	<link>/entries/Partial_Order_Reduction.html</link>
	<pubDate>Tue, 05 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Order_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal Binary Search Trees</title>
	<link>/entries/Optimal_BST.html</link>
	<pubDate>Sun, 27 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Optimal_BST.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden Markov Models</title>
	<link>/entries/Hidden_Markov_Models.html</link>
	<pubDate>Fri, 25 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hidden_Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Timed Automata</title>
	<link>/entries/Probabilistic_Timed_Automata.html</link>
	<pubDate>Thu, 24 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Axiom Systems for Category Theory in Free Logic</title>
	<link>/entries/AxiomaticCategoryTheory.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/AxiomaticCategoryTheory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational Rapidly Convergent Series</title>
	<link>/entries/Irrationality_J_Hancl.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationality_J_Hancl.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monadification, Memoization and Dynamic Programming</title>
	<link>/entries/Monad_Memo_DP.html</link>
	<pubDate>Tue, 22 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Memo_DP.html</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets: Sequential Specifications for Replicated Datatypes</title>
	<link>/entries/OpSets.html</link>
	<pubDate>Thu, 10 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/OpSets.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL Formalization of the Modular Assembly Kit for Security Properties</title>
	<link>/entries/Modular_Assembly_Kit_Security.html</link>
	<pubDate>Mon, 07 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_Assembly_Kit_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>WebAssembly</title>
	<link>/entries/WebAssembly.html</link>
	<pubDate>Sun, 29 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/WebAssembly.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2018 - Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2018.html</link>
	<pubDate>Fri, 27 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2018.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded Natural Functors with Covariance and Contravariance</title>
	<link>/entries/BNF_CC.html</link>
	<pubDate>Tue, 24 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/BNF_CC.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of Fishburn-Strategyproofness and Pareto-Efficiency</title>
	<link>/entries/Fishburn_Impossibility.html</link>
	<pubDate>Thu, 22 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Fishburn_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Weight-Balanced Trees</title>
	<link>/entries/Weight_Balanced_Trees.html</link>
	<pubDate>Tue, 13 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Weight_Balanced_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML</title>
	<link>/entries/CakeML.html</link>
	<pubDate>Mon, 12 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Theory of Architectural Design Patterns</title>
	<link>/entries/Architectural_Design_Patterns.html</link>
	<pubDate>Thu, 01 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Architectural_Design_Patterns.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hoare Logics for Time Bounds</title>
	<link>/entries/Hoare_Time.html</link>
	<pubDate>Mon, 26 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hoare_Time.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified factorization algorithm for integer polynomials with polynomial complexity</title>
	<link>/entries/LLL_Factorization.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Terms</title>
	<link>/entries/First_Order_Terms.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/First_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Error Function</title>
	<link>/entries/Error_Function.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Error_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>Treaps</title>
	<link>/entries/Treaps.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Treaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified LLL algorithm</title>
	<link>/entries/LLL_Basis_Reduction.html</link>
	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Bachmair and Ganzinger's Ordered Resolution Prover</title>
	<link>/entries/Ordered_Resolution_Prover.html</link>
	<pubDate>Thu, 18 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Ordered_Resolution_Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gromov Hyperbolicity</title>
	<link>/entries/Gromov_Hyperbolicity.html</link>
	<pubDate>Tue, 16 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Gromov_Hyperbolicity.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL formalisation of Green's Theorem</title>
	<link>/entries/Green.html</link>
	<pubDate>Thu, 11 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Green.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Falling Factorial of a Sum</title>
	<link>/entries/Falling_Factorial_Sum.html</link>
	<pubDate>Fri, 22 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Falling_Factorial_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet L-Functions and Dirichlet's Theorem</title>
	<link>/entries/Dirichlet_L.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_L.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Mason–Stothers Theorem</title>
	<link>/entries/Mason_Stothers.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Mason_Stothers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Median-of-Medians Selection Algorithm</title>
	<link>/entries/Median_Of_Medians_Selection.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Of_Medians_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>Operations on Bounded Natural Functors</title>
	<link>/entries/BNF_Operations.html</link>
	<pubDate>Tue, 19 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/BNF_Operations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The string search algorithm by Knuth, Morris and Pratt</title>
	<link>/entries/Knuth_Morris_Pratt.html</link>
	<pubDate>Mon, 18 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Morris_Pratt.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic Matrices and the Perron-Frobenius Theorem</title>
	<link>/entries/Stochastic_Matrices.html</link>
	<pubDate>Wed, 22 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stochastic_Matrices.html</guid>
	<description></description>
	</item>

	<item>
	<title>The IMAP CmRDT</title>
	<link>/entries/IMAP-CRDT.html</link>
	<pubDate>Thu, 09 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/IMAP-CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid Multi-Lane Spatial Logic</title>
	<link>/entries/Hybrid_Multi_Lane_Spatial_Logic.html</link>
	<pubDate>Mon, 06 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Multi_Lane_Spatial_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Kuratowski Closure-Complement Theorem</title>
	<link>/entries/Kuratowski_Closure_Complement.html</link>
	<pubDate>Thu, 26 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Kuratowski_Closure_Complement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Büchi Complementation</title>
	<link>/entries/Buchi_Complementation.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buchi_Complementation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transition Systems and Automata</title>
	<link>/entries/Transition_Systems_and_Automata.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Transition_Systems_and_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Count the Number of Complex Roots</title>
	<link>/entries/Count_Complex_Roots.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Count_Complex_Roots.html</guid>
	<description></description>
	</item>

	<item>
	<title>Evaluate Winding Numbers through Cauchy Indices</title>
	<link>/entries/Winding_Number_Eval.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Winding_Number_Eval.html</guid>
	<description></description>
	</item>

	<item>
	<title>Homogeneous Linear Diophantine Equations</title>
	<link>/entries/Diophantine_Eqns_Lin_Hom.html</link>
	<pubDate>Sat, 14 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Diophantine_Eqns_Lin_Hom.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet Series</title>
	<link>/entries/Dirichlet_Series.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Recurrences</title>
	<link>/entries/Linear_Recurrences.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Recurrences.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hurwitz and Riemann ζ Functions</title>
	<link>/entries/Zeta_Function.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>Computer-assisted Reconstruction and Assessment of E. J. Lowe's Modal Ontological Argument</title>
	<link>/entries/Lowe_Ontological_Argument.html</link>
	<pubDate>Thu, 21 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Lowe_Ontological_Argument.html</guid>
	<description></description>
	</item>

	<item>
	<title>Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL</title>
	<link>/entries/PLM.html</link>
	<pubDate>Sun, 17 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PLM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Anselm's God in Isabelle/HOL</title>
	<link>/entries/AnselmGod.html</link>
	<pubDate>Wed, 06 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/AnselmGod.html</guid>
	<description></description>
	</item>

	<item>
	<title>Microeconomics and the First Welfare Theorem</title>
	<link>/entries/First_Welfare_Theorem.html</link>
	<pubDate>Fri, 01 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/First_Welfare_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Orbit-Stabiliser Theorem with Application to Rotational Symmetries</title>
	<link>/entries/Orbit_Stabiliser.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Orbit_Stabiliser.html</guid>
	<description></description>
	</item>

	<item>
	<title>Root-Balanced Tree</title>
	<link>/entries/Root_Balanced_Tree.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Root_Balanced_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>The LambdaMu-calculus</title>
	<link>/entries/LambdaMu.html</link>
	<pubDate>Wed, 16 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaMu.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stewart's Theorem and Apollonius' Theorem</title>
	<link>/entries/Stewart_Apollonius.html</link>
	<pubDate>Mon, 31 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stewart_Apollonius.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dynamic Architectures</title>
	<link>/entries/DynamicArchitectures.html</link>
	<pubDate>Fri, 28 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/DynamicArchitectures.html</guid>
	<description></description>
	</item>

	<item>
	<title>Declarative Semantics for Functional Languages</title>
	<link>/entries/Decl_Sem_Fun_PL.html</link>
	<pubDate>Fri, 21 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Decl_Sem_Fun_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOLCF-Prelude</title>
	<link>/entries/HOLCF-Prelude.html</link>
	<pubDate>Sat, 15 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/HOLCF-Prelude.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowski's Theorem</title>
	<link>/entries/Minkowskis_Theorem.html</link>
	<pubDate>Thu, 13 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minkowskis_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Metatheory and Type Inference for a Name-Carrying Simply-Typed Lambda Calculus</title>
	<link>/entries/Name_Carrying_Type_Inference.html</link>
	<pubDate>Sun, 09 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Name_Carrying_Type_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</title>
	<link>/entries/CRDT.html</link>
	<pubDate>Fri, 07 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone-Kleene Relation Algebras</title>
	<link>/entries/Stone_Kleene_Relation_Algebras.html</link>
	<pubDate>Thu, 06 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Kleene_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional Proof Systems</title>
	<link>/entries/Propositional_Proof_Systems.html</link>
	<pubDate>Wed, 21 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Propositional_Proof_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Semigroups and Convolution Algebras</title>
	<link>/entries/PSemigroupsConvolution.html</link>
	<pubDate>Tue, 13 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PSemigroupsConvolution.html</guid>
	<description></description>
	</item>

	<item>
	<title>Buffon's Needle Problem</title>
	<link>/entries/Buffons_Needle.html</link>
	<pubDate>Tue, 06 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buffons_Needle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Flow Networks and the Min-Cut-Max-Flow Theorem</title>
	<link>/entries/Flow_Networks.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Flow_Networks.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Push-Relabel Algorithms</title>
	<link>/entries/Prpu_Maxflow.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Prpu_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/entries/Optics.html</link>
	<pubDate>Thu, 25 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Optics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Developing Security Protocols by Refinement</title>
	<link>/entries/Security_Protocol_Refinement.html</link>
	<pubDate>Wed, 24 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Security_Protocol_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dictionary Construction</title>
	<link>/entries/Dict_Construction.html</link>
	<pubDate>Wed, 24 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dict_Construction.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Floyd-Warshall Algorithm for Shortest Paths</title>
	<link>/entries/Floyd_Warshall.html</link>
	<pubDate>Mon, 08 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/entries/CryptHOL.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CryptHOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Effect polymorphism in higher-order logic</title>
	<link>/entries/Monomorphic_Monad.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monomorphic_Monad.html</guid>
	<description></description>
	</item>

	<item>
	<title>Game-based cryptography in HOL</title>
	<link>/entries/Game_Based_Crypto.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Game_Based_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monad normalisation</title>
	<link>/entries/Monad_Normalisation.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Normalisation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
	<link>/entries/Probabilistic_While.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_While.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monoidal Categories</title>
	<link>/entries/MonoidalCategory.html</link>
	<pubDate>Thu, 04 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/MonoidalCategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Types, Tableaus and Gödel’s God in Isabelle/HOL</title>
	<link>/entries/Types_Tableaus_and_Goedels_God.html</link>
	<pubDate>Mon, 01 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Types_Tableaus_and_Goedels_God.html</guid>
	<description></description>
	</item>

	<item>
	<title>Local Lexing</title>
	<link>/entries/LocalLexing.html</link>
	<pubDate>Fri, 28 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LocalLexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructor Functions</title>
	<link>/entries/Constructor_Funs.html</link>
	<pubDate>Wed, 19 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Constructor_Funs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lazifying case constants</title>
	<link>/entries/Lazy_Case.html</link>
	<pubDate>Tue, 18 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Lazy_Case.html</guid>
	<description></description>
	</item>

	<item>
	<title>Subresultants</title>
	<link>/entries/Subresultants.html</link>
	<pubDate>Thu, 06 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Subresultants.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expected Shape of Random Binary Search Trees</title>
	<link>/entries/Random_BSTs.html</link>
	<pubDate>Tue, 04 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Random_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower bound on comparison-based sorting algorithms</title>
	<link>/entries/Comparison_Sort_Lower_Bound.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Comparison_Sort_Lower_Bound.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
	<link>/entries/Quick_Sort_Cost.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Euler–MacLaurin Formula</title>
	<link>/entries/Euler_MacLaurin.html</link>
	<pubDate>Fri, 10 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_MacLaurin.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Group Law for Elliptic Curves</title>
	<link>/entries/Elliptic_Curves_Group_Law.html</link>
	<pubDate>Tue, 28 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Elliptic_Curves_Group_Law.html</guid>
	<description></description>
	</item>

	<item>
	<title>Menger's Theorem</title>
	<link>/entries/Menger.html</link>
	<pubDate>Sun, 26 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Menger.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Dynamic Logic</title>
	<link>/entries/Differential_Dynamic_Logic.html</link>
	<pubDate>Mon, 13 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Dynamic_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Soundness</title>
	<link>/entries/Abstract_Soundness.html</link>
	<pubDate>Fri, 10 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract_Soundness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Relation Algebras</title>
	<link>/entries/Stone_Relation_Algebras.html</link>
	<pubDate>Tue, 07 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refining Authenticated Key Agreement with Strong Adversaries</title>
	<link>/entries/Key_Agreement_Strong_Adversaries.html</link>
	<pubDate>Tue, 31 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Key_Agreement_Strong_Adversaries.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli Numbers</title>
	<link>/entries/Bernoulli.html</link>
	<pubDate>Tue, 24 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bernoulli.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bertrand's postulate</title>
	<link>/entries/Bertrands_Postulate.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bertrands_Postulate.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minimal Static Single Assignment Form</title>
	<link>/entries/Minimal_SSA.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minimal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of e</title>
	<link>/entries/E_Transcendental.html</link>
	<pubDate>Thu, 12 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/E_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Network Models and Their Application to Firewall Policies</title>
	<link>/entries/UPF_Firewall.html</link>
	<pubDate>Sun, 08 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/UPF_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method</title>
	<link>/entries/Password_Authentication_Protocol.html</link>
	<pubDate>Tue, 03 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Password_Authentication_Protocol.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Logic According to Harrison</title>
	<link>/entries/FOL_Harrison.html</link>
	<pubDate>Sun, 01 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/FOL_Harrison.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent Refinement Algebra and Rely Quotients</title>
	<link>/entries/Concurrent_Ref_Alg.html</link>
	<pubDate>Fri, 30 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Ref_Alg.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Twelvefold Way</title>
	<link>/entries/Twelvefold_Way.html</link>
	<pubDate>Thu, 29 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Twelvefold_Way.html</guid>
	<description></description>
	</item>

	<item>
	<title>Proof Strategy Language</title>
	<link>/entries/Proof_Strategy_Language.html</link>
	<pubDate>Tue, 20 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Proof_Strategy_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>Paraconsistency</title>
	<link>/entries/Paraconsistency.html</link>
	<pubDate>Wed, 07 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Paraconsistency.html</guid>
	<description></description>
	</item>

	<item>
	<title>COMPLX: A Verification Framework for Concurrent Imperative Programs</title>
	<link>/entries/Complx.html</link>
	<pubDate>Tue, 29 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Complx.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Interpretation of Annotated Commands</title>
	<link>/entries/Abs_Int_ITP2012.html</link>
	<pubDate>Wed, 23 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Abs_Int_ITP2012.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separata: Isabelle tactics for Separation Algebra</title>
	<link>/entries/Separata.html</link>
	<pubDate>Wed, 16 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Separata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Knuth–Bendix Orders for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_KBOs.html</link>
	<pubDate>Sat, 12 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_KBOs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Nested Multisets, Hereditary Multisets, and Syntactic Ordinals</title>
	<link>/entries/Nested_Multisets_Ordinals.html</link>
	<pubDate>Sat, 12 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Nested_Multisets_Ordinals.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expressiveness of Deep Learning</title>
	<link>/entries/Deep_Learning.html</link>
	<pubDate>Thu, 10 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Deep_Learning.html</guid>
	<description></description>
	</item>

	<item>
	<title>Modal Logics for Nominal Transition Systems</title>
	<link>/entries/Modal_Logics_for_NTS.html</link>
	<pubDate>Tue, 25 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Modal_Logics_for_NTS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stable Matching</title>
	<link>/entries/Stable_Matching.html</link>
	<pubDate>Mon, 24 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stable_Matching.html</guid>
	<description></description>
	</item>

	<item>
	<title>LOFT — Verified Migration of Linux Firewalls to SDN</title>
	<link>/entries/LOFT.html</link>
	<pubDate>Fri, 21 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LOFT.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</title>
	<link>/entries/SPARCv8.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SPARCv8.html</guid>
	<description></description>
	</item>

	<item>
	<title>Source Coding Theorem</title>
	<link>/entries/Source_Coding_Theorem.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Source_Coding_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Factorization Algorithm of Berlekamp and Zassenhaus</title>
	<link>/entries/Berlekamp_Zassenhaus.html</link>
	<pubDate>Fri, 14 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Berlekamp_Zassenhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Intersecting Chords Theorem</title>
	<link>/entries/Chord_Segments.html</link>
	<pubDate>Tue, 11 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Chord_Segments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lp spaces</title>
	<link>/entries/Lp.html</link>
	<pubDate>Wed, 05 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher–Yates shuffle</title>
	<link>/entries/Fisher_Yates.html</link>
	<pubDate>Fri, 30 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Fisher_Yates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Allen's Interval Calculus</title>
	<link>/entries/Allen_Calculus.html</link>
	<pubDate>Thu, 29 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Allen_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Recursive Path Orders for Lambda-Free Higher-Order Terms</title>
	<link>/entries/Lambda_Free_RPOs.html</link>
	<pubDate>Fri, 23 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lambda_Free_RPOs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables Semantics</title>
	<link>/entries/Iptables_Semantics.html</link>
	<pubDate>Fri, 09 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Iptables_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Variant of the Superposition Calculus</title>
	<link>/entries/SuperCalc.html</link>
	<pubDate>Tue, 06 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SuperCalc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Algebras</title>
	<link>/entries/Stone_Algebras.html</link>
	<pubDate>Tue, 06 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stirling's formula</title>
	<link>/entries/Stirling_Formula.html</link>
	<pubDate>Thu, 01 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stirling_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>Routing</title>
	<link>/entries/Routing.html</link>
	<pubDate>Wed, 31 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Routing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simple Firewall</title>
	<link>/entries/Simple_Firewall.html</link>
	<pubDate>Wed, 24 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Simple_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths</title>
	<link>/entries/InfPathElimination.html</link>
	<pubDate>Thu, 18 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/InfPathElimination.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Edmonds-Karp Algorithm</title>
	<link>/entries/EdmondsKarp_Maxflow.html</link>
	<pubDate>Fri, 12 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/EdmondsKarp_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Imperative Refinement Framework</title>
	<link>/entries/Refine_Imperative_HOL.html</link>
	<pubDate>Mon, 08 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ptolemy's Theorem</title>
	<link>/entries/Ptolemys_Theorem.html</link>
	<pubDate>Sun, 07 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Ptolemys_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Surprise Paradox</title>
	<link>/entries/Surprise_Paradox.html</link>
	<pubDate>Sun, 17 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Surprise_Paradox.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pairing Heap</title>
	<link>/entries/Pairing_Heap.html</link>
	<pubDate>Thu, 14 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Pairing_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Framework for Verifying Depth-First Search Algorithms</title>
	<link>/entries/DFS_Framework.html</link>
	<pubDate>Tue, 05 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/DFS_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Chamber Complexes, Coxeter Systems, and Buildings</title>
	<link>/entries/Buildings.html</link>
	<pubDate>Fri, 01 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Buildings.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Resolution Calculus for First-Order Logic</title>
	<link>/entries/Resolution_FOL.html</link>
	<pubDate>Thu, 30 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Resolution_FOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Z Property</title>
	<link>/entries/Rewriting_Z.html</link>
	<pubDate>Thu, 30 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Rewriting_Z.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Security-Preserving Refinement for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Refinement.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>IP Addresses</title>
	<link>/entries/IP_Addresses.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/IP_Addresses.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Multisets</title>
	<link>/entries/Card_Multisets.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Multisets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory with Adjunctions and Limits</title>
	<link>/entries/Category3.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Category3.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Dependent Security Type System for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Type_Systems.html</link>
	<pubDate>Sat, 25 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Catalan Numbers</title>
	<link>/entries/Catalan_Numbers.html</link>
	<pubDate>Tue, 21 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Catalan_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Program Construction and Verification Components Based on Kleene Algebra</title>
	<link>/entries/Algebraic_VCs.html</link>
	<pubDate>Sat, 18 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Concurrent Composition</title>
	<link>/entries/Noninterference_Concurrent_Composition.html</link>
	<pubDate>Mon, 13 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Concurrent_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Machine Word Library</title>
	<link>/entries/Word_Lib.html</link>
	<pubDate>Thu, 09 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Word_Lib.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Decomposition</title>
	<link>/entries/Tree_Decomposition.html</link>
	<pubDate>Tue, 31 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Tree_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Equivalence Relations</title>
	<link>/entries/Card_Equiv_Relations.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Equiv_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>POSIX Lexing with Derivatives of Regular Expressions</title>
	<link>/entries/Posix-Lexing.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Posix-Lexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Perron-Frobenius Theorem for Spectral Radius Analysis</title>
	<link>/entries/Perron_Frobenius.html</link>
	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Perron_Frobenius.html</guid>
	<description></description>
	</item>

	<item>
	<title>The meta theory of the Incredible Proof Machine</title>
	<link>/entries/Incredible_Proof_Machine.html</link>
	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Incredible_Proof_Machine.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Constructive Proof for FLP</title>
	<link>/entries/FLP.html</link>
	<pubDate>Wed, 18 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/FLP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks</title>
	<link>/entries/MFMC_Countable.html</link>
	<pubDate>Mon, 09 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/MFMC_Countable.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Social Choice Theory</title>
	<link>/entries/Randomised_Social_Choice.html</link>
	<pubDate>Thu, 05 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_Social_Choice.html</guid>
	<description></description>
	</item>

	<item>
	<title>Spivey's Generalized Recurrence for Bell Numbers</title>
	<link>/entries/Bell_Numbers_Spivey.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Bell_Numbers_Spivey.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of SD-Efficiency and SD-Strategy-Proofness</title>
	<link>/entries/SDS_Impossibility.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SDS_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases Theory</title>
	<link>/entries/Groebner_Bases.html</link>
	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Bases.html</guid>
	<description></description>
	</item>

	<item>
	<title>No Faster-Than-Light Observers</title>
	<link>/entries/No_FTL_observers.html</link>
	<pubDate>Thu, 28 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/No_FTL_observers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formalisation of the Cocke-Younger-Kasami algorithm</title>
	<link>/entries/CYK.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/CYK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algorithms for Reduced Ordered Binary Decision Diagrams</title>
	<link>/entries/ROBDD.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/ROBDD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Sequential Composition</title>
	<link>/entries/Noninterference_Sequential_Composition.html</link>
	<pubDate>Tue, 26 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Sequential_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebras with Domain</title>
	<link>/entries/KAD.html</link>
	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional Resolution and Prime Implicates Generation</title>
	<link>/entries/PropResPI.html</link>
	<pubDate>Fri, 11 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/PropResPI.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cartan Fixed Point Theorems</title>
	<link>/entries/Cartan_FP.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Cartan_FP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Timed Automata</title>
	<link>/entries/Timed_Automata.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Temporal Logic</title>
	<link>/entries/LTL.html</link>
	<pubDate>Tue, 01 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
	<link>/entries/List_Update.html</link>
	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/List_Update.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Construction of Static Single Assignment Form</title>
	<link>/entries/Formal_SSA.html</link>
	<pubDate>Fri, 05 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Factorization</title>
	<link>/entries/Polynomial_Factorization.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Interpolation</title>
	<link>/entries/Polynomial_Interpolation.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Interpolation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knot Theory</title>
	<link>/entries/Knot_Theory.html</link>
	<pubDate>Wed, 20 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Knot_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tensor Product of Matrices</title>
	<link>/entries/Matrix_Tensor.html</link>
	<pubDate>Mon, 18 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix_Tensor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Number Partitions</title>
	<link>/entries/Card_Number_Partitions.html</link>
	<pubDate>Thu, 14 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Number_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Basic Geometric Properties of Triangles</title>
	<link>/entries/Triangle.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Triangle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Descartes' Rule of Signs</title>
	<link>/entries/Descartes_Sign_Rule.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Descartes_Sign_Rule.html</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville numbers</title>
	<link>/entries/Liouville_Numbers.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Liouville_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Divergence of the Prime Harmonic Series</title>
	<link>/entries/Prime_Harmonic_Series.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Harmonic_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic Numbers in Isabelle/HOL</title>
	<link>/entries/Algebraic_Numbers.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Applicative Lifting</title>
	<link>/entries/Applicative_Lifting.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Applicative_Lifting.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Stern-Brocot Tree</title>
	<link>/entries/Stern_Brocot.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Stern_Brocot.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Set Partitions</title>
	<link>/entries/Card_Partitions.html</link>
	<pubDate>Sat, 12 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Latin Square</title>
	<link>/entries/Latin_Square.html</link>
	<pubDate>Wed, 02 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Latin_Square.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ergodic Theory</title>
	<link>/entries/Ergodic_Theory.html</link>
	<pubDate>Tue, 01 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Ergodic_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Euler's Partition Theorem</title>
	<link>/entries/Euler_Partition.html</link>
	<pubDate>Thu, 19 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_Partition.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Tortoise and Hare Algorithm</title>
	<link>/entries/TortoiseHare.html</link>
	<pubDate>Wed, 18 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/TortoiseHare.html</guid>
	<description></description>
	</item>

	<item>
	<title>Planarity Certificates</title>
	<link>/entries/Planarity_Certificates.html</link>
	<pubDate>Wed, 11 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Planarity_Certificates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Positional Determinacy of Parity Games</title>
	<link>/entries/Parity_Game.html</link>
	<pubDate>Mon, 02 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Parity_Game.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Meta-Model for the Isabelle API</title>
	<link>/entries/Isabelle_Meta_Model.html</link>
	<pubDate>Wed, 16 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Meta_Model.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear Temporal Logic to Deterministic (Generalized) Rabin Automata</title>
	<link>/entries/LTL_to_DRA.html</link>
	<pubDate>Fri, 04 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices, Jordan Normal Forms, and Spectral Radius Theory</title>
	<link>/entries/Jordan_Normal_Form.html</link>
	<pubDate>Fri, 21 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing Diagrams II</title>
	<link>/entries/Decreasing-Diagrams-II.html</link>
	<pubDate>Thu, 20 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Decreasing-Diagrams-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inductive Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Inductive_Unwinding.html</link>
	<pubDate>Tue, 18 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Inductive_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Representations of Finite Groups</title>
	<link>/entries/Rep_Fin_Groups.html</link>
	<pubDate>Wed, 12 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Rep_Fin_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysing and Comparing Encodability Criteria for Process Calculi</title>
	<link>/entries/Encodability_Process_Calculi.html</link>
	<pubDate>Mon, 10 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Encodability_Process_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating Cases from Labeled Subgoals</title>
	<link>/entries/Case_Labeling.html</link>
	<pubDate>Tue, 21 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Case_Labeling.html</guid>
	<description></description>
	</item>

	<item>
	<title>Landau Symbols</title>
	<link>/entries/Landau_Symbols.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Landau_Symbols.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Akra-Bazzi theorem and the Master theorem</title>
	<link>/entries/Akra_Bazzi.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Akra_Bazzi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite Normal Form</title>
	<link>/entries/Hermite.html</link>
	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derangements Formula</title>
	<link>/entries/Derangements.html</link>
	<pubDate>Sat, 27 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Derangements.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Multirelations</title>
	<link>/entries/Multirelations.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Multirelations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Reasoning about Lists via List Interleaving</title>
	<link>/entries/List_Interleaving.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/List_Interleaving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generic Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Generic_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Generic_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Ipurge Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Ipurge_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Ipurge_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Parameterized Dynamic Tables</title>
	<link>/entries/Dynamic_Tables.html</link>
	<pubDate>Sun, 07 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Dynamic_Tables.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derivatives of Logical Formulas</title>
	<link>/entries/Formula_Derivatives.html</link>
	<pubDate>Thu, 28 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Formula_Derivatives.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Zoo of Probabilistic Systems</title>
	<link>/entries/Probabilistic_System_Zoo.html</link>
	<pubDate>Wed, 27 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_System_Zoo.html</guid>
	<description></description>
	</item>

	<item>
	<title>VCG - Combinatorial Vickrey-Clarke-Groves Auctions</title>
	<link>/entries/Vickrey_Clarke_Groves.html</link>
	<pubDate>Thu, 30 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Vickrey_Clarke_Groves.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Lattices</title>
	<link>/entries/Residuated_Lattices.html</link>
	<pubDate>Wed, 15 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Residuated_Lattices.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent IMP</title>
	<link>/entries/ConcurrentIMP.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentIMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO</title>
	<link>/entries/ConcurrentGC.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentGC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Trie</title>
	<link>/entries/Trie.html</link>
	<pubDate>Mon, 30 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Trie.html</guid>
	<description></description>
	</item>

	<item>
	<title>Consensus Refined</title>
	<link>/entries/Consensus_Refined.html</link>
	<pubDate>Wed, 18 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Consensus_Refined.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving class instances for datatypes</title>
	<link>/entries/Deriving.html</link>
	<pubDate>Wed, 11 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safety of Call Arity</title>
	<link>/entries/Call_Arity.html</link>
	<pubDate>Fri, 20 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Call_Arity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Echelon Form</title>
	<link>/entries/Echelon_Form.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Echelon_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>QR Decomposition</title>
	<link>/entries/QR_Decomposition.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/QR_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Automata in Hereditarily Finite Set Theory</title>
	<link>/entries/Finite_Automata_HF.html</link>
	<pubDate>Thu, 05 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Automata_HF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the UpDown Scheme</title>
	<link>/entries/UpDown_Scheme.html</link>
	<pubDate>Wed, 28 Jan 2015 00:00:00 +0000</pubDate>

	<guid>/entries/UpDown_Scheme.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Unified Policy Framework (UPF)</title>
	<link>/entries/UPF.html</link>
	<pubDate>Fri, 28 Nov 2014 00:00:00 +0000</pubDate>

	<guid>/entries/UPF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Loop freedom of the (untimed) AODV routing protocol</title>
	<link>/entries/AODV.html</link>
	<pubDate>Thu, 23 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AODV.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting Definition Option</title>
	<link>/entries/Lifting_Definition_Option.html</link>
	<pubDate>Mon, 13 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_Definition_Option.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion in HOL with Code Generation</title>
	<link>/entries/Stream_Fusion_Code.html</link>
	<pubDate>Fri, 10 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Stream_Fusion_Code.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Compiler for Probability Density Functions</title>
	<link>/entries/Density_Compiler.html</link>
	<pubDate>Thu, 09 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Density_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Refinement Calculus for Reactive Systems</title>
	<link>/entries/RefinementReactive.html</link>
	<pubDate>Wed, 08 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/RefinementReactive.html</guid>
	<description></description>
	</item>

	<item>
	<title>Certification Monads</title>
	<link>/entries/Certification_Monads.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Certification_Monads.html</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/entries/XML.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/XML.html</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative Insertion Sort</title>
	<link>/entries/Imperative_Insertion_Sort.html</link>
	<pubDate>Thu, 25 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Imperative_Insertion_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sturm–Tarski Theorem</title>
	<link>/entries/Sturm_Tarski.html</link>
	<pubDate>Fri, 19 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Sturm_Tarski.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cayley-Hamilton Theorem</title>
	<link>/entries/Cayley_Hamilton.html</link>
	<pubDate>Mon, 15 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Cayley_Hamilton.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Jordan-Hölder Theorem</title>
	<link>/entries/Jordan_Hoelder.html</link>
	<pubDate>Tue, 09 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Hoelder.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Queues Based on Braun Trees</title>
	<link>/entries/Priority_Queue_Braun.html</link>
	<pubDate>Thu, 04 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Queue_Braun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Algorithm and Its Applications</title>
	<link>/entries/Gauss_Jordan.html</link>
	<pubDate>Wed, 03 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Jordan.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Valued Special Functions: Upper and Lower Bounds</title>
	<link>/entries/Special_Function_Bounds.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Special_Function_Bounds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Vector Spaces</title>
	<link>/entries/VectorSpace.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/VectorSpace.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skew Heap</title>
	<link>/entries/Skew_Heap.html</link>
	<pubDate>Wed, 13 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Skew_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Splay Tree</title>
	<link>/entries/Splay_Tree.html</link>
	<pubDate>Tue, 12 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Splay_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Haskell's Show Class in Isabelle/HOL</title>
	<link>/entries/Show.html</link>
	<pubDate>Tue, 29 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Show.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Specification of a Generic Separation Kernel</title>
	<link>/entries/CISC-Kernel.html</link>
	<pubDate>Fri, 18 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CISC-Kernel.html</guid>
	<description></description>
	</item>

	<item>
	<title>pGCL for Isabelle</title>
	<link>/entries/pGCL.html</link>
	<pubDate>Sun, 13 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/pGCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amortized Complexity Verified</title>
	<link>/entries/Amortized_Complexity.html</link>
	<pubDate>Mon, 07 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Amortized_Complexity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Network Security Policy Verification</title>
	<link>/entries/Network_Security_Policy_Verification.html</link>
	<pubDate>Fri, 04 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Network_Security_Policy_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pop-Refinement</title>
	<link>/entries/Pop_Refinement.html</link>
	<pubDate>Thu, 03 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Pop_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decision Procedures for MSO on Words Based on Derivatives of Regular Expressions</title>
	<link>/entries/MSO_Regex_Equivalence.html</link>
	<pubDate>Thu, 12 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/MSO_Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Boolean Expression Checkers</title>
	<link>/entries/Boolean_Expression_Checkers.html</link>
	<pubDate>Sun, 08 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Boolean_Expression_Checkers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Fully Verified Executable LTL Model Checker</title>
	<link>/entries/CAVA_LTL_Modelchecker.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_LTL_Modelchecker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear-Time Temporal Logic to Generalized Büchi Automata</title>
	<link>/entries/LTL_to_GBA.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_GBA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Promela Formalization</title>
	<link>/entries/Promela.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Promela.html</guid>
	<description></description>
	</item>

	<item>
	<title>The CAVA Automata Library</title>
	<link>/entries/CAVA_Automata.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</title>
	<link>/entries/Gabow_SCC.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gabow_SCC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference Security in Communicating Sequential Processes</title>
	<link>/entries/Noninterference_CSP.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive closure according to Roy-Floyd-Warshall</title>
	<link>/entries/Roy_Floyd_Warshall.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Roy_Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Algebras</title>
	<link>/entries/Regular_Algebras.html</link>
	<pubDate>Wed, 21 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation and Analysis of Component Dependencies</title>
	<link>/entries/ComponentDependencies.html</link>
	<pubDate>Mon, 28 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/ComponentDependencies.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Assumptions and Guarantees for Compositional Noninterference</title>
	<link>/entries/SIFUM_Type_Systems.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Declassification with WHAT-and-WHERE-Security</title>
	<link>/entries/WHATandWHERE_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/WHATandWHERE_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Strong Security</title>
	<link>/entries/Strong_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Strong_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded-Deducibility Security</title>
	<link>/entries/Bounded_Deducibility_Security.html</link>
	<pubDate>Tue, 22 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Bounded_Deducibility_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A shallow embedding of HyperCTL*</title>
	<link>/entries/HyperCTL.html</link>
	<pubDate>Wed, 16 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/HyperCTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Completeness</title>
	<link>/entries/Abstract_Completeness.html</link>
	<pubDate>Wed, 16 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract_Completeness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Discrete Summation</title>
	<link>/entries/Discrete_Summation.html</link>
	<pubDate>Sun, 13 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Discrete_Summation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax and semantics of a GPU kernel programming language</title>
	<link>/entries/GPU_Kernel_PL.html</link>
	<pubDate>Thu, 03 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/GPU_Kernel_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Noninterference</title>
	<link>/entries/Probabilistic_Noninterference.html</link>
	<pubDate>Tue, 11 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mechanization of the Algebra for Wireless Networks (AWN)</title>
	<link>/entries/AWN.html</link>
	<pubDate>Sat, 08 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AWN.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mutually Recursive Partial Functions</title>
	<link>/entries/Partial_Function_MR.html</link>
	<pubDate>Tue, 18 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Function_MR.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Random Graphs -- Subgraph Containment</title>
	<link>/entries/Random_Graph_Subgraph_Threshold.html</link>
	<pubDate>Thu, 13 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Random_Graph_Subgraph_Threshold.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of Selection and Heap Sort Using Locales</title>
	<link>/entries/Selection_Heap_Sort.html</link>
	<pubDate>Tue, 11 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Selection_Heap_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Affine Arithmetic</title>
	<link>/entries/Affine_Arithmetic.html</link>
	<pubDate>Fri, 07 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Affine_Arithmetic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Implementing field extensions of the form Q[sqrt(b)]</title>
	<link>/entries/Real_Impl.html</link>
	<pubDate>Thu, 06 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Impl.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unified Decision Procedures for Regular Expression Equivalence</title>
	<link>/entries/Regex_Equivalence.html</link>
	<pubDate>Thu, 30 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secondary Sylow Theorems</title>
	<link>/entries/Secondary_Sylow.html</link>
	<pubDate>Tue, 28 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Secondary_Sylow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relation Algebra</title>
	<link>/entries/Relation_Algebra.html</link>
	<pubDate>Sat, 25 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Relation_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
	<link>/entries/KAT_and_DRA.html</link>
	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/KAT_and_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5</title>
	<link>/entries/Featherweight_OCL.html</link>
	<pubDate>Thu, 16 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Featherweight_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Properties of Crypto-Based Components</title>
	<link>/entries/CryptoBasedCompositionalProperties.html</link>
	<pubDate>Sat, 11 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CryptoBasedCompositionalProperties.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm's Theorem</title>
	<link>/entries/Sturm_Sequences.html</link>
	<pubDate>Sat, 11 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Sturm_Sequences.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Method for the Proof of Theorems on Tail-recursive Functions</title>
	<link>/entries/Tail_Recursive_Functions.html</link>
	<pubDate>Sun, 01 Dec 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Tail_Recursive_Functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gödel's Incompleteness Theorems</title>
	<link>/entries/Incompleteness.html</link>
	<pubDate>Sun, 17 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Incompleteness.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hereditarily Finite Sets</title>
	<link>/entries/HereditarilyFinite.html</link>
	<pubDate>Sun, 17 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/HereditarilyFinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Codatatype of Formal Languages</title>
	<link>/entries/Coinductive_Languages.html</link>
	<pubDate>Fri, 15 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive_Languages.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Processing Components: Isabelle/HOL Formalisation and Case Studies</title>
	<link>/entries/FocusStreamsCaseStudies.html</link>
	<pubDate>Thu, 14 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/FocusStreamsCaseStudies.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gödel's God in Isabelle/HOL</title>
	<link>/entries/GoedelGod.html</link>
	<pubDate>Tue, 12 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/GoedelGod.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing Diagrams</title>
	<link>/entries/Decreasing-Diagrams.html</link>
	<pubDate>Fri, 01 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Decreasing-Diagrams.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automatic Data Refinement</title>
	<link>/entries/Automatic_Refinement.html</link>
	<pubDate>Wed, 02 Oct 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Automatic_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Native Word</title>
	<link>/entries/Native_Word.html</link>
	<pubDate>Tue, 17 Sep 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Native_Word.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of IEEE Floating Point Arithmetic</title>
	<link>/entries/IEEE_Floating_Point.html</link>
	<pubDate>Sat, 27 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/IEEE_Floating_Point.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lehmer's Theorem</title>
	<link>/entries/Lehmer.html</link>
	<pubDate>Mon, 22 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Lehmer.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pratt's Primality Certificates</title>
	<link>/entries/Pratt_Certificate.html</link>
	<pubDate>Mon, 22 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Pratt_Certificate.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Königsberg Bridge Problem and the Friendship Theorem</title>
	<link>/entries/Koenigsberg_Friendship.html</link>
	<pubDate>Fri, 19 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Koenigsberg_Friendship.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sound and Complete Sort Encodings for First-Order Logic</title>
	<link>/entries/Sort_Encodings.html</link>
	<pubDate>Thu, 27 Jun 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Sort_Encodings.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Axiomatic Characterization of the Single-Source Shortest Path Problem</title>
	<link>/entries/ShortestPath.html</link>
	<pubDate>Wed, 22 May 2013 00:00:00 +0000</pubDate>

	<guid>/entries/ShortestPath.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Theory</title>
	<link>/entries/Graph_Theory.html</link>
	<pubDate>Sun, 28 Apr 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Graph_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Light-weight Containers</title>
	<link>/entries/Containers.html</link>
	<pubDate>Mon, 15 Apr 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Containers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Nominal 2</title>
	<link>/entries/Nominal2.html</link>
	<pubDate>Thu, 21 Feb 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Nominal2.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</title>
	<link>/entries/Launchbury.html</link>
	<pubDate>Thu, 31 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Launchbury.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ribbon Proofs</title>
	<link>/entries/Ribbon_Proofs.html</link>
	<pubDate>Sat, 19 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Ribbon_Proofs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Rank-Nullity Theorem in Linear Algebra</title>
	<link>/entries/Rank_Nullity_Theorem.html</link>
	<pubDate>Wed, 16 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Rank_Nullity_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra</title>
	<link>/entries/Kleene_Algebra.html</link>
	<pubDate>Tue, 15 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Kleene_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Computing N-th Roots using the Babylonian Method</title>
	<link>/entries/Sqrt_Babylonian.html</link>
	<pubDate>Thu, 03 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Sqrt_Babylonian.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Separation Logic Framework for Imperative HOL</title>
	<link>/entries/Separation_Logic_Imperative_HOL.html</link>
	<pubDate>Wed, 14 Nov 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Open Induction</title>
	<link>/entries/Open_Induction.html</link>
	<pubDate>Fri, 02 Nov 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Open_Induction.html</guid>
	<description></description>
	</item>

	<item>
	<title>The independence of Tarski's Euclidean axiom</title>
	<link>/entries/Tarskis_Geometry.html</link>
	<pubDate>Tue, 30 Oct 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Tarskis_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bondy's Theorem</title>
	<link>/entries/Bondy.html</link>
	<pubDate>Sat, 27 Oct 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Bondy.html</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic Noninterference</title>
	<link>/entries/Possibilistic_Noninterference.html</link>
	<pubDate>Mon, 10 Sep 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Possibilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating linear orders for datatypes</title>
	<link>/entries/Datatype_Order_Generator.html</link>
	<pubDate>Tue, 07 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Datatype_Order_Generator.html</guid>
	<description></description>
	</item>

	<item>
	<title>Proving the Impossibility of Trisecting an Angle and Doubling the Cube</title>
	<link>/entries/Impossible_Geometry.html</link>
	<pubDate>Sun, 05 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Impossible_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</title>
	<link>/entries/Heard_Of.html</link>
	<pubDate>Fri, 27 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Heard_Of.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logical Relations for PCF</title>
	<link>/entries/PCF.html</link>
	<pubDate>Sun, 01 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/PCF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Type Constructor Classes and Monad Transformers</title>
	<link>/entries/Tycon.html</link>
	<pubDate>Tue, 26 Jun 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Tycon.html</guid>
	<description></description>
	</item>

	<item>
	<title>CCS in nominal logic</title>
	<link>/entries/CCS.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/CCS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Psi-calculi in Isabelle</title>
	<link>/entries/Psi_Calculi.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Psi_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>The pi-calculus in nominal logic</title>
	<link>/entries/Pi_Calculus.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Circus</title>
	<link>/entries/Circus.html</link>
	<pubDate>Sun, 27 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Circus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separation Algebra</title>
	<link>/entries/Separation_Algebra.html</link>
	<pubDate>Fri, 11 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stuttering Equivalence</title>
	<link>/entries/Stuttering_Equivalence.html</link>
	<pubDate>Mon, 07 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Stuttering_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive Study of Confidentiality</title>
	<link>/entries/Inductive_Confidentiality.html</link>
	<pubDate>Wed, 02 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Confidentiality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinary Differential Equations</title>
	<link>/entries/Ordinary_Differential_Equations.html</link>
	<pubDate>Thu, 26 Apr 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinary_Differential_Equations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Well-Quasi-Orders</title>
	<link>/entries/Well_Quasi_Orders.html</link>
	<pubDate>Fri, 13 Apr 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Well_Quasi_Orders.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abortable Linearizable Modules</title>
	<link>/entries/Abortable_Linearizable_Modules.html</link>
	<pubDate>Thu, 01 Mar 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Abortable_Linearizable_Modules.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures</title>
	<link>/entries/Transitive-Closure-II.html</link>
	<pubDate>Wed, 29 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Probabilistic Proof of the Girth-Chromatic Number Theorem</title>
	<link>/entries/Girth_Chromatic.html</link>
	<pubDate>Mon, 06 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Girth_Chromatic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dijkstra's Shortest Path Algorithm</title>
	<link>/entries/Dijkstra_Shortest_Path.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Dijkstra_Shortest_Path.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refinement for Monadic Programs</title>
	<link>/entries/Refine_Monadic.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Monadic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Models</title>
	<link>/entries/Markov_Models.html</link>
	<pubDate>Tue, 03 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Definitional Encoding of TLA* in Isabelle/HOL</title>
	<link>/entries/TLA.html</link>
	<pubDate>Sat, 19 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/TLA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient Mergesort</title>
	<link>/entries/Efficient-Mergesort.html</link>
	<pubDate>Wed, 09 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Efficient-Mergesort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebra of Monotonic Boolean Transformers</title>
	<link>/entries/MonoBoolTranAlgebra.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/MonoBoolTranAlgebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lattice Properties</title>
	<link>/entries/LatticeProperties.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/LatticeProperties.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pseudo Hoops</title>
	<link>/entries/PseudoHoops.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/PseudoHoops.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Myhill-Nerode Theorem Based on Regular Expressions</title>
	<link>/entries/Myhill-Nerode.html</link>
	<pubDate>Fri, 26 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Myhill-Nerode.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
	<link>/entries/Gauss-Jordan-Elim-Fun.html</link>
	<pubDate>Fri, 19 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss-Jordan-Elim-Fun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum Cardinality Matching</title>
	<link>/entries/Max-Card-Matching.html</link>
	<pubDate>Thu, 21 Jul 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Max-Card-Matching.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knowledge-based programs</title>
	<link>/entries/KBPs.html</link>
	<pubDate>Tue, 17 May 2011 00:00:00 +0000</pubDate>

	<guid>/entries/KBPs.html</guid>
	<description></description>
	</item>

	<item>
	<title>The General Triangle Is Unique</title>
	<link>/entries/General-Triangle.html</link>
	<pubDate>Fri, 01 Apr 2011 00:00:00 +0000</pubDate>

	<guid>/entries/General-Triangle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures of Finite Relations</title>
	<link>/entries/Transitive-Closure.html</link>
	<pubDate>Mon, 14 Mar 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure.html</guid>
	<description></description>
	</item>

	<item>
	<title>AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</title>
	<link>/entries/AutoFocus-Stream.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/AutoFocus-Stream.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infinite Lists</title>
	<link>/entries/List-Infinite.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/List-Infinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interval Temporal Logic on Natural Numbers</title>
	<link>/entries/Nat-Interval-Logic.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Nat-Interval-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lightweight Java</title>
	<link>/entries/LightweightJava.html</link>
	<pubDate>Mon, 07 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/LightweightJava.html</guid>
	<description></description>
	</item>

	<item>
	<title>RIPEMD-160</title>
	<link>/entries/RIPEMD-160-SPARK.html</link>
	<pubDate>Mon, 10 Jan 2011 00:00:00 +0000</pubDate>

	<guid>/entries/RIPEMD-160-SPARK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower Semicontinuous Functions</title>
	<link>/entries/Lower_Semicontinuous.html</link>
	<pubDate>Sat, 08 Jan 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Lower_Semicontinuous.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hall's Marriage Theorem</title>
	<link>/entries/Marriage.html</link>
	<pubDate>Fri, 17 Dec 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Marriage.html</guid>
	<description></description>
	</item>

	<item>
	<title>Shivers' Control Flow Analysis</title>
	<link>/entries/Shivers-CFA.html</link>
	<pubDate>Tue, 16 Nov 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Shivers-CFA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial Heaps and Skew Binomial Heaps</title>
	<link>/entries/Binomial-Heaps.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Heaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finger Trees</title>
	<link>/entries/Finger-Trees.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Finger-Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Functional Binomial Queues</title>
	<link>/entries/Binomial-Queues.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Queues.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Normalization of Moggis's Computational Metalanguage</title>
	<link>/entries/Lam-ml-Normalization.html</link>
	<pubDate>Sun, 29 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Lam-ml-Normalization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Multivariate Polynomials</title>
	<link>/entries/Polynomials.html</link>
	<pubDate>Tue, 10 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Statecharts using Hierarchical Automata</title>
	<link>/entries/Statecharts.html</link>
	<pubDate>Sun, 08 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Statecharts.html</guid>
	<description></description>
	</item>

	<item>
	<title>Free Groups</title>
	<link>/entries/Free-Groups.html</link>
	<pubDate>Thu, 24 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Free-Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory</title>
	<link>/entries/Category2.html</link>
	<pubDate>Sun, 20 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Category2.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Matrix Operations on Matrices of Arbitrary Dimensions</title>
	<link>/entries/Matrix.html</link>
	<pubDate>Thu, 17 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Rewriting</title>
	<link>/entries/Abstract-Rewriting.html</link>
	<pubDate>Mon, 14 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract-Rewriting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Semantics and Data Refinement of Invariant Based Programs</title>
	<link>/entries/DataRefinementIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/DataRefinementIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</title>
	<link>/entries/GraphMarkingIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/GraphMarkingIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Complete Proof of the Robbins Conjecture</title>
	<link>/entries/Robbins-Conjecture.html</link>
	<pubDate>Sat, 22 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Robbins-Conjecture.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Sets and Expressions</title>
	<link>/entries/Regular-Sets.html</link>
	<pubDate>Wed, 12 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Regular-Sets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Locally Nameless Sigma Calculus</title>
	<link>/entries/Locally-Nameless-Sigma.html</link>
	<pubDate>Fri, 30 Apr 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Locally-Nameless-Sigma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Free Boolean Algebra</title>
	<link>/entries/Free-Boolean-Algebra.html</link>
	<pubDate>Mon, 29 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Free-Boolean-Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inter-Procedural Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing_Inter.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing_Inter.html</guid>
	<description></description>
	</item>

	<item>
	<title>List Index</title>
	<link>/entries/List-Index.html</link>
	<pubDate>Sat, 20 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/List-Index.html</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive</title>
	<link>/entries/Coinductive.html</link>
	<pubDate>Fri, 12 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Fast SAT Solver for Isabelle in Standard ML</title>
	<link>/entries/DPT-SAT-Solver.html</link>
	<pubDate>Wed, 09 Dec 2009 00:00:00 +0000</pubDate>

	<guid>/entries/DPT-SAT-Solver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Logic-Automaton Connection</title>
	<link>/entries/Presburger-Automata.html</link>
	<pubDate>Thu, 03 Dec 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Presburger-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Collections Framework</title>
	<link>/entries/Collections.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Collections.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Automata</title>
	<link>/entries/Tree-Automata.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Tree-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Perfect Number Theorem</title>
	<link>/entries/Perfect-Number-Thm.html</link>
	<pubDate>Sun, 22 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Perfect-Number-Thm.html</guid>
	<description></description>
	</item>

	<item>
	<title>Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</title>
	<link>/entries/HRB-Slicing.html</link>
	<pubDate>Fri, 13 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/HRB-Slicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Worker/Wrapper Transformation</title>
	<link>/entries/WorkerWrapper.html</link>
	<pubDate>Fri, 30 Oct 2009 00:00:00 +0000</pubDate>

	<guid>/entries/WorkerWrapper.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinals and Cardinals</title>
	<link>/entries/Ordinals_and_Cardinals.html</link>
	<pubDate>Tue, 01 Sep 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinals_and_Cardinals.html</guid>
	<description></description>
	</item>

	<item>
	<title>Invertibility in Sequent Calculi</title>
	<link>/entries/SequentInvertibility.html</link>
	<pubDate>Fri, 28 Aug 2009 00:00:00 +0000</pubDate>

	<guid>/entries/SequentInvertibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Example of a Cofinitary Group in Isabelle/HOL</title>
	<link>/entries/CofGroups.html</link>
	<pubDate>Tue, 04 Aug 2009 00:00:00 +0000</pubDate>

	<guid>/entries/CofGroups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Code Generation for Functions as Data</title>
	<link>/entries/FinFun.html</link>
	<pubDate>Wed, 06 May 2009 00:00:00 +0000</pubDate>

	<guid>/entries/FinFun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion</title>
	<link>/entries/Stream-Fusion.html</link>
	<pubDate>Wed, 29 Apr 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Stream-Fusion.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Bytecode Logic for JML and Types</title>
	<link>/entries/BytecodeLogicJmlTypes.html</link>
	<pubDate>Fri, 12 Dec 2008 00:00:00 +0000</pubDate>

	<guid>/entries/BytecodeLogicJmlTypes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secure information flow and program logics</title>
	<link>/entries/SIFPL.html</link>
	<pubDate>Mon, 10 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SIFPL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Some classical results in Social Choice Theory</title>
	<link>/entries/SenSocialChoice.html</link>
	<pubDate>Sun, 09 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SenSocialChoice.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fun With Tilings</title>
	<link>/entries/FunWithTilings.html</link>
	<pubDate>Fri, 07 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/FunWithTilings.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Textbook Proof of Huffman's Algorithm</title>
	<link>/entries/Huffman.html</link>
	<pubDate>Wed, 15 Oct 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Huffman.html</guid>
	<description></description>
	</item>

	<item>
	<title>Towards Certified Slicing</title>
	<link>/entries/Slicing.html</link>
	<pubDate>Tue, 16 Sep 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Slicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Correctness Proof for the Volpano/Smith Security Typing System</title>
	<link>/entries/VolpanoSmith.html</link>
	<pubDate>Tue, 02 Sep 2008 00:00:00 +0000</pubDate>

	<guid>/entries/VolpanoSmith.html</guid>
	<description></description>
	</item>

	<item>
	<title>Arrow and Gibbard-Satterthwaite</title>
	<link>/entries/ArrowImpossibilityGS.html</link>
	<pubDate>Mon, 01 Sep 2008 00:00:00 +0000</pubDate>

	<guid>/entries/ArrowImpossibilityGS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fun With Functions</title>
	<link>/entries/FunWithFunctions.html</link>
	<pubDate>Tue, 26 Aug 2008 00:00:00 +0000</pubDate>

	<guid>/entries/FunWithFunctions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Verification of Modern SAT Solvers</title>
	<link>/entries/SATSolverVerification.html</link>
	<pubDate>Wed, 23 Jul 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SATSolverVerification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion Theory I</title>
	<link>/entries/Recursion-Theory-I.html</link>
	<pubDate>Sat, 05 Apr 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Recursion-Theory-I.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</title>
	<link>/entries/Simpl.html</link>
	<pubDate>Fri, 29 Feb 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Simpl.html</guid>
	<description></description>
	</item>

	<item>
	<title>BDD Normalisation</title>
	<link>/entries/BDD.html</link>
	<pubDate>Fri, 29 Feb 2008 00:00:00 +0000</pubDate>

	<guid>/entries/BDD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Normalization by Evaluation</title>
	<link>/entries/NormByEval.html</link>
	<pubDate>Mon, 18 Feb 2008 00:00:00 +0000</pubDate>

	<guid>/entries/NormByEval.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantifier Elimination for Linear Arithmetic</title>
	<link>/entries/LinearQuantifierElim.html</link>
	<pubDate>Fri, 11 Jan 2008 00:00:00 +0000</pubDate>

	<guid>/entries/LinearQuantifierElim.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors</title>
	<link>/entries/Program-Conflict-Analysis.html</link>
	<pubDate>Fri, 14 Dec 2007 00:00:00 +0000</pubDate>

	<guid>/entries/Program-Conflict-Analysis.html</guid>
	<description></description>
	</item>

	<item>
	<title>Jinja with Threads</title>
	<link>/entries/JinjaThreads.html</link>
	<pubDate>Mon, 03 Dec 2007 00:00:00 +0000</pubDate>

	<guid>/entries/JinjaThreads.html</guid>
	<description></description>
	</item>

	<item>
	<title>Much Ado About Two</title>
	<link>/entries/MuchAdoAboutTwo.html</link>
	<pubDate>Tue, 06 Nov 2007 00:00:00 +0000</pubDate>

	<guid>/entries/MuchAdoAboutTwo.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</title>
	<link>/entries/Fermat3_4.html</link>
	<pubDate>Sun, 12 Aug 2007 00:00:00 +0000</pubDate>

	<guid>/entries/Fermat3_4.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sums of Two and Four Squares</title>
	<link>/entries/SumSquares.html</link>
	<pubDate>Sun, 12 Aug 2007 00:00:00 +0000</pubDate>

	<guid>/entries/SumSquares.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fundamental Properties of Valuation Theory and Hensel's Lemma</title>
	<link>/entries/Valuation.html</link>
	<pubDate>Wed, 08 Aug 2007 00:00:00 +0000</pubDate>

	<guid>/entries/Valuation.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Logic According to Fitting</title>
	<link>/entries/FOL-Fitting.html</link>
	<pubDate>Thu, 02 Aug 2007 00:00:00 +0000</pubDate>

	<guid>/entries/FOL-Fitting.html</guid>
	<description></description>
	</item>

	<item>
	<title>POPLmark Challenge Via de Bruijn Indices</title>
	<link>/entries/POPLmark-deBruijn.html</link>
	<pubDate>Thu, 02 Aug 2007 00:00:00 +0000</pubDate>

	<guid>/entries/POPLmark-deBruijn.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hotel Key Card System</title>
	<link>/entries/HotelKeyCards.html</link>
	<pubDate>Sat, 09 Sep 2006 00:00:00 +0000</pubDate>

	<guid>/entries/HotelKeyCards.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Hoare Logics</title>
	<link>/entries/Abstract-Hoare-Logics.html</link>
	<pubDate>Tue, 08 Aug 2006 00:00:00 +0000</pubDate>

	<guid>/entries/Abstract-Hoare-Logics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Flyspeck I: Tame Graphs</title>
	<link>/entries/Flyspeck-Tame.html</link>
	<pubDate>Mon, 22 May 2006 00:00:00 +0000</pubDate>

	<guid>/entries/Flyspeck-Tame.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoreC++</title>
	<link>/entries/CoreC++.html</link>
	<pubDate>Mon, 15 May 2006 00:00:00 +0000</pubDate>

	<guid>/entries/CoreC++.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Theory of Featherweight Java in Isabelle/HOL</title>
	<link>/entries/FeatherweightJava.html</link>
	<pubDate>Fri, 31 Mar 2006 00:00:00 +0000</pubDate>

	<guid>/entries/FeatherweightJava.html</guid>
	<description></description>
	</item>

	<item>
	<title>Instances of Schneider's generalized protocol of clock synchronization</title>
	<link>/entries/ClockSynchInst.html</link>
	<pubDate>Wed, 15 Mar 2006 00:00:00 +0000</pubDate>

	<guid>/entries/ClockSynchInst.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality</title>
	<link>/entries/Cauchy.html</link>
	<pubDate>Tue, 14 Mar 2006 00:00:00 +0000</pubDate>

	<guid>/entries/Cauchy.html</guid>
	<description></description>
	</item>

	<item>
	<title>Countable Ordinals</title>
	<link>/entries/Ordinal.html</link>
	<pubDate>Fri, 11 Nov 2005 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinal.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fast Fourier Transform</title>
	<link>/entries/FFT.html</link>
	<pubDate>Wed, 12 Oct 2005 00:00:00 +0000</pubDate>

	<guid>/entries/FFT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Generalized Protocol for Clock Synchronization</title>
	<link>/entries/GenClock.html</link>
	<pubDate>Fri, 24 Jun 2005 00:00:00 +0000</pubDate>

	<guid>/entries/GenClock.html</guid>
	<description></description>
	</item>

	<item>
	<title>Proving the Correctness of Disk Paxos</title>
	<link>/entries/DiskPaxos.html</link>
	<pubDate>Wed, 22 Jun 2005 00:00:00 +0000</pubDate>

	<guid>/entries/DiskPaxos.html</guid>
	<description></description>
	</item>

	<item>
	<title>Jive Data and Store Model</title>
	<link>/entries/JiveDataStoreModel.html</link>
	<pubDate>Mon, 20 Jun 2005 00:00:00 +0000</pubDate>

	<guid>/entries/JiveDataStoreModel.html</guid>
	<description></description>
	</item>

	<item>
	<title>Jinja is not Java</title>
	<link>/entries/Jinja.html</link>
	<pubDate>Wed, 01 Jun 2005 00:00:00 +0000</pubDate>

	<guid>/entries/Jinja.html</guid>
	<description></description>
	</item>

	<item>
	<title>SHA1, RSA, PSS and more</title>
	<link>/entries/RSAPSS.html</link>
	<pubDate>Mon, 02 May 2005 00:00:00 +0000</pubDate>

	<guid>/entries/RSAPSS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory to Yoneda's Lemma</title>
	<link>/entries/Category.html</link>
	<pubDate>Thu, 21 Apr 2005 00:00:00 +0000</pubDate>

	<guid>/entries/Category.html</guid>
	<description></description>
	</item>

	<item>
	<title>File Refinement</title>
	<link>/entries/FileRefinement.html</link>
	<pubDate>Thu, 09 Dec 2004 00:00:00 +0000</pubDate>

	<guid>/entries/FileRefinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Integration theory and random variables</title>
	<link>/entries/Integration.html</link>
	<pubDate>Fri, 19 Nov 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Integration.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic</title>
	<link>/entries/Verified-Prover.html</link>
	<pubDate>Tue, 28 Sep 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Verified-Prover.html</guid>
	<description></description>
	</item>

	<item>
	<title>Completeness theorem</title>
	<link>/entries/Completeness.html</link>
	<pubDate>Mon, 20 Sep 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Completeness.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ramsey's theorem, infinitary version</title>
	<link>/entries/Ramsey-Infinite.html</link>
	<pubDate>Mon, 20 Sep 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Ramsey-Infinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compiling Exceptions Correctly</title>
	<link>/entries/Compiling-Exceptions-Correctly.html</link>
	<pubDate>Fri, 09 Jul 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Compiling-Exceptions-Correctly.html</guid>
	<description></description>
	</item>

	<item>
	<title>Depth First Search</title>
	<link>/entries/Depth-First-Search.html</link>
	<pubDate>Thu, 24 Jun 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Depth-First-Search.html</guid>
	<description></description>
	</item>

	<item>
	<title>Groups, Rings and Modules</title>
	<link>/entries/Group-Ring-Module.html</link>
	<pubDate>Tue, 18 May 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Group-Ring-Module.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy Lists II</title>
	<link>/entries/Lazy-Lists-II.html</link>
	<pubDate>Mon, 26 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Lazy-Lists-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>Topology</title>
	<link>/entries/Topology.html</link>
	<pubDate>Mon, 26 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Topology.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Search Trees</title>
	<link>/entries/BinarySearchTree.html</link>
	<pubDate>Mon, 05 Apr 2004 00:00:00 +0000</pubDate>

	<guid>/entries/BinarySearchTree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Functional Automata</title>
	<link>/entries/Functional-Automata.html</link>
	<pubDate>Tue, 30 Mar 2004 00:00:00 +0000</pubDate>

	<guid>/entries/Functional-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>AVL Trees</title>
	<link>/entries/AVL-Trees.html</link>
	<pubDate>Fri, 19 Mar 2004 00:00:00 +0000</pubDate>

	<guid>/entries/AVL-Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mini ML</title>
	<link>/entries/MiniML.html</link>
	<pubDate>Fri, 19 Mar 2004 00:00:00 +0000</pubDate>

	<guid>/entries/MiniML.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abortable_Linearizable_Modules</title>
	<link>/theories/abortable_linearizable_modules/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abortable_linearizable_modules/</guid>
	<description></description>
	</item>

	<item>
	<title>Abs_Int_ITP2012</title>
	<link>/theories/abs_int_itp2012/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abs_int_itp2012/</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract-Hoare-Logics</title>
	<link>/theories/abstract-hoare-logics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abstract-hoare-logics/</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract-Rewriting</title>
	<link>/theories/abstract-rewriting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abstract-rewriting/</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract_Completeness</title>
	<link>/theories/abstract_completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abstract_completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract_Soundness</title>
	<link>/theories/abstract_soundness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/abstract_soundness/</guid>
	<description></description>
	</item>

	<item>
	<title>ABY3_Protocols</title>
	<link>/theories/aby3_protocols/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aby3_protocols/</guid>
	<description></description>
	</item>

	<item>
	<title>Ackermanns_not_PR</title>
	<link>/theories/ackermanns_not_pr/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ackermanns_not_pr/</guid>
	<description></description>
	</item>

	<item>
	<title>Actuarial_Mathematics</title>
	<link>/theories/actuarial_mathematics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/actuarial_mathematics/</guid>
	<description></description>
	</item>

	<item>
	<title>Adaptive_State_Counting</title>
	<link>/theories/adaptive_state_counting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/adaptive_state_counting/</guid>
	<description></description>
	</item>

	<item>
	<title>ADS_Functor</title>
	<link>/theories/ads_functor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ads_functor/</guid>
	<description></description>
	</item>

	<item>
	<title>Affine_Arithmetic</title>
	<link>/theories/affine_arithmetic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/affine_arithmetic/</guid>
	<description></description>
	</item>

	<item>
	<title>Aggregation_Algebras</title>
	<link>/theories/aggregation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aggregation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>AI_Planning_Languages_Semantics</title>
	<link>/theories/ai_planning_languages_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ai_planning_languages_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Akra_Bazzi</title>
	<link>/theories/akra_bazzi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/akra_bazzi/</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic_Numbers</title>
	<link>/theories/algebraic_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/algebraic_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic_VCs</title>
	<link>/theories/algebraic_vcs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/algebraic_vcs/</guid>
	<description></description>
	</item>

	<item>
	<title>Allen_Calculus</title>
	<link>/theories/allen_calculus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/allen_calculus/</guid>
	<description></description>
	</item>

	<item>
	<title>Amicable_Numbers</title>
	<link>/theories/amicable_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/amicable_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Amortized_Complexity</title>
	<link>/theories/amortized_complexity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/amortized_complexity/</guid>
	<description></description>
	</item>

	<item>
	<title>AnselmGod</title>
	<link>/theories/anselmgod/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/anselmgod/</guid>
	<description></description>
	</item>

	<item>
	<title>AODV</title>
	<link>/theories/aodv/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aodv/</guid>
	<description></description>
	</item>

	<item>
	<title>AOT</title>
	<link>/theories/aot/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aot/</guid>
	<description></description>
	</item>

	<item>
	<title>Applicative_Lifting</title>
	<link>/theories/applicative_lifting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/applicative_lifting/</guid>
	<description></description>
	</item>

	<item>
	<title>Approximation_Algorithms</title>
	<link>/theories/approximation_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/approximation_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Architectural_Design_Patterns</title>
	<link>/theories/architectural_design_patterns/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/architectural_design_patterns/</guid>
	<description></description>
	</item>

	<item>
	<title>Aristotles_Assertoric_Syllogistic</title>
	<link>/theories/aristotles_assertoric_syllogistic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aristotles_assertoric_syllogistic/</guid>
	<description></description>
	</item>

	<item>
	<title>Arith_Prog_Rel_Primes</title>
	<link>/theories/arith_prog_rel_primes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/arith_prog_rel_primes/</guid>
	<description></description>
	</item>

	<item>
	<title>ArrowImpossibilityGS</title>
	<link>/theories/arrowimpossibilitygs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/arrowimpossibilitygs/</guid>
	<description></description>
	</item>

	<item>
	<title>Attack_Trees</title>
	<link>/theories/attack_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/attack_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2_HOL</title>
	<link>/theories/auto2_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/auto2_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2_Imperative_HOL</title>
	<link>/theories/auto2_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/auto2_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>AutoFocus-Stream</title>
	<link>/theories/autofocus-stream/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/autofocus-stream/</guid>
	<description></description>
	</item>

	<item>
	<title>Automated_Stateful_Protocol_Verification</title>
	<link>/theories/automated_stateful_protocol_verification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/automated_stateful_protocol_verification/</guid>
	<description></description>
	</item>

	<item>
	<title>Automatic_Refinement</title>
	<link>/theories/automatic_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/automatic_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>AVL-Trees</title>
	<link>/theories/avl-trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/avl-trees/</guid>
	<description></description>
	</item>

	<item>
	<title>AWN</title>
	<link>/theories/awn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/awn/</guid>
	<description></description>
	</item>

	<item>
	<title>AxiomaticCategoryTheory</title>
	<link>/theories/axiomaticcategorytheory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/axiomaticcategorytheory/</guid>
	<description></description>
	</item>

	<item>
	<title>Balog_Szemeredi_Gowers</title>
	<link>/theories/balog_szemeredi_gowers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/balog_szemeredi_gowers/</guid>
	<description></description>
	</item>

	<item>
	<title>Banach_Steinhaus</title>
	<link>/theories/banach_steinhaus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/banach_steinhaus/</guid>
	<description></description>
	</item>

	<item>
	<title>BD_Security_Compositional</title>
	<link>/theories/bd_security_compositional/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bd_security_compositional/</guid>
	<description></description>
	</item>

	<item>
	<title>BDD</title>
	<link>/theories/bdd/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bdd/</guid>
	<description></description>
	</item>

	<item>
	<title>Belief_Revision</title>
	<link>/theories/belief_revision/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/belief_revision/</guid>
	<description></description>
	</item>

	<item>
	<title>Bell_Numbers_Spivey</title>
	<link>/theories/bell_numbers_spivey/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bell_numbers_spivey/</guid>
	<description></description>
	</item>

	<item>
	<title>BenOr_Kozen_Reif</title>
	<link>/theories/benor_kozen_reif/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/benor_kozen_reif/</guid>
	<description></description>
	</item>

	<item>
	<title>Berlekamp_Zassenhaus</title>
	<link>/theories/berlekamp_zassenhaus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/berlekamp_zassenhaus/</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli</title>
	<link>/theories/bernoulli/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bernoulli/</guid>
	<description></description>
	</item>

	<item>
	<title>Bertrands_Postulate</title>
	<link>/theories/bertrands_postulate/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bertrands_postulate/</guid>
	<description></description>
	</item>

	<item>
	<title>Bicategory</title>
	<link>/theories/bicategory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bicategory/</guid>
	<description></description>
	</item>

	<item>
	<title>Binary_Code_Imprimitive</title>
	<link>/theories/binary_code_imprimitive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binary_code_imprimitive/</guid>
	<description></description>
	</item>

	<item>
	<title>BinarySearchTree</title>
	<link>/theories/binarysearchtree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binarysearchtree/</guid>
	<description></description>
	</item>

	<item>
	<title>Binding_Syntax_Theory</title>
	<link>/theories/binding_syntax_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binding_syntax_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial-Heaps</title>
	<link>/theories/binomial-heaps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binomial-heaps/</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial-Queues</title>
	<link>/theories/binomial-queues/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binomial-queues/</guid>
	<description></description>
	</item>

	<item>
	<title>BirdKMP</title>
	<link>/theories/birdkmp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/birdkmp/</guid>
	<description></description>
	</item>

	<item>
	<title>Birkhoff_Finite_Distributive_Lattices</title>
	<link>/theories/birkhoff_finite_distributive_lattices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/birkhoff_finite_distributive_lattices/</guid>
	<description></description>
	</item>

	<item>
	<title>Blue_Eyes</title>
	<link>/theories/blue_eyes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/blue_eyes/</guid>
	<description></description>
	</item>

	<item>
	<title>BNF_CC</title>
	<link>/theories/bnf_cc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bnf_cc/</guid>
	<description></description>
	</item>

	<item>
	<title>BNF_Operations</title>
	<link>/theories/bnf_operations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bnf_operations/</guid>
	<description></description>
	</item>

	<item>
	<title>Bondy</title>
	<link>/theories/bondy/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bondy/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolean_Expression_Checkers</title>
	<link>/theories/boolean_expression_checkers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolean_expression_checkers/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos_Curious_Inference</title>
	<link>/theories/boolos_curious_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolos_curious_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos_Curious_Inference_Automated</title>
	<link>/theories/boolos_curious_inference_automated/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolos_curious_inference_automated/</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded_Deducibility_Security</title>
	<link>/theories/bounded_deducibility_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bounded_deducibility_security/</guid>
	<description></description>
	</item>

	<item>
	<title>BTree</title>
	<link>/theories/btree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/btree/</guid>
	<description></description>
	</item>

	<item>
	<title>Buchi_Complementation</title>
	<link>/theories/buchi_complementation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buchi_complementation/</guid>
	<description></description>
	</item>

	<item>
	<title>Budan_Fourier</title>
	<link>/theories/budan_fourier/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/budan_fourier/</guid>
	<description></description>
	</item>

	<item>
	<title>Buffons_Needle</title>
	<link>/theories/buffons_needle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buffons_needle/</guid>
	<description></description>
	</item>

	<item>
	<title>Buildings</title>
	<link>/theories/buildings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buildings/</guid>
	<description></description>
	</item>

	<item>
	<title>BytecodeLogicJmlTypes</title>
	<link>/theories/bytecodelogicjmltypes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bytecodelogicjmltypes/</guid>
	<description></description>
	</item>

	<item>
	<title>C2KA_DistributedSystems</title>
	<link>/theories/c2ka_distributedsystems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/c2ka_distributedsystems/</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML</title>
	<link>/theories/cakeml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cakeml/</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML_Codegen</title>
	<link>/theories/cakeml_codegen/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cakeml_codegen/</guid>
	<description></description>
	</item>

	<item>
	<title>Call_Arity</title>
	<link>/theories/call_arity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/call_arity/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Equiv_Relations</title>
	<link>/theories/card_equiv_relations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_equiv_relations/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Multisets</title>
	<link>/theories/card_multisets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_multisets/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Number_Partitions</title>
	<link>/theories/card_number_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_number_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Partitions</title>
	<link>/theories/card_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Cartan_FP</title>
	<link>/theories/cartan_fp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cartan_fp/</guid>
	<description></description>
	</item>

	<item>
	<title>Case_Labeling</title>
	<link>/theories/case_labeling/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/case_labeling/</guid>
	<description></description>
	</item>

	<item>
	<title>Catalan_Numbers</title>
	<link>/theories/catalan_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/catalan_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Category</title>
	<link>/theories/category/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category/</guid>
	<description></description>
	</item>

	<item>
	<title>Category2</title>
	<link>/theories/category2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category2/</guid>
	<description></description>
	</item>

	<item>
	<title>Category3</title>
	<link>/theories/category3/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category3/</guid>
	<description></description>
	</item>

	<item>
	<title>Cauchy</title>
	<link>/theories/cauchy/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cauchy/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Automata</title>
	<link>/theories/cava_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Base</title>
	<link>/theories/cava_base/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_base/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_LTL_Modelchecker</title>
	<link>/theories/cava_ltl_modelchecker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_ltl_modelchecker/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Setup</title>
	<link>/theories/cava_setup/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_setup/</guid>
	<description></description>
	</item>

	<item>
	<title>Cayley_Hamilton</title>
	<link>/theories/cayley_hamilton/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cayley_hamilton/</guid>
	<description></description>
	</item>

	<item>
	<title>CCS</title>
	<link>/theories/ccs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ccs/</guid>
	<description></description>
	</item>

	<item>
	<title>Certification_Monads</title>
	<link>/theories/certification_monads/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/certification_monads/</guid>
	<description></description>
	</item>

	<item>
	<title>Chandy_Lamport</title>
	<link>/theories/chandy_lamport/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/chandy_lamport/</guid>
	<description></description>
	</item>

	<item>
	<title>CHERI-C_Memory_Model</title>
	<link>/theories/cheri-c_memory_model/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cheri-c_memory_model/</guid>
	<description></description>
	</item>

	<item>
	<title>Chord_Segments</title>
	<link>/theories/chord_segments/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/chord_segments/</guid>
	<description></description>
	</item>

	<item>
	<title>Circus</title>
	<link>/theories/circus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/circus/</guid>
	<description></description>
	</item>

	<item>
	<title>CISC-Kernel</title>
	<link>/theories/cisc-kernel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cisc-kernel/</guid>
	<description></description>
	</item>

	<item>
	<title>Clean</title>
	<link>/theories/clean/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clean/</guid>
	<description></description>
	</item>

	<item>
	<title>Clique_and_Monotone_Circuits</title>
	<link>/theories/clique_and_monotone_circuits/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clique_and_monotone_circuits/</guid>
	<description></description>
	</item>

	<item>
	<title>ClockSynchInst</title>
	<link>/theories/clocksynchinst/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clocksynchinst/</guid>
	<description></description>
	</item>

	<item>
	<title>Closest_Pair_Points</title>
	<link>/theories/closest_pair_points/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/closest_pair_points/</guid>
	<description></description>
	</item>

	<item>
	<title>CoCon</title>
	<link>/theories/cocon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cocon/</guid>
	<description></description>
	</item>

	<item>
	<title>CofGroups</title>
	<link>/theories/cofgroups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cofgroups/</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive</title>
	<link>/theories/coinductive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/coinductive/</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive_Languages</title>
	<link>/theories/coinductive_languages/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/coinductive_languages/</guid>
	<description></description>
	</item>

	<item>
	<title>Collections</title>
	<link>/theories/collections/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/collections/</guid>
	<description></description>
	</item>

	<item>
	<title>Collections_Examples</title>
	<link>/theories/collections_examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/collections_examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinable_Wands</title>
	<link>/theories/combinable_wands/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinable_wands/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial_Enumeration_Algorithms</title>
	<link>/theories/combinatorial_enumeration_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorial_enumeration_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words</title>
	<link>/theories/combinatorics_words/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words_Graph_Lemma</title>
	<link>/theories/combinatorics_words_graph_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words_graph_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words_Lyndon</title>
	<link>/theories/combinatorics_words_lyndon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words_lyndon/</guid>
	<description></description>
	</item>

	<item>
	<title>CommCSL</title>
	<link>/theories/commcsl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/commcsl/</guid>
	<description></description>
	</item>

	<item>
	<title>Commuting_Hermitian</title>
	<link>/theories/commuting_hermitian/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/commuting_hermitian/</guid>
	<description></description>
	</item>

	<item>
	<title>Comparison_Sort_Lower_Bound</title>
	<link>/theories/comparison_sort_lower_bound/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/comparison_sort_lower_bound/</guid>
	<description></description>
	</item>

	<item>
	<title>Compiling-Exceptions-Correctly</title>
	<link>/theories/compiling-exceptions-correctly/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/compiling-exceptions-correctly/</guid>
	<description></description>
	</item>

	<item>
	<title>Complete_Non_Orders</title>
	<link>/theories/complete_non_orders/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complete_non_orders/</guid>
	<description></description>
	</item>

	<item>
	<title>Completeness</title>
	<link>/theories/completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Bounded_Operators</title>
	<link>/theories/complex_bounded_operators/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_bounded_operators/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Bounded_Operators_Dependencies</title>
	<link>/theories/complex_bounded_operators_dependencies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_bounded_operators_dependencies/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Geometry</title>
	<link>/theories/complex_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Complx</title>
	<link>/theories/complx/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complx/</guid>
	<description></description>
	</item>

	<item>
	<title>ComponentDependencies</title>
	<link>/theories/componentdependencies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/componentdependencies/</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent_Ref_Alg</title>
	<link>/theories/concurrent_ref_alg/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrent_ref_alg/</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent_Revisions</title>
	<link>/theories/concurrent_revisions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrent_revisions/</guid>
	<description></description>
	</item>

	<item>
	<title>ConcurrentGC</title>
	<link>/theories/concurrentgc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrentgc/</guid>
	<description></description>
	</item>

	<item>
	<title>ConcurrentIMP</title>
	<link>/theories/concurrentimp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrentimp/</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional_Simplification</title>
	<link>/theories/conditional_simplification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/conditional_simplification/</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional_Transfer_Rule</title>
	<link>/theories/conditional_transfer_rule/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/conditional_transfer_rule/</guid>
	<description></description>
	</item>

	<item>
	<title>Consensus_Refined</title>
	<link>/theories/consensus_refined/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/consensus_refined/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive_Cryptography</title>
	<link>/theories/constructive_cryptography/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructive_cryptography/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive_Cryptography_CM</title>
	<link>/theories/constructive_cryptography_cm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructive_cryptography_cm/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructor_Funs</title>
	<link>/theories/constructor_funs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructor_funs/</guid>
	<description></description>
	</item>

	<item>
	<title>Containers</title>
	<link>/theories/containers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/containers/</guid>
	<description></description>
	</item>

	<item>
	<title>Containers-Benchmarks</title>
	<link>/theories/containers-benchmarks/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/containers-benchmarks/</guid>
	<description></description>
	</item>

	<item>
	<title>Cook_Levin</title>
	<link>/theories/cook_levin/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cook_levin/</guid>
	<description></description>
	</item>

	<item>
	<title>Core_DOM</title>
	<link>/theories/core_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/core_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Core_SC_DOM</title>
	<link>/theories/core_sc_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/core_sc_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>CoreC++</title>
	<link>/theories/corec++/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/corec++/</guid>
	<description></description>
	</item>

	<item>
	<title>Correctness_Algebras</title>
	<link>/theories/correctness_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/correctness_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMed</title>
	<link>/theories/cosmed/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cosmed/</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMeDis</title>
	<link>/theories/cosmedis/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cosmedis/</guid>
	<description></description>
	</item>

	<item>
	<title>Cotangent_PFD_Formula</title>
	<link>/theories/cotangent_pfd_formula/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cotangent_pfd_formula/</guid>
	<description></description>
	</item>

	<item>
	<title>Count_Complex_Roots</title>
	<link>/theories/count_complex_roots/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/count_complex_roots/</guid>
	<description></description>
	</item>

	<item>
	<title>CRDT</title>
	<link>/theories/crdt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crdt/</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/theories/crypthol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crypthol/</guid>
	<description></description>
	</item>

	<item>
	<title>Crypto_Standards</title>
	<link>/theories/crypto_standards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crypto_standards/</guid>
	<description></description>
	</item>

	<item>
	<title>CryptoBasedCompositionalProperties</title>
	<link>/theories/cryptobasedcompositionalproperties/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cryptobasedcompositionalproperties/</guid>
	<description></description>
	</item>

	<item>
	<title>CRYSTALS-Kyber</title>
	<link>/theories/crystals-kyber/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crystals-kyber/</guid>
	<description></description>
	</item>

	<item>
	<title>CSP_RefTK</title>
	<link>/theories/csp_reftk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/csp_reftk/</guid>
	<description></description>
	</item>

	<item>
	<title>Cubic_Quartic_Equations</title>
	<link>/theories/cubic_quartic_equations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cubic_quartic_equations/</guid>
	<description></description>
	</item>

	<item>
	<title>CVP_Hardness</title>
	<link>/theories/cvp_hardness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cvp_hardness/</guid>
	<description></description>
	</item>

	<item>
	<title>CYK</title>
	<link>/theories/cyk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cyk/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Elementary_Categories</title>
	<link>/theories/czh_elementary_categories/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_elementary_categories/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Foundations</title>
	<link>/theories/czh_foundations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_foundations/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Universal_Constructions</title>
	<link>/theories/czh_universal_constructions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_universal_constructions/</guid>
	<description></description>
	</item>

	<item>
	<title>DataRefinementIBP</title>
	<link>/theories/datarefinementibp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/datarefinementibp/</guid>
	<description></description>
	</item>

	<item>
	<title>Datatype_Order_Generator</title>
	<link>/theories/datatype_order_generator/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/datatype_order_generator/</guid>
	<description></description>
	</item>

	<item>
	<title>DCR-ExecutionEquivalence</title>
	<link>/theories/dcr-executionequivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dcr-executionequivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Decl_Sem_Fun_PL</title>
	<link>/theories/decl_sem_fun_pl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decl_sem_fun_pl/</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing-Diagrams</title>
	<link>/theories/decreasing-diagrams/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decreasing-diagrams/</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing-Diagrams-II</title>
	<link>/theories/decreasing-diagrams-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decreasing-diagrams-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Dedekind_Real</title>
	<link>/theories/dedekind_real/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dedekind_real/</guid>
	<description></description>
	</item>

	<item>
	<title>Deep_Learning</title>
	<link>/theories/deep_learning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/deep_learning/</guid>
	<description></description>
	</item>

	<item>
	<title>Delta_System_Lemma</title>
	<link>/theories/delta_system_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/delta_system_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>Density_Compiler</title>
	<link>/theories/density_compiler/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/density_compiler/</guid>
	<description></description>
	</item>

	<item>
	<title>Dependent_SIFUM_Refinement</title>
	<link>/theories/dependent_sifum_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dependent_sifum_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>Dependent_SIFUM_Type_Systems</title>
	<link>/theories/dependent_sifum_type_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dependent_sifum_type_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>Depth-First-Search</title>
	<link>/theories/depth-first-search/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/depth-first-search/</guid>
	<description></description>
	</item>

	<item>
	<title>Derangements</title>
	<link>/theories/derangements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/derangements/</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving</title>
	<link>/theories/deriving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/deriving/</guid>
	<description></description>
	</item>

	<item>
	<title>Descartes_Sign_Rule</title>
	<link>/theories/descartes_sign_rule/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/descartes_sign_rule/</guid>
	<description></description>
	</item>

	<item>
	<title>Design_Theory</title>
	<link>/theories/design_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/design_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>DFS_Framework</title>
	<link>/theories/dfs_framework/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dfs_framework/</guid>
	<description></description>
	</item>

	<item>
	<title>Dict_Construction</title>
	<link>/theories/dict_construction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dict_construction/</guid>
	<description></description>
	</item>

	<item>
	<title>Differential_Dynamic_Logic</title>
	<link>/theories/differential_dynamic_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/differential_dynamic_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Differential_Game_Logic</title>
	<link>/theories/differential_game_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/differential_game_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Digit_Expansions</title>
	<link>/theories/digit_expansions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/digit_expansions/</guid>
	<description></description>
	</item>

	<item>
	<title>DigitsInBase</title>
	<link>/theories/digitsinbase/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/digitsinbase/</guid>
	<description></description>
	</item>

	<item>
	<title>Dijkstra_Shortest_Path</title>
	<link>/theories/dijkstra_shortest_path/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dijkstra_shortest_path/</guid>
	<description></description>
	</item>

	<item>
	<title>Diophantine_Eqns_Lin_Hom</title>
	<link>/theories/diophantine_eqns_lin_hom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/diophantine_eqns_lin_hom/</guid>
	<description></description>
	</item>

	<item>
	<title>Directed_Sets</title>
	<link>/theories/directed_sets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/directed_sets/</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet_L</title>
	<link>/theories/dirichlet_l/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dirichlet_l/</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet_Series</title>
	<link>/theories/dirichlet_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dirichlet_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Discrete_Summation</title>
	<link>/theories/discrete_summation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/discrete_summation/</guid>
	<description></description>
	</item>

	<item>
	<title>DiscretePricing</title>
	<link>/theories/discretepricing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/discretepricing/</guid>
	<description></description>
	</item>

	<item>
	<title>DiskPaxos</title>
	<link>/theories/diskpaxos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/diskpaxos/</guid>
	<description></description>
	</item>

	<item>
	<title>Distributed_Distinct_Elements</title>
	<link>/theories/distributed_distinct_elements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/distributed_distinct_elements/</guid>
	<description></description>
	</item>

	<item>
	<title>DOM_Components</title>
	<link>/theories/dom_components/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dom_components/</guid>
	<description></description>
	</item>

	<item>
	<title>Dominance_CHK</title>
	<link>/theories/dominance_chk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dominance_chk/</guid>
	<description></description>
	</item>

	<item>
	<title>DPRM_Theorem</title>
	<link>/theories/dprm_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dprm_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>DPT-SAT-Solver</title>
	<link>/theories/dpt-sat-solver/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dpt-sat-solver/</guid>
	<description></description>
	</item>

	<item>
	<title>Dynamic_Tables</title>
	<link>/theories/dynamic_tables/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dynamic_tables/</guid>
	<description></description>
	</item>

	<item>
	<title>DynamicArchitectures</title>
	<link>/theories/dynamicarchitectures/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dynamicarchitectures/</guid>
	<description></description>
	</item>

	<item>
	<title>E_Transcendental</title>
	<link>/theories/e_transcendental/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/e_transcendental/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Earley_Parser</title>
	+ <link>/theories/earley_parser/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/earley_parser/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Echelon_Form</title>
	<link>/theories/echelon_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/echelon_form/</guid>
	<description></description>
	</item>

	<item>
	<title>EdmondsKarp_Maxflow</title>
	<link>/theories/edmondskarp_maxflow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/edmondskarp_maxflow/</guid>
	<description></description>
	</item>

	<item>
	<title>Edwards_Elliptic_Curves_Group</title>
	<link>/theories/edwards_elliptic_curves_group/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/edwards_elliptic_curves_group/</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient-Mergesort</title>
	<link>/theories/efficient-mergesort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/efficient-mergesort/</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient_Weighted_Path_Order</title>
	<link>/theories/efficient_weighted_path_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/efficient_weighted_path_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Elliptic_Curves_Group_Law</title>
	<link>/theories/elliptic_curves_group_law/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/elliptic_curves_group_law/</guid>
	<description></description>
	</item>

	<item>
	<title>Encodability_Process_Calculi</title>
	<link>/theories/encodability_process_calculi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/encodability_process_calculi/</guid>
	<description></description>
	</item>

	<item>
	<title>Epistemic_Logic</title>
	<link>/theories/epistemic_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/epistemic_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Equivalence_Relation_Enumeration</title>
	<link>/theories/equivalence_relation_enumeration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/equivalence_relation_enumeration/</guid>
	<description></description>
	</item>

	<item>
	<title>Ergodic_Theory</title>
	<link>/theories/ergodic_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ergodic_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Error_Function</title>
	<link>/theories/error_function/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/error_function/</guid>
	<description></description>
	</item>

	<item>
	<title>Euler_MacLaurin</title>
	<link>/theories/euler_maclaurin/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/euler_maclaurin/</guid>
	<description></description>
	</item>

	<item>
	<title>Euler_Partition</title>
	<link>/theories/euler_partition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/euler_partition/</guid>
	<description></description>
	</item>

	<item>
	<title>Eval_FO</title>
	<link>/theories/eval_fo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/eval_fo/</guid>
	<description></description>
	</item>

	<item>
	<title>Example-Submission</title>
	<link>/theories/example-submission/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/example-submission/</guid>
	<description></description>
	</item>

	<item>
	<title>Executable_Randomized_Algorithms</title>
	<link>/theories/executable_randomized_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/executable_randomized_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Expander_Graphs</title>
	<link>/theories/expander_graphs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/expander_graphs/</guid>
	<description></description>
	</item>

	<item>
	<title>Extended_Finite_State_Machine_Inference</title>
	<link>/theories/extended_finite_state_machine_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/extended_finite_state_machine_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Extended_Finite_State_Machines</title>
	<link>/theories/extended_finite_state_machines/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/extended_finite_state_machines/</guid>
	<description></description>
	</item>

	<item>
	<title>Factor_Algebraic_Polynomial</title>
	<link>/theories/factor_algebraic_polynomial/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/factor_algebraic_polynomial/</guid>
	<description></description>
	</item>

	<item>
	<title>Factored_Transition_System_Bounding</title>
	<link>/theories/factored_transition_system_bounding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/factored_transition_system_bounding/</guid>
	<description></description>
	</item>

	<item>
	<title>Falling_Factorial_Sum</title>
	<link>/theories/falling_factorial_sum/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/falling_factorial_sum/</guid>
	<description></description>
	</item>

	<item>
	<title>Farkas</title>
	<link>/theories/farkas/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/farkas/</guid>
	<description></description>
	</item>

	<item>
	<title>Featherweight_OCL</title>
	<link>/theories/featherweight_ocl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/featherweight_ocl/</guid>
	<description></description>
	</item>

	<item>
	<title>FeatherweightJava</title>
	<link>/theories/featherweightjava/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/featherweightjava/</guid>
	<description></description>
	</item>

	<item>
	<title>Fermat3_4</title>
	<link>/theories/fermat3_4/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fermat3_4/</guid>
	<description></description>
	</item>

	<item>
	<title>FFT</title>
	<link>/theories/fft/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fft/</guid>
	<description></description>
	</item>

	<item>
	<title>FileRefinement</title>
	<link>/theories/filerefinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/filerefinement/</guid>
	<description></description>
	</item>

	<item>
	<title>FinFun</title>
	<link>/theories/finfun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finfun/</guid>
	<description></description>
	</item>

	<item>
	<title>Finger-Trees</title>
	<link>/theories/finger-trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finger-trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite-Map-Extras</title>
	<link>/theories/finite-map-extras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite-map-extras/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite_Automata_HF</title>
	<link>/theories/finite_automata_hf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite_automata_hf/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite_Fields</title>
	<link>/theories/finite_fields/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite_fields/</guid>
	<description></description>
	</item>

	<item>
	<title>Finitely_Generated_Abelian_Groups</title>
	<link>/theories/finitely_generated_abelian_groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finitely_generated_abelian_groups/</guid>
	<description></description>
	</item>

	<item>
	<title>First_Order_Terms</title>
	<link>/theories/first_order_terms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/first_order_terms/</guid>
	<description></description>
	</item>

	<item>
	<title>First_Welfare_Theorem</title>
	<link>/theories/first_welfare_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/first_welfare_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Fishburn_Impossibility</title>
	<link>/theories/fishburn_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fishburn_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher_Yates</title>
	<link>/theories/fisher_yates/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fisher_yates/</guid>
	<description></description>
	</item>

	<item>
	<title>Fishers_Inequality</title>
	<link>/theories/fishers_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fishers_inequality/</guid>
	<description></description>
	</item>

	<item>
	<title>Flow_Networks</title>
	<link>/theories/flow_networks/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flow_networks/</guid>
	<description></description>
	</item>

	<item>
	<title>Floyd_Warshall</title>
	<link>/theories/floyd_warshall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/floyd_warshall/</guid>
	<description></description>
	</item>

	<item>
	<title>FLP</title>
	<link>/theories/flp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flp/</guid>
	<description></description>
	</item>

	<item>
	<title>Flyspeck-Tame</title>
	<link>/theories/flyspeck-tame/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flyspeck-tame/</guid>
	<description></description>
	</item>

	<item>
	<title>Flyspeck-Tame-Computation</title>
	<link>/theories/flyspeck-tame-computation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flyspeck-tame-computation/</guid>
	<description></description>
	</item>

	<item>
	<title>FO_Theory_Rewriting</title>
	<link>/theories/fo_theory_rewriting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fo_theory_rewriting/</guid>
	<description></description>
	</item>

	<item>
	<title>FocusStreamsCaseStudies</title>
	<link>/theories/focusstreamscasestudies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/focusstreamscasestudies/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL-Fitting</title>
	<link>/theories/fol-fitting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol-fitting/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Axiomatic</title>
	<link>/theories/fol_axiomatic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_axiomatic/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Harrison</title>
	<link>/theories/fol_harrison/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_harrison/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc1</title>
	<link>/theories/fol_seq_calc1/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc1/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc2</title>
	<link>/theories/fol_seq_calc2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc2/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc3</title>
	<link>/theories/fol_seq_calc3/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc3/</guid>
	<description></description>
	</item>

	<item>
	<title>Forcing</title>
	<link>/theories/forcing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/forcing/</guid>
	<description></description>
	</item>

	<item>
	<title>Formal_Puiseux_Series</title>
	<link>/theories/formal_puiseux_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formal_puiseux_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Formal_SSA</title>
	<link>/theories/formal_ssa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formal_ssa/</guid>
	<description></description>
	</item>

	<item>
	<title>Formula_Derivatives</title>
	<link>/theories/formula_derivatives/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formula_derivatives/</guid>
	<description></description>
	</item>

	<item>
	<title>Formula_Derivatives-Examples</title>
	<link>/theories/formula_derivatives-examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formula_derivatives-examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Foundation_of_geometry</title>
	<link>/theories/foundation_of_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/foundation_of_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Fourier</title>
	<link>/theories/fourier/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fourier/</guid>
	<description></description>
	</item>

	<item>
	<title>Free-Boolean-Algebra</title>
	<link>/theories/free-boolean-algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/free-boolean-algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Free-Groups</title>
	<link>/theories/free-groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/free-groups/</guid>
	<description></description>
	</item>

	<item>
	<title>Frequency_Moments</title>
	<link>/theories/frequency_moments/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/frequency_moments/</guid>
	<description></description>
	</item>

	<item>
	<title>Fresh_Identifiers</title>
	<link>/theories/fresh_identifiers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fresh_identifiers/</guid>
	<description></description>
	</item>

	<item>
	<title>FSM_Tests</title>
	<link>/theories/fsm_tests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fsm_tests/</guid>
	<description></description>
	</item>

	<item>
	<title>Functional-Automata</title>
	<link>/theories/functional-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/functional-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Functional_Ordered_Resolution_Prover</title>
	<link>/theories/functional_ordered_resolution_prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/functional_ordered_resolution_prover/</guid>
	<description></description>
	</item>

	<item>
	<title>FunWithFunctions</title>
	<link>/theories/funwithfunctions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/funwithfunctions/</guid>
	<description></description>
	</item>

	<item>
	<title>FunWithTilings</title>
	<link>/theories/funwithtilings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/funwithtilings/</guid>
	<description></description>
	</item>

	<item>
	<title>Furstenberg_Topology</title>
	<link>/theories/furstenberg_topology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/furstenberg_topology/</guid>
	<description></description>
	</item>

	<item>
	<title>Gabow_SCC</title>
	<link>/theories/gabow_scc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gabow_scc/</guid>
	<description></description>
	</item>

	<item>
	<title>Gale_Shapley</title>
	<link>/theories/gale_shapley/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gale_shapley/</guid>
	<description></description>
	</item>

	<item>
	<title>GaleStewart_Games</title>
	<link>/theories/galestewart_games/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/galestewart_games/</guid>
	<description></description>
	</item>

	<item>
	<title>Game_Based_Crypto</title>
	<link>/theories/game_based_crypto/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/game_based_crypto/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan-Elim-Fun</title>
	<link>/theories/gauss-jordan-elim-fun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss-jordan-elim-fun/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss_Jordan</title>
	<link>/theories/gauss_jordan/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss_jordan/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss_Sums</title>
	<link>/theories/gauss_sums/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss_sums/</guid>
	<description></description>
	</item>

	<item>
	<title>Gaussian_Integers</title>
	<link>/theories/gaussian_integers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gaussian_integers/</guid>
	<description></description>
	</item>

	<item>
	<title>GenClock</title>
	<link>/theories/genclock/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/genclock/</guid>
	<description></description>
	</item>

	<item>
	<title>General-Triangle</title>
	<link>/theories/general-triangle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/general-triangle/</guid>
	<description></description>
	</item>

	<item>
	<title>Generalized_Counting_Sort</title>
	<link>/theories/generalized_counting_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generalized_counting_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>Generic_Deriving</title>
	<link>/theories/generic_deriving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generic_deriving/</guid>
	<description></description>
	</item>

	<item>
	<title>Generic_Join</title>
	<link>/theories/generic_join/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generic_join/</guid>
	<description></description>
	</item>

	<item>
	<title>GewirthPGCProof</title>
	<link>/theories/gewirthpgcproof/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gewirthpgcproof/</guid>
	<description></description>
	</item>

	<item>
	<title>Girth_Chromatic</title>
	<link>/theories/girth_chromatic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/girth_chromatic/</guid>
	<description></description>
	</item>

	<item>
	<title>Given_Clause_Loops</title>
	<link>/theories/given_clause_loops/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/given_clause_loops/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_HFSet_Semantic</title>
	<link>/theories/goedel_hfset_semantic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_hfset_semantic/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_HFSet_Semanticless</title>
	<link>/theories/goedel_hfset_semanticless/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_hfset_semanticless/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_Incompleteness</title>
	<link>/theories/goedel_incompleteness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_incompleteness/</guid>
	<description></description>
	</item>

	<item>
	<title>GoedelGod</title>
	<link>/theories/goedelgod/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedelgod/</guid>
	<description></description>
	</item>

	<item>
	<title>Goodstein_Lambda</title>
	<link>/theories/goodstein_lambda/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goodstein_lambda/</guid>
	<description></description>
	</item>

	<item>
	<title>GPU_Kernel_PL</title>
	<link>/theories/gpu_kernel_pl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gpu_kernel_pl/</guid>
	<description></description>
	</item>

	<item>
	<title>Graph_Saturation</title>
	<link>/theories/graph_saturation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graph_saturation/</guid>
	<description></description>
	</item>

	<item>
	<title>Graph_Theory</title>
	<link>/theories/graph_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graph_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>GraphMarkingIBP</title>
	<link>/theories/graphmarkingibp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graphmarkingibp/</guid>
	<description></description>
	</item>

	<item>
	<title>Gray_Codes</title>
	<link>/theories/gray_codes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gray_codes/</guid>
	<description></description>
	</item>

	<item>
	<title>Green</title>
	<link>/theories/green/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/green/</guid>
	<description></description>
	</item>

	<item>
	<title>Groebner_Bases</title>
	<link>/theories/groebner_bases/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/groebner_bases/</guid>
	<description></description>
	</item>

	<item>
	<title>Groebner_Macaulay</title>
	<link>/theories/groebner_macaulay/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/groebner_macaulay/</guid>
	<description></description>
	</item>

	<item>
	<title>Gromov_Hyperbolicity</title>
	<link>/theories/gromov_hyperbolicity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gromov_hyperbolicity/</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck_Schemes</title>
	<link>/theories/grothendieck_schemes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/grothendieck_schemes/</guid>
	<description></description>
	</item>

	<item>
	<title>Group-Ring-Module</title>
	<link>/theories/group-ring-module/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/group-ring-module/</guid>
	<description></description>
	</item>

	<item>
	<title>Hahn_Jordan_Decomposition</title>
	<link>/theories/hahn_jordan_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hahn_jordan_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Hales_Jewett</title>
	<link>/theories/hales_jewett/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hales_jewett/</guid>
	<description></description>
	</item>

	<item>
	<title>Heard_Of</title>
	<link>/theories/heard_of/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/heard_of/</guid>
	<description></description>
	</item>

	<item>
	<title>Hello_World</title>
	<link>/theories/hello_world/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hello_world/</guid>
	<description></description>
	</item>

	<item>
	<title>HereditarilyFinite</title>
	<link>/theories/hereditarilyfinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hereditarilyfinite/</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite</title>
	<link>/theories/hermite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hermite/</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite_Lindemann</title>
	<link>/theories/hermite_lindemann/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hermite_lindemann/</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden_Markov_Models</title>
	<link>/theories/hidden_markov_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hidden_markov_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Higher_Order_Terms</title>
	<link>/theories/higher_order_terms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/higher_order_terms/</guid>
	<description></description>
	</item>

	<item>
	<title>Hoare_Time</title>
	<link>/theories/hoare_time/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hoare_time/</guid>
	<description></description>
	</item>

	<item>
	<title>HoareForDivergence</title>
	<link>/theories/hoarefordivergence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hoarefordivergence/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP</title>
	<link>/theories/hol-csp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-csp/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-ARCH-COMP</title>
	<link>/theories/hol-ode-arch-comp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-arch-comp/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-Examples</title>
	<link>/theories/hol-ode-examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-examples/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-Numerics</title>
	<link>/theories/hol-ode-numerics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-numerics/</guid>
	<description></description>
	</item>

	<item>
	<title>HOLCF-Prelude</title>
	<link>/theories/holcf-prelude/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/holcf-prelude/</guid>
	<description></description>
	</item>

	<item>
	<title>Hood_Melville_Queue</title>
	<link>/theories/hood_melville_queue/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hood_melville_queue/</guid>
	<description></description>
	</item>

	<item>
	<title>HotelKeyCards</title>
	<link>/theories/hotelkeycards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hotelkeycards/</guid>
	<description></description>
	</item>

	<item>
	<title>HRB-Slicing</title>
	<link>/theories/hrb-slicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hrb-slicing/</guid>
	<description></description>
	</item>

	<item>
	<title>Huffman</title>
	<link>/theories/huffman/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/huffman/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Logic</title>
	<link>/theories/hybrid_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Multi_Lane_Spatial_Logic</title>
	<link>/theories/hybrid_multi_lane_spatial_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_multi_lane_spatial_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Systems_VCs</title>
	<link>/theories/hybrid_systems_vcs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_systems_vcs/</guid>
	<description></description>
	</item>

	<item>
	<title>HyperCTL</title>
	<link>/theories/hyperctl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperctl/</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual</title>
	<link>/theories/hyperdual/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperdual/</guid>
	<description></description>
	</item>

	<item>
	<title>HyperHoareLogic</title>
	<link>/theories/hyperhoarelogic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperhoarelogic/</guid>
	<description></description>
	</item>

	<item>
	<title>IEEE_Floating_Point</title>
	<link>/theories/ieee_floating_point/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ieee_floating_point/</guid>
	<description></description>
	</item>

	<item>
	<title>IFC_Tracking</title>
	<link>/theories/ifc_tracking/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ifc_tracking/</guid>
	<description></description>
	</item>

	<item>
	<title>IMAP-CRDT</title>
	<link>/theories/imap-crdt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imap-crdt/</guid>
	<description></description>
	</item>

	<item>
	<title>IMO2019</title>
	<link>/theories/imo2019/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imo2019/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2</title>
	<link>/theories/imp2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp2/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2_Binary_Heap</title>
	<link>/theories/imp2_binary_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp2_binary_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP_Compiler</title>
	<link>/theories/imp_compiler/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp_compiler/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP_Compiler_Reuse</title>
	<link>/theories/imp_compiler_reuse/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp_compiler_reuse/</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative_Insertion_Sort</title>
	<link>/theories/imperative_insertion_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imperative_insertion_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>Implicational_Logic</title>
	<link>/theories/implicational_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/implicational_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Impossible_Geometry</title>
	<link>/theories/impossible_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/impossible_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Incompleteness</title>
	<link>/theories/incompleteness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/incompleteness/</guid>
	<description></description>
	</item>

	<item>
	<title>Incredible_Proof_Machine</title>
	<link>/theories/incredible_proof_machine/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/incredible_proof_machine/</guid>
	<description></description>
	</item>

	<item>
	<title>Independence_CH</title>
	<link>/theories/independence_ch/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/independence_ch/</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive_Confidentiality</title>
	<link>/theories/inductive_confidentiality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/inductive_confidentiality/</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive_Inference</title>
	<link>/theories/inductive_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/inductive_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>InformationFlowSlicing</title>
	<link>/theories/informationflowslicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/informationflowslicing/</guid>
	<description></description>
	</item>

	<item>
	<title>InformationFlowSlicing_Inter</title>
	<link>/theories/informationflowslicing_inter/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/informationflowslicing_inter/</guid>
	<description></description>
	</item>

	<item>
	<title>InfPathElimination</title>
	<link>/theories/infpathelimination/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/infpathelimination/</guid>
	<description></description>
	</item>

	<item>
	<title>Integration</title>
	<link>/theories/integration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/integration/</guid>
	<description></description>
	</item>

	<item>
	<title>Interpolation_Polynomials_HOL_Algebra</title>
	<link>/theories/interpolation_polynomials_hol_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interpolation_polynomials_hol_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Interpreter_Optimizations</title>
	<link>/theories/interpreter_optimizations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interpreter_optimizations/</guid>
	<description></description>
	</item>

	<item>
	<title>Interval_Arithmetic_Word32</title>
	<link>/theories/interval_arithmetic_word32/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interval_arithmetic_word32/</guid>
	<description></description>
	</item>

	<item>
	<title>Intro_Dest_Elim</title>
	<link>/theories/intro_dest_elim/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/intro_dest_elim/</guid>
	<description></description>
	</item>

	<item>
	<title>Involutions2Squares</title>
	<link>/theories/involutions2squares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/involutions2squares/</guid>
	<description></description>
	</item>

	<item>
	<title>IP_Addresses</title>
	<link>/theories/ip_addresses/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ip_addresses/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics</title>
	<link>/theories/iptables_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics_Examples</title>
	<link>/theories/iptables_semantics_examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics_examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics_Examples_Big</title>
	<link>/theories/iptables_semantics_examples_big/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics_examples_big/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational_Series_Erdos_Straus</title>
	<link>/theories/irrational_series_erdos_straus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrational_series_erdos_straus/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality_J_Hancl</title>
	<link>/theories/irrationality_j_hancl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrationality_j_hancl/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationals_From_THEBOOK</title>
	<link>/theories/irrationals_from_thebook/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrationals_from_thebook/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_C</title>
	<link>/theories/isabelle_c/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_c/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_Marries_Dirac</title>
	<link>/theories/isabelle_marries_dirac/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_marries_dirac/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_Meta_Model</title>
	<link>/theories/isabelle_meta_model/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_meta_model/</guid>
	<description></description>
	</item>

	<item>
	<title>IsaGeoCoq</title>
	<link>/theories/isageocoq/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isageocoq/</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet</title>
	<link>/theories/isanet/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isanet/</guid>
	<description></description>
	</item>

	<item>
	<title>Jacobson_Basic_Algebra</title>
	<link>/theories/jacobson_basic_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jacobson_basic_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Jinja</title>
	<link>/theories/jinja/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinja/</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaDCI</title>
	<link>/theories/jinjadci/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinjadci/</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaThreads</title>
	<link>/theories/jinjathreads/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinjathreads/</guid>
	<description></description>
	</item>

	<item>
	<title>JiveDataStoreModel</title>
	<link>/theories/jivedatastoremodel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jivedatastoremodel/</guid>
	<description></description>
	</item>

	<item>
	<title>Jordan_Hoelder</title>
	<link>/theories/jordan_hoelder/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jordan_hoelder/</guid>
	<description></description>
	</item>

	<item>
	<title>Jordan_Normal_Form</title>
	<link>/theories/jordan_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jordan_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>KAD</title>
	<link>/theories/kad/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kad/</guid>
	<description></description>
	</item>

	<item>
	<title>KAT_and_DRA</title>
	<link>/theories/kat_and_dra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kat_and_dra/</guid>
	<description></description>
	</item>

	<item>
	<title>KBPs</title>
	<link>/theories/kbps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kbps/</guid>
	<description></description>
	</item>

	<item>
	<title>KD_Tree</title>
	<link>/theories/kd_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kd_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Key_Agreement_Strong_Adversaries</title>
	<link>/theories/key_agreement_strong_adversaries/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/key_agreement_strong_adversaries/</guid>
	<description></description>
	</item>

	<item>
	<title>Khovanskii_Theorem</title>
	<link>/theories/khovanskii_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/khovanskii_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene_Algebra</title>
	<link>/theories/kleene_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kleene_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Kneser_Cauchy_Davenport</title>
	<link>/theories/kneser_cauchy_davenport/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kneser_cauchy_davenport/</guid>
	<description></description>
	</item>

	<item>
	<title>Knights_Tour</title>
	<link>/theories/knights_tour/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knights_tour/</guid>
	<description></description>
	</item>

	<item>
	<title>Knot_Theory</title>
	<link>/theories/knot_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knot_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Knuth_Bendix_Order</title>
	<link>/theories/knuth_bendix_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knuth_bendix_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Knuth_Morris_Pratt</title>
	<link>/theories/knuth_morris_pratt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knuth_morris_pratt/</guid>
	<description></description>
	</item>

	<item>
	<title>Koenigsberg_Friendship</title>
	<link>/theories/koenigsberg_friendship/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/koenigsberg_friendship/</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal</title>
	<link>/theories/kruskal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kruskal/</guid>
	<description></description>
	</item>

	<item>
	<title>Kuratowski_Closure_Complement</title>
	<link>/theories/kuratowski_closure_complement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kuratowski_closure_complement/</guid>
	<description></description>
	</item>

	<item>
	<title>Lam-ml-Normalization</title>
	<link>/theories/lam-ml-normalization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lam-ml-normalization/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_EPO</title>
	<link>/theories/lambda_free_epo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_epo/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_KBOs</title>
	<link>/theories/lambda_free_kbos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_kbos/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_RPOs</title>
	<link>/theories/lambda_free_rpos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_rpos/</guid>
	<description></description>
	</item>

	<item>
	<title>LambdaAuth</title>
	<link>/theories/lambdaauth/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambdaauth/</guid>
	<description></description>
	</item>

	<item>
	<title>LambdaMu</title>
	<link>/theories/lambdamu/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambdamu/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambert_W</title>
	<link>/theories/lambert_w/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambert_w/</guid>
	<description></description>
	</item>

	<item>
	<title>Landau_Symbols</title>
	<link>/theories/landau_symbols/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/landau_symbols/</guid>
	<description></description>
	</item>

	<item>
	<title>Laplace_Transform</title>
	<link>/theories/laplace_transform/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/laplace_transform/</guid>
	<description></description>
	</item>

	<item>
	<title>Latin_Square</title>
	<link>/theories/latin_square/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/latin_square/</guid>
	<description></description>
	</item>

	<item>
	<title>LatticeProperties</title>
	<link>/theories/latticeproperties/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/latticeproperties/</guid>
	<description></description>
	</item>

	<item>
	<title>Launchbury</title>
	<link>/theories/launchbury/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/launchbury/</guid>
	<description></description>
	</item>

	<item>
	<title>Laws_of_Large_Numbers</title>
	<link>/theories/laws_of_large_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/laws_of_large_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy-Lists-II</title>
	<link>/theories/lazy-lists-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lazy-lists-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy_Case</title>
	<link>/theories/lazy_case/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lazy_case/</guid>
	<description></description>
	</item>

	<item>
	<title>Lehmer</title>
	<link>/theories/lehmer/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lehmer/</guid>
	<description></description>
	</item>

	<item>
	<title>LEM</title>
	<link>/theories/lem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lem/</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting_Definition_Option</title>
	<link>/theories/lifting_definition_option/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lifting_definition_option/</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting_the_Exponent</title>
	<link>/theories/lifting_the_exponent/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lifting_the_exponent/</guid>
	<description></description>
	</item>

	<item>
	<title>LightweightJava</title>
	<link>/theories/lightweightjava/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lightweightjava/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Inequalities</title>
	<link>/theories/linear_inequalities/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_inequalities/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Programming</title>
	<link>/theories/linear_programming/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_programming/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Recurrences</title>
	<link>/theories/linear_recurrences/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_recurrences/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Recurrences_Solver</title>
	<link>/theories/linear_recurrences_solver/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_recurrences_solver/</guid>
	<description></description>
	</item>

	<item>
	<title>LinearQuantifierElim</title>
	<link>/theories/linearquantifierelim/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linearquantifierelim/</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville_Numbers</title>
	<link>/theories/liouville_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/liouville_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>List-Index</title>
	<link>/theories/list-index/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list-index/</guid>
	<description></description>
	</item>

	<item>
	<title>List-Infinite</title>
	<link>/theories/list-infinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list-infinite/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Interleaving</title>
	<link>/theories/list_interleaving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_interleaving/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Inversions</title>
	<link>/theories/list_inversions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_inversions/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Update</title>
	<link>/theories/list_update/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_update/</guid>
	<description></description>
	</item>

	<item>
	<title>LLL_Basis_Reduction</title>
	<link>/theories/lll_basis_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lll_basis_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>LLL_Factorization</title>
	<link>/theories/lll_factorization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lll_factorization/</guid>
	<description></description>
	</item>

	<item>
	<title>Localization_Ring</title>
	<link>/theories/localization_ring/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/localization_ring/</guid>
	<description></description>
	</item>

	<item>
	<title>LocalLexing</title>
	<link>/theories/locallexing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/locallexing/</guid>
	<description></description>
	</item>

	<item>
	<title>Locally-Nameless-Sigma</title>
	<link>/theories/locally-nameless-sigma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/locally-nameless-sigma/</guid>
	<description></description>
	</item>

	<item>
	<title>LOFT</title>
	<link>/theories/loft/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/loft/</guid>
	<description></description>
	</item>

	<item>
	<title>Logging_Independent_Anonymity</title>
	<link>/theories/logging_independent_anonymity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/logging_independent_anonymity/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_Approximation</title>
	<link>/theories/lorenz_approximation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_approximation/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_C0</title>
	<link>/theories/lorenz_c0/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_c0/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_C1</title>
	<link>/theories/lorenz_c1/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_c1/</guid>
	<description></description>
	</item>

	<item>
	<title>Lowe_Ontological_Argument</title>
	<link>/theories/lowe_ontological_argument/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lowe_ontological_argument/</guid>
	<description></description>
	</item>

	<item>
	<title>Lower_Semicontinuous</title>
	<link>/theories/lower_semicontinuous/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lower_semicontinuous/</guid>
	<description></description>
	</item>

	<item>
	<title>Lp</title>
	<link>/theories/lp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lp/</guid>
	<description></description>
	</item>

	<item>
	<title>LP_Duality</title>
	<link>/theories/lp_duality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lp_duality/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL</title>
	<link>/theories/ltl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_Master_Theorem</title>
	<link>/theories/ltl_master_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_master_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_Normal_Form</title>
	<link>/theories/ltl_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_to_DRA</title>
	<link>/theories/ltl_to_dra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_to_dra/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_to_GBA</title>
	<link>/theories/ltl_to_gba/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_to_gba/</guid>
	<description></description>
	</item>

	<item>
	<title>Lucas_Theorem</title>
	<link>/theories/lucas_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lucas_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Markov_Models</title>
	<link>/theories/markov_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/markov_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Marriage</title>
	<link>/theories/marriage/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/marriage/</guid>
	<description></description>
	</item>

	<item>
	<title>Mason_Stothers</title>
	<link>/theories/mason_stothers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mason_stothers/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices_for_ODEs</title>
	<link>/theories/matrices_for_odes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrices_for_odes/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrix</title>
	<link>/theories/matrix/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrix/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrix_Tensor</title>
	<link>/theories/matrix_tensor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrix_tensor/</guid>
	<description></description>
	</item>

	<item>
	<title>Matroids</title>
	<link>/theories/matroids/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matroids/</guid>
	<description></description>
	</item>

	<item>
	<title>Max-Card-Matching</title>
	<link>/theories/max-card-matching/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/max-card-matching/</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum_Segment_Sum</title>
	<link>/theories/maximum_segment_sum/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/maximum_segment_sum/</guid>
	<description></description>
	</item>

	<item>
	<title>MDP-Algorithms</title>
	<link>/theories/mdp-algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mdp-algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>MDP-Rewards</title>
	<link>/theories/mdp-rewards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mdp-rewards/</guid>
	<description></description>
	</item>

	<item>
	<title>Median_Method</title>
	<link>/theories/median_method/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/median_method/</guid>
	<description></description>
	</item>

	<item>
	<title>Median_Of_Medians_Selection</title>
	<link>/theories/median_of_medians_selection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/median_of_medians_selection/</guid>
	<description></description>
	</item>

	<item>
	<title>Menger</title>
	<link>/theories/menger/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/menger/</guid>
	<description></description>
	</item>

	<item>
	<title>Mereology</title>
	<link>/theories/mereology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mereology/</guid>
	<description></description>
	</item>

	<item>
	<title>Mersenne_Primes</title>
	<link>/theories/mersenne_primes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mersenne_primes/</guid>
	<description></description>
	</item>

	<item>
	<title>Metalogic_ProofChecker</title>
	<link>/theories/metalogic_proofchecker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/metalogic_proofchecker/</guid>
	<description></description>
	</item>

	<item>
	<title>MFMC_Countable</title>
	<link>/theories/mfmc_countable/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfmc_countable/</guid>
	<description></description>
	</item>

	<item>
	<title>MFODL_Monitor_Optimized</title>
	<link>/theories/mfodl_monitor_optimized/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfodl_monitor_optimized/</guid>
	<description></description>
	</item>

	<item>
	<title>MFOTL_Monitor</title>
	<link>/theories/mfotl_monitor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfotl_monitor/</guid>
	<description></description>
	</item>

	<item>
	<title>MHComputation</title>
	<link>/theories/mhcomputation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mhcomputation/</guid>
	<description></description>
	</item>

	<item>
	<title>Minimal_SSA</title>
	<link>/theories/minimal_ssa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minimal_ssa/</guid>
	<description></description>
	</item>

	<item>
	<title>MiniML</title>
	<link>/theories/miniml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/miniml/</guid>
	<description></description>
	</item>

	<item>
	<title>MiniSail</title>
	<link>/theories/minisail/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minisail/</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowskis_Theorem</title>
	<link>/theories/minkowskis_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minkowskis_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Minsky_Machines</title>
	<link>/theories/minsky_machines/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minsky_machines/</guid>
	<description></description>
	</item>

	<item>
	<title>MLSS_Decision_Proc</title>
	<link>/theories/mlss_decision_proc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mlss_decision_proc/</guid>
	<description></description>
	</item>

	<item>
	<title>Modal_Logics_for_NTS</title>
	<link>/theories/modal_logics_for_nts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modal_logics_for_nts/</guid>
	<description></description>
	</item>

	<item>
	<title>Modular_arithmetic_LLL_and_HNF_algorithms</title>
	<link>/theories/modular_arithmetic_lll_and_hnf_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modular_arithmetic_lll_and_hnf_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Modular_Assembly_Kit_Security</title>
	<link>/theories/modular_assembly_kit_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modular_assembly_kit_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Monad_Memo_DP</title>
	<link>/theories/monad_memo_dp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monad_memo_dp/</guid>
	<description></description>
	</item>

	<item>
	<title>Monad_Normalisation</title>
	<link>/theories/monad_normalisation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monad_normalisation/</guid>
	<description></description>
	</item>

	<item>
	<title>MonoBoolTranAlgebra</title>
	<link>/theories/monobooltranalgebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monobooltranalgebra/</guid>
	<description></description>
	</item>

	<item>
	<title>MonoidalCategory</title>
	<link>/theories/monoidalcategory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monoidalcategory/</guid>
	<description></description>
	</item>

	<item>
	<title>Monomorphic_Monad</title>
	<link>/theories/monomorphic_monad/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monomorphic_monad/</guid>
	<description></description>
	</item>

	<item>
	<title>MSO_Regex_Equivalence</title>
	<link>/theories/mso_regex_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mso_regex_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>MuchAdoAboutTwo</title>
	<link>/theories/muchadoabouttwo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/muchadoabouttwo/</guid>
	<description></description>
	</item>

	<item>
	<title>Multi_Party_Computation</title>
	<link>/theories/multi_party_computation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multi_party_computation/</guid>
	<description></description>
	</item>

	<item>
	<title>Multirelations</title>
	<link>/theories/multirelations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multirelations/</guid>
	<description></description>
	</item>

	<item>
	<title>Multirelations_Heterogeneous</title>
	<link>/theories/multirelations_heterogeneous/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multirelations_heterogeneous/</guid>
	<description></description>
	</item>

	<item>
	<title>Multiset_Ordering_NPC</title>
	<link>/theories/multiset_ordering_npc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multiset_ordering_npc/</guid>
	<description></description>
	</item>

	<item>
	<title>Multitape_To_Singletape_TM</title>
	<link>/theories/multitape_to_singletape_tm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multitape_to_singletape_tm/</guid>
	<description></description>
	</item>

	<item>
	<title>Myhill-Nerode</title>
	<link>/theories/myhill-nerode/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/myhill-nerode/</guid>
	<description></description>
	</item>

	<item>
	<title>Name_Carrying_Type_Inference</title>
	<link>/theories/name_carrying_type_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/name_carrying_type_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Nano_JSON</title>
	<link>/theories/nano_json/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nano_json/</guid>
	<description></description>
	</item>

	<item>
	<title>Nash_Williams</title>
	<link>/theories/nash_williams/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nash_williams/</guid>
	<description></description>
	</item>

	<item>
	<title>Nat-Interval-Logic</title>
	<link>/theories/nat-interval-logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nat-interval-logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Native_Word</title>
	<link>/theories/native_word/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/native_word/</guid>
	<description></description>
	</item>

	<item>
	<title>Nested_Multisets_Ordinals</title>
	<link>/theories/nested_multisets_ordinals/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nested_multisets_ordinals/</guid>
	<description></description>
	</item>

	<item>
	<title>Network_Security_Policy_Verification</title>
	<link>/theories/network_security_policy_verification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/network_security_policy_verification/</guid>
	<description></description>
	</item>

	<item>
	<title>Neumann_Morgenstern_Utility</title>
	<link>/theories/neumann_morgenstern_utility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/neumann_morgenstern_utility/</guid>
	<description></description>
	</item>

	<item>
	<title>No_FTL_observers</title>
	<link>/theories/no_ftl_observers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/no_ftl_observers/</guid>
	<description></description>
	</item>

	<item>
	<title>No_FTL_observers_Gen_Rel</title>
	<link>/theories/no_ftl_observers_gen_rel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/no_ftl_observers_gen_rel/</guid>
	<description></description>
	</item>

	<item>
	<title>Nominal2</title>
	<link>/theories/nominal2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nominal2/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Concurrent_Composition</title>
	<link>/theories/noninterference_concurrent_composition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_concurrent_composition/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_CSP</title>
	<link>/theories/noninterference_csp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_csp/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Generic_Unwinding</title>
	<link>/theories/noninterference_generic_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_generic_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Inductive_Unwinding</title>
	<link>/theories/noninterference_inductive_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_inductive_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Ipurge_Unwinding</title>
	<link>/theories/noninterference_ipurge_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_ipurge_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Sequential_Composition</title>
	<link>/theories/noninterference_sequential_composition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_sequential_composition/</guid>
	<description></description>
	</item>

	<item>
	<title>NormByEval</title>
	<link>/theories/normbyeval/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/normbyeval/</guid>
	<description></description>
	</item>

	<item>
	<title>Nullstellensatz</title>
	<link>/theories/nullstellensatz/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nullstellensatz/</guid>
	<description></description>
	</item>

	<item>
	<title>Number_Theoretic_Transform</title>
	<link>/theories/number_theoretic_transform/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/number_theoretic_transform/</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/theories/octonions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/octonions/</guid>
	<description></description>
	</item>

	<item>
	<title>Old_Datatype_Show</title>
	<link>/theories/old_datatype_show/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/old_datatype_show/</guid>
	<description></description>
	</item>

	<item>
	<title>Open_Induction</title>
	<link>/theories/open_induction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/open_induction/</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets</title>
	<link>/theories/opsets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/opsets/</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/theories/optics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/optics/</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal_BST</title>
	<link>/theories/optimal_bst/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/optimal_bst/</guid>
	<description></description>
	</item>

	<item>
	<title>Orbit_Stabiliser</title>
	<link>/theories/orbit_stabiliser/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/orbit_stabiliser/</guid>
	<description></description>
	</item>

	<item>
	<title>Order_Lattice_Props</title>
	<link>/theories/order_lattice_props/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/order_lattice_props/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordered_Resolution_Prover</title>
	<link>/theories/ordered_resolution_prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordered_resolution_prover/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal</title>
	<link>/theories/ordinal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinal/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal_Partitions</title>
	<link>/theories/ordinal_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinal_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinals_and_Cardinals</title>
	<link>/theories/ordinals_and_cardinals/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinals_and_cardinals/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinary_Differential_Equations</title>
	<link>/theories/ordinary_differential_equations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinary_differential_equations/</guid>
	<description></description>
	</item>

	<item>
	<title>PAC_Checker</title>
	<link>/theories/pac_checker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pac_checker/</guid>
	<description></description>
	</item>

	<item>
	<title>Package_logic</title>
	<link>/theories/package_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/package_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Padic_Field</title>
	<link>/theories/padic_field/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/padic_field/</guid>
	<description></description>
	</item>

	<item>
	<title>Padic_Ints</title>
	<link>/theories/padic_ints/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/padic_ints/</guid>
	<description></description>
	</item>

	<item>
	<title>Pairing_Heap</title>
	<link>/theories/pairing_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pairing_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>PAL</title>
	<link>/theories/pal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pal/</guid>
	<description></description>
	</item>

	<item>
	<title>PAPP_Impossibility</title>
	<link>/theories/papp_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/papp_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Paraconsistency</title>
	<link>/theories/paraconsistency/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/paraconsistency/</guid>
	<description></description>
	</item>

	<item>
	<title>Parity_Game</title>
	<link>/theories/parity_game/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/parity_game/</guid>
	<description></description>
	</item>

	<item>
	<title>Partial_Function_MR</title>
	<link>/theories/partial_function_mr/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/partial_function_mr/</guid>
	<description></description>
	</item>

	<item>
	<title>Partial_Order_Reduction</title>
	<link>/theories/partial_order_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/partial_order_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Password_Authentication_Protocol</title>
	<link>/theories/password_authentication_protocol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/password_authentication_protocol/</guid>
	<description></description>
	</item>

	<item>
	<title>PCF</title>
	<link>/theories/pcf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pcf/</guid>
	<description></description>
	</item>

	<item>
	<title>Pell</title>
	<link>/theories/pell/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pell/</guid>
	<description></description>
	</item>

	<item>
	<title>Perfect-Number-Thm</title>
	<link>/theories/perfect-number-thm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/perfect-number-thm/</guid>
	<description></description>
	</item>

	<item>
	<title>Perron_Frobenius</title>
	<link>/theories/perron_frobenius/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/perron_frobenius/</guid>
	<description></description>
	</item>

	<item>
	<title>pGCL</title>
	<link>/theories/pgcl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pgcl/</guid>
	<description></description>
	</item>

	<item>
	<title>Physical_Quantities</title>
	<link>/theories/physical_quantities/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/physical_quantities/</guid>
	<description></description>
	</item>

	<item>
	<title>Pi_Calculus</title>
	<link>/theories/pi_calculus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pi_calculus/</guid>
	<description></description>
	</item>

	<item>
	<title>Pi_Transcendental</title>
	<link>/theories/pi_transcendental/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pi_transcendental/</guid>
	<description></description>
	</item>

	<item>
	<title>Planarity_Certificates</title>
	<link>/theories/planarity_certificates/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/planarity_certificates/</guid>
	<description></description>
	</item>

	<item>
	<title>PLM</title>
	<link>/theories/plm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/plm/</guid>
	<description></description>
	</item>

	<item>
	<title>Pluennecke_Ruzsa_Inequality</title>
	<link>/theories/pluennecke_ruzsa_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pluennecke_ruzsa_inequality/</guid>
	<description></description>
	</item>

	<item>
	<title>Poincare_Bendixson</title>
	<link>/theories/poincare_bendixson/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poincare_bendixson/</guid>
	<description></description>
	</item>

	<item>
	<title>Poincare_Disc</title>
	<link>/theories/poincare_disc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poincare_disc/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Polygonal_Number_Theorem</title>
	+ <link>/theories/polygonal_number_theorem/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/polygonal_number_theorem/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Polynomial_Factorization</title>
	<link>/theories/polynomial_factorization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomial_factorization/</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial_Interpolation</title>
	<link>/theories/polynomial_interpolation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomial_interpolation/</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomials</title>
	<link>/theories/polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Pop_Refinement</title>
	<link>/theories/pop_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pop_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>POPLmark-deBruijn</title>
	<link>/theories/poplmark-debruijn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poplmark-debruijn/</guid>
	<description></description>
	</item>

	<item>
	<title>Posix-Lexing</title>
	<link>/theories/posix-lexing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/posix-lexing/</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic_Noninterference</title>
	<link>/theories/possibilistic_noninterference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/possibilistic_noninterference/</guid>
	<description></description>
	</item>

	<item>
	<title>Power_Sum_Polynomials</title>
	<link>/theories/power_sum_polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/power_sum_polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Pratt_Certificate</title>
	<link>/theories/pratt_certificate/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pratt_certificate/</guid>
	<description></description>
	</item>

	<item>
	<title>Prefix_Free_Code_Combinators</title>
	<link>/theories/prefix_free_code_combinators/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prefix_free_code_combinators/</guid>
	<description></description>
	</item>

	<item>
	<title>Presburger-Automata</title>
	<link>/theories/presburger-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/presburger-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Prim_Dijkstra_Simple</title>
	<link>/theories/prim_dijkstra_simple/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prim_dijkstra_simple/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Distribution_Elementary</title>
	<link>/theories/prime_distribution_elementary/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_distribution_elementary/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Harmonic_Series</title>
	<link>/theories/prime_harmonic_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_harmonic_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Number_Theorem</title>
	<link>/theories/prime_number_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_number_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Priority_Queue_Braun</title>
	<link>/theories/priority_queue_braun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/priority_queue_braun/</guid>
	<description></description>
	</item>

	<item>
	<title>Priority_Search_Trees</title>
	<link>/theories/priority_search_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/priority_search_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Noninterference</title>
	<link>/theories/probabilistic_noninterference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_noninterference/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Prime_Tests</title>
	<link>/theories/probabilistic_prime_tests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_prime_tests/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_System_Zoo</title>
	<link>/theories/probabilistic_system_zoo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_system_zoo/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Timed_Automata</title>
	<link>/theories/probabilistic_timed_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_timed_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_While</title>
	<link>/theories/probabilistic_while/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_while/</guid>
	<description></description>
	</item>

	<item>
	<title>Probability_Inequality_Completeness</title>
	<link>/theories/probability_inequality_completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probability_inequality_completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Program-Conflict-Analysis</title>
	<link>/theories/program-conflict-analysis/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/program-conflict-analysis/</guid>
	<description></description>
	</item>

	<item>
	<title>Progress_Tracking</title>
	<link>/theories/progress_tracking/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/progress_tracking/</guid>
	<description></description>
	</item>

	<item>
	<title>Projective_Geometry</title>
	<link>/theories/projective_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/projective_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Projective_Measurements</title>
	<link>/theories/projective_measurements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/projective_measurements/</guid>
	<description></description>
	</item>

	<item>
	<title>Promela</title>
	<link>/theories/promela/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/promela/</guid>
	<description></description>
	</item>

	<item>
	<title>Proof_Strategy_Language</title>
	<link>/theories/proof_strategy_language/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/proof_strategy_language/</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional_Logic_Class</title>
	<link>/theories/propositional_logic_class/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/propositional_logic_class/</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional_Proof_Systems</title>
	<link>/theories/propositional_proof_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/propositional_proof_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>PropResPI</title>
	<link>/theories/proprespi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/proprespi/</guid>
	<description></description>
	</item>

	<item>
	<title>Prpu_Maxflow</title>
	<link>/theories/prpu_maxflow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prpu_maxflow/</guid>
	<description></description>
	</item>

	<item>
	<title>PSemigroupsConvolution</title>
	<link>/theories/psemigroupsconvolution/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/psemigroupsconvolution/</guid>
	<description></description>
	</item>

	<item>
	<title>PseudoHoops</title>
	<link>/theories/pseudohoops/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pseudohoops/</guid>
	<description></description>
	</item>

	<item>
	<title>Psi_Calculi</title>
	<link>/theories/psi_calculi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/psi_calculi/</guid>
	<description></description>
	</item>

	<item>
	<title>Ptolemys_Theorem</title>
	<link>/theories/ptolemys_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ptolemys_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Public_Announcement_Logic</title>
	<link>/theories/public_announcement_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/public_announcement_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>QHLProver</title>
	<link>/theories/qhlprover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/qhlprover/</guid>
	<description></description>
	</item>

	<item>
	<title>QR_Decomposition</title>
	<link>/theories/qr_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/qr_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Quantales</title>
	<link>/theories/quantales/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quantales/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Quantales_Converse</title>
	+ <link>/theories/quantales_converse/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/quantales_converse/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Quantifier_Elimination_Hybrid</title>
	<link>/theories/quantifier_elimination_hybrid/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quantifier_elimination_hybrid/</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi_Borel_Spaces</title>
	<link>/theories/quasi_borel_spaces/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quasi_borel_spaces/</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/theories/quaternions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quaternions/</guid>
	<description></description>
	</item>

	<item>
	<title>Query_Optimization</title>
	<link>/theories/query_optimization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/query_optimization/</guid>
	<description></description>
	</item>

	<item>
	<title>Quick_Sort_Cost</title>
	<link>/theories/quick_sort_cost/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quick_sort_cost/</guid>
	<description></description>
	</item>

	<item>
	<title>Ramsey-Infinite</title>
	<link>/theories/ramsey-infinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ramsey-infinite/</guid>
	<description></description>
	</item>

	<item>
	<title>Random_BSTs</title>
	<link>/theories/random_bsts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/random_bsts/</guid>
	<description></description>
	</item>

	<item>
	<title>Random_Graph_Subgraph_Threshold</title>
	<link>/theories/random_graph_subgraph_threshold/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/random_graph_subgraph_threshold/</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised_BSTs</title>
	<link>/theories/randomised_bsts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/randomised_bsts/</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised_Social_Choice</title>
	<link>/theories/randomised_social_choice/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/randomised_social_choice/</guid>
	<description></description>
	</item>

	<item>
	<title>Rank_Nullity_Theorem</title>
	<link>/theories/rank_nullity_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rank_nullity_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Impl</title>
	<link>/theories/real_impl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_impl/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Power</title>
	<link>/theories/real_power/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_power/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Time_Deque</title>
	<link>/theories/real_time_deque/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_time_deque/</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion-Addition</title>
	<link>/theories/recursion-addition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/recursion-addition/</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion-Theory-I</title>
	<link>/theories/recursion-theory-i/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/recursion-theory-i/</guid>
	<description></description>
	</item>

	<item>
	<title>Refine_Imperative_HOL</title>
	<link>/theories/refine_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refine_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Refine_Monadic</title>
	<link>/theories/refine_monadic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refine_monadic/</guid>
	<description></description>
	</item>

	<item>
	<title>RefinementReactive</title>
	<link>/theories/refinementreactive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refinementreactive/</guid>
	<description></description>
	</item>

	<item>
	<title>Regex_Equivalence</title>
	<link>/theories/regex_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regex_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Registers</title>
	<link>/theories/registers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/registers/</guid>
	<description></description>
	</item>

	<item>
	<title>Regression_Test_Selection</title>
	<link>/theories/regression_test_selection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regression_test_selection/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular-Sets</title>
	<link>/theories/regular-sets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular-sets/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular_Algebras</title>
	<link>/theories/regular_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular_Tree_Relations</title>
	<link>/theories/regular_tree_relations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular_tree_relations/</guid>
	<description></description>
	</item>

	<item>
	<title>Relation_Algebra</title>
	<link>/theories/relation_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relation_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational-Incorrectness-Logic</title>
	<link>/theories/relational-incorrectness-logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational-incorrectness-logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Disjoint_Set_Forests</title>
	<link>/theories/relational_disjoint_set_forests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_disjoint_set_forests/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Forests</title>
	<link>/theories/relational_forests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_forests/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Method</title>
	<link>/theories/relational_method/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_method/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Minimum_Spanning_Trees</title>
	<link>/theories/relational_minimum_spanning_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_minimum_spanning_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Paths</title>
	<link>/theories/relational_paths/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_paths/</guid>
	<description></description>
	</item>

	<item>
	<title>Rensets</title>
	<link>/theories/rensets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rensets/</guid>
	<description></description>
	</item>

	<item>
	<title>Rep_Fin_Groups</title>
	<link>/theories/rep_fin_groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rep_fin_groups/</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated_Lattices</title>
	<link>/theories/residuated_lattices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/residuated_lattices/</guid>
	<description></description>
	</item>

	<item>
	<title>ResiduatedTransitionSystem</title>
	<link>/theories/residuatedtransitionsystem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/residuatedtransitionsystem/</guid>
	<description></description>
	</item>

	<item>
	<title>Resolution_FOL</title>
	<link>/theories/resolution_fol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/resolution_fol/</guid>
	<description></description>
	</item>

	<item>
	<title>Rewrite_Properties_Reduction</title>
	<link>/theories/rewrite_properties_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rewrite_properties_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Rewriting_Z</title>
	<link>/theories/rewriting_z/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rewriting_z/</guid>
	<description></description>
	</item>

	<item>
	<title>Ribbon_Proofs</title>
	<link>/theories/ribbon_proofs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ribbon_proofs/</guid>
	<description></description>
	</item>

	<item>
	<title>RIPEMD-160-SPARK</title>
	<link>/theories/ripemd-160-spark/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ripemd-160-spark/</guid>
	<description></description>
	</item>

	<item>
	<title>Risk_Free_Lending</title>
	<link>/theories/risk_free_lending/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/risk_free_lending/</guid>
	<description></description>
	</item>

	<item>
	<title>Robbins-Conjecture</title>
	<link>/theories/robbins-conjecture/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robbins-conjecture/</guid>
	<description></description>
	</item>

	<item>
	<title>ROBDD</title>
	<link>/theories/robdd/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robdd/</guid>
	<description></description>
	</item>

	<item>
	<title>Robinson_Arithmetic</title>
	<link>/theories/robinson_arithmetic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robinson_arithmetic/</guid>
	<description></description>
	</item>

	<item>
	<title>Root_Balanced_Tree</title>
	<link>/theories/root_balanced_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/root_balanced_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Roth_Arithmetic_Progressions</title>
	<link>/theories/roth_arithmetic_progressions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/roth_arithmetic_progressions/</guid>
	<description></description>
	</item>

	<item>
	<title>Routing</title>
	<link>/theories/routing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/routing/</guid>
	<description></description>
	</item>

	<item>
	<title>Roy_Floyd_Warshall</title>
	<link>/theories/roy_floyd_warshall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/roy_floyd_warshall/</guid>
	<description></description>
	</item>

	<item>
	<title>RSAPSS</title>
	<link>/theories/rsapss/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rsapss/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_Distance</title>
	<link>/theories/safe_distance/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_distance/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_OCL</title>
	<link>/theories/safe_ocl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_ocl/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_Range_RC</title>
	<link>/theories/safe_range_rc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_range_rc/</guid>
	<description></description>
	</item>

	<item>
	<title>SATSolverVerification</title>
	<link>/theories/satsolververification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/satsolververification/</guid>
	<description></description>
	</item>

	<item>
	<title>Saturation_Framework</title>
	<link>/theories/saturation_framework/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/saturation_framework/</guid>
	<description></description>
	</item>

	<item>
	<title>Saturation_Framework_Extensions</title>
	<link>/theories/saturation_framework_extensions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/saturation_framework_extensions/</guid>
	<description></description>
	</item>

	<item>
	<title>Sauer_Shelah_Lemma</title>
	<link>/theories/sauer_shelah_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sauer_shelah_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>SC_DOM_Components</title>
	<link>/theories/sc_dom_components/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sc_dom_components/</guid>
	<description></description>
	</item>

	<item>
	<title>SCC_Bloemen_Sequential</title>
	<link>/theories/scc_bloemen_sequential/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/scc_bloemen_sequential/</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz_Spacetime</title>
	<link>/theories/schutz_spacetime/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/schutz_spacetime/</guid>
	<description></description>
	</item>

	<item>
	<title>Schwartz_Zippel</title>
	<link>/theories/schwartz_zippel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/schwartz_zippel/</guid>
	<description></description>
	</item>

	<item>
	<title>SDS_Impossibility</title>
	<link>/theories/sds_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sds_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Secondary_Sylow</title>
	<link>/theories/secondary_sylow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/secondary_sylow/</guid>
	<description></description>
	</item>

	<item>
	<title>Security_Protocol_Refinement</title>
	<link>/theories/security_protocol_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/security_protocol_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>Selection_Heap_Sort</title>
	<link>/theories/selection_heap_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/selection_heap_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>SenSocialChoice</title>
	<link>/theories/sensocialchoice/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sensocialchoice/</guid>
	<description></description>
	</item>

	<item>
	<title>Separata</title>
	<link>/theories/separata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separata/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Algebra</title>
	<link>/theories/separation_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Logic_Imperative_HOL</title>
	<link>/theories/separation_logic_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_logic_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Logic_Unbounded</title>
	<link>/theories/separation_logic_unbounded/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_logic_unbounded/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_Basic</title>
	<link>/theories/sepref_basic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_basic/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_IICF</title>
	<link>/theories/sepref_iicf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_iicf/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_Prereq</title>
	<link>/theories/sepref_prereq/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_prereq/</guid>
	<description></description>
	</item>

	<item>
	<title>SequentInvertibility</title>
	<link>/theories/sequentinvertibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sequentinvertibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Shadow_DOM</title>
	<link>/theories/shadow_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shadow_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Shadow_SC_DOM</title>
	<link>/theories/shadow_sc_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shadow_sc_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Shivers-CFA</title>
	<link>/theories/shivers-cfa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shivers-cfa/</guid>
	<description></description>
	</item>

	<item>
	<title>ShortestPath</title>
	<link>/theories/shortestpath/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shortestpath/</guid>
	<description></description>
	</item>

	<item>
	<title>Show</title>
	<link>/theories/show/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/show/</guid>
	<description></description>
	</item>

	<item>
	<title>SIFPL</title>
	<link>/theories/sifpl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sifpl/</guid>
	<description></description>
	</item>

	<item>
	<title>SIFUM_Type_Systems</title>
	<link>/theories/sifum_type_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sifum_type_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>Sigma_Commit_Crypto</title>
	<link>/theories/sigma_commit_crypto/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sigma_commit_crypto/</guid>
	<description></description>
	</item>

	<item>
	<title>Signature_Groebner</title>
	<link>/theories/signature_groebner/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/signature_groebner/</guid>
	<description></description>
	</item>

	<item>
	<title>Simpl</title>
	<link>/theories/simpl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simpl/</guid>
	<description></description>
	</item>

	<item>
	<title>Simple_Clause_Learning</title>
	<link>/theories/simple_clause_learning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simple_clause_learning/</guid>
	<description></description>
	</item>

	<item>
	<title>Simple_Firewall</title>
	<link>/theories/simple_firewall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simple_firewall/</guid>
	<description></description>
	</item>

	<item>
	<title>Simplex</title>
	<link>/theories/simplex/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplex/</guid>
	<description></description>
	</item>

	<item>
	<title>Simplicial_complexes_and_boolean_functions</title>
	<link>/theories/simplicial_complexes_and_boolean_functions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplicial_complexes_and_boolean_functions/</guid>
	<description></description>
	</item>

	<item>
	<title>SimplifiedOntologicalArgument</title>
	<link>/theories/simplifiedontologicalargument/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplifiedontologicalargument/</guid>
	<description></description>
	</item>

	<item>
	<title>Skew_Heap</title>
	<link>/theories/skew_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/skew_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>Skip_Lists</title>
	<link>/theories/skip_lists/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/skip_lists/</guid>
	<description></description>
	</item>

	<item>
	<title>Slicing</title>
	<link>/theories/slicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/slicing/</guid>
	<description></description>
	</item>

	<item>
	<title>Sliding_Window_Algorithm</title>
	<link>/theories/sliding_window_algorithm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sliding_window_algorithm/</guid>
	<description></description>
	</item>

	<item>
	<title>SM</title>
	<link>/theories/sm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sm/</guid>
	<description></description>
	</item>

	<item>
	<title>SM_Base</title>
	<link>/theories/sm_base/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sm_base/</guid>
	<description></description>
	</item>

	<item>
	<title>Smith_Normal_Form</title>
	<link>/theories/smith_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/smith_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth_Manifolds</title>
	<link>/theories/smooth_manifolds/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/smooth_manifolds/</guid>
	<description></description>
	</item>

	<item>
	<title>Solidity</title>
	<link>/theories/solidity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/solidity/</guid>
	<description></description>
	</item>

	<item>
	<title>Sophomores_Dream</title>
	<link>/theories/sophomores_dream/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sophomores_dream/</guid>
	<description></description>
	</item>

	<item>
	<title>Sort_Encodings</title>
	<link>/theories/sort_encodings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sort_encodings/</guid>
	<description></description>
	</item>

	<item>
	<title>Source_Coding_Theorem</title>
	<link>/theories/source_coding_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/source_coding_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>SPARCv8</title>
	<link>/theories/sparcv8/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sparcv8/</guid>
	<description></description>
	</item>

	<item>
	<title>SpecCheck</title>
	<link>/theories/speccheck/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/speccheck/</guid>
	<description></description>
	</item>

	<item>
	<title>Special_Function_Bounds</title>
	<link>/theories/special_function_bounds/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/special_function_bounds/</guid>
	<description></description>
	</item>

	<item>
	<title>Splay_Tree</title>
	<link>/theories/splay_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/splay_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Sqrt_Babylonian</title>
	<link>/theories/sqrt_babylonian/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sqrt_babylonian/</guid>
	<description></description>
	</item>

	<item>
	<title>Stable_Matching</title>
	<link>/theories/stable_matching/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stable_matching/</guid>
	<description></description>
	</item>

	<item>
	<title>Stalnaker_Logic</title>
	<link>/theories/stalnaker_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stalnaker_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Statecharts</title>
	<link>/theories/statecharts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/statecharts/</guid>
	<description></description>
	</item>

	<item>
	<title>Stateful_Protocol_Composition_and_Typing</title>
	<link>/theories/stateful_protocol_composition_and_typing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stateful_protocol_composition_and_typing/</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar_Quorums</title>
	<link>/theories/stellar_quorums/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stellar_quorums/</guid>
	<description></description>
	</item>

	<item>
	<title>Stern_Brocot</title>
	<link>/theories/stern_brocot/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stern_brocot/</guid>
	<description></description>
	</item>

	<item>
	<title>Stewart_Apollonius</title>
	<link>/theories/stewart_apollonius/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stewart_apollonius/</guid>
	<description></description>
	</item>

	<item>
	<title>Stirling_Formula</title>
	<link>/theories/stirling_formula/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stirling_formula/</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic_Matrices</title>
	<link>/theories/stochastic_matrices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stochastic_matrices/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Algebras</title>
	<link>/theories/stone_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Kleene_Relation_Algebras</title>
	<link>/theories/stone_kleene_relation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_kleene_relation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Relation_Algebras</title>
	<link>/theories/stone_relation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_relation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Store_Buffer_Reduction</title>
	<link>/theories/store_buffer_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/store_buffer_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Stream-Fusion</title>
	<link>/theories/stream-fusion/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stream-fusion/</guid>
	<description></description>
	</item>

	<item>
	<title>Stream_Fusion_Code</title>
	<link>/theories/stream_fusion_code/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stream_fusion_code/</guid>
	<description></description>
	</item>

	<item>
	<title>StrictOmegaCategories</title>
	<link>/theories/strictomegacategories/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/strictomegacategories/</guid>
	<description></description>
	</item>

	<item>
	<title>Strong_Security</title>
	<link>/theories/strong_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/strong_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm_Sequences</title>
	<link>/theories/sturm_sequences/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sturm_sequences/</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm_Tarski</title>
	<link>/theories/sturm_tarski/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sturm_tarski/</guid>
	<description></description>
	</item>

	<item>
	<title>Stuttering_Equivalence</title>
	<link>/theories/stuttering_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stuttering_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Subresultants</title>
	<link>/theories/subresultants/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/subresultants/</guid>
	<description></description>
	</item>

	<item>
	<title>Subset_Boolean_Algebras</title>
	<link>/theories/subset_boolean_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/subset_boolean_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>SumSquares</title>
	<link>/theories/sumsquares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sumsquares/</guid>
	<description></description>
	</item>

	<item>
	<title>Sunflowers</title>
	<link>/theories/sunflowers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sunflowers/</guid>
	<description></description>
	</item>

	<item>
	<title>SuperCalc</title>
	<link>/theories/supercalc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/supercalc/</guid>
	<description></description>
	</item>

	<item>
	<title>Suppes_Theorem</title>
	<link>/theories/suppes_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/suppes_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Surprise_Paradox</title>
	<link>/theories/surprise_paradox/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/surprise_paradox/</guid>
	<description></description>
	</item>

	<item>
	<title>Symmetric_Polynomials</title>
	<link>/theories/symmetric_polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/symmetric_polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax_Independent_Logic</title>
	<link>/theories/syntax_independent_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/syntax_independent_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Synthetic_Completeness</title>
	<link>/theories/synthetic_completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/synthetic_completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Szemeredi_Regularity</title>
	<link>/theories/szemeredi_regularity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/szemeredi_regularity/</guid>
	<description></description>
	</item>

	<item>
	<title>Szpilrajn</title>
	<link>/theories/szpilrajn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/szpilrajn/</guid>
	<description></description>
	</item>

	<item>
	<title>Tail_Recursive_Functions</title>
	<link>/theories/tail_recursive_functions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tail_recursive_functions/</guid>
	<description></description>
	</item>

	<item>
	<title>Tarskis_Geometry</title>
	<link>/theories/tarskis_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tarskis_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor_Models</title>
	<link>/theories/taylor_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/taylor_models/</guid>
	<description></description>
	</item>

	<item>
	<title>TESL_Language</title>
	<link>/theories/tesl_language/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tesl_language/</guid>
	<description></description>
	</item>

	<item>
	<title>Three_Circles</title>
	<link>/theories/three_circles/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/three_circles/</guid>
	<description></description>
	</item>

	<item>
	<title>Three_Squares</title>
	<link>/theories/three_squares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/three_squares/</guid>
	<description></description>
	</item>

	<item>
	<title>Timed_Automata</title>
	<link>/theories/timed_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/timed_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>TLA</title>
	<link>/theories/tla/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tla/</guid>
	<description></description>
	</item>

	<item>
	<title>Topological_Semantics</title>
	<link>/theories/topological_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/topological_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Topology</title>
	<link>/theories/topology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/topology/</guid>
	<description></description>
	</item>

	<item>
	<title>TortoiseHare</title>
	<link>/theories/tortoisehare/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tortoisehare/</guid>
	<description></description>
	</item>

	<item>
	<title>Transcendence_Series_Hancl_Rucki</title>
	<link>/theories/transcendence_series_hancl_rucki/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transcendence_series_hancl_rucki/</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer_Semantics</title>
	<link>/theories/transformer_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transformer_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Transition_Systems_and_Automata</title>
	<link>/theories/transition_systems_and_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transition_systems_and_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive-Closure</title>
	<link>/theories/transitive-closure/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive-closure/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive-Closure-II</title>
	<link>/theories/transitive-closure-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive-closure-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive_Models</title>
	<link>/theories/transitive_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Treaps</title>
	<link>/theories/treaps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/treaps/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree-Automata</title>
	<link>/theories/tree-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree_Decomposition</title>
	<link>/theories/tree_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree_Enumeration</title>
	<link>/theories/tree_enumeration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree_enumeration/</guid>
	<description></description>
	</item>

	<item>
	<title>Triangle</title>
	<link>/theories/triangle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/triangle/</guid>
	<description></description>
	</item>

	<item>
	<title>Trie</title>
	<link>/theories/trie/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/trie/</guid>
	<description></description>
	</item>

	<item>
	<title>TsirelsonBound</title>
	<link>/theories/tsirelsonbound/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tsirelsonbound/</guid>
	<description></description>
	</item>

	<item>
	<title>Turans_Graph_Theorem</title>
	<link>/theories/turans_graph_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/turans_graph_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Twelvefold_Way</title>
	<link>/theories/twelvefold_way/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/twelvefold_way/</guid>
	<description></description>
	</item>

	<item>
	<title>Two_Generated_Word_Monoids_Intersection</title>
	<link>/theories/two_generated_word_monoids_intersection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/two_generated_word_monoids_intersection/</guid>
	<description></description>
	</item>

	<item>
	<title>Tycon</title>
	<link>/theories/tycon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tycon/</guid>
	<description></description>
	</item>

	<item>
	<title>Types_Tableaus_and_Goedels_God</title>
	<link>/theories/types_tableaus_and_goedels_god/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/types_tableaus_and_goedels_god/</guid>
	<description></description>
	</item>

	<item>
	<title>Types_To_Sets_Extension</title>
	<link>/theories/types_to_sets_extension/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/types_to_sets_extension/</guid>
	<description></description>
	</item>

	<item>
	<title>Undirected_Graph_Theory</title>
	<link>/theories/undirected_graph_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/undirected_graph_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Universal_Hash_Families</title>
	<link>/theories/universal_hash_families/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/universal_hash_families/</guid>
	<description></description>
	</item>

	<item>
	<title>Universal_Turing_Machine</title>
	<link>/theories/universal_turing_machine/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/universal_turing_machine/</guid>
	<description></description>
	</item>

	<item>
	<title>UpDown_Scheme</title>
	<link>/theories/updown_scheme/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/updown_scheme/</guid>
	<description></description>
	</item>

	<item>
	<title>UPF</title>
	<link>/theories/upf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/upf/</guid>
	<description></description>
	</item>

	<item>
	<title>UPF_Firewall</title>
	<link>/theories/upf_firewall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/upf_firewall/</guid>
	<description></description>
	</item>

	<item>
	<title>UTP</title>
	<link>/theories/utp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/utp/</guid>
	<description></description>
	</item>

	<item>
	<title>UTP-Toolkit</title>
	<link>/theories/utp-toolkit/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/utp-toolkit/</guid>
	<description></description>
	</item>

	<item>
	<title>Valuation</title>
	<link>/theories/valuation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/valuation/</guid>
	<description></description>
	</item>

	<item>
	<title>Van_der_Waerden</title>
	<link>/theories/van_der_waerden/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/van_der_waerden/</guid>
	<description></description>
	</item>

	<item>
	<title>Van_Emde_Boas_Trees</title>
	<link>/theories/van_emde_boas_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/van_emde_boas_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>VectorSpace</title>
	<link>/theories/vectorspace/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vectorspace/</guid>
	<description></description>
	</item>

	<item>
	<title>VeriComp</title>
	<link>/theories/vericomp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vericomp/</guid>
	<description></description>
	</item>

	<item>
	<title>Verified-Prover</title>
	<link>/theories/verified-prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verified-prover/</guid>
	<description></description>
	</item>

	<item>
	<title>Verified_SAT_Based_AI_Planning</title>
	<link>/theories/verified_sat_based_ai_planning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verified_sat_based_ai_planning/</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis2018</title>
	<link>/theories/verifythis2018/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verifythis2018/</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis2019</title>
	<link>/theories/verifythis2019/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verifythis2019/</guid>
	<description></description>
	</item>

	<item>
	<title>Vickrey_Clarke_Groves</title>
	<link>/theories/vickrey_clarke_groves/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vickrey_clarke_groves/</guid>
	<description></description>
	</item>

	<item>
	<title>Virtual_Substitution</title>
	<link>/theories/virtual_substitution/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/virtual_substitution/</guid>
	<description></description>
	</item>

	<item>
	<title>VolpanoSmith</title>
	<link>/theories/volpanosmith/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/volpanosmith/</guid>
	<description></description>
	</item>

	<item>
	<title>VYDRA_MDL</title>
	<link>/theories/vydra_mdl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vydra_mdl/</guid>
	<description></description>
	</item>

	<item>
	<title>WebAssembly</title>
	<link>/theories/webassembly/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/webassembly/</guid>
	<description></description>
	</item>

	<item>
	<title>Weight_Balanced_Trees</title>
	<link>/theories/weight_balanced_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weight_balanced_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Weighted_Arithmetic_Geometric_Mean</title>
	<link>/theories/weighted_arithmetic_geometric_mean/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weighted_arithmetic_geometric_mean/</guid>
	<description></description>
	</item>

	<item>
	<title>Weighted_Path_Order</title>
	<link>/theories/weighted_path_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weighted_path_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Well_Quasi_Orders</title>
	<link>/theories/well_quasi_orders/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/well_quasi_orders/</guid>
	<description></description>
	</item>

	<item>
	<title>Wetzels_Problem</title>
	<link>/theories/wetzels_problem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/wetzels_problem/</guid>
	<description></description>
	</item>

	<item>
	<title>WHATandWHERE_Security</title>
	<link>/theories/whatandwhere_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/whatandwhere_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Winding_Number_Eval</title>
	<link>/theories/winding_number_eval/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/winding_number_eval/</guid>
	<description></description>
	</item>

	<item>
	<title>WOOT_Strong_Eventual_Consistency</title>
	<link>/theories/woot_strong_eventual_consistency/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/woot_strong_eventual_consistency/</guid>
	<description></description>
	</item>

	<item>
	<title>Word_Lib</title>
	<link>/theories/word_lib/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/word_lib/</guid>
	<description></description>
	</item>

	<item>
	<title>WorkerWrapper</title>
	<link>/theories/workerwrapper/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/workerwrapper/</guid>
	<description></description>
	</item>

	<item>
	<title>X86_Semantics</title>
	<link>/theories/x86_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/x86_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/theories/xml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/xml/</guid>
	<description></description>
	</item>

	<item>
	<title>Youngs_Inequality</title>
	<link>/theories/youngs_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/youngs_inequality/</guid>
	<description></description>
	</item>

	<item>
	<title>Zeckendorf</title>
	<link>/theories/zeckendorf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/zeckendorf/</guid>
	<description></description>
	</item>

	<item>
	<title>Zeta_3_Irrational</title>
	<link>/theories/zeta_3_irrational/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/zeta_3_irrational/</guid>
	<description></description>
	</item>

	<item>
	<title>Zeta_Function</title>
	<link>/theories/zeta_function/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/zeta_function/</guid>
	<description></description>
	</item>

	<item>
	<title>ZFC_in_HOL</title>
	<link>/theories/zfc_in_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/zfc_in_hol/</guid>
	<description></description>
	</item>

	</channel>
	</rss>
	diff --git a/web/js/theory.js b/web/js/theory.js
	--- a/web/js/theory.js
	+++ b/web/js/theory.js
	@@ -1,279 +1,283 @@
	/* constants */

	const ID_THEORY_LIST = 'theories'
	const CLASS_LOADER = 'loader'
	const CLASS_ANIMATION = 'animation'
	const ATTRIBUTE_THEORY_SRC = 'theory-src'
	const CLASS_NAVBAR_TYPE = 'theory-navbar-type'
	const CLASS_THY_NAV = 'thy-nav'
	const PARAM_NAVBAR_TYPE = 'theory-navbar-type'
	const ID_NAVBAR_TYPE_SELECTOR = 'navbar-type-selector'
	const ID_NAVBAR = 'theory-navbar'
	const NAVBAR_TYPES = ['fact', 'type', 'const']


	/* routing */

	function target(base_href, rel_href) {
	const href_parts = rel_href.split('/')

	- if (href_parts.length === 1) return `#${href_parts[0]}`
	- else if (href_parts.length === 3 && href_parts[0] === '..' && href_parts[1] !== '..') {
	- return `../${href_parts[1].toLowerCase()}/#${href_parts[2]}`
	+ if (href_parts.length === 1) {
	+ const ref_parts = href_parts[0].split('.html')
	+ const file_parts = ref_parts[0].split('.')
	+ if (file_parts.length == 1) return `#${href_parts[0]}`
	+ else {
	+ return `../${file_parts[0].toLowerCase()}/#${file_parts.slice(1).join('.')}.html${ref_parts[1]}`
	+ }
	}
	else return `${base_href}/../${rel_href}`
	}

	function to_id(thy_name, ref) {
	if (ref) return `${thy_name}.html#${ref}`
	else return `${thy_name}.html`
	}

	const to_fresh_id = (id) => `${id}#`
	const to_svg_id = (id) => `${id}#svg`
	const to_container_id = (id) => `${id}#container`
	const to_collapsible_id = (id) => `${id}#collapsible`
	const to_spinner_id = (id) => `${id}#spinner`
	const to_nav_id = (id) => `${id}#nav`
	const to_ul_id = (id) => `${id}#ul`
	const of_ul_id = (id) => id.split('#').slice(0, -1).join('#')
	const to_a_id = (id) => `${id}#a`

	/* document translation */

	function translate(base_href, thy_name, thy_body) {
	const thy_refs = [...thy_body.getElementsByTagName('span')].map((span) => {
	let ref = span.getAttribute('id')
	if (ref) {
	span.setAttribute('id', to_fresh_id(to_id(thy_name, ref)))
	}
	return ref
	}).filter(e => e)
	for (const link of thy_body.getElementsByTagName('a')) {
	const rel_href = link.getAttribute('href')
	link.setAttribute('href', target(base_href, rel_href))
	}
	return thy_refs
	}


	/* theory lazy-loading */

	async function fetch_theory_body(href) {
	const html_str = await fetch(href).then((http_res) => {
	if (http_res.status !== 200) return Promise.resolve(`<body>${http_res.statusText}</body>`)
	else return http_res.text()
	}).catch((_) => {
	console.log(`Could not load theory at '${href}'. Redirecting...`)
	window.location.replace(href)
	})

	const parser = new DOMParser()
	const html = parser.parseFromString(html_str, 'text/html')
	return html.getElementsByTagName('body')[0]
	}

	async function load_theory(thy_name, href) {
	const thy_body = await fetch_theory_body(href)
	const refs = translate(href, thy_name, thy_body)

	const collapse = document.getElementById(to_collapsible_id(thy_name))
	collapse.append(...Array(...thy_body.children).slice(1))
	return refs
	}

	async function open_theory(thy_name) {
	const container = document.getElementById(to_container_id(thy_name))

	if (container) {
	if (document.getElementById(to_collapsible_id(thy_name))) open(container)
	else {
	const collapsible = parse_elem(`
	<div id="${to_collapsible_id(thy_name)}" class="${CLASS_COLLAPSIBLE}">
	<div id="${to_spinner_id(thy_name)}" class=${CLASS_LOADER}><div class=${CLASS_ANIMATION}></div></div>
	</div>`)
	container.appendChild(collapsible)
	open(container)
	let refs = await load_theory(thy_name, container.getAttribute(ATTRIBUTE_THEORY_SRC))
	await load_theory_nav(thy_name, refs)
	const spinner = document.getElementById(to_spinner_id(thy_name))
	spinner.parentNode.removeChild(spinner)
	}
	}
	}

	function nav_tree_rec(thy_name, path, key, ref_parts, type) {
	const rec_ref = ref_parts.filter(e => e.length > 0)
	const ref = `${path.join('.')}.${key}\|${type}`
	const id = to_id(thy_name, ref)
	let res
	if (rec_ref.length < ref_parts.length) {
	res = `<a id="${to_a_id(id)}" class="${CLASS_SPY_LINK}" href="#${id}">${escape_html(key)}</a>`
	} else {
	const head_id = to_id(thy_name, `${[...path, key, ...ref_parts[0]].join('.')}\|${type}`)
	res = `<a id="${to_a_id(id)}" class="${CLASS_SPY_LINK}" href="#${head_id}">${escape_html(key)}</a>`
	}

	if (rec_ref.length > 1) {
	const by_key = group_by(rec_ref)
	const children = Object.keys(by_key).map((key1) => `
	<li>${nav_tree_rec(thy_name, [...path, key], key1, by_key[key1], type)}</li>`)
	return `
	${res}
	<ul id="${to_ul_id(id)}" class="${CLASS_COLLAPSIBLE} ${CLASS_COLLAPSED}">
	${children.join('')}
	</ul>`
	} else return res
	}

	function nav_tree(thy_name, refs, type) {
	let trees = Object.entries(group_by(refs \|\| [])).map(([key, parts]) =>
	`<li>${nav_tree_rec(thy_name, [thy_name], key, parts, type)}</li>`)

	return parse_elem(`
	<ul id="${to_ul_id(thy_name)}" class="${CLASS_NAVBAR_TYPE} ${CLASS_COLLAPSIBLE} ${CLASS_COLLAPSED}">
	${trees.join('')}
	</ul>`)
	}


	const cached_refs = Object.fromEntries(NAVBAR_TYPES.map(t => [t, {}]))
	const load_theory_nav = (thy_name, refs) => {
	let selected = get_query(PARAM_NAVBAR_TYPE) \|\| NAVBAR_TYPES[0]

	let by_type = group_by(refs.filter(ref => ref.includes('\|')).map((id) => id.split('\|').reverse()))
	let type_selector = document.getElementById(ID_NAVBAR_TYPE_SELECTOR)
	let options = [...type_selector.options].map(e => e.value)

	for (let [type, elems] of Object.entries(by_type)) {
	if (NAVBAR_TYPES.includes(type) && !options.includes(type)) {
	type_selector.appendChild(parse_elem(`<option value=${type}>${type}</option>`))
	}

	let parts_by_thy = group_by(elems.map((s) => s[0].split('.')))
	if (NAVBAR_TYPES.includes(type)) cached_refs[type][thy_name] = parts_by_thy[thy_name]
	}

	let tree = nav_tree(thy_name, cached_refs[selected][thy_name], selected)
	document.getElementById(to_nav_id(thy_name)).appendChild(tree)

	ScrollSpy.instance.refresh()
	}

	/* state */

	let navbar_last_opened = []


	/* controls */

	const follow_theory_hash = async () => {
	let hash = window.location.hash

	if (hash.length > 1) {
	const id = hash.slice(1)

	const thy_name = strip_suffix(id.split('#')[0], '.html')
	await open_theory(thy_name)

	ScrollSpy.instance.scroll_to(id)
	}
	}

	const toggle_theory = async (thy_name) => {
	const hash = `#${to_id(thy_name)}`
	const collapsible = document.getElementById(to_container_id(thy_name))
	if (collapsible) {
	if (!collapse(collapsible)) {
	if (window.location.hash === hash) open(collapsible)
	else window.location.hash = hash
	}
	} else window.location.hash = hash
	}

	const change_selector = (type) => {
	let old_type = get_query(PARAM_NAVBAR_TYPE)
	if (!old_type \|\| old_type !== type) {
	set_query(PARAM_NAVBAR_TYPE, type)

	for (const elem of document.getElementsByClassName(CLASS_NAVBAR_TYPE)) {
	let thy_name = of_ul_id(elem.id)
	elem.replaceWith(nav_tree(thy_name, cached_refs[type][thy_name], type))
	}

	ScrollSpy.instance.refresh()
	}
	}

	const open_tree = (elem) => {
	if (elem.classList.contains(CLASS_COLLAPSIBLE)) {
	if (open(elem)) navbar_last_opened.push(elem)
	}
	if (elem.parentElement) open_tree(elem.parentElement)
	}

	const sync_navbar = (link) => {
	for (const elem of navbar_last_opened){
	collapse(elem)
	}

	open_tree(link.parentElement)

	link.scrollIntoView({block: "center"})
	}


	/* setup */

	const init = async () => {
	const theory_list = document.getElementById(ID_THEORY_LIST)
	const navbar = document.getElementById(ID_NAVBAR)

	if (theory_list && navbar) {
	const thy_names = []

	for (const theory of theory_list.children) {
	thy_names.push(theory.id)

	const href = theory.getAttribute('href')
	const thy_name = theory.id

	const thy_collapsible = parse_elem(`
	<div id="${to_container_id(thy_name)}" theory-src="${href}" style="min-width: fit-content" class="${CLASS_COLLAPSE_CONTAINER} ${CLASS_COLLAPSED}">
	<h2 id="${to_id(thy_name)}" style="cursor: pointer" onclick="toggle_theory('${thy_name}')">
	${thy_name}
	<svg id="${to_svg_id(thy_name)}" viewBox="0 0 10 10" aria-hidden="true" focusable="false"
	class="${CLASS_INVERTIBLE}">
	<path d="m1.6953 6.7407 3.3047-3.3929 3.3047 3.3927" fill="none" stroke-linecap="round"
	stroke-linejoin="round" stroke-width="2"/>
	</svg>
	</h2>
	</div>`)
	theory.replaceWith(thy_collapsible)
	}

	const type = get_query(PARAM_NAVBAR_TYPE) ? get_query(PARAM_NAVBAR_TYPE) : NAVBAR_TYPES[0]
	navbar.appendChild(parse_elem(`
	<li>
	<select id=${ID_NAVBAR_TYPE_SELECTOR} onchange="change_selector(this.options[this.selectedIndex].value)">
	<option value=${type} selected="selected">${type}</option>
	</select>
	</li>`))
	navbar.append(...thy_names.map((thy_name) => parse_elem(`
	<li id="${to_nav_id(thy_name)}">
	<a id="${to_a_id(thy_name)}" class="${CLASS_SPY_LINK} ${CLASS_THY_NAV}" href="#${to_id(thy_name)}">
	${thy_name}
	</a>
	</li>`)))
	navbar.insertAdjacentElement('afterend', document.createElement('hr'));

	window.onhashchange = follow_theory_hash
	window.addEventListener(EVENT_SPY_ACTIVATE, (e) => sync_navbar(e.relatedTarget))

	new ScrollSpy(document.body, ID_NAVBAR, "#")

	await follow_theory_hash()
	}
	}

	document.addEventListener('DOMContentLoaded', init)
	diff --git a/web/sitemap.xml b/web/sitemap.xml
	--- a/web/sitemap.xml
	+++ b/web/sitemap.xml
	@@ -1,6305 +1,6332 @@
	<?xml version="1.0" encoding="utf-8" standalone="yes"?>
	<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9"
	xmlns:xhtml="http://www.w3.org/1999/xhtml">
	<url>
	<loc>/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/argyraki/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/dependencies/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/entries/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/leek/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/topics/mathematics/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/entries/Polygonal_Number_Theorem.html</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/dependencies/three_squares/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/topics/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/yez/</loc>
	+ <lastmod>2023-08-10T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/calk/</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/topics/computer-science/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	- </url><url>
	- <loc>/entries/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/dependencies/kad/</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/dependencies/kleene_algebra/</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/entries/Quantales_Converse.html</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/dependencies/quantales/</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/struth/</loc>
	+ <lastmod>2023-07-25T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/entries/Earley_Parser.html</loc>
	+ <lastmod>2023-07-16T00:00:00+00:00</lastmod>
	+ </url><url>
	+ <loc>/authors/rau/</loc>
	+ <lastmod>2023-07-16T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Gray_Codes.html</loc>
	<lastmod>2023-07-11T00:00:00+00:00</lastmod>
	</url><url>
	- <loc>/topics/mathematics/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	- </url><url>
	<loc>/authors/spitz/</loc>
	<lastmod>2023-07-11T00:00:00+00:00</lastmod>
	</url><url>
	- <loc>/topics/</loc>
	- <lastmod>2023-07-11T00:00:00+00:00</lastmod>
	- </url><url>
	- <loc>/dependencies/</loc>
	- <lastmod>2023-06-19T00:00:00+00:00</lastmod>
	- </url><url>
	<loc>/authors/eberl/</loc>
	<lastmod>2023-06-19T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Executable_Randomized_Algorithms.html</loc>
	<lastmod>2023-06-19T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/karayel/</loc>
	<lastmod>2023-06-19T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/probabilistic_while/</loc>
	<lastmod>2023-06-19T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/zeta_function/</loc>
	<lastmod>2023-06-19T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/christfort/</loc>
	<lastmod>2023-06-16T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/DCR-ExecutionEquivalence.html</loc>
	<lastmod>2023-06-16T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/debois/</loc>
	<lastmod>2023-06-16T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/dalvit/</loc>
	<lastmod>2023-06-12T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Zeckendorf.html</loc>
	<lastmod>2023-06-12T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Crypto_Standards.html</loc>
	<lastmod>2023-06-06T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/elliptic_curves_group_law/</loc>
	<lastmod>2023-06-06T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/whitley/</loc>
	<lastmod>2023-06-06T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Efficient_Weighted_Path_Order.html</loc>
	<lastmod>2023-06-01T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/topics/logic/</loc>
	<lastmod>2023-06-01T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/thiemann/</loc>
	<lastmod>2023-06-01T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/weighted_path_order/</loc>
	<lastmod>2023-06-01T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/wenninger/</loc>
	<lastmod>2023-06-01T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/complete_non_orders/</loc>
	<lastmod>2023-05-24T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/dubut/</loc>
	<lastmod>2023-05-24T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Directed_Sets.html</loc>
	<lastmod>2023-05-24T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/yamada/</loc>
	<lastmod>2023-05-24T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/authors/guttmann/</loc>
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	</url><url>
	<loc>/entries/Multirelations_Heterogeneous.html</loc>
	<lastmod>2023-05-22T00:00:00+00:00</lastmod>
	</url><url>
	- <loc>/authors/struth/</loc>
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	- </url><url>
	<loc>/authors/cremer/</loc>
	<lastmod>2023-05-09T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Tree_Enumeration.html</loc>
	<lastmod>2023-05-09T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/dependencies/undirected_graph_theory/</loc>
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	<loc>/entries/MLSS_Decision_Proc.html</loc>
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	<loc>/authors/stevens/</loc>
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	<loc>/dependencies/dirichlet_l/</loc>
	<lastmod>2023-05-03T00:00:00+00:00</lastmod>
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	<loc>/entries/Three_Squares.html</loc>
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	<loc>/authors/stannett/</loc>
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	<loc>/entries/MHComputation.html</loc>
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	<loc>/dependencies/factor_algebraic_polynomial/</loc>
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	<loc>/authors/kim/</loc>
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	<loc>/dependencies/skip_lists/</loc>
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	<loc>/authors/tan/</loc>
	<lastmod>2023-04-27T00:00:00+00:00</lastmod>
	</url><url>
	<loc>/entries/Schwartz_Zippel.html</loc>
	<lastmod>2023-04-27T00:00:00+00:00</lastmod>
	</url><url>
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	+<meta property="og:description" content="759 Entries 465 Authors ~243,500 Lemmas ~3,952,800 Lines of Code Most used AFP entries: Name Used by ? entries 1. List-Index 23 2. Collections 19 3. Show 16 4. Jordan_Normal_Form 15 5. Coinductive 13 6. Deriving 13 7. Polynomial_Factorization 12 8. Regular-Sets 12 9." />
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	- <td class="statsnumber">756</td>
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	<td><a href="../">Entries</a></td>
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	<td><a href="../authors/">Authors</a></td>
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	- <td class="statsnumber">~242,800</td>
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	<td>1.</td>
	<td><a href="../entries/List-Index.html">List-Index</a></td>
	<td><a href="../dependencies/list-index/">23</a></td>
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	<td>2.</td>
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	<td><a href="../dependencies/collections/">19</a></td>
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	<td><a href="../dependencies/show/">16</a></td>
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	<tr>
	<td>4.</td>
	<td><a href="../entries/Jordan_Normal_Form.html">Jordan_Normal_Form</a></td>
	<td><a href="../dependencies/jordan_normal_form/">15</a></td>
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	<td>5.</td>
	<td><a href="../entries/Coinductive.html">Coinductive</a></td>
	<td><a href="../dependencies/coinductive/">13</a></td>
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	<td><a href="../entries/Deriving.html">Deriving</a></td>
	<td><a href="../dependencies/deriving/">13</a></td>
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	<tr>
	<td>7.</td>
	<td><a href="../entries/Polynomial_Factorization.html">Polynomial_Factorization</a></td>
	<td><a href="../dependencies/polynomial_factorization/">12</a></td>
	</tr>

	<tr>
	<td>8.</td>
	<td><a href="../entries/Regular-Sets.html">Regular-Sets</a></td>
	<td><a href="../dependencies/regular-sets/">12</a></td>
	</tr>

	<tr>
	<td>9.</td>
	<td><a href="../entries/Landau_Symbols.html">Landau_Symbols</a></td>
	<td><a href="../dependencies/landau_symbols/">11</a></td>
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	<tr>
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	<td><a href="../entries/Abstract-Rewriting.html">Abstract-Rewriting</a></td>
	<td><a href="../dependencies/abstract-rewriting/">10</a></td>
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	<tr>
	<td>11.</td>
	<td><a href="../entries/Automatic_Refinement.html">Automatic_Refinement</a></td>
	<td><a href="../dependencies/automatic_refinement/">10</a></td>
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	<td>12.</td>
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	<td><a href="../dependencies/native_word/">10</a></td>
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orithm','ADS_Functor','Matrices_for_ODEs','Gaussian_Integers','Lambert_W','Power_Sum_Polynomials','Attack_Trees','Banach_Steinhaus','Forcing','LTL_Normal_Form','Recursion-Addition','Irrational_Series_Erdos_Straus','Knuth_Bendix_Order','Nash_Williams','Smith_Normal_Form','Safe_Distance','Relational_Paths','Chandy_Lamport','Ordinal_Partitions','Amicable_Numbers','BirdKMP','Saturation_Framework_Extensions','Relational_Disjoint_Set_Forests','Inductive_Inference','PAC_Checker','Extended_Finite_State_Machine_Inference','Extended_Finite_State_Machines','Goedel_HFSet_Semantic','Goedel_HFSet_Semanticless','Goedel_Incompleteness','Robinson_Arithmetic','Syntax_Independent_Logic','Core_SC_DOM','DOM_Components','SC_DOM_Components','Shadow_DOM','Shadow_SC_DOM','Finite-Map-Extras','Physical_Quantities','AI_Planning_Languages_Semantics','Verified_SAT_Based_AI_Planning','CSP_RefTK','Isabelle_Marries_Dirac','Relational_Method','Interpreter_Optimizations','Relational_Minimum_Spanning_Trees','Topological_Semantics','Delta_System_Lemma','JinjaDCI','Hood_Melville_Queue','Blue_Eyes','IsaGeoCoq','Laws_of_Large_Numbers','Formal_Puiseux_Series','BTree','Sunflowers','Mereology','Hermite_Lindemann','Projective_Measurements','Modular_arithmetic_LLL_and_HNF_algorithms','Constructive_Cryptography_CM','Padic_Ints','Grothendieck_Schemes','IFC_Tracking','Progress_Tracking','GaleStewart_Games','BenOr_Kozen_Reif','Lifting_the_Exponent','Metalogic_ProofChecker','Regression_Test_Selection','Combinatorics_Words','Combinatorics_Words_Graph_Lemma','Combinatorics_Words_Lyndon','IMP_Compiler','Public_Announcement_Logic','MiniSail','Van_der_Waerden','SpecCheck','Finitely_Generated_Abelian_Groups','Schutz_Spacetime','Relational_Forests','Design_Theory','BD_Security_Compositional','CoCon','CoSMeDis','CoSMed','Fresh_Identifiers','Three_Circles','Logging_Independent_Anonymity','Cubic_Quartic_Equations','Dominance_CHK','CZH_Elementary_Categories','CZH_Foundations','CZH_Universal_Constructions','Conditional_Simplification','Conditional_Transfer_Rule','Intro_Dest_Elim','Types_To_Sets_Extension','Weighted_Path_Order','Complex_Bounded_Operators','FOL_Axiomatic','Virtual_Substitution','Correctness_Algebras','X86_Semantics','Belief_Revision','Registers','Szemeredi_Regularity','Factor_Algebraic_Polynomial','PAL','Real_Power','SimplifiedOntologicalArgument','Hahn_Jordan_Decomposition','Foundation_of_geometry','Van_Emde_Boas_Trees','Simplicial_complexes_and_boolean_functions','Regular_Tree_Relations','MDP-Algorithms','MDP-Rewards','Roth_Arithmetic_Progressions','Gale_Shapley','Hyperdual','Knights_Tour','Irrationals_From_THEBOOK','Actuarial_Mathematics','Median_Method','Interpolation_Polynomials_HOL_Algebra','FOL_Seq_Calc2','Youngs_Inequality','FO_Theory_Rewriting','LP_Duality','Quasi_Borel_Spaces','Equivalence_Relation_Enumeration','VYDRA_MDL','Eval_FO','Wetzels_Problem','Universal_Hash_Families','ResiduatedTransitionSystem','Transitive_Models','Independence_CH','Cotangent_PFD_Formula','FOL_Seq_Calc3','Ackermanns_not_PR','Dedekind_Real','Frequency_Moments','Prefix_Free_Code_Combinators','Sophomores_Dream','Digit_Expansions','Multiset_Ordering_NPC','Fishers_Inequality','Clique_and_Monotone_Circuits','Package_logic','Pluennecke_Ruzsa_Inequality','Combinable_Wands','Rewrite_Properties_Reduction','DPRM_Theorem','Finite_Fields','IsaNet','Boolos_Curious_Inference','Real_Time_Deque','IMP_Compiler_Reuse','Weighted_Arithmetic_Geometric_Mean','Commuting_Hermitian','Solidity','Nano_JSON','FSM_Tests','Involutions2Squares','SCC_Bloemen_Sequential','Number_Theoretic_Transform','Hales_Jewett','Khovanskii_Theorem','Separation_Logic_Unbounded','CRYSTALS-Kyber','Implicational_Logic','Risk_Free_Lending','Padic_Field','Stalnaker_Logic','Safe_Range_RC','Maximum_Segment_Sum','Undirected_Graph_Theory','Query_Optimization','PAPP_Impossibility','Balog_Szemeredi_Gowers','Combinatorial_Enumeration_Algorithms','Turans_Graph_Theorem','Kneser_Cauchy_Davenport','Sauer_Shelah_Lemma','CHERI-C_Memory_Model','AOT','Multitape_To_Singletape_TM','Boolos_Curious_Inference_Automated','Birkhoff_Finite_Distributive_Lattices','Propositional_Logic_Class','Quantifier_Elimination_Hybrid','Binary_Code_Imprimitive','Two_Generated_Word_Monoids_Intersection','Cook_Levin','Synthetic_Completeness','StrictOmegaCategories','HoareForDivergence','Suppes_Theorem','Given_Clause_Loops','ABY3_Protocols','CVP_Hardness','Edwards_Elliptic_Curves_Group','Probability_Inequality_Completeness','Rensets','Expander_Graphs','No_FTL_observers_Gen_Rel','CommCSL','DigitsInBase','Distributed_Distinct_Elements','HyperHoareLogic','TsirelsonBound','Simple_Clause_Learning','Schwartz_Zippel','MHComputation','Three_Squares','MLSS_Decision_Proc','Tree_Enumeration','Multirelations_Heterogeneous','Directed_Sets','Efficient_Weighted_Path_Order','Crypto_Standards','Zeckendorf','DCR-ExecutionEquivalence','Executable_Randomized_Algorithms','Gray_Codes']
	- const article_years_unique = ['2004','','','','','','','','','','','','','','2005','','','','','','','','2006','','','','','','','2007','','','','','','','','2008','','','','','','','','','','','','','','','2009','','','','','','','','','','','','2010','','','','','','','','','','','','','','','','','','','','','','2011','','','','','','','','','','','','','','','','','2012','','','','','','','','','','','','','','','','','','','','','','','','','2013','','','','','','','','','','','','','','','','','','','','','','','2014','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2015','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2016','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2017','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2018','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2019','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2020','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2021','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2022','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2023','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','']
	- const loc_articles = [839,1507,1542,1096,1058,2419,44195,205,142,1974,209,1110,3792,506,1141,3766,17615,3119,6430,1145,447,2081,1275,1583,1838,12832,13047,2685,1228,3556,4238,9647,2847,970,1740,79712,4738,3396,2185,10664,31022,6726,30332,180,793,1047,14413,2032,254,2220,5959,3463,799,1540,684,6654,8,2625,27490,264,32555,5025,4380,208,9538,447,2380,3399,606,6305,2060,840,713,1024,5632,1427,4078,2230,5997,22604,1602,3370,1587,2451,2591,260,1617,16,2937,6557,7804,6381,992,125,10130,332,239,1831,999,1756,4420,434,4461,11862,2835,8575,1045,408,2940,2612,37929,3207,1480,2612,3141,2580,23005,27588,2266,4107,7701,1249,260,5309,73,9729,663,6748,1512,4355,1249,1916,6214,4972,10086,7261,538,3830,4591,202,853,2003,5274,10320,1524,150,5292,706,2217,10739,1463,1958,3067,11487,1860,1190,1219,2174,1144,14860,2212,1957,166,10685,6420,572,590,465,1698,1909,4134,1403,2138,2280,1959,2467,220,4429,9396,5430,3999,4472,406,5935,1829,12828,3234,9486,4560,926,659,63,1653,2338,9085,822,1302,875,2563,627,945,1405,1296,7880,1922,90,28055,2879,2796,1116,5259,4863,8842,1356,5986,527,6650,2584,1772,5327,1092,4121,952,1064,2362,1089,2446,477,2074,3763,1954,710,16080,8267,908,1056,21228,9679,8665,3142,9156,695,435,13986,478,898,10416,2724,1162,741,405,495,498,838,3622,4616,6184,4106,8166,12091,3188,518,17581,2876,2418,5496,885,2453,1159,17387,576,703,5047,10687,4287,3811,5337,329,4108,1057,15095,3257,2582,553,8478,206,27172,8773,3324,2960,398,12811,9483,370,173,384,19029,2545,6119,3774,645,2415,4344,9355,20053,3963,3419,319,3174,169,19414,14667,541,2652,7059,7595,3898,3243,4510,855,2289,5027,1349,276,4339,1475,3482,7119,9662,592,1728,852,2194,12198,4382,590,13558,1695,4484,1644,694,835,737,3394,105,68,10492,1127,8628,11278,4135,378,4711,1200,2103,639,14042,3160,3930,4869,468,1531,5570,5721,1993,4205,478,4124,3147,3472,88,480,1090,1883,2194,250,10669,822,7432,3105,5166,2782,8208,8127,2324,6164,940,6514,992,489,810,8949,3434,338,854,493,4593,9455,15962,6462,10342,787,3757,3253,1820,8442,3278,12945,672,843,3402,3638,11570,13548,3734,6261,528,965,7711,1042,1221,5017,1390,2755,1622,2173,13357,805,9900,2667,541,1271,2825,4896,9770,2765,937,11918,2205,1743,7906,1386,449,685,1812,1235,3559,3578,2951,2218,1644,5182,4968,2767,17370,34354,3235,6018,1891,373,9969,30917,3018,3298,5306,4590,10509,986,15793,4437,11122,5556,3301,1264,2973,805,10229,2606,5735,3365,472,4879,3199,13317,987,787,4455,527,713,782,2321,2134,9936,2090,3734,5801,2350,4212,3808,176,1726,9694,7217,5069,5729,4562,14098,10292,6404,4470,1909,68341,2345,3937,3485,1706,3154,944,1033,597,370,691,764,2564,332,27828,24161,10943,3063,744,2353,1560,2537,1609,1242,1939,1338,12000,1034,1444,1902,2124,729,13319,3036,5074,9793,6176,1261,2899,2101,5904,12873,9015,4257,4725,426,11477,3546,1295,8100,16379,3523,7792,12655,15363,648,1761,2352,16434,1435,7700,3995,6542,4579,1788,3304,24539,745,365,26525,290,2404,15792,615,4039,3767,4958,17079,8442,15846,6578,4131,7221,1015,10340,384,9264,4094,13798,488,837,666,840,19737,1088,233,4435,7684,1962,4943,3003,11165,14638,5913,303,3758,817,1329,3157,40867,29090,22137,225,1053,169,17516,4304,18653,1306,21773,13533,2621,1324,5953,913,2961,133,2203,455,2619,6076,20605,1882,9607,7685,4271,1385,2302,2946,3108,264,6316,777,757,2594,656,9228,633,8738,502,7222,5937,337,829,18644,18383,14310,730,719,201,1411,4183,816,401,1312,1820,5168,2337,2169,654,2031,3338,8754,4622,7330,95,3657,1527,414,5785,16082,1041,53749,351,3175,1961,2144,1373,2325,3656,431,2239,39749,534,3097,144,2696,19190,2366,2961,1593,799,2051,419,4072,23190,1844,135,888,3535,8765,1937,1294,51217,2400,1105,2001,1036,6140,3226,4780,6881,7843,1739,7492,8371,8943,818,6726,3268,6198,9443,1034,193,3098,5281,3002,7219,618,1263,43708,292,157,3889,497]
	+ const no_articles = [14,22,29,37,52,64,86,103,128,151,208,253,326,396,455,511,577,650,722,759]
	+ const no_loc = [60600,96300,130800,238300,353000,435200,516300,567400,737300,824600,1036200,1216500,1600500,1856500,2129000,2448500,2848900,3364700,3729900,3952800]
	+ const no_authors = [14,11,6,6,10,6,24,11,17,16,37,20,63,32,26,38,30,43,32,23]
	+ const no_authors_series = [14,25,31,37,47,53,77,88,105,121,158,178,241,273,299,337,367,410,442,465]
	+ const all_articles = ['AVL-Trees','MiniML','Functional-Automata','BinarySearchTree','Lazy-Lists-II','Topology','Group-Ring-Module','Depth-First-Search','Compiling-Exceptions-Correctly','Completeness','Ramsey-Infinite','Verified-Prover','Integration','FileRefinement','Category','RSAPSS','Jinja','JiveDataStoreModel','DiskPaxos','GenClock','FFT','Ordinal','Cauchy','ClockSynchInst','FeatherweightJava','CoreC++','Flyspeck-Tame','Abstract-Hoare-Logics','HotelKeyCards','FOL-Fitting','POPLmark-deBruijn','Valuation','Fermat3_4','SumSquares','MuchAdoAboutTwo','JinjaThreads','Program-Conflict-Analysis','LinearQuantifierElim','NormByEval','BDD','Simpl','Recursion-Theory-I','SATSolverVerification','FunWithFunctions','ArrowImpossibilityGS','VolpanoSmith','Slicing','Huffman','FunWithTilings','SenSocialChoice','SIFPL','BytecodeLogicJmlTypes','Stream-Fusion','FinFun','CofGroups','SequentInvertibility','Ordinals_and_Cardinals','WorkerWrapper','HRB-Slicing','Perfect-Number-Thm','Collections','Tree-Automata','Presburger-Automata','DPT-SAT-Solver','Coinductive','List-Index','InformationFlowSlicing','InformationFlowSlicing_Inter','Free-Boolean-Algebra','Locally-Nameless-Sigma','Regular-Sets','Robbins-Conjecture','DataRefinementIBP','GraphMarkingIBP','Abstract-Rewriting','Matrix','Category2','Free-Groups','Statecharts','Polynomials','Lam-ml-Normalization','Binomial-Heaps','Binomial-Queues','Finger-Trees','Shivers-CFA','Marriage','Lower_Semicontinuous','RIPEMD-160-SPARK','LightweightJava','AutoFocus-Stream','List-Infinite','Nat-Interval-Logic','Transitive-Closure','General-Triangle','KBPs','Max-Card-Matching','Gauss-Jordan-Elim-Fun','Myhill-Nerode','LatticeProperties','MonoBoolTranAlgebra','PseudoHoops','Efficient-Mergesort','TLA','Markov_Models','Dijkstra_Shortest_Path','Refine_Monadic','Girth_Chromatic','Transitive-Closure-II','Abortable_Linearizable_Modules','Well_Quasi_Orders','Ordinary_Differential_Equations','Inductive_Confidentiality','Stuttering_Equivalence','Separation_Algebra','Circus','CCS','Pi_Calculus','Psi_Calculi','Tycon','PCF','Heard_Of','Impossible_Geometry','Datatype_Order_Generator','Possibilistic_Noninterference','Bondy','Tarskis_Geometry','Open_Induction','Separation_Logic_Imperative_HOL','Sqrt_Babylonian','Kleene_Algebra','Rank_Nullity_Theorem','Ribbon_Proofs','Launchbury','Nominal2','Containers','Graph_Theory','ShortestPath','Sort_Encodings','Koenigsberg_Friendship','Lehmer','Pratt_Certificate','IEEE_Floating_Point','Native_Word','Automatic_Refinement','Decreasing-Diagrams','GoedelGod','FocusStreamsCaseStudies','Coinductive_Languages','HereditarilyFinite','Incompleteness','Tail_Recursive_Functions','CryptoBasedCompositionalProperties','Sturm_Sequences','Featherweight_OCL','KAT_and_DRA','Relation_Algebra','Secondary_Sylow','Regex_Equivalence','Real_Impl','Affine_Arithmetic','Selection_Heap_Sort','Random_Graph_Subgraph_Threshold','Partial_Function_MR','AWN','Probabilistic_Noninterference','GPU_Kernel_PL','Discrete_Summation','Abstract_Completeness','HyperCTL','Bounded_Deducibility_Security','SIFUM_Type_Systems','Strong_Security','WHATandWHERE_Security','ComponentDependencies','Regular_Algebras','Noninterference_CSP','Roy_Floyd_Warshall','CAVA_Automata','CAVA_LTL_Modelchecker','Gabow_SCC','LTL_to_GBA','Promela','Boolean_Expression_Checkers','MSO_Regex_Equivalence','Pop_Refinement','Network_Security_Policy_Verification','Amortized_Complexity','pGCL','CISC-Kernel','Show','Splay_Tree','Skew_Heap','Special_Function_Bounds','VectorSpace','Gauss_Jordan','Priority_Queue_Braun','Jordan_Hoelder','Cayley_Hamilton','Sturm_Tarski','Imperative_Insertion_Sort','Certification_Monads','XML','RefinementReactive','Density_Compiler','Stream_Fusion_Code','Lifting_Definition_Option','AODV','UPF','UpDown_Scheme','Finite_Automata_HF','Echelon_Form','QR_Decomposition','Call_Arity','Deriving','Consensus_Refined','Trie','ConcurrentGC','ConcurrentIMP','Residuated_Lattices','Vickrey_Clarke_Groves','Probabilistic_System_Zoo','Formula_Derivatives','Dynamic_Tables','List_Interleaving','Multirelations','Noninterference_Generic_Unwinding','Noninterference_Ipurge_Unwinding','Derangements','Hermite','Akra_Bazzi','Landau_Symbols','Case_Labeling','Encodability_Process_Calculi','Rep_Fin_Groups','Noninterference_Inductive_Unwinding','Decreasing-Diagrams-II','Jordan_Normal_Form','LTL_to_DRA','Isabelle_Meta_Model','Parity_Game','Planarity_Certificates','TortoiseHare','Euler_Partition','Ergodic_Theory','Latin_Square','Card_Partitions','Algebraic_Numbers','Applicative_Lifting','Stern_Brocot','Descartes_Sign_Rule','Liouville_Numbers','Prime_Harmonic_Series','Triangle','Card_Number_Partitions','Matrix_Tensor','Knot_Theory','Polynomial_Factorization','Polynomial_Interpolation','Formal_SSA','List_Update','LTL','Cartan_FP','Timed_Automata','PropResPI','KAD','Noninterference_Sequential_Composition','CYK','ROBDD','No_FTL_observers','Groebner_Bases','Bell_Numbers_Spivey','SDS_Impossibility','Randomised_Social_Choice','MFMC_Countable','FLP','Incredible_Proof_Machine','Perron_Frobenius','Card_Equiv_Relations','Posix-Lexing','Tree_Decomposition','Word_Lib','Noninterference_Concurrent_Composition','Algebraic_VCs','Catalan_Numbers','Dependent_SIFUM_Type_Systems','Card_Multisets','Category3','Dependent_SIFUM_Refinement','IP_Addresses','Resolution_FOL','Rewriting_Z','Buildings','DFS_Framework','Pairing_Heap','Surprise_Paradox','Ptolemys_Theorem','Refine_Imperative_HOL','EdmondsKarp_Maxflow','InfPathElimination','Simple_Firewall','Routing','Stirling_Formula','Stone_Algebras','SuperCalc','Iptables_Semantics','Lambda_Free_RPOs','Allen_Calculus','Fisher_Yates','Lp','Chord_Segments','Berlekamp_Zassenhaus','SPARCv8','Source_Coding_Theorem','LOFT','Stable_Matching','Modal_Logics_for_NTS','Deep_Learning','Lambda_Free_KBOs','Nested_Multisets_Ordinals','Separata','Abs_Int_ITP2012','Complx','Paraconsistency','Proof_Strategy_Language','Twelvefold_Way','Concurrent_Ref_Alg','FOL_Harrison','Password_Authentication_Protocol','UPF_Firewall','E_Transcendental','Bertrands_Postulate','Minimal_SSA','Bernoulli','Key_Agreement_Strong_Adversaries','Stone_Relation_Algebras','Abstract_Soundness','Differential_Dynamic_Logic','Menger','Elliptic_Curves_Group_Law','Euler_MacLaurin','Comparison_Sort_Lower_Bound','Quick_Sort_Cost','Random_BSTs','Subresultants','Lazy_Case','Constructor_Funs','LocalLexing','Types_Tableaus_and_Goedels_God','MonoidalCategory','CryptHOL','Game_Based_Crypto','Monad_Normalisation','Monomorphic_Monad','Probabilistic_While','Floyd_Warshall','Dict_Construction','Security_Protocol_Refinement','Optics','Flow_Networks','Prpu_Maxflow','Buffons_Needle','PSemigroupsConvolution','Propositional_Proof_Systems','Stone_Kleene_Relation_Algebras','CRDT','Name_Carrying_Type_Inference','Minkowskis_Theorem','HOLCF-Prelude','Decl_Sem_Fun_PL','DynamicArchitectures','Stewart_Apollonius','LambdaMu','Orbit_Stabiliser','Root_Balanced_Tree','First_Welfare_Theorem','AnselmGod','PLM','Lowe_Ontological_Argument','Dirichlet_Series','Linear_Recurrences','Zeta_Function','Diophantine_Eqns_Lin_Hom','Count_Complex_Roots','Winding_Number_Eval','Buchi_Complementation','Transition_Systems_and_Automata','Kuratowski_Closure_Complement','Hybrid_Multi_Lane_Spatial_Logic','IMAP-CRDT','Stochastic_Matrices','Knuth_Morris_Pratt','BNF_Operations','Dirichlet_L','Mason_Stothers','Median_Of_Medians_Selection','Falling_Factorial_Sum','Taylor_Models','Green','Gromov_Hyperbolicity','Ordered_Resolution_Prover','LLL_Basis_Reduction','Error_Function','First_Order_Terms','LLL_Factorization','Treaps','Hoare_Time','Architectural_Design_Patterns','CakeML','Weight_Balanced_Trees','Fishburn_Impossibility','BNF_CC','VerifyThis2018','WebAssembly','Modular_Assembly_Kit_Security','OpSets','Monad_Memo_DP','AxiomaticCategoryTheory','Irrationality_J_Hancl','Probabilistic_Timed_Automata','Hidden_Markov_Models','Optimal_BST','Partial_Order_Reduction','Localization_Ring','Projective_Geometry','Pell','Neumann_Morgenstern_Utility','DiscretePricing','Minsky_Machines','Simplex','Budan_Fourier','Quaternions','Octonions','Aggregation_Algebras','Prime_Number_Theorem','Signature_Groebner','Symmetric_Polynomials','Pi_Transcendental','Factored_Transition_System_Bounding','Lambda_Free_EPO','Randomised_BSTs','Smooth_Manifolds','Epistemic_Logic','GewirthPGCProof','Generic_Deriving','Matroids','Auto2_HOL','Functional_Ordered_Resolution_Prover','Graph_Saturation','Order_Lattice_Props','Quantales','Transformer_Semantics','Constructive_Cryptography','Auto2_Imperative_HOL','Concurrent_Revisions','Core_DOM','Store_Buffer_Reduction','Higher_Order_Terms','IMP2','Farkas','List_Inversions','UTP','Universal_Turing_Machine','Probabilistic_Prime_Tests','Kruskal','Prime_Distribution_Elementary','Safe_OCL','QHLProver','Transcendence_Series_Hancl_Rucki','Binding_Syntax_Theory','LTL_Master_Theorem','HOL-CSP','Multi_Party_Computation','LambdaAuth','KD_Tree','Differential_Game_Logic','IMP2_Binary_Heap','Groebner_Macaulay','Nullstellensatz','Linear_Inequalities','Prim_Dijkstra_Simple','Priority_Search_Trees','Complete_Non_Orders','MFOTL_Monitor','CakeML_Codegen','FOL_Seq_Calc1','Szpilrajn','TESL_Language','Stellar_Quorums','IMO2019','C2KA_DistributedSystems','Linear_Programming','Laplace_Transform','Adaptive_State_Counting','Jacobson_Basic_Algebra','Fourier','Hybrid_Systems_VCs','Generic_Join','Clean','Sigma_Commit_Crypto','Aristotles_Assertoric_Syllogistic','VerifyThis2019','Isabelle_C','ZFC_in_HOL','Interval_Arithmetic_Word32','Generalized_Counting_Sort','Gauss_Sums','Complex_Geometry','Poincare_Disc','Poincare_Bendixson','Hybrid_Logic','Zeta_3_Irrational','Bicategory','Skip_Lists','Closest_Pair_Points','Approximation_Algorithms','Mersenne_Primes','Subset_Boolean_Algebras','Arith_Prog_Rel_Primes','VeriComp','Goodstein_Lambda','Hello_World','Relational-Incorrectness-Logic','Furstenberg_Topology','WOOT_Strong_Eventual_Consistency','Lucas_Theorem','Automated_Stateful_Protocol_Verification','Stateful_Protocol_Composition_and_Typing','MFODL_Monitor_Optimized','Saturation_Framework','Sliding_Window_Algorithm','ADS_Functor','Matrices_for_ODEs','Gaussian_Integers','Lambert_W','Power_Sum_Polynomials','Attack_Trees','Banach_Steinhaus','Forcing','LTL_Normal_Form','Recursion-Addition','Irrational_Series_Erdos_Straus','Knuth_Bendix_Order','Nash_Williams','Smith_Normal_Form','Safe_Distance','Relational_Paths','Chandy_Lamport','Ordinal_Partitions','Amicable_Numbers','BirdKMP','Saturation_Framework_Extensions','Relational_Disjoint_Set_Forests','Inductive_Inference','PAC_Checker','Extended_Finite_State_Machine_Inference','Extended_Finite_State_Machines','Goedel_HFSet_Semantic','Goedel_HFSet_Semanticless','Goedel_Incompleteness','Robinson_Arithmetic','Syntax_Independent_Logic','Core_SC_DOM','DOM_Components','SC_DOM_Components','Shadow_DOM','Shadow_SC_DOM','Finite-Map-Extras','Physical_Quantities','AI_Planning_Languages_Semantics','Verified_SAT_Based_AI_Planning','CSP_RefTK','Isabelle_Marries_Dirac','Relational_Method','Interpreter_Optimizations','Relational_Minimum_Spanning_Trees','Topological_Semantics','Delta_System_Lemma','JinjaDCI','Hood_Melville_Queue','Blue_Eyes','IsaGeoCoq','Laws_of_Large_Numbers','Formal_Puiseux_Series','BTree','Sunflowers','Mereology','Hermite_Lindemann','Projective_Measurements','Modular_arithmetic_LLL_and_HNF_algorithms','Constructive_Cryptography_CM','Padic_Ints','Grothendieck_Schemes','IFC_Tracking','Progress_Tracking','GaleStewart_Games','BenOr_Kozen_Reif','Lifting_the_Exponent','Metalogic_ProofChecker','Regression_Test_Selection','Combinatorics_Words','Combinatorics_Words_Graph_Lemma','Combinatorics_Words_Lyndon','IMP_Compiler','Public_Announcement_Logic','MiniSail','Van_der_Waerden','SpecCheck','Finitely_Generated_Abelian_Groups','Schutz_Spacetime','Relational_Forests','Design_Theory','BD_Security_Compositional','CoCon','CoSMeDis','CoSMed','Fresh_Identifiers','Three_Circles','Logging_Independent_Anonymity','Cubic_Quartic_Equations','Dominance_CHK','CZH_Elementary_Categories','CZH_Foundations','CZH_Universal_Constructions','Conditional_Simplification','Conditional_Transfer_Rule','Intro_Dest_Elim','Types_To_Sets_Extension','Weighted_Path_Order','Complex_Bounded_Operators','FOL_Axiomatic','Virtual_Substitution','Correctness_Algebras','X86_Semantics','Belief_Revision','Registers','Szemeredi_Regularity','Factor_Algebraic_Polynomial','PAL','Real_Power','SimplifiedOntologicalArgument','Hahn_Jordan_Decomposition','Foundation_of_geometry','Van_Emde_Boas_Trees','Simplicial_complexes_and_boolean_functions','Regular_Tree_Relations','MDP-Algorithms','MDP-Rewards','Roth_Arithmetic_Progressions','Gale_Shapley','Hyperdual','Knights_Tour','Irrationals_From_THEBOOK','Actuarial_Mathematics','Median_Method','Interpolation_Polynomials_HOL_Algebra','FOL_Seq_Calc2','Youngs_Inequality','FO_Theory_Rewriting','LP_Duality','Quasi_Borel_Spaces','Equivalence_Relation_Enumeration','VYDRA_MDL','Eval_FO','Wetzels_Problem','Universal_Hash_Families','ResiduatedTransitionSystem','Transitive_Models','Independence_CH','Cotangent_PFD_Formula','FOL_Seq_Calc3','Ackermanns_not_PR','Dedekind_Real','Frequency_Moments','Prefix_Free_Code_Combinators','Sophomores_Dream','Digit_Expansions','Multiset_Ordering_NPC','Fishers_Inequality','Clique_and_Monotone_Circuits','Package_logic','Pluennecke_Ruzsa_Inequality','Combinable_Wands','Rewrite_Properties_Reduction','DPRM_Theorem','Finite_Fields','IsaNet','Boolos_Curious_Inference','Real_Time_Deque','IMP_Compiler_Reuse','Weighted_Arithmetic_Geometric_Mean','Commuting_Hermitian','Solidity','Nano_JSON','FSM_Tests','Involutions2Squares','SCC_Bloemen_Sequential','Number_Theoretic_Transform','Hales_Jewett','Khovanskii_Theorem','Separation_Logic_Unbounded','CRYSTALS-Kyber','Implicational_Logic','Risk_Free_Lending','Padic_Field','Stalnaker_Logic','Safe_Range_RC','Maximum_Segment_Sum','Undirected_Graph_Theory','Query_Optimization','PAPP_Impossibility','Balog_Szemeredi_Gowers','Combinatorial_Enumeration_Algorithms','Turans_Graph_Theorem','Kneser_Cauchy_Davenport','Sauer_Shelah_Lemma','CHERI-C_Memory_Model','AOT','Multitape_To_Singletape_TM','Boolos_Curious_Inference_Automated','Birkhoff_Finite_Distributive_Lattices','Propositional_Logic_Class','Quantifier_Elimination_Hybrid','Binary_Code_Imprimitive','Two_Generated_Word_Monoids_Intersection','Cook_Levin','Synthetic_Completeness','StrictOmegaCategories','HoareForDivergence','Suppes_Theorem','Given_Clause_Loops','ABY3_Protocols','CVP_Hardness','Edwards_Elliptic_Curves_Group','Probability_Inequality_Completeness','Rensets','Expander_Graphs','No_FTL_observers_Gen_Rel','CommCSL','DigitsInBase','Distributed_Distinct_Elements','HyperHoareLogic','TsirelsonBound','Simple_Clause_Learning','Schwartz_Zippel','MHComputation','Three_Squares','MLSS_Decision_Proc','Tree_Enumeration','Multirelations_Heterogeneous','Directed_Sets','Efficient_Weighted_Path_Order','Crypto_Standards','Zeckendorf','DCR-ExecutionEquivalence','Executable_Randomized_Algorithms','Gray_Codes','Earley_Parser','Quantales_Converse','Polygonal_Number_Theorem']
	+ const article_years_unique = ['2004','','','','','','','','','','','','','','2005','','','','','','','','2006','','','','','','','2007','','','','','','','','2008','','','','','','','','','','','','','','','2009','','','','','','','','','','','','2010','','','','','','','','','','','','','','','','','','','','','','2011','','','','','','','','','','','','','','','','','2012','','','','','','','','','','','','','','','','','','','','','','','','','2013','','','','','','','','','','','','','','','','','','','','','','','2014','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2015','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2016','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2017','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2018','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2019','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2020','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2021','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2022','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','2023','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','','']
	+ const loc_articles = [839,1507,1542,1096,1058,2419,44195,205,142,1974,209,1110,3792,506,1141,3766,17615,3119,6430,1145,447,2081,1275,1583,1838,12832,13047,2685,1228,3556,4238,9647,2847,970,1740,79712,4738,3396,2185,10664,31022,6726,30332,180,793,1047,14413,2032,254,2220,5959,3463,799,1540,684,6654,8,2625,27490,264,32555,5025,4380,208,9538,447,2380,3399,606,6305,2060,840,713,1024,5632,1427,4078,2230,5997,22604,1602,3370,1587,2451,2591,260,1617,16,2937,6557,7804,6381,992,125,10130,332,239,1831,999,1756,4420,434,4461,11862,2835,8575,1045,408,2940,2612,37929,3207,1480,2612,3141,2580,23005,27588,2266,4107,7701,1249,260,5309,73,9729,663,6748,1512,4355,1249,1916,6214,4972,10086,7261,538,3830,4591,202,853,2003,5274,10320,1524,150,5292,706,2217,10739,1463,1958,3067,11487,1860,1190,1219,2174,1144,14860,2212,1957,166,10685,6420,572,590,465,1698,1909,4134,1403,2138,2280,1959,2467,220,4429,9396,5430,3999,4472,406,5935,1829,12828,3234,9486,4560,926,659,63,1653,2338,9085,822,1302,875,2563,627,945,1405,1296,7880,1922,90,28055,2879,2796,1116,5259,4863,8842,1356,5986,527,6650,2584,1772,5327,1092,4121,952,1064,2362,1089,2446,477,2074,3763,1954,710,16080,8267,908,1056,21228,9679,8665,3142,9156,695,435,13986,478,898,10416,2724,1162,741,405,495,498,838,3622,4616,6184,4106,8166,12091,3188,518,17581,2876,2418,5496,885,2453,1159,17387,576,703,5047,10687,4287,3811,5337,329,4108,1057,15095,3257,2582,553,8478,206,27172,8773,3324,2960,398,12811,9483,370,173,384,19029,2545,6119,3774,645,2415,4344,9355,20053,3963,3419,319,3174,169,19414,14667,541,2652,7059,7595,3898,3243,4510,855,2289,5027,1349,276,4339,1475,3482,7119,9662,592,1728,852,2194,12198,4382,590,13558,1695,4484,1644,694,835,737,3394,105,68,10492,1127,8628,11278,4135,378,4711,1200,2103,639,14042,3160,3930,4869,468,1531,5570,5721,1993,4205,478,4124,3147,3472,88,480,1090,1883,2194,250,10669,822,7432,3105,5166,2782,8208,8127,2324,6164,940,6514,992,489,810,8949,3434,338,854,493,4593,9455,15962,6462,10342,787,3757,3253,1820,8442,3278,12945,672,843,3402,3638,11570,13548,3734,6261,528,965,7711,1042,1221,5017,1390,2755,1622,2173,13357,805,9900,2667,541,1271,2825,4896,9770,2765,937,11918,2205,1743,7906,1386,449,685,1812,1235,3559,3578,2951,2218,1644,5182,4968,2767,17370,34354,3235,6018,1891,373,9969,30917,3018,3298,5306,4590,10509,986,15793,4437,11122,5556,3301,1264,2973,805,10229,2606,5735,3365,472,4879,3199,13317,987,787,4455,527,713,782,2321,2134,9936,2090,3734,5801,2350,4212,3808,176,1726,9694,7217,5069,5729,4562,14098,10292,6404,4470,1909,68341,2345,3937,3485,1706,3154,944,1033,597,370,691,764,2564,332,27828,24161,10943,3063,744,2353,1560,2537,1609,1242,1939,1338,12000,1034,1444,1902,2124,729,13319,3036,5074,9793,6176,1261,2899,2101,5904,12873,9015,4257,4725,426,11477,3546,1295,8100,16379,3523,7792,12655,15363,648,1761,2352,16434,1435,7700,3995,6542,4579,1788,3304,24539,745,365,26525,290,2404,15792,615,4039,3767,4958,17079,8442,15846,6578,4131,7221,1015,10340,384,9264,4094,13798,488,837,666,840,19737,1088,233,4435,7684,1962,4943,3003,11165,14638,5913,303,3758,817,1329,3157,40867,29090,22137,225,1053,169,17516,4304,18653,1306,21773,13533,2621,1324,5953,913,2961,133,2203,455,2619,6076,20605,1882,9607,7685,4271,1385,2302,2946,3108,264,6316,777,757,2594,656,9228,633,8738,502,7222,5937,337,829,18644,18383,14310,730,719,201,1411,4183,816,401,1312,1820,5168,2337,2169,654,2031,3338,8754,4622,7330,95,3657,1527,414,5785,16082,1041,53749,351,3175,1961,2144,1373,2325,3656,431,2239,39749,534,3097,144,2696,19190,2366,2961,1593,799,2051,419,4072,23190,1844,135,888,3535,8765,1937,1294,51217,2400,1105,2001,1036,6140,3226,4780,6881,7843,1739,7492,8371,8943,818,6726,3268,6198,9443,1034,193,3098,5281,3002,7219,618,1263,43708,292,157,3889,497,6003,1765,1936]
	</script>
	<h4>Growth in number of entries:</h4>
	<script src="../js/Chart.js"></script>
	<div class="chart">
	<canvas id="num-articles-canvas"></canvas>
	</div>
	<script>
	new Chart(document.getElementById("num-articles-canvas"), {
	type: 'bar',
	data: {
	labels: years,
	datasets: [{
	label: 'Size of the AFP in # of entries',
	data: no_articles,
	backgroundColor: "rgba(46, 45, 78, 1)"
	}],
	},
	options: {
	responsive: true,
	maintainAspectRatio: false,
	scales: {
	yAxes: [{
	ticks: {
	beginAtZero: true
	}
	}]
	},
	}
	})
	</script>

	<h4>Growth in lines of code:</h4>
	<div class="chart">
	<canvas id="loc-years-canvas"></canvas>
	</div>
	<script>
	new Chart(document.getElementById("loc-years-canvas"), {
	type: 'bar',
	data: {
	labels: years,
	datasets: [{
	label: 'size of the AFP in lines of code',
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	scales: {
	yAxes: [{
	ticks: {
	beginAtZero: true
	}
	}]
	},
	}
	})
	</script>

	<h4>Growth in number of authors:</h4>
	<div class="chart">
	<canvas id="author-years-canvas"></canvas>
	</div>
	<script>
	new Chart(document.getElementById("author-years-canvas"), {
	type: 'bar',
	data: {
	labels: years,
	datasets: [{
	label: 'new authors per year',
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	backgroundColor: "rgba(101, 99, 136, 1)"
	},
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	data: no_authors_series,
	backgroundColor: "rgba(0, 15, 48, 1)"
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	},
	options: {
	responsive: true,
	maintainAspectRatio: false,
	scales: {
	yAxes: [{
	ticks: {
	beginAtZero: true
	}
	}]
	},
	}
	})
	</script>

	<h4>Size of entries:</h4>
	<div class="chart">
	<canvas id="loc-articles-canvas"></canvas>
	</div>
	<script>
	new Chart(document.getElementById("loc-articles-canvas"), {
	type: 'bar',
	data: {
	labels: article_years_unique,
	datasets: [{
	label: 'loc per article',
	data: loc_articles,
	backgroundColor: 'rgba(101, 99, 136, 1)'
	}]
	},
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	responsive: true,
	maintainAspectRatio: false,
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	</td>
	</tr>

	</tbody>
	</table>

	</div>
	</div>
	</div>
	</body>

	</html>
	\ No newline at end of file
	diff --git a/web/submission/index.html b/web/submission/index.html
	--- a/web/submission/index.html
	+++ b/web/submission/index.html
	@@ -1,126 +1,127 @@
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	<meta property="og:description" content="Submission Guidelines The submission must follow the following Isabelle style rules. For additional guidelines on Isabelle proofs, also see the this guide (feel free to follow all of these; only the below are mandatory). Technical details about the submission process and the format of the submission are explained on the submission site.
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	No use of the commands sorry or back. Instantiations must not use Isabelle-generated names such as xa — use Isar, the subgoal command or rename_tac to avoid such names."/>


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	<h1 >
	<span class='first'>E</span>ntry <span class='first'>S</span>ubmission</h1>
	<div>


	</div>
	</header><div><div>
	<h2 id="submission-guidelines">Submission Guidelines</h2>
	<p><strong>The submission must follow the following Isabelle style rules.</strong> For additional guidelines on Isabelle proofs, also see the this <a href="https://proofcraft.org/blog/isabelle-style.html">guide</a> (feel free to follow all of these; only the below are mandatory). <strong>Technical details about the submission process and the format of the submission are explained on the submission site.</strong></p>
	<ul>
	<li>No use of the commands <code>sorry</code> or <code>back</code>.</li>
	<li>Instantiations must not use Isabelle-generated names such as <code>xa</code> — use Isar, the <code>subgoal</code> command or <code>rename_tac</code> to avoid such names.</li>
	<li>No use of the command <code>smt_oracle</code>.</li>
	<li>If your theories contain calls to <code>nitpick</code>, <code>quickcheck</code>, or <code>nunchaku</code> those calls must include the <code>expect</code> parameter. Alternatively the <code>expect</code> parameter must be set globally via, e.g. <code>nitpick_params</code>.</li>
	<li><code>apply</code> scripts should be indented by subgoal as in the Isabelle distribution. If an <code>apply</code> command is applied to a state with <code>n+1</code> subgoals, it must be indented by <code>n</code> spaces relative to the first <code>apply</code> in the sequence.</li>
	<li>Only named lemmas should carry attributes such as <code>[simp]</code>.</li>
	<li>We prefer structured Isar proofs over apply style, but do not mandate them.</li>
	<li>If there are proof steps that take significant time, i.e. longer than roughly 1 min, please add a short comment to that step, so maintainers will know what to expect.</li>
	<li>The entry must contain a ROOT file with one session that has the name of the entry. We strongly encourage precisely one session per entry, but exceptions can be made. All sessions must be in chapter AFP, and all theory files of the submission must be contained in at least one session. See also the example <a href="https://foss.heptapod.net/isa-afp/afp-2020/-/blob/branch/default/thys/Example-Submission/ROOT">ROOT</a> file in the <a href="../entries/Example-Submission.html">Example submission</a>.</li>
	<li>The entry should cite all sources that the theories are based on, for example textbooks or research articles containing informal versions of the proofs.</li>
	+<li>All sessions must have a timeout after which they are assumed to be in a non-terminating state. Timeouts may be no finer than five-minute increments, i.e. must be divisible by 300.</li>
	</ul>
	<p>Your submission must contain an abstract to be displayed on the web site – usually this will be the same as the abstract of your proof document in the root.tex file. You can use LaTeX formulae in this web site abstract, either inline formulae in the form $a+b$ or $a+b$ or display formulae in the form $$a + b$$ or \[a + b\]. Other occurrences of these characters must be escaped (e.g. \$ or \\(). Note that LaTeX in the title of an entry is <em>not</em> allowed. Most basic LaTeX functionality should be supported. For details on what parts of LaTeX are supported, see the <a href="https://docs.mathjax.org/en/v2.7-latest/tex.html">MathJax documentation.</a></p>
	<p>It is possible and encouraged to build on other archive entries in your submission. There is a standardised way to <a href="../help">refer to other AFP entries</a> in your theories.</p>
	<h2 id="submission-form">Submission Form</h2>
	<p><a href="../webapp/submit">Submission form</a></p>
	<p>Your submission will be refereed and you will receive notification as soon as possible. If accepted, you must agree to maintain your archive entry or nominate someone else to maintain it. The Isabelle development team will assist with maintenance, but it does not have the resources to fully maintain the complete archive.</p>
	<p>If you have questions regarding your submission, please email <a href="mailto:afp-submit@in.tum.de">afp-submit@in.tum.de</a>. If you need help with Isabelle, please use the <a href="mailto:isabelle-users@cl.cam.ac.uk">isabelle-users@cl.cam.ac.uk</a> mailing list. It is always a good idea to <a href="https://lists.cam.ac.uk/mailman/listinfo/cl-isabelle-users">subscribe</a>.</p>
	<h1 id="updating-entries">Updating Entries</h1>
	<h2 id="change">Change</h2>
	<p>The Archive of Formal Proofs is an online resource and therefore more dynamic than a normal scientific journal. Existing entries can and do evolve and can also be updated significantly by their authors.</p>
	<p>This conflicts with the purpose of archiving and preserving entries as they have been submitted and with the purpose of providing a clear and simple interface to readers.</p>
	<p>The AFP deals with this by synchronizing such updates with Isabelle releases:</p>
	<ul>
	<li>The entries released and visible on the main site are always working with the most recent stable Isabelle version and do not change.</li>
	<li>In the background, the archive maintainers evolve all entries to be up to date with the current Isabelle development version. Authors can contribute changes to this version which is available as a <a href="https://foss.heptapod.net/isa-afp/afp-devel/">Heptapod mercurial repository</a> or as tar.gz package on the <a href="../download">download page</a>.</li>
	<li>When a new Isabelle version is released, the above mentioned development version of AFP is frozen and turns into the main version displayed on the front page. Older versions (including the original submission) of all entries are archived and remain accessible.</li>
	</ul>
	<p>Significant changes of an entry should be recorded in the metadata of the entry using the keyword “extra-history”. The resulting web page should look <a href="https://www.isa-afp.org/entries/JinjaThreads.html">something like this</a>.</p>
	<h2 id="monotonicity">Monotonicity</h2>
	<p>Updating an entry should be mostly monotone: you add new material, but you do not modify existing material in a major way. Ideally, entries (by other people) that build on yours should not be affected. Otherwise you have to liaise with them first. If you intend to carry out major non-monotone changes, you will need to submit a completely new entry (with a description of how it relates to the old one). This should be required only very rarely: AFP entries should be mature enough not to require major changes to their interface (i.e. the main functions and theorems provided).</p>
	<p>Major monotone changes, e.g. adding a new concept rather than more results on existing concepts, may also call for a new entry, but one that builds on the existing one. This depends on how you would like to organize your entries.</p>
	<h2 id="if-you-are-an-author">If you are an Author</h2>
	<p>The above means that if you are an author and would like to provide a new, better version of your AFP entry, you can do so.</p>
	<p>To achieve this, you should base your changes on the <a href="https:/foss.heptapod.net/isa-afp/afp-devel/">mercurial development version</a> of your AFP entry and test it against the current <a href="https://isabelle.in.tum.de/devel/">Isabelle development version</a>.</p>
	<p>If you would like to get write access to your entry in the mercurial repository or if you need assistance, please contact the <a href="../about#editors">editors</a>.</p>

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	</html>
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	diff --git a/web/theories/earley_parser/index.html b/web/theories/earley_parser/index.html
	new file mode 100644
	--- /dev/null
	+++ b/web/theories/earley_parser/index.html
	@@ -0,0 +1,90 @@
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	+ <a id="Limit" href="https://www.isa-afp.org/browser_info/current/AFP/Earley_Parser/Limit.html"><h2>Limit</h2></a>
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	+ <a id="Earley_Recognizer" href="https://www.isa-afp.org/browser_info/current/AFP/Earley_Parser/Earley_Recognizer.html"><h2>Earley_Recognizer</h2></a>
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	\ No newline at end of file
	diff --git a/web/theories/example-submission/index.html b/web/theories/example-submission/index.html
	new file mode 100644
	--- /dev/null
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	\ No newline at end of file
	diff --git a/web/theories/index.xml b/web/theories/index.xml
	--- a/web/theories/index.xml
	+++ b/web/theories/index.xml
	@@ -1,7039 +1,7066 @@
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	<title>Abstract-Rewriting</title>
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	<item>
	<title>ABY3_Protocols</title>
	<link>/theories/aby3_protocols/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aby3_protocols/</guid>
	<description></description>
	</item>

	<item>
	<title>Ackermanns_not_PR</title>
	<link>/theories/ackermanns_not_pr/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ackermanns_not_pr/</guid>
	<description></description>
	</item>

	<item>
	<title>Actuarial_Mathematics</title>
	<link>/theories/actuarial_mathematics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/actuarial_mathematics/</guid>
	<description></description>
	</item>

	<item>
	<title>Adaptive_State_Counting</title>
	<link>/theories/adaptive_state_counting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/adaptive_state_counting/</guid>
	<description></description>
	</item>

	<item>
	<title>ADS_Functor</title>
	<link>/theories/ads_functor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ads_functor/</guid>
	<description></description>
	</item>

	<item>
	<title>Affine_Arithmetic</title>
	<link>/theories/affine_arithmetic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/affine_arithmetic/</guid>
	<description></description>
	</item>

	<item>
	<title>Aggregation_Algebras</title>
	<link>/theories/aggregation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aggregation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>AI_Planning_Languages_Semantics</title>
	<link>/theories/ai_planning_languages_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ai_planning_languages_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Akra_Bazzi</title>
	<link>/theories/akra_bazzi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/akra_bazzi/</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic_Numbers</title>
	<link>/theories/algebraic_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/algebraic_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic_VCs</title>
	<link>/theories/algebraic_vcs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/algebraic_vcs/</guid>
	<description></description>
	</item>

	<item>
	<title>Allen_Calculus</title>
	<link>/theories/allen_calculus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/allen_calculus/</guid>
	<description></description>
	</item>

	<item>
	<title>Amicable_Numbers</title>
	<link>/theories/amicable_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/amicable_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Amortized_Complexity</title>
	<link>/theories/amortized_complexity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/amortized_complexity/</guid>
	<description></description>
	</item>

	<item>
	<title>AnselmGod</title>
	<link>/theories/anselmgod/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/anselmgod/</guid>
	<description></description>
	</item>

	<item>
	<title>AODV</title>
	<link>/theories/aodv/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aodv/</guid>
	<description></description>
	</item>

	<item>
	<title>AOT</title>
	<link>/theories/aot/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aot/</guid>
	<description></description>
	</item>

	<item>
	<title>Applicative_Lifting</title>
	<link>/theories/applicative_lifting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/applicative_lifting/</guid>
	<description></description>
	</item>

	<item>
	<title>Approximation_Algorithms</title>
	<link>/theories/approximation_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/approximation_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Architectural_Design_Patterns</title>
	<link>/theories/architectural_design_patterns/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/architectural_design_patterns/</guid>
	<description></description>
	</item>

	<item>
	<title>Aristotles_Assertoric_Syllogistic</title>
	<link>/theories/aristotles_assertoric_syllogistic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/aristotles_assertoric_syllogistic/</guid>
	<description></description>
	</item>

	<item>
	<title>Arith_Prog_Rel_Primes</title>
	<link>/theories/arith_prog_rel_primes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/arith_prog_rel_primes/</guid>
	<description></description>
	</item>

	<item>
	<title>ArrowImpossibilityGS</title>
	<link>/theories/arrowimpossibilitygs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/arrowimpossibilitygs/</guid>
	<description></description>
	</item>

	<item>
	<title>Attack_Trees</title>
	<link>/theories/attack_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/attack_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2_HOL</title>
	<link>/theories/auto2_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/auto2_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Auto2_Imperative_HOL</title>
	<link>/theories/auto2_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/auto2_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>AutoFocus-Stream</title>
	<link>/theories/autofocus-stream/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/autofocus-stream/</guid>
	<description></description>
	</item>

	<item>
	<title>Automated_Stateful_Protocol_Verification</title>
	<link>/theories/automated_stateful_protocol_verification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/automated_stateful_protocol_verification/</guid>
	<description></description>
	</item>

	<item>
	<title>Automatic_Refinement</title>
	<link>/theories/automatic_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/automatic_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>AVL-Trees</title>
	<link>/theories/avl-trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/avl-trees/</guid>
	<description></description>
	</item>

	<item>
	<title>AWN</title>
	<link>/theories/awn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/awn/</guid>
	<description></description>
	</item>

	<item>
	<title>AxiomaticCategoryTheory</title>
	<link>/theories/axiomaticcategorytheory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/axiomaticcategorytheory/</guid>
	<description></description>
	</item>

	<item>
	<title>Balog_Szemeredi_Gowers</title>
	<link>/theories/balog_szemeredi_gowers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/balog_szemeredi_gowers/</guid>
	<description></description>
	</item>

	<item>
	<title>Banach_Steinhaus</title>
	<link>/theories/banach_steinhaus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/banach_steinhaus/</guid>
	<description></description>
	</item>

	<item>
	<title>BD_Security_Compositional</title>
	<link>/theories/bd_security_compositional/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bd_security_compositional/</guid>
	<description></description>
	</item>

	<item>
	<title>BDD</title>
	<link>/theories/bdd/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bdd/</guid>
	<description></description>
	</item>

	<item>
	<title>Belief_Revision</title>
	<link>/theories/belief_revision/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/belief_revision/</guid>
	<description></description>
	</item>

	<item>
	<title>Bell_Numbers_Spivey</title>
	<link>/theories/bell_numbers_spivey/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bell_numbers_spivey/</guid>
	<description></description>
	</item>

	<item>
	<title>BenOr_Kozen_Reif</title>
	<link>/theories/benor_kozen_reif/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/benor_kozen_reif/</guid>
	<description></description>
	</item>

	<item>
	<title>Berlekamp_Zassenhaus</title>
	<link>/theories/berlekamp_zassenhaus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/berlekamp_zassenhaus/</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli</title>
	<link>/theories/bernoulli/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bernoulli/</guid>
	<description></description>
	</item>

	<item>
	<title>Bertrands_Postulate</title>
	<link>/theories/bertrands_postulate/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bertrands_postulate/</guid>
	<description></description>
	</item>

	<item>
	<title>Bicategory</title>
	<link>/theories/bicategory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bicategory/</guid>
	<description></description>
	</item>

	<item>
	<title>Binary_Code_Imprimitive</title>
	<link>/theories/binary_code_imprimitive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binary_code_imprimitive/</guid>
	<description></description>
	</item>

	<item>
	<title>BinarySearchTree</title>
	<link>/theories/binarysearchtree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binarysearchtree/</guid>
	<description></description>
	</item>

	<item>
	<title>Binding_Syntax_Theory</title>
	<link>/theories/binding_syntax_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binding_syntax_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial-Heaps</title>
	<link>/theories/binomial-heaps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binomial-heaps/</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial-Queues</title>
	<link>/theories/binomial-queues/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/binomial-queues/</guid>
	<description></description>
	</item>

	<item>
	<title>BirdKMP</title>
	<link>/theories/birdkmp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/birdkmp/</guid>
	<description></description>
	</item>

	<item>
	<title>Birkhoff_Finite_Distributive_Lattices</title>
	<link>/theories/birkhoff_finite_distributive_lattices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/birkhoff_finite_distributive_lattices/</guid>
	<description></description>
	</item>

	<item>
	<title>Blue_Eyes</title>
	<link>/theories/blue_eyes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/blue_eyes/</guid>
	<description></description>
	</item>

	<item>
	<title>BNF_CC</title>
	<link>/theories/bnf_cc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bnf_cc/</guid>
	<description></description>
	</item>

	<item>
	<title>BNF_Operations</title>
	<link>/theories/bnf_operations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bnf_operations/</guid>
	<description></description>
	</item>

	<item>
	<title>Bondy</title>
	<link>/theories/bondy/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bondy/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolean_Expression_Checkers</title>
	<link>/theories/boolean_expression_checkers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolean_expression_checkers/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos_Curious_Inference</title>
	<link>/theories/boolos_curious_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolos_curious_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Boolos_Curious_Inference_Automated</title>
	<link>/theories/boolos_curious_inference_automated/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/boolos_curious_inference_automated/</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded_Deducibility_Security</title>
	<link>/theories/bounded_deducibility_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bounded_deducibility_security/</guid>
	<description></description>
	</item>

	<item>
	<title>BTree</title>
	<link>/theories/btree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/btree/</guid>
	<description></description>
	</item>

	<item>
	<title>Buchi_Complementation</title>
	<link>/theories/buchi_complementation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buchi_complementation/</guid>
	<description></description>
	</item>

	<item>
	<title>Budan_Fourier</title>
	<link>/theories/budan_fourier/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/budan_fourier/</guid>
	<description></description>
	</item>

	<item>
	<title>Buffons_Needle</title>
	<link>/theories/buffons_needle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buffons_needle/</guid>
	<description></description>
	</item>

	<item>
	<title>Buildings</title>
	<link>/theories/buildings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/buildings/</guid>
	<description></description>
	</item>

	<item>
	<title>BytecodeLogicJmlTypes</title>
	<link>/theories/bytecodelogicjmltypes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/bytecodelogicjmltypes/</guid>
	<description></description>
	</item>

	<item>
	<title>C2KA_DistributedSystems</title>
	<link>/theories/c2ka_distributedsystems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/c2ka_distributedsystems/</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML</title>
	<link>/theories/cakeml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cakeml/</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML_Codegen</title>
	<link>/theories/cakeml_codegen/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cakeml_codegen/</guid>
	<description></description>
	</item>

	<item>
	<title>Call_Arity</title>
	<link>/theories/call_arity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/call_arity/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Equiv_Relations</title>
	<link>/theories/card_equiv_relations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_equiv_relations/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Multisets</title>
	<link>/theories/card_multisets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_multisets/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Number_Partitions</title>
	<link>/theories/card_number_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_number_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Card_Partitions</title>
	<link>/theories/card_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/card_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Cartan_FP</title>
	<link>/theories/cartan_fp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cartan_fp/</guid>
	<description></description>
	</item>

	<item>
	<title>Case_Labeling</title>
	<link>/theories/case_labeling/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/case_labeling/</guid>
	<description></description>
	</item>

	<item>
	<title>Catalan_Numbers</title>
	<link>/theories/catalan_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/catalan_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Category</title>
	<link>/theories/category/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category/</guid>
	<description></description>
	</item>

	<item>
	<title>Category2</title>
	<link>/theories/category2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category2/</guid>
	<description></description>
	</item>

	<item>
	<title>Category3</title>
	<link>/theories/category3/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/category3/</guid>
	<description></description>
	</item>

	<item>
	<title>Cauchy</title>
	<link>/theories/cauchy/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cauchy/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Automata</title>
	<link>/theories/cava_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Base</title>
	<link>/theories/cava_base/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_base/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_LTL_Modelchecker</title>
	<link>/theories/cava_ltl_modelchecker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_ltl_modelchecker/</guid>
	<description></description>
	</item>

	<item>
	<title>CAVA_Setup</title>
	<link>/theories/cava_setup/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cava_setup/</guid>
	<description></description>
	</item>

	<item>
	<title>Cayley_Hamilton</title>
	<link>/theories/cayley_hamilton/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cayley_hamilton/</guid>
	<description></description>
	</item>

	<item>
	<title>CCS</title>
	<link>/theories/ccs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ccs/</guid>
	<description></description>
	</item>

	<item>
	<title>Certification_Monads</title>
	<link>/theories/certification_monads/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/certification_monads/</guid>
	<description></description>
	</item>

	<item>
	<title>Chandy_Lamport</title>
	<link>/theories/chandy_lamport/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/chandy_lamport/</guid>
	<description></description>
	</item>

	<item>
	<title>CHERI-C_Memory_Model</title>
	<link>/theories/cheri-c_memory_model/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cheri-c_memory_model/</guid>
	<description></description>
	</item>

	<item>
	<title>Chord_Segments</title>
	<link>/theories/chord_segments/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/chord_segments/</guid>
	<description></description>
	</item>

	<item>
	<title>Circus</title>
	<link>/theories/circus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/circus/</guid>
	<description></description>
	</item>

	<item>
	<title>CISC-Kernel</title>
	<link>/theories/cisc-kernel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cisc-kernel/</guid>
	<description></description>
	</item>

	<item>
	<title>Clean</title>
	<link>/theories/clean/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clean/</guid>
	<description></description>
	</item>

	<item>
	<title>Clique_and_Monotone_Circuits</title>
	<link>/theories/clique_and_monotone_circuits/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clique_and_monotone_circuits/</guid>
	<description></description>
	</item>

	<item>
	<title>ClockSynchInst</title>
	<link>/theories/clocksynchinst/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/clocksynchinst/</guid>
	<description></description>
	</item>

	<item>
	<title>Closest_Pair_Points</title>
	<link>/theories/closest_pair_points/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/closest_pair_points/</guid>
	<description></description>
	</item>

	<item>
	<title>CoCon</title>
	<link>/theories/cocon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cocon/</guid>
	<description></description>
	</item>

	<item>
	<title>CofGroups</title>
	<link>/theories/cofgroups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cofgroups/</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive</title>
	<link>/theories/coinductive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/coinductive/</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive_Languages</title>
	<link>/theories/coinductive_languages/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/coinductive_languages/</guid>
	<description></description>
	</item>

	<item>
	<title>Collections</title>
	<link>/theories/collections/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/collections/</guid>
	<description></description>
	</item>

	<item>
	<title>Collections_Examples</title>
	<link>/theories/collections_examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/collections_examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinable_Wands</title>
	<link>/theories/combinable_wands/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinable_wands/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial_Enumeration_Algorithms</title>
	<link>/theories/combinatorial_enumeration_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorial_enumeration_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words</title>
	<link>/theories/combinatorics_words/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words_Graph_Lemma</title>
	<link>/theories/combinatorics_words_graph_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words_graph_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics_Words_Lyndon</title>
	<link>/theories/combinatorics_words_lyndon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/combinatorics_words_lyndon/</guid>
	<description></description>
	</item>

	<item>
	<title>CommCSL</title>
	<link>/theories/commcsl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/commcsl/</guid>
	<description></description>
	</item>

	<item>
	<title>Commuting_Hermitian</title>
	<link>/theories/commuting_hermitian/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/commuting_hermitian/</guid>
	<description></description>
	</item>

	<item>
	<title>Comparison_Sort_Lower_Bound</title>
	<link>/theories/comparison_sort_lower_bound/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/comparison_sort_lower_bound/</guid>
	<description></description>
	</item>

	<item>
	<title>Compiling-Exceptions-Correctly</title>
	<link>/theories/compiling-exceptions-correctly/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/compiling-exceptions-correctly/</guid>
	<description></description>
	</item>

	<item>
	<title>Complete_Non_Orders</title>
	<link>/theories/complete_non_orders/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complete_non_orders/</guid>
	<description></description>
	</item>

	<item>
	<title>Completeness</title>
	<link>/theories/completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Bounded_Operators</title>
	<link>/theories/complex_bounded_operators/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_bounded_operators/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Bounded_Operators_Dependencies</title>
	<link>/theories/complex_bounded_operators_dependencies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_bounded_operators_dependencies/</guid>
	<description></description>
	</item>

	<item>
	<title>Complex_Geometry</title>
	<link>/theories/complex_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complex_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Complx</title>
	<link>/theories/complx/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/complx/</guid>
	<description></description>
	</item>

	<item>
	<title>ComponentDependencies</title>
	<link>/theories/componentdependencies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/componentdependencies/</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent_Ref_Alg</title>
	<link>/theories/concurrent_ref_alg/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrent_ref_alg/</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent_Revisions</title>
	<link>/theories/concurrent_revisions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrent_revisions/</guid>
	<description></description>
	</item>

	<item>
	<title>ConcurrentGC</title>
	<link>/theories/concurrentgc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrentgc/</guid>
	<description></description>
	</item>

	<item>
	<title>ConcurrentIMP</title>
	<link>/theories/concurrentimp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/concurrentimp/</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional_Simplification</title>
	<link>/theories/conditional_simplification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/conditional_simplification/</guid>
	<description></description>
	</item>

	<item>
	<title>Conditional_Transfer_Rule</title>
	<link>/theories/conditional_transfer_rule/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/conditional_transfer_rule/</guid>
	<description></description>
	</item>

	<item>
	<title>Consensus_Refined</title>
	<link>/theories/consensus_refined/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/consensus_refined/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive_Cryptography</title>
	<link>/theories/constructive_cryptography/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructive_cryptography/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive_Cryptography_CM</title>
	<link>/theories/constructive_cryptography_cm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructive_cryptography_cm/</guid>
	<description></description>
	</item>

	<item>
	<title>Constructor_Funs</title>
	<link>/theories/constructor_funs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/constructor_funs/</guid>
	<description></description>
	</item>

	<item>
	<title>Containers</title>
	<link>/theories/containers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/containers/</guid>
	<description></description>
	</item>

	<item>
	<title>Containers-Benchmarks</title>
	<link>/theories/containers-benchmarks/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/containers-benchmarks/</guid>
	<description></description>
	</item>

	<item>
	<title>Cook_Levin</title>
	<link>/theories/cook_levin/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cook_levin/</guid>
	<description></description>
	</item>

	<item>
	<title>Core_DOM</title>
	<link>/theories/core_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/core_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Core_SC_DOM</title>
	<link>/theories/core_sc_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/core_sc_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>CoreC++</title>
	<link>/theories/corec++/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/corec++/</guid>
	<description></description>
	</item>

	<item>
	<title>Correctness_Algebras</title>
	<link>/theories/correctness_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/correctness_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMed</title>
	<link>/theories/cosmed/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cosmed/</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMeDis</title>
	<link>/theories/cosmedis/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cosmedis/</guid>
	<description></description>
	</item>

	<item>
	<title>Cotangent_PFD_Formula</title>
	<link>/theories/cotangent_pfd_formula/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cotangent_pfd_formula/</guid>
	<description></description>
	</item>

	<item>
	<title>Count_Complex_Roots</title>
	<link>/theories/count_complex_roots/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/count_complex_roots/</guid>
	<description></description>
	</item>

	<item>
	<title>CRDT</title>
	<link>/theories/crdt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crdt/</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/theories/crypthol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crypthol/</guid>
	<description></description>
	</item>

	<item>
	<title>Crypto_Standards</title>
	<link>/theories/crypto_standards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crypto_standards/</guid>
	<description></description>
	</item>

	<item>
	<title>CryptoBasedCompositionalProperties</title>
	<link>/theories/cryptobasedcompositionalproperties/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cryptobasedcompositionalproperties/</guid>
	<description></description>
	</item>

	<item>
	<title>CRYSTALS-Kyber</title>
	<link>/theories/crystals-kyber/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/crystals-kyber/</guid>
	<description></description>
	</item>

	<item>
	<title>CSP_RefTK</title>
	<link>/theories/csp_reftk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/csp_reftk/</guid>
	<description></description>
	</item>

	<item>
	<title>Cubic_Quartic_Equations</title>
	<link>/theories/cubic_quartic_equations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cubic_quartic_equations/</guid>
	<description></description>
	</item>

	<item>
	<title>CVP_Hardness</title>
	<link>/theories/cvp_hardness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cvp_hardness/</guid>
	<description></description>
	</item>

	<item>
	<title>CYK</title>
	<link>/theories/cyk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/cyk/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Elementary_Categories</title>
	<link>/theories/czh_elementary_categories/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_elementary_categories/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Foundations</title>
	<link>/theories/czh_foundations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_foundations/</guid>
	<description></description>
	</item>

	<item>
	<title>CZH_Universal_Constructions</title>
	<link>/theories/czh_universal_constructions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/czh_universal_constructions/</guid>
	<description></description>
	</item>

	<item>
	<title>DataRefinementIBP</title>
	<link>/theories/datarefinementibp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/datarefinementibp/</guid>
	<description></description>
	</item>

	<item>
	<title>Datatype_Order_Generator</title>
	<link>/theories/datatype_order_generator/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/datatype_order_generator/</guid>
	<description></description>
	</item>

	<item>
	<title>DCR-ExecutionEquivalence</title>
	<link>/theories/dcr-executionequivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dcr-executionequivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Decl_Sem_Fun_PL</title>
	<link>/theories/decl_sem_fun_pl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decl_sem_fun_pl/</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing-Diagrams</title>
	<link>/theories/decreasing-diagrams/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decreasing-diagrams/</guid>
	<description></description>
	</item>

	<item>
	<title>Decreasing-Diagrams-II</title>
	<link>/theories/decreasing-diagrams-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/decreasing-diagrams-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Dedekind_Real</title>
	<link>/theories/dedekind_real/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dedekind_real/</guid>
	<description></description>
	</item>

	<item>
	<title>Deep_Learning</title>
	<link>/theories/deep_learning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/deep_learning/</guid>
	<description></description>
	</item>

	<item>
	<title>Delta_System_Lemma</title>
	<link>/theories/delta_system_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/delta_system_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>Density_Compiler</title>
	<link>/theories/density_compiler/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/density_compiler/</guid>
	<description></description>
	</item>

	<item>
	<title>Dependent_SIFUM_Refinement</title>
	<link>/theories/dependent_sifum_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dependent_sifum_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>Dependent_SIFUM_Type_Systems</title>
	<link>/theories/dependent_sifum_type_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dependent_sifum_type_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>Depth-First-Search</title>
	<link>/theories/depth-first-search/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/depth-first-search/</guid>
	<description></description>
	</item>

	<item>
	<title>Derangements</title>
	<link>/theories/derangements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/derangements/</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving</title>
	<link>/theories/deriving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/deriving/</guid>
	<description></description>
	</item>

	<item>
	<title>Descartes_Sign_Rule</title>
	<link>/theories/descartes_sign_rule/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/descartes_sign_rule/</guid>
	<description></description>
	</item>

	<item>
	<title>Design_Theory</title>
	<link>/theories/design_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/design_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>DFS_Framework</title>
	<link>/theories/dfs_framework/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dfs_framework/</guid>
	<description></description>
	</item>

	<item>
	<title>Dict_Construction</title>
	<link>/theories/dict_construction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dict_construction/</guid>
	<description></description>
	</item>

	<item>
	<title>Differential_Dynamic_Logic</title>
	<link>/theories/differential_dynamic_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/differential_dynamic_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Differential_Game_Logic</title>
	<link>/theories/differential_game_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/differential_game_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Digit_Expansions</title>
	<link>/theories/digit_expansions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/digit_expansions/</guid>
	<description></description>
	</item>

	<item>
	<title>DigitsInBase</title>
	<link>/theories/digitsinbase/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/digitsinbase/</guid>
	<description></description>
	</item>

	<item>
	<title>Dijkstra_Shortest_Path</title>
	<link>/theories/dijkstra_shortest_path/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dijkstra_shortest_path/</guid>
	<description></description>
	</item>

	<item>
	<title>Diophantine_Eqns_Lin_Hom</title>
	<link>/theories/diophantine_eqns_lin_hom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/diophantine_eqns_lin_hom/</guid>
	<description></description>
	</item>

	<item>
	<title>Directed_Sets</title>
	<link>/theories/directed_sets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/directed_sets/</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet_L</title>
	<link>/theories/dirichlet_l/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dirichlet_l/</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet_Series</title>
	<link>/theories/dirichlet_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dirichlet_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Discrete_Summation</title>
	<link>/theories/discrete_summation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/discrete_summation/</guid>
	<description></description>
	</item>

	<item>
	<title>DiscretePricing</title>
	<link>/theories/discretepricing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/discretepricing/</guid>
	<description></description>
	</item>

	<item>
	<title>DiskPaxos</title>
	<link>/theories/diskpaxos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/diskpaxos/</guid>
	<description></description>
	</item>

	<item>
	<title>Distributed_Distinct_Elements</title>
	<link>/theories/distributed_distinct_elements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/distributed_distinct_elements/</guid>
	<description></description>
	</item>

	<item>
	<title>DOM_Components</title>
	<link>/theories/dom_components/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dom_components/</guid>
	<description></description>
	</item>

	<item>
	<title>Dominance_CHK</title>
	<link>/theories/dominance_chk/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dominance_chk/</guid>
	<description></description>
	</item>

	<item>
	<title>DPRM_Theorem</title>
	<link>/theories/dprm_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dprm_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>DPT-SAT-Solver</title>
	<link>/theories/dpt-sat-solver/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dpt-sat-solver/</guid>
	<description></description>
	</item>

	<item>
	<title>Dynamic_Tables</title>
	<link>/theories/dynamic_tables/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dynamic_tables/</guid>
	<description></description>
	</item>

	<item>
	<title>DynamicArchitectures</title>
	<link>/theories/dynamicarchitectures/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/dynamicarchitectures/</guid>
	<description></description>
	</item>

	<item>
	<title>E_Transcendental</title>
	<link>/theories/e_transcendental/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/e_transcendental/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Earley_Parser</title>
	+ <link>/theories/earley_parser/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/earley_parser/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Echelon_Form</title>
	<link>/theories/echelon_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/echelon_form/</guid>
	<description></description>
	</item>

	<item>
	<title>EdmondsKarp_Maxflow</title>
	<link>/theories/edmondskarp_maxflow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/edmondskarp_maxflow/</guid>
	<description></description>
	</item>

	<item>
	<title>Edwards_Elliptic_Curves_Group</title>
	<link>/theories/edwards_elliptic_curves_group/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/edwards_elliptic_curves_group/</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient-Mergesort</title>
	<link>/theories/efficient-mergesort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/efficient-mergesort/</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient_Weighted_Path_Order</title>
	<link>/theories/efficient_weighted_path_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/efficient_weighted_path_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Elliptic_Curves_Group_Law</title>
	<link>/theories/elliptic_curves_group_law/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/elliptic_curves_group_law/</guid>
	<description></description>
	</item>

	<item>
	<title>Encodability_Process_Calculi</title>
	<link>/theories/encodability_process_calculi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/encodability_process_calculi/</guid>
	<description></description>
	</item>

	<item>
	<title>Epistemic_Logic</title>
	<link>/theories/epistemic_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/epistemic_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Equivalence_Relation_Enumeration</title>
	<link>/theories/equivalence_relation_enumeration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/equivalence_relation_enumeration/</guid>
	<description></description>
	</item>

	<item>
	<title>Ergodic_Theory</title>
	<link>/theories/ergodic_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ergodic_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Error_Function</title>
	<link>/theories/error_function/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/error_function/</guid>
	<description></description>
	</item>

	<item>
	<title>Euler_MacLaurin</title>
	<link>/theories/euler_maclaurin/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/euler_maclaurin/</guid>
	<description></description>
	</item>

	<item>
	<title>Euler_Partition</title>
	<link>/theories/euler_partition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/euler_partition/</guid>
	<description></description>
	</item>

	<item>
	<title>Eval_FO</title>
	<link>/theories/eval_fo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/eval_fo/</guid>
	<description></description>
	</item>

	<item>
	<title>Example-Submission</title>
	<link>/theories/example-submission/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/example-submission/</guid>
	<description></description>
	</item>

	<item>
	<title>Executable_Randomized_Algorithms</title>
	<link>/theories/executable_randomized_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/executable_randomized_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Expander_Graphs</title>
	<link>/theories/expander_graphs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/expander_graphs/</guid>
	<description></description>
	</item>

	<item>
	<title>Extended_Finite_State_Machine_Inference</title>
	<link>/theories/extended_finite_state_machine_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/extended_finite_state_machine_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Extended_Finite_State_Machines</title>
	<link>/theories/extended_finite_state_machines/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/extended_finite_state_machines/</guid>
	<description></description>
	</item>

	<item>
	<title>Factor_Algebraic_Polynomial</title>
	<link>/theories/factor_algebraic_polynomial/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/factor_algebraic_polynomial/</guid>
	<description></description>
	</item>

	<item>
	<title>Factored_Transition_System_Bounding</title>
	<link>/theories/factored_transition_system_bounding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/factored_transition_system_bounding/</guid>
	<description></description>
	</item>

	<item>
	<title>Falling_Factorial_Sum</title>
	<link>/theories/falling_factorial_sum/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/falling_factorial_sum/</guid>
	<description></description>
	</item>

	<item>
	<title>Farkas</title>
	<link>/theories/farkas/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/farkas/</guid>
	<description></description>
	</item>

	<item>
	<title>Featherweight_OCL</title>
	<link>/theories/featherweight_ocl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/featherweight_ocl/</guid>
	<description></description>
	</item>

	<item>
	<title>FeatherweightJava</title>
	<link>/theories/featherweightjava/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/featherweightjava/</guid>
	<description></description>
	</item>

	<item>
	<title>Fermat3_4</title>
	<link>/theories/fermat3_4/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fermat3_4/</guid>
	<description></description>
	</item>

	<item>
	<title>FFT</title>
	<link>/theories/fft/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fft/</guid>
	<description></description>
	</item>

	<item>
	<title>FileRefinement</title>
	<link>/theories/filerefinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/filerefinement/</guid>
	<description></description>
	</item>

	<item>
	<title>FinFun</title>
	<link>/theories/finfun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finfun/</guid>
	<description></description>
	</item>

	<item>
	<title>Finger-Trees</title>
	<link>/theories/finger-trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finger-trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite-Map-Extras</title>
	<link>/theories/finite-map-extras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite-map-extras/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite_Automata_HF</title>
	<link>/theories/finite_automata_hf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite_automata_hf/</guid>
	<description></description>
	</item>

	<item>
	<title>Finite_Fields</title>
	<link>/theories/finite_fields/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finite_fields/</guid>
	<description></description>
	</item>

	<item>
	<title>Finitely_Generated_Abelian_Groups</title>
	<link>/theories/finitely_generated_abelian_groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/finitely_generated_abelian_groups/</guid>
	<description></description>
	</item>

	<item>
	<title>First_Order_Terms</title>
	<link>/theories/first_order_terms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/first_order_terms/</guid>
	<description></description>
	</item>

	<item>
	<title>First_Welfare_Theorem</title>
	<link>/theories/first_welfare_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/first_welfare_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Fishburn_Impossibility</title>
	<link>/theories/fishburn_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fishburn_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher_Yates</title>
	<link>/theories/fisher_yates/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fisher_yates/</guid>
	<description></description>
	</item>

	<item>
	<title>Fishers_Inequality</title>
	<link>/theories/fishers_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fishers_inequality/</guid>
	<description></description>
	</item>

	<item>
	<title>Flow_Networks</title>
	<link>/theories/flow_networks/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flow_networks/</guid>
	<description></description>
	</item>

	<item>
	<title>Floyd_Warshall</title>
	<link>/theories/floyd_warshall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/floyd_warshall/</guid>
	<description></description>
	</item>

	<item>
	<title>FLP</title>
	<link>/theories/flp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flp/</guid>
	<description></description>
	</item>

	<item>
	<title>Flyspeck-Tame</title>
	<link>/theories/flyspeck-tame/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flyspeck-tame/</guid>
	<description></description>
	</item>

	<item>
	<title>Flyspeck-Tame-Computation</title>
	<link>/theories/flyspeck-tame-computation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/flyspeck-tame-computation/</guid>
	<description></description>
	</item>

	<item>
	<title>FO_Theory_Rewriting</title>
	<link>/theories/fo_theory_rewriting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fo_theory_rewriting/</guid>
	<description></description>
	</item>

	<item>
	<title>FocusStreamsCaseStudies</title>
	<link>/theories/focusstreamscasestudies/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/focusstreamscasestudies/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL-Fitting</title>
	<link>/theories/fol-fitting/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol-fitting/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Axiomatic</title>
	<link>/theories/fol_axiomatic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_axiomatic/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Harrison</title>
	<link>/theories/fol_harrison/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_harrison/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc1</title>
	<link>/theories/fol_seq_calc1/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc1/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc2</title>
	<link>/theories/fol_seq_calc2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc2/</guid>
	<description></description>
	</item>

	<item>
	<title>FOL_Seq_Calc3</title>
	<link>/theories/fol_seq_calc3/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fol_seq_calc3/</guid>
	<description></description>
	</item>

	<item>
	<title>Forcing</title>
	<link>/theories/forcing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/forcing/</guid>
	<description></description>
	</item>

	<item>
	<title>Formal_Puiseux_Series</title>
	<link>/theories/formal_puiseux_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formal_puiseux_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Formal_SSA</title>
	<link>/theories/formal_ssa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formal_ssa/</guid>
	<description></description>
	</item>

	<item>
	<title>Formula_Derivatives</title>
	<link>/theories/formula_derivatives/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formula_derivatives/</guid>
	<description></description>
	</item>

	<item>
	<title>Formula_Derivatives-Examples</title>
	<link>/theories/formula_derivatives-examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/formula_derivatives-examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Foundation_of_geometry</title>
	<link>/theories/foundation_of_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/foundation_of_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Fourier</title>
	<link>/theories/fourier/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fourier/</guid>
	<description></description>
	</item>

	<item>
	<title>Free-Boolean-Algebra</title>
	<link>/theories/free-boolean-algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/free-boolean-algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Free-Groups</title>
	<link>/theories/free-groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/free-groups/</guid>
	<description></description>
	</item>

	<item>
	<title>Frequency_Moments</title>
	<link>/theories/frequency_moments/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/frequency_moments/</guid>
	<description></description>
	</item>

	<item>
	<title>Fresh_Identifiers</title>
	<link>/theories/fresh_identifiers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fresh_identifiers/</guid>
	<description></description>
	</item>

	<item>
	<title>FSM_Tests</title>
	<link>/theories/fsm_tests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/fsm_tests/</guid>
	<description></description>
	</item>

	<item>
	<title>Functional-Automata</title>
	<link>/theories/functional-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/functional-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Functional_Ordered_Resolution_Prover</title>
	<link>/theories/functional_ordered_resolution_prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/functional_ordered_resolution_prover/</guid>
	<description></description>
	</item>

	<item>
	<title>FunWithFunctions</title>
	<link>/theories/funwithfunctions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/funwithfunctions/</guid>
	<description></description>
	</item>

	<item>
	<title>FunWithTilings</title>
	<link>/theories/funwithtilings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/funwithtilings/</guid>
	<description></description>
	</item>

	<item>
	<title>Furstenberg_Topology</title>
	<link>/theories/furstenberg_topology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/furstenberg_topology/</guid>
	<description></description>
	</item>

	<item>
	<title>Gabow_SCC</title>
	<link>/theories/gabow_scc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gabow_scc/</guid>
	<description></description>
	</item>

	<item>
	<title>Gale_Shapley</title>
	<link>/theories/gale_shapley/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gale_shapley/</guid>
	<description></description>
	</item>

	<item>
	<title>GaleStewart_Games</title>
	<link>/theories/galestewart_games/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/galestewart_games/</guid>
	<description></description>
	</item>

	<item>
	<title>Game_Based_Crypto</title>
	<link>/theories/game_based_crypto/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/game_based_crypto/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan-Elim-Fun</title>
	<link>/theories/gauss-jordan-elim-fun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss-jordan-elim-fun/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss_Jordan</title>
	<link>/theories/gauss_jordan/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss_jordan/</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss_Sums</title>
	<link>/theories/gauss_sums/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gauss_sums/</guid>
	<description></description>
	</item>

	<item>
	<title>Gaussian_Integers</title>
	<link>/theories/gaussian_integers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gaussian_integers/</guid>
	<description></description>
	</item>

	<item>
	<title>GenClock</title>
	<link>/theories/genclock/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/genclock/</guid>
	<description></description>
	</item>

	<item>
	<title>General-Triangle</title>
	<link>/theories/general-triangle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/general-triangle/</guid>
	<description></description>
	</item>

	<item>
	<title>Generalized_Counting_Sort</title>
	<link>/theories/generalized_counting_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generalized_counting_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>Generic_Deriving</title>
	<link>/theories/generic_deriving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generic_deriving/</guid>
	<description></description>
	</item>

	<item>
	<title>Generic_Join</title>
	<link>/theories/generic_join/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/generic_join/</guid>
	<description></description>
	</item>

	<item>
	<title>GewirthPGCProof</title>
	<link>/theories/gewirthpgcproof/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gewirthpgcproof/</guid>
	<description></description>
	</item>

	<item>
	<title>Girth_Chromatic</title>
	<link>/theories/girth_chromatic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/girth_chromatic/</guid>
	<description></description>
	</item>

	<item>
	<title>Given_Clause_Loops</title>
	<link>/theories/given_clause_loops/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/given_clause_loops/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_HFSet_Semantic</title>
	<link>/theories/goedel_hfset_semantic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_hfset_semantic/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_HFSet_Semanticless</title>
	<link>/theories/goedel_hfset_semanticless/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_hfset_semanticless/</guid>
	<description></description>
	</item>

	<item>
	<title>Goedel_Incompleteness</title>
	<link>/theories/goedel_incompleteness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedel_incompleteness/</guid>
	<description></description>
	</item>

	<item>
	<title>GoedelGod</title>
	<link>/theories/goedelgod/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goedelgod/</guid>
	<description></description>
	</item>

	<item>
	<title>Goodstein_Lambda</title>
	<link>/theories/goodstein_lambda/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/goodstein_lambda/</guid>
	<description></description>
	</item>

	<item>
	<title>GPU_Kernel_PL</title>
	<link>/theories/gpu_kernel_pl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gpu_kernel_pl/</guid>
	<description></description>
	</item>

	<item>
	<title>Graph_Saturation</title>
	<link>/theories/graph_saturation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graph_saturation/</guid>
	<description></description>
	</item>

	<item>
	<title>Graph_Theory</title>
	<link>/theories/graph_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graph_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>GraphMarkingIBP</title>
	<link>/theories/graphmarkingibp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/graphmarkingibp/</guid>
	<description></description>
	</item>

	<item>
	<title>Gray_Codes</title>
	<link>/theories/gray_codes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gray_codes/</guid>
	<description></description>
	</item>

	<item>
	<title>Green</title>
	<link>/theories/green/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/green/</guid>
	<description></description>
	</item>

	<item>
	<title>Groebner_Bases</title>
	<link>/theories/groebner_bases/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/groebner_bases/</guid>
	<description></description>
	</item>

	<item>
	<title>Groebner_Macaulay</title>
	<link>/theories/groebner_macaulay/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/groebner_macaulay/</guid>
	<description></description>
	</item>

	<item>
	<title>Gromov_Hyperbolicity</title>
	<link>/theories/gromov_hyperbolicity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/gromov_hyperbolicity/</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck_Schemes</title>
	<link>/theories/grothendieck_schemes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/grothendieck_schemes/</guid>
	<description></description>
	</item>

	<item>
	<title>Group-Ring-Module</title>
	<link>/theories/group-ring-module/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/group-ring-module/</guid>
	<description></description>
	</item>

	<item>
	<title>Hahn_Jordan_Decomposition</title>
	<link>/theories/hahn_jordan_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hahn_jordan_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Hales_Jewett</title>
	<link>/theories/hales_jewett/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hales_jewett/</guid>
	<description></description>
	</item>

	<item>
	<title>Heard_Of</title>
	<link>/theories/heard_of/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/heard_of/</guid>
	<description></description>
	</item>

	<item>
	<title>Hello_World</title>
	<link>/theories/hello_world/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hello_world/</guid>
	<description></description>
	</item>

	<item>
	<title>HereditarilyFinite</title>
	<link>/theories/hereditarilyfinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hereditarilyfinite/</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite</title>
	<link>/theories/hermite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hermite/</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite_Lindemann</title>
	<link>/theories/hermite_lindemann/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hermite_lindemann/</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden_Markov_Models</title>
	<link>/theories/hidden_markov_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hidden_markov_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Higher_Order_Terms</title>
	<link>/theories/higher_order_terms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/higher_order_terms/</guid>
	<description></description>
	</item>

	<item>
	<title>Hoare_Time</title>
	<link>/theories/hoare_time/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hoare_time/</guid>
	<description></description>
	</item>

	<item>
	<title>HoareForDivergence</title>
	<link>/theories/hoarefordivergence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hoarefordivergence/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP</title>
	<link>/theories/hol-csp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-csp/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-ARCH-COMP</title>
	<link>/theories/hol-ode-arch-comp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-arch-comp/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-Examples</title>
	<link>/theories/hol-ode-examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-examples/</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-ODE-Numerics</title>
	<link>/theories/hol-ode-numerics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hol-ode-numerics/</guid>
	<description></description>
	</item>

	<item>
	<title>HOLCF-Prelude</title>
	<link>/theories/holcf-prelude/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/holcf-prelude/</guid>
	<description></description>
	</item>

	<item>
	<title>Hood_Melville_Queue</title>
	<link>/theories/hood_melville_queue/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hood_melville_queue/</guid>
	<description></description>
	</item>

	<item>
	<title>HotelKeyCards</title>
	<link>/theories/hotelkeycards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hotelkeycards/</guid>
	<description></description>
	</item>

	<item>
	<title>HRB-Slicing</title>
	<link>/theories/hrb-slicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hrb-slicing/</guid>
	<description></description>
	</item>

	<item>
	<title>Huffman</title>
	<link>/theories/huffman/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/huffman/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Logic</title>
	<link>/theories/hybrid_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Multi_Lane_Spatial_Logic</title>
	<link>/theories/hybrid_multi_lane_spatial_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_multi_lane_spatial_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Hybrid_Systems_VCs</title>
	<link>/theories/hybrid_systems_vcs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hybrid_systems_vcs/</guid>
	<description></description>
	</item>

	<item>
	<title>HyperCTL</title>
	<link>/theories/hyperctl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperctl/</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual</title>
	<link>/theories/hyperdual/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperdual/</guid>
	<description></description>
	</item>

	<item>
	<title>HyperHoareLogic</title>
	<link>/theories/hyperhoarelogic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/hyperhoarelogic/</guid>
	<description></description>
	</item>

	<item>
	<title>IEEE_Floating_Point</title>
	<link>/theories/ieee_floating_point/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ieee_floating_point/</guid>
	<description></description>
	</item>

	<item>
	<title>IFC_Tracking</title>
	<link>/theories/ifc_tracking/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ifc_tracking/</guid>
	<description></description>
	</item>

	<item>
	<title>IMAP-CRDT</title>
	<link>/theories/imap-crdt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imap-crdt/</guid>
	<description></description>
	</item>

	<item>
	<title>IMO2019</title>
	<link>/theories/imo2019/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imo2019/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2</title>
	<link>/theories/imp2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp2/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2_Binary_Heap</title>
	<link>/theories/imp2_binary_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp2_binary_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP_Compiler</title>
	<link>/theories/imp_compiler/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp_compiler/</guid>
	<description></description>
	</item>

	<item>
	<title>IMP_Compiler_Reuse</title>
	<link>/theories/imp_compiler_reuse/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imp_compiler_reuse/</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative_Insertion_Sort</title>
	<link>/theories/imperative_insertion_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/imperative_insertion_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>Implicational_Logic</title>
	<link>/theories/implicational_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/implicational_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Impossible_Geometry</title>
	<link>/theories/impossible_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/impossible_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Incompleteness</title>
	<link>/theories/incompleteness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/incompleteness/</guid>
	<description></description>
	</item>

	<item>
	<title>Incredible_Proof_Machine</title>
	<link>/theories/incredible_proof_machine/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/incredible_proof_machine/</guid>
	<description></description>
	</item>

	<item>
	<title>Independence_CH</title>
	<link>/theories/independence_ch/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/independence_ch/</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive_Confidentiality</title>
	<link>/theories/inductive_confidentiality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/inductive_confidentiality/</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive_Inference</title>
	<link>/theories/inductive_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/inductive_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>InformationFlowSlicing</title>
	<link>/theories/informationflowslicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/informationflowslicing/</guid>
	<description></description>
	</item>

	<item>
	<title>InformationFlowSlicing_Inter</title>
	<link>/theories/informationflowslicing_inter/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/informationflowslicing_inter/</guid>
	<description></description>
	</item>

	<item>
	<title>InfPathElimination</title>
	<link>/theories/infpathelimination/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/infpathelimination/</guid>
	<description></description>
	</item>

	<item>
	<title>Integration</title>
	<link>/theories/integration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/integration/</guid>
	<description></description>
	</item>

	<item>
	<title>Interpolation_Polynomials_HOL_Algebra</title>
	<link>/theories/interpolation_polynomials_hol_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interpolation_polynomials_hol_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Interpreter_Optimizations</title>
	<link>/theories/interpreter_optimizations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interpreter_optimizations/</guid>
	<description></description>
	</item>

	<item>
	<title>Interval_Arithmetic_Word32</title>
	<link>/theories/interval_arithmetic_word32/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/interval_arithmetic_word32/</guid>
	<description></description>
	</item>

	<item>
	<title>Intro_Dest_Elim</title>
	<link>/theories/intro_dest_elim/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/intro_dest_elim/</guid>
	<description></description>
	</item>

	<item>
	<title>Involutions2Squares</title>
	<link>/theories/involutions2squares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/involutions2squares/</guid>
	<description></description>
	</item>

	<item>
	<title>IP_Addresses</title>
	<link>/theories/ip_addresses/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ip_addresses/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics</title>
	<link>/theories/iptables_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics_Examples</title>
	<link>/theories/iptables_semantics_examples/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics_examples/</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables_Semantics_Examples_Big</title>
	<link>/theories/iptables_semantics_examples_big/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/iptables_semantics_examples_big/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational_Series_Erdos_Straus</title>
	<link>/theories/irrational_series_erdos_straus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrational_series_erdos_straus/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality_J_Hancl</title>
	<link>/theories/irrationality_j_hancl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrationality_j_hancl/</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationals_From_THEBOOK</title>
	<link>/theories/irrationals_from_thebook/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/irrationals_from_thebook/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_C</title>
	<link>/theories/isabelle_c/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_c/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_Marries_Dirac</title>
	<link>/theories/isabelle_marries_dirac/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_marries_dirac/</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle_Meta_Model</title>
	<link>/theories/isabelle_meta_model/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isabelle_meta_model/</guid>
	<description></description>
	</item>

	<item>
	<title>IsaGeoCoq</title>
	<link>/theories/isageocoq/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isageocoq/</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet</title>
	<link>/theories/isanet/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/isanet/</guid>
	<description></description>
	</item>

	<item>
	<title>Jacobson_Basic_Algebra</title>
	<link>/theories/jacobson_basic_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jacobson_basic_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Jinja</title>
	<link>/theories/jinja/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinja/</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaDCI</title>
	<link>/theories/jinjadci/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinjadci/</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaThreads</title>
	<link>/theories/jinjathreads/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jinjathreads/</guid>
	<description></description>
	</item>

	<item>
	<title>JiveDataStoreModel</title>
	<link>/theories/jivedatastoremodel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jivedatastoremodel/</guid>
	<description></description>
	</item>

	<item>
	<title>Jordan_Hoelder</title>
	<link>/theories/jordan_hoelder/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jordan_hoelder/</guid>
	<description></description>
	</item>

	<item>
	<title>Jordan_Normal_Form</title>
	<link>/theories/jordan_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/jordan_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>KAD</title>
	<link>/theories/kad/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kad/</guid>
	<description></description>
	</item>

	<item>
	<title>KAT_and_DRA</title>
	<link>/theories/kat_and_dra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kat_and_dra/</guid>
	<description></description>
	</item>

	<item>
	<title>KBPs</title>
	<link>/theories/kbps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kbps/</guid>
	<description></description>
	</item>

	<item>
	<title>KD_Tree</title>
	<link>/theories/kd_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kd_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Key_Agreement_Strong_Adversaries</title>
	<link>/theories/key_agreement_strong_adversaries/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/key_agreement_strong_adversaries/</guid>
	<description></description>
	</item>

	<item>
	<title>Khovanskii_Theorem</title>
	<link>/theories/khovanskii_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/khovanskii_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene_Algebra</title>
	<link>/theories/kleene_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kleene_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Kneser_Cauchy_Davenport</title>
	<link>/theories/kneser_cauchy_davenport/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kneser_cauchy_davenport/</guid>
	<description></description>
	</item>

	<item>
	<title>Knights_Tour</title>
	<link>/theories/knights_tour/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knights_tour/</guid>
	<description></description>
	</item>

	<item>
	<title>Knot_Theory</title>
	<link>/theories/knot_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knot_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Knuth_Bendix_Order</title>
	<link>/theories/knuth_bendix_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knuth_bendix_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Knuth_Morris_Pratt</title>
	<link>/theories/knuth_morris_pratt/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/knuth_morris_pratt/</guid>
	<description></description>
	</item>

	<item>
	<title>Koenigsberg_Friendship</title>
	<link>/theories/koenigsberg_friendship/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/koenigsberg_friendship/</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal</title>
	<link>/theories/kruskal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kruskal/</guid>
	<description></description>
	</item>

	<item>
	<title>Kuratowski_Closure_Complement</title>
	<link>/theories/kuratowski_closure_complement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/kuratowski_closure_complement/</guid>
	<description></description>
	</item>

	<item>
	<title>Lam-ml-Normalization</title>
	<link>/theories/lam-ml-normalization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lam-ml-normalization/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_EPO</title>
	<link>/theories/lambda_free_epo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_epo/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_KBOs</title>
	<link>/theories/lambda_free_kbos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_kbos/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambda_Free_RPOs</title>
	<link>/theories/lambda_free_rpos/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambda_free_rpos/</guid>
	<description></description>
	</item>

	<item>
	<title>LambdaAuth</title>
	<link>/theories/lambdaauth/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambdaauth/</guid>
	<description></description>
	</item>

	<item>
	<title>LambdaMu</title>
	<link>/theories/lambdamu/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambdamu/</guid>
	<description></description>
	</item>

	<item>
	<title>Lambert_W</title>
	<link>/theories/lambert_w/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lambert_w/</guid>
	<description></description>
	</item>

	<item>
	<title>Landau_Symbols</title>
	<link>/theories/landau_symbols/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/landau_symbols/</guid>
	<description></description>
	</item>

	<item>
	<title>Laplace_Transform</title>
	<link>/theories/laplace_transform/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/laplace_transform/</guid>
	<description></description>
	</item>

	<item>
	<title>Latin_Square</title>
	<link>/theories/latin_square/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/latin_square/</guid>
	<description></description>
	</item>

	<item>
	<title>LatticeProperties</title>
	<link>/theories/latticeproperties/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/latticeproperties/</guid>
	<description></description>
	</item>

	<item>
	<title>Launchbury</title>
	<link>/theories/launchbury/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/launchbury/</guid>
	<description></description>
	</item>

	<item>
	<title>Laws_of_Large_Numbers</title>
	<link>/theories/laws_of_large_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/laws_of_large_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy-Lists-II</title>
	<link>/theories/lazy-lists-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lazy-lists-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Lazy_Case</title>
	<link>/theories/lazy_case/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lazy_case/</guid>
	<description></description>
	</item>

	<item>
	<title>Lehmer</title>
	<link>/theories/lehmer/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lehmer/</guid>
	<description></description>
	</item>

	<item>
	<title>LEM</title>
	<link>/theories/lem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lem/</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting_Definition_Option</title>
	<link>/theories/lifting_definition_option/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lifting_definition_option/</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting_the_Exponent</title>
	<link>/theories/lifting_the_exponent/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lifting_the_exponent/</guid>
	<description></description>
	</item>

	<item>
	<title>LightweightJava</title>
	<link>/theories/lightweightjava/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lightweightjava/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Inequalities</title>
	<link>/theories/linear_inequalities/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_inequalities/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Programming</title>
	<link>/theories/linear_programming/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_programming/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Recurrences</title>
	<link>/theories/linear_recurrences/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_recurrences/</guid>
	<description></description>
	</item>

	<item>
	<title>Linear_Recurrences_Solver</title>
	<link>/theories/linear_recurrences_solver/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linear_recurrences_solver/</guid>
	<description></description>
	</item>

	<item>
	<title>LinearQuantifierElim</title>
	<link>/theories/linearquantifierelim/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/linearquantifierelim/</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville_Numbers</title>
	<link>/theories/liouville_numbers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/liouville_numbers/</guid>
	<description></description>
	</item>

	<item>
	<title>List-Index</title>
	<link>/theories/list-index/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list-index/</guid>
	<description></description>
	</item>

	<item>
	<title>List-Infinite</title>
	<link>/theories/list-infinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list-infinite/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Interleaving</title>
	<link>/theories/list_interleaving/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_interleaving/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Inversions</title>
	<link>/theories/list_inversions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_inversions/</guid>
	<description></description>
	</item>

	<item>
	<title>List_Update</title>
	<link>/theories/list_update/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/list_update/</guid>
	<description></description>
	</item>

	<item>
	<title>LLL_Basis_Reduction</title>
	<link>/theories/lll_basis_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lll_basis_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>LLL_Factorization</title>
	<link>/theories/lll_factorization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lll_factorization/</guid>
	<description></description>
	</item>

	<item>
	<title>Localization_Ring</title>
	<link>/theories/localization_ring/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/localization_ring/</guid>
	<description></description>
	</item>

	<item>
	<title>LocalLexing</title>
	<link>/theories/locallexing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/locallexing/</guid>
	<description></description>
	</item>

	<item>
	<title>Locally-Nameless-Sigma</title>
	<link>/theories/locally-nameless-sigma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/locally-nameless-sigma/</guid>
	<description></description>
	</item>

	<item>
	<title>LOFT</title>
	<link>/theories/loft/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/loft/</guid>
	<description></description>
	</item>

	<item>
	<title>Logging_Independent_Anonymity</title>
	<link>/theories/logging_independent_anonymity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/logging_independent_anonymity/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_Approximation</title>
	<link>/theories/lorenz_approximation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_approximation/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_C0</title>
	<link>/theories/lorenz_c0/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_c0/</guid>
	<description></description>
	</item>

	<item>
	<title>Lorenz_C1</title>
	<link>/theories/lorenz_c1/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lorenz_c1/</guid>
	<description></description>
	</item>

	<item>
	<title>Lowe_Ontological_Argument</title>
	<link>/theories/lowe_ontological_argument/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lowe_ontological_argument/</guid>
	<description></description>
	</item>

	<item>
	<title>Lower_Semicontinuous</title>
	<link>/theories/lower_semicontinuous/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lower_semicontinuous/</guid>
	<description></description>
	</item>

	<item>
	<title>Lp</title>
	<link>/theories/lp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lp/</guid>
	<description></description>
	</item>

	<item>
	<title>LP_Duality</title>
	<link>/theories/lp_duality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lp_duality/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL</title>
	<link>/theories/ltl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_Master_Theorem</title>
	<link>/theories/ltl_master_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_master_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_Normal_Form</title>
	<link>/theories/ltl_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_to_DRA</title>
	<link>/theories/ltl_to_dra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_to_dra/</guid>
	<description></description>
	</item>

	<item>
	<title>LTL_to_GBA</title>
	<link>/theories/ltl_to_gba/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ltl_to_gba/</guid>
	<description></description>
	</item>

	<item>
	<title>Lucas_Theorem</title>
	<link>/theories/lucas_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/lucas_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Markov_Models</title>
	<link>/theories/markov_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/markov_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Marriage</title>
	<link>/theories/marriage/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/marriage/</guid>
	<description></description>
	</item>

	<item>
	<title>Mason_Stothers</title>
	<link>/theories/mason_stothers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mason_stothers/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices_for_ODEs</title>
	<link>/theories/matrices_for_odes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrices_for_odes/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrix</title>
	<link>/theories/matrix/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrix/</guid>
	<description></description>
	</item>

	<item>
	<title>Matrix_Tensor</title>
	<link>/theories/matrix_tensor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matrix_tensor/</guid>
	<description></description>
	</item>

	<item>
	<title>Matroids</title>
	<link>/theories/matroids/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/matroids/</guid>
	<description></description>
	</item>

	<item>
	<title>Max-Card-Matching</title>
	<link>/theories/max-card-matching/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/max-card-matching/</guid>
	<description></description>
	</item>

	<item>
	<title>Maximum_Segment_Sum</title>
	<link>/theories/maximum_segment_sum/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/maximum_segment_sum/</guid>
	<description></description>
	</item>

	<item>
	<title>MDP-Algorithms</title>
	<link>/theories/mdp-algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mdp-algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>MDP-Rewards</title>
	<link>/theories/mdp-rewards/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mdp-rewards/</guid>
	<description></description>
	</item>

	<item>
	<title>Median_Method</title>
	<link>/theories/median_method/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/median_method/</guid>
	<description></description>
	</item>

	<item>
	<title>Median_Of_Medians_Selection</title>
	<link>/theories/median_of_medians_selection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/median_of_medians_selection/</guid>
	<description></description>
	</item>

	<item>
	<title>Menger</title>
	<link>/theories/menger/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/menger/</guid>
	<description></description>
	</item>

	<item>
	<title>Mereology</title>
	<link>/theories/mereology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mereology/</guid>
	<description></description>
	</item>

	<item>
	<title>Mersenne_Primes</title>
	<link>/theories/mersenne_primes/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mersenne_primes/</guid>
	<description></description>
	</item>

	<item>
	<title>Metalogic_ProofChecker</title>
	<link>/theories/metalogic_proofchecker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/metalogic_proofchecker/</guid>
	<description></description>
	</item>

	<item>
	<title>MFMC_Countable</title>
	<link>/theories/mfmc_countable/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfmc_countable/</guid>
	<description></description>
	</item>

	<item>
	<title>MFODL_Monitor_Optimized</title>
	<link>/theories/mfodl_monitor_optimized/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfodl_monitor_optimized/</guid>
	<description></description>
	</item>

	<item>
	<title>MFOTL_Monitor</title>
	<link>/theories/mfotl_monitor/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mfotl_monitor/</guid>
	<description></description>
	</item>

	<item>
	<title>MHComputation</title>
	<link>/theories/mhcomputation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mhcomputation/</guid>
	<description></description>
	</item>

	<item>
	<title>Minimal_SSA</title>
	<link>/theories/minimal_ssa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minimal_ssa/</guid>
	<description></description>
	</item>

	<item>
	<title>MiniML</title>
	<link>/theories/miniml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/miniml/</guid>
	<description></description>
	</item>

	<item>
	<title>MiniSail</title>
	<link>/theories/minisail/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minisail/</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowskis_Theorem</title>
	<link>/theories/minkowskis_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minkowskis_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Minsky_Machines</title>
	<link>/theories/minsky_machines/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/minsky_machines/</guid>
	<description></description>
	</item>

	<item>
	<title>MLSS_Decision_Proc</title>
	<link>/theories/mlss_decision_proc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mlss_decision_proc/</guid>
	<description></description>
	</item>

	<item>
	<title>Modal_Logics_for_NTS</title>
	<link>/theories/modal_logics_for_nts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modal_logics_for_nts/</guid>
	<description></description>
	</item>

	<item>
	<title>Modular_arithmetic_LLL_and_HNF_algorithms</title>
	<link>/theories/modular_arithmetic_lll_and_hnf_algorithms/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modular_arithmetic_lll_and_hnf_algorithms/</guid>
	<description></description>
	</item>

	<item>
	<title>Modular_Assembly_Kit_Security</title>
	<link>/theories/modular_assembly_kit_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/modular_assembly_kit_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Monad_Memo_DP</title>
	<link>/theories/monad_memo_dp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monad_memo_dp/</guid>
	<description></description>
	</item>

	<item>
	<title>Monad_Normalisation</title>
	<link>/theories/monad_normalisation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monad_normalisation/</guid>
	<description></description>
	</item>

	<item>
	<title>MonoBoolTranAlgebra</title>
	<link>/theories/monobooltranalgebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monobooltranalgebra/</guid>
	<description></description>
	</item>

	<item>
	<title>MonoidalCategory</title>
	<link>/theories/monoidalcategory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monoidalcategory/</guid>
	<description></description>
	</item>

	<item>
	<title>Monomorphic_Monad</title>
	<link>/theories/monomorphic_monad/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/monomorphic_monad/</guid>
	<description></description>
	</item>

	<item>
	<title>MSO_Regex_Equivalence</title>
	<link>/theories/mso_regex_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/mso_regex_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>MuchAdoAboutTwo</title>
	<link>/theories/muchadoabouttwo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/muchadoabouttwo/</guid>
	<description></description>
	</item>

	<item>
	<title>Multi_Party_Computation</title>
	<link>/theories/multi_party_computation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multi_party_computation/</guid>
	<description></description>
	</item>

	<item>
	<title>Multirelations</title>
	<link>/theories/multirelations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multirelations/</guid>
	<description></description>
	</item>

	<item>
	<title>Multirelations_Heterogeneous</title>
	<link>/theories/multirelations_heterogeneous/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multirelations_heterogeneous/</guid>
	<description></description>
	</item>

	<item>
	<title>Multiset_Ordering_NPC</title>
	<link>/theories/multiset_ordering_npc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multiset_ordering_npc/</guid>
	<description></description>
	</item>

	<item>
	<title>Multitape_To_Singletape_TM</title>
	<link>/theories/multitape_to_singletape_tm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/multitape_to_singletape_tm/</guid>
	<description></description>
	</item>

	<item>
	<title>Myhill-Nerode</title>
	<link>/theories/myhill-nerode/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/myhill-nerode/</guid>
	<description></description>
	</item>

	<item>
	<title>Name_Carrying_Type_Inference</title>
	<link>/theories/name_carrying_type_inference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/name_carrying_type_inference/</guid>
	<description></description>
	</item>

	<item>
	<title>Nano_JSON</title>
	<link>/theories/nano_json/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nano_json/</guid>
	<description></description>
	</item>

	<item>
	<title>Nash_Williams</title>
	<link>/theories/nash_williams/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nash_williams/</guid>
	<description></description>
	</item>

	<item>
	<title>Nat-Interval-Logic</title>
	<link>/theories/nat-interval-logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nat-interval-logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Native_Word</title>
	<link>/theories/native_word/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/native_word/</guid>
	<description></description>
	</item>

	<item>
	<title>Nested_Multisets_Ordinals</title>
	<link>/theories/nested_multisets_ordinals/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nested_multisets_ordinals/</guid>
	<description></description>
	</item>

	<item>
	<title>Network_Security_Policy_Verification</title>
	<link>/theories/network_security_policy_verification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/network_security_policy_verification/</guid>
	<description></description>
	</item>

	<item>
	<title>Neumann_Morgenstern_Utility</title>
	<link>/theories/neumann_morgenstern_utility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/neumann_morgenstern_utility/</guid>
	<description></description>
	</item>

	<item>
	<title>No_FTL_observers</title>
	<link>/theories/no_ftl_observers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/no_ftl_observers/</guid>
	<description></description>
	</item>

	<item>
	<title>No_FTL_observers_Gen_Rel</title>
	<link>/theories/no_ftl_observers_gen_rel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/no_ftl_observers_gen_rel/</guid>
	<description></description>
	</item>

	<item>
	<title>Nominal2</title>
	<link>/theories/nominal2/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nominal2/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Concurrent_Composition</title>
	<link>/theories/noninterference_concurrent_composition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_concurrent_composition/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_CSP</title>
	<link>/theories/noninterference_csp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_csp/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Generic_Unwinding</title>
	<link>/theories/noninterference_generic_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_generic_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Inductive_Unwinding</title>
	<link>/theories/noninterference_inductive_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_inductive_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Ipurge_Unwinding</title>
	<link>/theories/noninterference_ipurge_unwinding/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_ipurge_unwinding/</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference_Sequential_Composition</title>
	<link>/theories/noninterference_sequential_composition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/noninterference_sequential_composition/</guid>
	<description></description>
	</item>

	<item>
	<title>NormByEval</title>
	<link>/theories/normbyeval/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/normbyeval/</guid>
	<description></description>
	</item>

	<item>
	<title>Nullstellensatz</title>
	<link>/theories/nullstellensatz/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/nullstellensatz/</guid>
	<description></description>
	</item>

	<item>
	<title>Number_Theoretic_Transform</title>
	<link>/theories/number_theoretic_transform/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/number_theoretic_transform/</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/theories/octonions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/octonions/</guid>
	<description></description>
	</item>

	<item>
	<title>Old_Datatype_Show</title>
	<link>/theories/old_datatype_show/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/old_datatype_show/</guid>
	<description></description>
	</item>

	<item>
	<title>Open_Induction</title>
	<link>/theories/open_induction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/open_induction/</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets</title>
	<link>/theories/opsets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/opsets/</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/theories/optics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/optics/</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal_BST</title>
	<link>/theories/optimal_bst/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/optimal_bst/</guid>
	<description></description>
	</item>

	<item>
	<title>Orbit_Stabiliser</title>
	<link>/theories/orbit_stabiliser/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/orbit_stabiliser/</guid>
	<description></description>
	</item>

	<item>
	<title>Order_Lattice_Props</title>
	<link>/theories/order_lattice_props/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/order_lattice_props/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordered_Resolution_Prover</title>
	<link>/theories/ordered_resolution_prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordered_resolution_prover/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal</title>
	<link>/theories/ordinal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinal/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal_Partitions</title>
	<link>/theories/ordinal_partitions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinal_partitions/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinals_and_Cardinals</title>
	<link>/theories/ordinals_and_cardinals/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinals_and_cardinals/</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinary_Differential_Equations</title>
	<link>/theories/ordinary_differential_equations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ordinary_differential_equations/</guid>
	<description></description>
	</item>

	<item>
	<title>PAC_Checker</title>
	<link>/theories/pac_checker/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pac_checker/</guid>
	<description></description>
	</item>

	<item>
	<title>Package_logic</title>
	<link>/theories/package_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/package_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Padic_Field</title>
	<link>/theories/padic_field/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/padic_field/</guid>
	<description></description>
	</item>

	<item>
	<title>Padic_Ints</title>
	<link>/theories/padic_ints/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/padic_ints/</guid>
	<description></description>
	</item>

	<item>
	<title>Pairing_Heap</title>
	<link>/theories/pairing_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pairing_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>PAL</title>
	<link>/theories/pal/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pal/</guid>
	<description></description>
	</item>

	<item>
	<title>PAPP_Impossibility</title>
	<link>/theories/papp_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/papp_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Paraconsistency</title>
	<link>/theories/paraconsistency/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/paraconsistency/</guid>
	<description></description>
	</item>

	<item>
	<title>Parity_Game</title>
	<link>/theories/parity_game/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/parity_game/</guid>
	<description></description>
	</item>

	<item>
	<title>Partial_Function_MR</title>
	<link>/theories/partial_function_mr/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/partial_function_mr/</guid>
	<description></description>
	</item>

	<item>
	<title>Partial_Order_Reduction</title>
	<link>/theories/partial_order_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/partial_order_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Password_Authentication_Protocol</title>
	<link>/theories/password_authentication_protocol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/password_authentication_protocol/</guid>
	<description></description>
	</item>

	<item>
	<title>PCF</title>
	<link>/theories/pcf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pcf/</guid>
	<description></description>
	</item>

	<item>
	<title>Pell</title>
	<link>/theories/pell/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pell/</guid>
	<description></description>
	</item>

	<item>
	<title>Perfect-Number-Thm</title>
	<link>/theories/perfect-number-thm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/perfect-number-thm/</guid>
	<description></description>
	</item>

	<item>
	<title>Perron_Frobenius</title>
	<link>/theories/perron_frobenius/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/perron_frobenius/</guid>
	<description></description>
	</item>

	<item>
	<title>pGCL</title>
	<link>/theories/pgcl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pgcl/</guid>
	<description></description>
	</item>

	<item>
	<title>Physical_Quantities</title>
	<link>/theories/physical_quantities/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/physical_quantities/</guid>
	<description></description>
	</item>

	<item>
	<title>Pi_Calculus</title>
	<link>/theories/pi_calculus/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pi_calculus/</guid>
	<description></description>
	</item>

	<item>
	<title>Pi_Transcendental</title>
	<link>/theories/pi_transcendental/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pi_transcendental/</guid>
	<description></description>
	</item>

	<item>
	<title>Planarity_Certificates</title>
	<link>/theories/planarity_certificates/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/planarity_certificates/</guid>
	<description></description>
	</item>

	<item>
	<title>PLM</title>
	<link>/theories/plm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/plm/</guid>
	<description></description>
	</item>

	<item>
	<title>Pluennecke_Ruzsa_Inequality</title>
	<link>/theories/pluennecke_ruzsa_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pluennecke_ruzsa_inequality/</guid>
	<description></description>
	</item>

	<item>
	<title>Poincare_Bendixson</title>
	<link>/theories/poincare_bendixson/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poincare_bendixson/</guid>
	<description></description>
	</item>

	<item>
	<title>Poincare_Disc</title>
	<link>/theories/poincare_disc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poincare_disc/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Polygonal_Number_Theorem</title>
	+ <link>/theories/polygonal_number_theorem/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/polygonal_number_theorem/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Polynomial_Factorization</title>
	<link>/theories/polynomial_factorization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomial_factorization/</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial_Interpolation</title>
	<link>/theories/polynomial_interpolation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomial_interpolation/</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomials</title>
	<link>/theories/polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Pop_Refinement</title>
	<link>/theories/pop_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pop_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>POPLmark-deBruijn</title>
	<link>/theories/poplmark-debruijn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/poplmark-debruijn/</guid>
	<description></description>
	</item>

	<item>
	<title>Posix-Lexing</title>
	<link>/theories/posix-lexing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/posix-lexing/</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic_Noninterference</title>
	<link>/theories/possibilistic_noninterference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/possibilistic_noninterference/</guid>
	<description></description>
	</item>

	<item>
	<title>Power_Sum_Polynomials</title>
	<link>/theories/power_sum_polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/power_sum_polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Pratt_Certificate</title>
	<link>/theories/pratt_certificate/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pratt_certificate/</guid>
	<description></description>
	</item>

	<item>
	<title>Prefix_Free_Code_Combinators</title>
	<link>/theories/prefix_free_code_combinators/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prefix_free_code_combinators/</guid>
	<description></description>
	</item>

	<item>
	<title>Presburger-Automata</title>
	<link>/theories/presburger-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/presburger-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Prim_Dijkstra_Simple</title>
	<link>/theories/prim_dijkstra_simple/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prim_dijkstra_simple/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Distribution_Elementary</title>
	<link>/theories/prime_distribution_elementary/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_distribution_elementary/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Harmonic_Series</title>
	<link>/theories/prime_harmonic_series/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_harmonic_series/</guid>
	<description></description>
	</item>

	<item>
	<title>Prime_Number_Theorem</title>
	<link>/theories/prime_number_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prime_number_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Priority_Queue_Braun</title>
	<link>/theories/priority_queue_braun/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/priority_queue_braun/</guid>
	<description></description>
	</item>

	<item>
	<title>Priority_Search_Trees</title>
	<link>/theories/priority_search_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/priority_search_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Noninterference</title>
	<link>/theories/probabilistic_noninterference/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_noninterference/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Prime_Tests</title>
	<link>/theories/probabilistic_prime_tests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_prime_tests/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_System_Zoo</title>
	<link>/theories/probabilistic_system_zoo/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_system_zoo/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_Timed_Automata</title>
	<link>/theories/probabilistic_timed_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_timed_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic_While</title>
	<link>/theories/probabilistic_while/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probabilistic_while/</guid>
	<description></description>
	</item>

	<item>
	<title>Probability_Inequality_Completeness</title>
	<link>/theories/probability_inequality_completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/probability_inequality_completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Program-Conflict-Analysis</title>
	<link>/theories/program-conflict-analysis/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/program-conflict-analysis/</guid>
	<description></description>
	</item>

	<item>
	<title>Progress_Tracking</title>
	<link>/theories/progress_tracking/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/progress_tracking/</guid>
	<description></description>
	</item>

	<item>
	<title>Projective_Geometry</title>
	<link>/theories/projective_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/projective_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Projective_Measurements</title>
	<link>/theories/projective_measurements/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/projective_measurements/</guid>
	<description></description>
	</item>

	<item>
	<title>Promela</title>
	<link>/theories/promela/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/promela/</guid>
	<description></description>
	</item>

	<item>
	<title>Proof_Strategy_Language</title>
	<link>/theories/proof_strategy_language/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/proof_strategy_language/</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional_Logic_Class</title>
	<link>/theories/propositional_logic_class/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/propositional_logic_class/</guid>
	<description></description>
	</item>

	<item>
	<title>Propositional_Proof_Systems</title>
	<link>/theories/propositional_proof_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/propositional_proof_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>PropResPI</title>
	<link>/theories/proprespi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/proprespi/</guid>
	<description></description>
	</item>

	<item>
	<title>Prpu_Maxflow</title>
	<link>/theories/prpu_maxflow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/prpu_maxflow/</guid>
	<description></description>
	</item>

	<item>
	<title>PSemigroupsConvolution</title>
	<link>/theories/psemigroupsconvolution/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/psemigroupsconvolution/</guid>
	<description></description>
	</item>

	<item>
	<title>PseudoHoops</title>
	<link>/theories/pseudohoops/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/pseudohoops/</guid>
	<description></description>
	</item>

	<item>
	<title>Psi_Calculi</title>
	<link>/theories/psi_calculi/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/psi_calculi/</guid>
	<description></description>
	</item>

	<item>
	<title>Ptolemys_Theorem</title>
	<link>/theories/ptolemys_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ptolemys_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Public_Announcement_Logic</title>
	<link>/theories/public_announcement_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/public_announcement_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>QHLProver</title>
	<link>/theories/qhlprover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/qhlprover/</guid>
	<description></description>
	</item>

	<item>
	<title>QR_Decomposition</title>
	<link>/theories/qr_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/qr_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Quantales</title>
	<link>/theories/quantales/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quantales/</guid>
	<description></description>
	</item>

	<item>
	+ <title>Quantales_Converse</title>
	+ <link>/theories/quantales_converse/</link>
	+ <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
	+
	+ <guid>/theories/quantales_converse/</guid>
	+ <description></description>
	+ </item>
	+
	+ <item>
	<title>Quantifier_Elimination_Hybrid</title>
	<link>/theories/quantifier_elimination_hybrid/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quantifier_elimination_hybrid/</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi_Borel_Spaces</title>
	<link>/theories/quasi_borel_spaces/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quasi_borel_spaces/</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/theories/quaternions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quaternions/</guid>
	<description></description>
	</item>

	<item>
	<title>Query_Optimization</title>
	<link>/theories/query_optimization/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/query_optimization/</guid>
	<description></description>
	</item>

	<item>
	<title>Quick_Sort_Cost</title>
	<link>/theories/quick_sort_cost/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/quick_sort_cost/</guid>
	<description></description>
	</item>

	<item>
	<title>Ramsey-Infinite</title>
	<link>/theories/ramsey-infinite/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ramsey-infinite/</guid>
	<description></description>
	</item>

	<item>
	<title>Random_BSTs</title>
	<link>/theories/random_bsts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/random_bsts/</guid>
	<description></description>
	</item>

	<item>
	<title>Random_Graph_Subgraph_Threshold</title>
	<link>/theories/random_graph_subgraph_threshold/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/random_graph_subgraph_threshold/</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised_BSTs</title>
	<link>/theories/randomised_bsts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/randomised_bsts/</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised_Social_Choice</title>
	<link>/theories/randomised_social_choice/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/randomised_social_choice/</guid>
	<description></description>
	</item>

	<item>
	<title>Rank_Nullity_Theorem</title>
	<link>/theories/rank_nullity_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rank_nullity_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Impl</title>
	<link>/theories/real_impl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_impl/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Power</title>
	<link>/theories/real_power/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_power/</guid>
	<description></description>
	</item>

	<item>
	<title>Real_Time_Deque</title>
	<link>/theories/real_time_deque/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/real_time_deque/</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion-Addition</title>
	<link>/theories/recursion-addition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/recursion-addition/</guid>
	<description></description>
	</item>

	<item>
	<title>Recursion-Theory-I</title>
	<link>/theories/recursion-theory-i/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/recursion-theory-i/</guid>
	<description></description>
	</item>

	<item>
	<title>Refine_Imperative_HOL</title>
	<link>/theories/refine_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refine_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Refine_Monadic</title>
	<link>/theories/refine_monadic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refine_monadic/</guid>
	<description></description>
	</item>

	<item>
	<title>RefinementReactive</title>
	<link>/theories/refinementreactive/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/refinementreactive/</guid>
	<description></description>
	</item>

	<item>
	<title>Regex_Equivalence</title>
	<link>/theories/regex_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regex_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Registers</title>
	<link>/theories/registers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/registers/</guid>
	<description></description>
	</item>

	<item>
	<title>Regression_Test_Selection</title>
	<link>/theories/regression_test_selection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regression_test_selection/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular-Sets</title>
	<link>/theories/regular-sets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular-sets/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular_Algebras</title>
	<link>/theories/regular_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Regular_Tree_Relations</title>
	<link>/theories/regular_tree_relations/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/regular_tree_relations/</guid>
	<description></description>
	</item>

	<item>
	<title>Relation_Algebra</title>
	<link>/theories/relation_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relation_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational-Incorrectness-Logic</title>
	<link>/theories/relational-incorrectness-logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational-incorrectness-logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Disjoint_Set_Forests</title>
	<link>/theories/relational_disjoint_set_forests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_disjoint_set_forests/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Forests</title>
	<link>/theories/relational_forests/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_forests/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Method</title>
	<link>/theories/relational_method/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_method/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Minimum_Spanning_Trees</title>
	<link>/theories/relational_minimum_spanning_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_minimum_spanning_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Relational_Paths</title>
	<link>/theories/relational_paths/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/relational_paths/</guid>
	<description></description>
	</item>

	<item>
	<title>Rensets</title>
	<link>/theories/rensets/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rensets/</guid>
	<description></description>
	</item>

	<item>
	<title>Rep_Fin_Groups</title>
	<link>/theories/rep_fin_groups/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rep_fin_groups/</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated_Lattices</title>
	<link>/theories/residuated_lattices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/residuated_lattices/</guid>
	<description></description>
	</item>

	<item>
	<title>ResiduatedTransitionSystem</title>
	<link>/theories/residuatedtransitionsystem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/residuatedtransitionsystem/</guid>
	<description></description>
	</item>

	<item>
	<title>Resolution_FOL</title>
	<link>/theories/resolution_fol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/resolution_fol/</guid>
	<description></description>
	</item>

	<item>
	<title>Rewrite_Properties_Reduction</title>
	<link>/theories/rewrite_properties_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rewrite_properties_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Rewriting_Z</title>
	<link>/theories/rewriting_z/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rewriting_z/</guid>
	<description></description>
	</item>

	<item>
	<title>Ribbon_Proofs</title>
	<link>/theories/ribbon_proofs/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ribbon_proofs/</guid>
	<description></description>
	</item>

	<item>
	<title>RIPEMD-160-SPARK</title>
	<link>/theories/ripemd-160-spark/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/ripemd-160-spark/</guid>
	<description></description>
	</item>

	<item>
	<title>Risk_Free_Lending</title>
	<link>/theories/risk_free_lending/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/risk_free_lending/</guid>
	<description></description>
	</item>

	<item>
	<title>Robbins-Conjecture</title>
	<link>/theories/robbins-conjecture/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robbins-conjecture/</guid>
	<description></description>
	</item>

	<item>
	<title>ROBDD</title>
	<link>/theories/robdd/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robdd/</guid>
	<description></description>
	</item>

	<item>
	<title>Robinson_Arithmetic</title>
	<link>/theories/robinson_arithmetic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/robinson_arithmetic/</guid>
	<description></description>
	</item>

	<item>
	<title>Root_Balanced_Tree</title>
	<link>/theories/root_balanced_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/root_balanced_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Roth_Arithmetic_Progressions</title>
	<link>/theories/roth_arithmetic_progressions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/roth_arithmetic_progressions/</guid>
	<description></description>
	</item>

	<item>
	<title>Routing</title>
	<link>/theories/routing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/routing/</guid>
	<description></description>
	</item>

	<item>
	<title>Roy_Floyd_Warshall</title>
	<link>/theories/roy_floyd_warshall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/roy_floyd_warshall/</guid>
	<description></description>
	</item>

	<item>
	<title>RSAPSS</title>
	<link>/theories/rsapss/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/rsapss/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_Distance</title>
	<link>/theories/safe_distance/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_distance/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_OCL</title>
	<link>/theories/safe_ocl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_ocl/</guid>
	<description></description>
	</item>

	<item>
	<title>Safe_Range_RC</title>
	<link>/theories/safe_range_rc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/safe_range_rc/</guid>
	<description></description>
	</item>

	<item>
	<title>SATSolverVerification</title>
	<link>/theories/satsolververification/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/satsolververification/</guid>
	<description></description>
	</item>

	<item>
	<title>Saturation_Framework</title>
	<link>/theories/saturation_framework/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/saturation_framework/</guid>
	<description></description>
	</item>

	<item>
	<title>Saturation_Framework_Extensions</title>
	<link>/theories/saturation_framework_extensions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/saturation_framework_extensions/</guid>
	<description></description>
	</item>

	<item>
	<title>Sauer_Shelah_Lemma</title>
	<link>/theories/sauer_shelah_lemma/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sauer_shelah_lemma/</guid>
	<description></description>
	</item>

	<item>
	<title>SC_DOM_Components</title>
	<link>/theories/sc_dom_components/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sc_dom_components/</guid>
	<description></description>
	</item>

	<item>
	<title>SCC_Bloemen_Sequential</title>
	<link>/theories/scc_bloemen_sequential/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/scc_bloemen_sequential/</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz_Spacetime</title>
	<link>/theories/schutz_spacetime/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/schutz_spacetime/</guid>
	<description></description>
	</item>

	<item>
	<title>Schwartz_Zippel</title>
	<link>/theories/schwartz_zippel/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/schwartz_zippel/</guid>
	<description></description>
	</item>

	<item>
	<title>SDS_Impossibility</title>
	<link>/theories/sds_impossibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sds_impossibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Secondary_Sylow</title>
	<link>/theories/secondary_sylow/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/secondary_sylow/</guid>
	<description></description>
	</item>

	<item>
	<title>Security_Protocol_Refinement</title>
	<link>/theories/security_protocol_refinement/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/security_protocol_refinement/</guid>
	<description></description>
	</item>

	<item>
	<title>Selection_Heap_Sort</title>
	<link>/theories/selection_heap_sort/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/selection_heap_sort/</guid>
	<description></description>
	</item>

	<item>
	<title>SenSocialChoice</title>
	<link>/theories/sensocialchoice/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sensocialchoice/</guid>
	<description></description>
	</item>

	<item>
	<title>Separata</title>
	<link>/theories/separata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separata/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Algebra</title>
	<link>/theories/separation_algebra/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_algebra/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Logic_Imperative_HOL</title>
	<link>/theories/separation_logic_imperative_hol/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_logic_imperative_hol/</guid>
	<description></description>
	</item>

	<item>
	<title>Separation_Logic_Unbounded</title>
	<link>/theories/separation_logic_unbounded/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/separation_logic_unbounded/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_Basic</title>
	<link>/theories/sepref_basic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_basic/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_IICF</title>
	<link>/theories/sepref_iicf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_iicf/</guid>
	<description></description>
	</item>

	<item>
	<title>Sepref_Prereq</title>
	<link>/theories/sepref_prereq/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sepref_prereq/</guid>
	<description></description>
	</item>

	<item>
	<title>SequentInvertibility</title>
	<link>/theories/sequentinvertibility/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sequentinvertibility/</guid>
	<description></description>
	</item>

	<item>
	<title>Shadow_DOM</title>
	<link>/theories/shadow_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shadow_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Shadow_SC_DOM</title>
	<link>/theories/shadow_sc_dom/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shadow_sc_dom/</guid>
	<description></description>
	</item>

	<item>
	<title>Shivers-CFA</title>
	<link>/theories/shivers-cfa/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shivers-cfa/</guid>
	<description></description>
	</item>

	<item>
	<title>ShortestPath</title>
	<link>/theories/shortestpath/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/shortestpath/</guid>
	<description></description>
	</item>

	<item>
	<title>Show</title>
	<link>/theories/show/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/show/</guid>
	<description></description>
	</item>

	<item>
	<title>SIFPL</title>
	<link>/theories/sifpl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sifpl/</guid>
	<description></description>
	</item>

	<item>
	<title>SIFUM_Type_Systems</title>
	<link>/theories/sifum_type_systems/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sifum_type_systems/</guid>
	<description></description>
	</item>

	<item>
	<title>Sigma_Commit_Crypto</title>
	<link>/theories/sigma_commit_crypto/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sigma_commit_crypto/</guid>
	<description></description>
	</item>

	<item>
	<title>Signature_Groebner</title>
	<link>/theories/signature_groebner/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/signature_groebner/</guid>
	<description></description>
	</item>

	<item>
	<title>Simpl</title>
	<link>/theories/simpl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simpl/</guid>
	<description></description>
	</item>

	<item>
	<title>Simple_Clause_Learning</title>
	<link>/theories/simple_clause_learning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simple_clause_learning/</guid>
	<description></description>
	</item>

	<item>
	<title>Simple_Firewall</title>
	<link>/theories/simple_firewall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simple_firewall/</guid>
	<description></description>
	</item>

	<item>
	<title>Simplex</title>
	<link>/theories/simplex/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplex/</guid>
	<description></description>
	</item>

	<item>
	<title>Simplicial_complexes_and_boolean_functions</title>
	<link>/theories/simplicial_complexes_and_boolean_functions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplicial_complexes_and_boolean_functions/</guid>
	<description></description>
	</item>

	<item>
	<title>SimplifiedOntologicalArgument</title>
	<link>/theories/simplifiedontologicalargument/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/simplifiedontologicalargument/</guid>
	<description></description>
	</item>

	<item>
	<title>Skew_Heap</title>
	<link>/theories/skew_heap/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/skew_heap/</guid>
	<description></description>
	</item>

	<item>
	<title>Skip_Lists</title>
	<link>/theories/skip_lists/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/skip_lists/</guid>
	<description></description>
	</item>

	<item>
	<title>Slicing</title>
	<link>/theories/slicing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/slicing/</guid>
	<description></description>
	</item>

	<item>
	<title>Sliding_Window_Algorithm</title>
	<link>/theories/sliding_window_algorithm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sliding_window_algorithm/</guid>
	<description></description>
	</item>

	<item>
	<title>SM</title>
	<link>/theories/sm/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sm/</guid>
	<description></description>
	</item>

	<item>
	<title>SM_Base</title>
	<link>/theories/sm_base/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sm_base/</guid>
	<description></description>
	</item>

	<item>
	<title>Smith_Normal_Form</title>
	<link>/theories/smith_normal_form/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/smith_normal_form/</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth_Manifolds</title>
	<link>/theories/smooth_manifolds/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/smooth_manifolds/</guid>
	<description></description>
	</item>

	<item>
	<title>Solidity</title>
	<link>/theories/solidity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/solidity/</guid>
	<description></description>
	</item>

	<item>
	<title>Sophomores_Dream</title>
	<link>/theories/sophomores_dream/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sophomores_dream/</guid>
	<description></description>
	</item>

	<item>
	<title>Sort_Encodings</title>
	<link>/theories/sort_encodings/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sort_encodings/</guid>
	<description></description>
	</item>

	<item>
	<title>Source_Coding_Theorem</title>
	<link>/theories/source_coding_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/source_coding_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>SPARCv8</title>
	<link>/theories/sparcv8/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sparcv8/</guid>
	<description></description>
	</item>

	<item>
	<title>SpecCheck</title>
	<link>/theories/speccheck/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/speccheck/</guid>
	<description></description>
	</item>

	<item>
	<title>Special_Function_Bounds</title>
	<link>/theories/special_function_bounds/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/special_function_bounds/</guid>
	<description></description>
	</item>

	<item>
	<title>Splay_Tree</title>
	<link>/theories/splay_tree/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/splay_tree/</guid>
	<description></description>
	</item>

	<item>
	<title>Sqrt_Babylonian</title>
	<link>/theories/sqrt_babylonian/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sqrt_babylonian/</guid>
	<description></description>
	</item>

	<item>
	<title>Stable_Matching</title>
	<link>/theories/stable_matching/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stable_matching/</guid>
	<description></description>
	</item>

	<item>
	<title>Stalnaker_Logic</title>
	<link>/theories/stalnaker_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stalnaker_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Statecharts</title>
	<link>/theories/statecharts/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/statecharts/</guid>
	<description></description>
	</item>

	<item>
	<title>Stateful_Protocol_Composition_and_Typing</title>
	<link>/theories/stateful_protocol_composition_and_typing/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stateful_protocol_composition_and_typing/</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar_Quorums</title>
	<link>/theories/stellar_quorums/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stellar_quorums/</guid>
	<description></description>
	</item>

	<item>
	<title>Stern_Brocot</title>
	<link>/theories/stern_brocot/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stern_brocot/</guid>
	<description></description>
	</item>

	<item>
	<title>Stewart_Apollonius</title>
	<link>/theories/stewart_apollonius/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stewart_apollonius/</guid>
	<description></description>
	</item>

	<item>
	<title>Stirling_Formula</title>
	<link>/theories/stirling_formula/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stirling_formula/</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic_Matrices</title>
	<link>/theories/stochastic_matrices/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stochastic_matrices/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Algebras</title>
	<link>/theories/stone_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Kleene_Relation_Algebras</title>
	<link>/theories/stone_kleene_relation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_kleene_relation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Stone_Relation_Algebras</title>
	<link>/theories/stone_relation_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stone_relation_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>Store_Buffer_Reduction</title>
	<link>/theories/store_buffer_reduction/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/store_buffer_reduction/</guid>
	<description></description>
	</item>

	<item>
	<title>Stream-Fusion</title>
	<link>/theories/stream-fusion/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stream-fusion/</guid>
	<description></description>
	</item>

	<item>
	<title>Stream_Fusion_Code</title>
	<link>/theories/stream_fusion_code/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stream_fusion_code/</guid>
	<description></description>
	</item>

	<item>
	<title>StrictOmegaCategories</title>
	<link>/theories/strictomegacategories/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/strictomegacategories/</guid>
	<description></description>
	</item>

	<item>
	<title>Strong_Security</title>
	<link>/theories/strong_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/strong_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm_Sequences</title>
	<link>/theories/sturm_sequences/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sturm_sequences/</guid>
	<description></description>
	</item>

	<item>
	<title>Sturm_Tarski</title>
	<link>/theories/sturm_tarski/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sturm_tarski/</guid>
	<description></description>
	</item>

	<item>
	<title>Stuttering_Equivalence</title>
	<link>/theories/stuttering_equivalence/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/stuttering_equivalence/</guid>
	<description></description>
	</item>

	<item>
	<title>Subresultants</title>
	<link>/theories/subresultants/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/subresultants/</guid>
	<description></description>
	</item>

	<item>
	<title>Subset_Boolean_Algebras</title>
	<link>/theories/subset_boolean_algebras/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/subset_boolean_algebras/</guid>
	<description></description>
	</item>

	<item>
	<title>SumSquares</title>
	<link>/theories/sumsquares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sumsquares/</guid>
	<description></description>
	</item>

	<item>
	<title>Sunflowers</title>
	<link>/theories/sunflowers/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/sunflowers/</guid>
	<description></description>
	</item>

	<item>
	<title>SuperCalc</title>
	<link>/theories/supercalc/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/supercalc/</guid>
	<description></description>
	</item>

	<item>
	<title>Suppes_Theorem</title>
	<link>/theories/suppes_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/suppes_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Surprise_Paradox</title>
	<link>/theories/surprise_paradox/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/surprise_paradox/</guid>
	<description></description>
	</item>

	<item>
	<title>Symmetric_Polynomials</title>
	<link>/theories/symmetric_polynomials/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/symmetric_polynomials/</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax_Independent_Logic</title>
	<link>/theories/syntax_independent_logic/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/syntax_independent_logic/</guid>
	<description></description>
	</item>

	<item>
	<title>Synthetic_Completeness</title>
	<link>/theories/synthetic_completeness/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/synthetic_completeness/</guid>
	<description></description>
	</item>

	<item>
	<title>Szemeredi_Regularity</title>
	<link>/theories/szemeredi_regularity/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/szemeredi_regularity/</guid>
	<description></description>
	</item>

	<item>
	<title>Szpilrajn</title>
	<link>/theories/szpilrajn/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/szpilrajn/</guid>
	<description></description>
	</item>

	<item>
	<title>Tail_Recursive_Functions</title>
	<link>/theories/tail_recursive_functions/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tail_recursive_functions/</guid>
	<description></description>
	</item>

	<item>
	<title>Tarskis_Geometry</title>
	<link>/theories/tarskis_geometry/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tarskis_geometry/</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor_Models</title>
	<link>/theories/taylor_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/taylor_models/</guid>
	<description></description>
	</item>

	<item>
	<title>TESL_Language</title>
	<link>/theories/tesl_language/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tesl_language/</guid>
	<description></description>
	</item>

	<item>
	<title>Three_Circles</title>
	<link>/theories/three_circles/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/three_circles/</guid>
	<description></description>
	</item>

	<item>
	<title>Three_Squares</title>
	<link>/theories/three_squares/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/three_squares/</guid>
	<description></description>
	</item>

	<item>
	<title>Timed_Automata</title>
	<link>/theories/timed_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/timed_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>TLA</title>
	<link>/theories/tla/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tla/</guid>
	<description></description>
	</item>

	<item>
	<title>Topological_Semantics</title>
	<link>/theories/topological_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/topological_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Topology</title>
	<link>/theories/topology/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/topology/</guid>
	<description></description>
	</item>

	<item>
	<title>TortoiseHare</title>
	<link>/theories/tortoisehare/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tortoisehare/</guid>
	<description></description>
	</item>

	<item>
	<title>Transcendence_Series_Hancl_Rucki</title>
	<link>/theories/transcendence_series_hancl_rucki/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transcendence_series_hancl_rucki/</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer_Semantics</title>
	<link>/theories/transformer_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transformer_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>Transition_Systems_and_Automata</title>
	<link>/theories/transition_systems_and_automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transition_systems_and_automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive-Closure</title>
	<link>/theories/transitive-closure/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive-closure/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive-Closure-II</title>
	<link>/theories/transitive-closure-ii/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive-closure-ii/</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive_Models</title>
	<link>/theories/transitive_models/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/transitive_models/</guid>
	<description></description>
	</item>

	<item>
	<title>Treaps</title>
	<link>/theories/treaps/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/treaps/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree-Automata</title>
	<link>/theories/tree-automata/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree-automata/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree_Decomposition</title>
	<link>/theories/tree_decomposition/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree_decomposition/</guid>
	<description></description>
	</item>

	<item>
	<title>Tree_Enumeration</title>
	<link>/theories/tree_enumeration/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tree_enumeration/</guid>
	<description></description>
	</item>

	<item>
	<title>Triangle</title>
	<link>/theories/triangle/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/triangle/</guid>
	<description></description>
	</item>

	<item>
	<title>Trie</title>
	<link>/theories/trie/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/trie/</guid>
	<description></description>
	</item>

	<item>
	<title>TsirelsonBound</title>
	<link>/theories/tsirelsonbound/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tsirelsonbound/</guid>
	<description></description>
	</item>

	<item>
	<title>Turans_Graph_Theorem</title>
	<link>/theories/turans_graph_theorem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/turans_graph_theorem/</guid>
	<description></description>
	</item>

	<item>
	<title>Twelvefold_Way</title>
	<link>/theories/twelvefold_way/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/twelvefold_way/</guid>
	<description></description>
	</item>

	<item>
	<title>Two_Generated_Word_Monoids_Intersection</title>
	<link>/theories/two_generated_word_monoids_intersection/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/two_generated_word_monoids_intersection/</guid>
	<description></description>
	</item>

	<item>
	<title>Tycon</title>
	<link>/theories/tycon/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/tycon/</guid>
	<description></description>
	</item>

	<item>
	<title>Types_Tableaus_and_Goedels_God</title>
	<link>/theories/types_tableaus_and_goedels_god/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/types_tableaus_and_goedels_god/</guid>
	<description></description>
	</item>

	<item>
	<title>Types_To_Sets_Extension</title>
	<link>/theories/types_to_sets_extension/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/types_to_sets_extension/</guid>
	<description></description>
	</item>

	<item>
	<title>Undirected_Graph_Theory</title>
	<link>/theories/undirected_graph_theory/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/undirected_graph_theory/</guid>
	<description></description>
	</item>

	<item>
	<title>Universal_Hash_Families</title>
	<link>/theories/universal_hash_families/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/universal_hash_families/</guid>
	<description></description>
	</item>

	<item>
	<title>Universal_Turing_Machine</title>
	<link>/theories/universal_turing_machine/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/universal_turing_machine/</guid>
	<description></description>
	</item>

	<item>
	<title>UpDown_Scheme</title>
	<link>/theories/updown_scheme/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/updown_scheme/</guid>
	<description></description>
	</item>

	<item>
	<title>UPF</title>
	<link>/theories/upf/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/upf/</guid>
	<description></description>
	</item>

	<item>
	<title>UPF_Firewall</title>
	<link>/theories/upf_firewall/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/upf_firewall/</guid>
	<description></description>
	</item>

	<item>
	<title>UTP</title>
	<link>/theories/utp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/utp/</guid>
	<description></description>
	</item>

	<item>
	<title>UTP-Toolkit</title>
	<link>/theories/utp-toolkit/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/utp-toolkit/</guid>
	<description></description>
	</item>

	<item>
	<title>Valuation</title>
	<link>/theories/valuation/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/valuation/</guid>
	<description></description>
	</item>

	<item>
	<title>Van_der_Waerden</title>
	<link>/theories/van_der_waerden/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/van_der_waerden/</guid>
	<description></description>
	</item>

	<item>
	<title>Van_Emde_Boas_Trees</title>
	<link>/theories/van_emde_boas_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/van_emde_boas_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>VectorSpace</title>
	<link>/theories/vectorspace/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vectorspace/</guid>
	<description></description>
	</item>

	<item>
	<title>VeriComp</title>
	<link>/theories/vericomp/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vericomp/</guid>
	<description></description>
	</item>

	<item>
	<title>Verified-Prover</title>
	<link>/theories/verified-prover/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verified-prover/</guid>
	<description></description>
	</item>

	<item>
	<title>Verified_SAT_Based_AI_Planning</title>
	<link>/theories/verified_sat_based_ai_planning/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verified_sat_based_ai_planning/</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis2018</title>
	<link>/theories/verifythis2018/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verifythis2018/</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis2019</title>
	<link>/theories/verifythis2019/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/verifythis2019/</guid>
	<description></description>
	</item>

	<item>
	<title>Vickrey_Clarke_Groves</title>
	<link>/theories/vickrey_clarke_groves/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vickrey_clarke_groves/</guid>
	<description></description>
	</item>

	<item>
	<title>Virtual_Substitution</title>
	<link>/theories/virtual_substitution/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/virtual_substitution/</guid>
	<description></description>
	</item>

	<item>
	<title>VolpanoSmith</title>
	<link>/theories/volpanosmith/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/volpanosmith/</guid>
	<description></description>
	</item>

	<item>
	<title>VYDRA_MDL</title>
	<link>/theories/vydra_mdl/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/vydra_mdl/</guid>
	<description></description>
	</item>

	<item>
	<title>WebAssembly</title>
	<link>/theories/webassembly/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/webassembly/</guid>
	<description></description>
	</item>

	<item>
	<title>Weight_Balanced_Trees</title>
	<link>/theories/weight_balanced_trees/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weight_balanced_trees/</guid>
	<description></description>
	</item>

	<item>
	<title>Weighted_Arithmetic_Geometric_Mean</title>
	<link>/theories/weighted_arithmetic_geometric_mean/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weighted_arithmetic_geometric_mean/</guid>
	<description></description>
	</item>

	<item>
	<title>Weighted_Path_Order</title>
	<link>/theories/weighted_path_order/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/weighted_path_order/</guid>
	<description></description>
	</item>

	<item>
	<title>Well_Quasi_Orders</title>
	<link>/theories/well_quasi_orders/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/well_quasi_orders/</guid>
	<description></description>
	</item>

	<item>
	<title>Wetzels_Problem</title>
	<link>/theories/wetzels_problem/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/wetzels_problem/</guid>
	<description></description>
	</item>

	<item>
	<title>WHATandWHERE_Security</title>
	<link>/theories/whatandwhere_security/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/whatandwhere_security/</guid>
	<description></description>
	</item>

	<item>
	<title>Winding_Number_Eval</title>
	<link>/theories/winding_number_eval/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/winding_number_eval/</guid>
	<description></description>
	</item>

	<item>
	<title>WOOT_Strong_Eventual_Consistency</title>
	<link>/theories/woot_strong_eventual_consistency/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/woot_strong_eventual_consistency/</guid>
	<description></description>
	</item>

	<item>
	<title>Word_Lib</title>
	<link>/theories/word_lib/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/word_lib/</guid>
	<description></description>
	</item>

	<item>
	<title>WorkerWrapper</title>
	<link>/theories/workerwrapper/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/workerwrapper/</guid>
	<description></description>
	</item>

	<item>
	<title>X86_Semantics</title>
	<link>/theories/x86_semantics/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/x86_semantics/</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/theories/xml/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/xml/</guid>
	<description></description>
	</item>

	<item>
	<title>Youngs_Inequality</title>
	<link>/theories/youngs_inequality/</link>
	<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>

	<guid>/theories/youngs_inequality/</guid>
	<description></description>
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	<h1 >
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	</header><div><h2>Subject Classification</h2><p>ACM: <a href="https://dl.acm.org/topic/ccs2012/10003752.10003809">Theory of computation~Design and analysis of algorithms</a></p><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=68Wxx">Computer science / Algorithms in computer science</a></p><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../../../entries/Earley_Parser.html">Earley Parser</a></h5> <br>by <a href="../../../authors/rau">Martin Rau</a></div>
	+ <span class="date">
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	+<article class="entry">
	+ <div class="item-text">
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	<article class="entry">
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	<h5><a class="title" href="../../../entries/Virtual_Substitution.html">Verified Quadratic Virtual Substitution for Real Arithmetic</a></h5> <br>by <a href="../../../authors/scharager">Matias Scharager</a>, <a href="../../../authors/cordwell">Katherine Kosaian</a>, <a href="../../../authors/mitsch">Stefan Mitsch</a> and <a href="../../../authors/platzer">André Platzer</a></div>
	<span class="date">
	Oct 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Regression_Test_Selection.html">Regression Test Selection</a></h5> <br>by <a href="../../../authors/mansky">Susannah Mansky</a></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/BenOr_Kozen_Reif.html">The BKR Decision Procedure for Univariate Real Arithmetic</a></h5> <br>by <a href="../../../authors/cordwell">Katherine Kosaian</a>, <a href="../../../authors/tan">Yong Kiam Tan</a> and <a href="../../../authors/platzer">André Platzer</a></div>
	<span class="date">
	Apr 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Progress_Tracking.html">Formalization of Timely Dataflow's Progress Tracking Protocol</a></h5> <br>by <a href="../../../authors/brun">Matthias Brun</a>, <a href="../../../authors/decova">Sára Decova</a>, <a href="../../../authors/lattuada">Andrea Lattuada</a> and <a href="../../../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Modular_arithmetic_LLL_and_HNF_algorithms.html">Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation</a></h5> <br>by <a href="../../../authors/bottesch">Ralph Bottesch</a>, <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Mar 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Projective_Measurements.html">Quantum projective measurements and the CHSH inequality</a></h5> <br>by <a href="../../../authors/echenim">Mnacho Echenim</a></div>
	<span class="date">
	Mar 03
	</span>
	</article>


	<h2 class="head">2020</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Relational_Minimum_Spanning_Trees.html">Relational Minimum Spanning Tree Algorithms</a></h5> <br>by <a href="../../../authors/guttmann">Walter Guttmann</a> and <a href="../../../authors/brien">Nicolas Robinson-O’Brien</a></div>
	<span class="date">
	Dec 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Isabelle_Marries_Dirac.html">Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information</a></h5> <br>by <a href="../../../authors/bordg">Anthony Bordg</a>, <a href="../../../authors/lachnitt">Hanna Lachnitt</a> and <a href="../../../authors/he">Yijun He</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/PAC_Checker.html">Practical Algebraic Calculus Checker</a></h5> <br>by <a href="../../../authors/fleury">Mathias Fleury</a> and <a href="../../../authors/kaufmann">Daniela Kaufmann</a></div>
	<span class="date">
	Aug 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Chandy_Lamport.html">A Formal Proof of The Chandy--Lamport Distributed Snapshot Algorithm</a></h5> <br>by <a href="../../../authors/fiedler">Ben Fiedler</a> and <a href="../../../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Jul 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Safe_Distance.html">A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles</a></h5> <br>by <a href="../../../authors/rizaldi">Albert Rizaldi</a> and <a href="../../../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Smith_Normal_Form.html">A verified algorithm for computing the Smith normal form of a matrix</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a></div>
	<span class="date">
	May 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Sliding_Window_Algorithm.html">Formalization of an Algorithm for Greedily Computing Associative Aggregations on Sliding Windows</a></h5> <br>by <a href="../../../authors/heimes">Lukas Heimes</a>, <a href="../../../authors/traytel">Dmitriy Traytel</a> and <a href="../../../authors/schneider">Joshua Schneider</a></div>
	<span class="date">
	Apr 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/MFODL_Monitor_Optimized.html">Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations</a></h5> <br>by <a href="../../../authors/dardinier">Thibault Dardinier</a>, <a href="../../../authors/heimes">Lukas Heimes</a>, <a href="../../../authors/raszyk">Martin Raszyk</a>, <a href="../../../authors/schneider">Joshua Schneider</a> and <a href="../../../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Apr 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/WOOT_Strong_Eventual_Consistency.html">Strong Eventual Consistency of the Collaborative Editing Framework WOOT</a></h5> <br>by <a href="../../../authors/karayel">Emin Karayel</a> and <a href="../../../authors/gonzalez">Edgar Gonzàlez</a></div>
	<span class="date">
	Mar 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Approximation_Algorithms.html">Verified Approximation Algorithms</a></h5> <br>by <a href="../../../authors/essmann">Robin Eßmann</a>, <a href="../../../authors/nipkow">Tobias Nipkow</a>, <a href="../../../authors/robillard">Simon Robillard</a> and <a href="../../../authors/sulejmani">Ujkan Sulejmani</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Closest_Pair_Points.html">Closest Pair of Points Algorithms</a></h5> <br>by <a href="../../../authors/rau">Martin Rau</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Skip_Lists.html">Skip Lists</a></h5> <br>by <a href="../../../authors/haslbeck">Max W. Haslbeck</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 09
	</span>
	</article>


	<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Generalized_Counting_Sort.html">An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</a></h5> <br>by <a href="../../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Dec 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/VerifyThis2019.html">VerifyThis 2019 -- Polished Isabelle Solutions</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/wimmer">Simon Wimmer</a></div>
	<span class="date">
	Oct 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Generic_Join.html">Formalization of Multiway-Join Algorithms</a></h5> <br>by <a href="../../../authors/dardinier">Thibault Dardinier</a></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Adaptive_State_Counting.html">Formalisation of an Adaptive State Counting Algorithm</a></h5> <br>by <a href="../../../authors/sachtleben">Robert Sachtleben</a></div>
	<span class="date">
	Aug 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stellar_Quorums.html">Stellar Quorum Systems</a></h5> <br>by <a href="../../../authors/losa">Giuliano Losa</a></div>
	<span class="date">
	Aug 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/MFOTL_Monitor.html">Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</a></h5> <br>by <a href="../../../authors/schneider">Joshua Schneider</a> and <a href="../../../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Jul 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Prim_Dijkstra_Simple.html">Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/IMP2_Binary_Heap.html">Binary Heaps for IMP2</a></h5> <br>by <a href="../../../authors/griebel">Simon Griebel</a></div>
	<span class="date">
	Jun 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Kruskal.html">Kruskal's Algorithm for Minimum Spanning Forest</a></h5> <br>by <a href="../../../authors/haslbeckm">Maximilian P. L. Haslbeck</a>, <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/biendarra">Julian Biendarra</a></div>
	<span class="date">
	Feb 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Probabilistic_Prime_Tests.html">Probabilistic Primality Testing</a></h5> <br>by <a href="../../../authors/stuewe">Daniel Stüwe</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Feb 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/List_Inversions.html">The Inversions of a List</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Feb 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/IMP2.html">IMP2 – Simple Program Verification in Isabelle/HOL</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/wimmer">Simon Wimmer</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Auto2_Imperative_HOL.html">Verifying Imperative Programs using Auto2</a></h5> <br>by <a href="../../../authors/zhan">Bohua Zhan</a></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Randomised_BSTs.html">Randomised Binary Search Trees</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Oct 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Signature_Groebner.html">Signature-Based Gröbner Basis Algorithms</a></h5> <br>by <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Simplex.html">An Incremental Simplex Algorithm with Unsatisfiable Core Generation</a></h5> <br>by <a href="../../../authors/maricf">Filip Marić</a>, <a href="../../../authors/spasic">Mirko Spasić</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Aug 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Optimal_BST.html">Optimal Binary Search Trees</a></h5> <br>by <a href="../../../authors/nipkow">Tobias Nipkow</a> and <a href="../../../authors/somogyi">Dániel Somogyi</a></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Hidden_Markov_Models.html">Hidden Markov Models</a></h5> <br>by <a href="../../../authors/wimmer">Simon Wimmer</a></div>
	<span class="date">
	May 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Monad_Memo_DP.html">Monadification, Memoization and Dynamic Programming</a></h5> <br>by <a href="../../../authors/wimmer">Simon Wimmer</a>, <a href="../../../authors/hu">Shuwei Hu</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/OpSets.html">OpSets: Sequential Specifications for Replicated Datatypes</a></h5> <br>by <a href="../../../authors/kleppmann">Martin Kleppmann</a>, <a href="../../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../../authors/mulligan">Dominic P. Mulligan</a> and <a href="../../../authors/beresford">Alastair R. Beresford</a></div>
	<span class="date">
	May 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/VerifyThis2018.html">VerifyThis 2018 - Polished Isabelle Solutions</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/wimmer">Simon Wimmer</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/First_Order_Terms.html">First-Order Terms</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/LLL_Basis_Reduction.html">A verified LLL algorithm</a></h5> <br>by <a href="../../../authors/bottesch">Ralph Bottesch</a>, <a href="../../../authors/divason">Jose Divasón</a>, <a href="../../../authors/haslbeck">Max W. Haslbeck</a>, <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Feb 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Taylor_Models.html">Taylor Models</a></h5> <br>by <a href="../../../authors/traut">Christoph Traut</a> and <a href="../../../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2017</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Median_Of_Medians_Selection.html">The Median-of-Medians Selection Algorithm</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Knuth_Morris_Pratt.html">The string search algorithm by Knuth, Morris and Pratt</a></h5> <br>by <a href="../../../authors/hellauer">Fabian Hellauer</a> and <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Dec 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/IMAP-CRDT.html">The IMAP CmRDT</a></h5> <br>by <a href="../../../authors/jungnickel">Tim Jungnickel</a>, <a href="../../../authors/oldenburg">Lennart Oldenburg</a> and <a href="../../../authors/loibl">Matthias Loibl</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Diophantine_Eqns_Lin_Hom.html">Homogeneous Linear Diophantine Equations</a></h5> <br>by <a href="../../../authors/messner">Florian Messner</a>, <a href="../../../authors/parsert">Julian Parsert</a>, <a href="../../../authors/schoepf">Jonas Schöpf</a> and <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/CRDT.html">A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</a></h5> <br>by <a href="../../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../../authors/kleppmann">Martin Kleppmann</a>, <a href="../../../authors/mulligan">Dominic P. Mulligan</a> and <a href="../../../authors/beresford">Alastair R. Beresford</a></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Prpu_Maxflow.html">Formalizing Push-Relabel Algorithms</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Jun 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Floyd_Warshall.html">The Floyd-Warshall Algorithm for Shortest Paths</a></h5> <br>by <a href="../../../authors/wimmer">Simon Wimmer</a> and <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Probabilistic_While.html">Probabilistic while loop</a></h5> <br>by <a href="../../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Random_BSTs.html">Expected Shape of Random Binary Search Trees</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Apr 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Quick_Sort_Cost.html">The number of comparisons in QuickSort</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Quick_Sort_Cost.html">The number of comparisons in QuickSort</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Comparison_Sort_Lower_Bound.html">Lower bound on comparison-based sorting algorithms</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Fisher_Yates.html">Fisher–Yates shuffle</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/EdmondsKarp_Maxflow.html">Formalizing the Edmonds-Karp Algorithm</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/DFS_Framework.html">A Framework for Verifying Depth-First Search Algorithms</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/neumann">René Neumann</a></div>
	<span class="date">
	Jul 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Groebner_Bases.html">Gröbner Bases Theory</a></h5> <br>by <a href="../../../authors/immler">Fabian Immler</a> and <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/ROBDD.html">Algorithms for Reduced Ordered Binary Decision Diagrams</a></h5> <br>by <a href="../../../authors/michaelis">Julius Michaelis</a>, <a href="../../../authors/haslbeck">Max W. Haslbeck</a>, <a href="../../../authors/lammich">Peter Lammich</a> and <a href="../../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/CYK.html">A formalisation of the Cocke-Younger-Kasami algorithm</a></h5> <br>by <a href="../../../authors/bortin">Maksym Bortin</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/List_Update.html">Analysis of List Update Algorithms</a></h5> <br>by <a href="../../../authors/haslbeckm">Maximilian P. L. Haslbeck</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/List_Update.html">Analysis of List Update Algorithms</a></h5> <br>by <a href="../../../authors/haslbeckm">Maximilian P. L. Haslbeck</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 17
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/TortoiseHare.html">The Tortoise and Hare Algorithm</a></h5> <br>by <a href="../../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Nov 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Hermite.html">Hermite Normal Form</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/ConcurrentGC.html">Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO</a></h5> <br>by <a href="../../../authors/gammie">Peter Gammie</a>, <a href="../../../authors/hosking">Tony Hosking</a> and <a href="../../../authors/engelhardt">Kai Engelhardt</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Consensus_Refined.html">Consensus Refined</a></h5> <br>by <a href="../../../authors/maric">Ognjen Marić</a> and <a href="../../../authors/sprenger">Christoph Sprenger</a></div>
	<span class="date">
	Mar 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/QR_Decomposition.html">QR Decomposition</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Echelon_Form.html">Echelon Form</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/UpDown_Scheme.html">Verification of the UpDown Scheme</a></h5> <br>by <a href="../../../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Jan 28
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Imperative_Insertion_Sort.html">Imperative Insertion Sort</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Sep 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gauss_Jordan.html">Gauss-Jordan Algorithm and Its Applications</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Sep 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Boolean_Expression_Checkers.html">Boolean Expression Checkers</a></h5> <br>by <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jun 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gabow_SCC.html">Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Roy_Floyd_Warshall.html">Transitive closure according to Roy-Floyd-Warshall</a></h5> <br>by <a href="../../../authors/wenzel">Makarius Wenzel</a></div>
	<span class="date">
	May 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Selection_Heap_Sort.html">Verification of Selection and Heap Sort Using Locales</a></h5> <br>by <a href="../../../authors/petrovic">Danijela Petrovic</a></div>
	<span class="date">
	Feb 11
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Heard_Of.html">Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</a></h5> <br>by <a href="../../../authors/debrat">Henri Debrat</a> and <a href="../../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Abortable_Linearizable_Modules.html">Abortable Linearizable Modules</a></h5> <br>by <a href="../../../authors/guerraoui">Rachid Guerraoui</a>, <a href="../../../authors/kuncak">Viktor Kuncak</a> and <a href="../../../authors/losa">Giuliano Losa</a></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Transitive-Closure-II.html">Executable Transitive Closures</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Dijkstra_Shortest_Path.html">Dijkstra's Shortest Path Algorithm</a></h5> <br>by <a href="../../../authors/nordhoff">Benedikt Nordhoff</a> and <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Efficient-Mergesort.html">Efficient Mergesort</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>by <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Transitive-Closure.html">Executable Transitive Closures of Finite Relations</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a>, <a href="../../../authors/thiemann">René Thiemann</a>, <a href="../../../authors/maletzky">Alexander Maletzky</a>, <a href="../../../authors/immler">Fabian Immler</a>, <a href="../../../authors/haftmann">Florian Haftmann</a>, <a href="../../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/GraphMarkingIBP.html">Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</a></h5> <br>by <a href="../../../authors/preoteasa">Viorel Preoteasa</a> and <a href="../../../authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>


	<h2 class="head">2008</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/SATSolverVerification.html">Formal Verification of Modern SAT Solvers</a></h5> <br>by <a href="../../../authors/maricf">Filip Marić</a></div>
	<span class="date">
	Jul 23
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/MuchAdoAboutTwo.html">Much Ado About Two</a></h5> <br>by <a href="../../../authors/boehme">Sascha Böhme</a></div>
	<span class="date">
	Nov 06
	</span>
	</article>


	<h2 class="head">2006</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/ClockSynchInst.html">Instances of Schneider's generalized protocol of clock synchronization</a></h5> <br>by <a href="../../../authors/barsotti">Damián Barsotti</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>


	<h2 class="head">2005</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/FFT.html">Fast Fourier Transform</a></h5> <br>by <a href="../../../authors/ballarin">Clemens Ballarin</a></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/GenClock.html">Formalization of a Generalized Protocol for Clock Synchronization</a></h5> <br>by <a href="../../../authors/tiu">Alwen Tiu</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/DiskPaxos.html">Proving the Correctness of Disk Paxos</a></h5> <br>by <a href="../../../authors/jaskelioff">Mauro Jaskelioff</a> and <a href="../../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jun 22
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Depth-First-Search.html">Depth First Search</a></h5> <br>by <a href="../../../authors/nishihara">Toshiaki Nishihara</a> and <a href="../../../authors/minamide">Yasuhiko Minamide</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>




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	<title>A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL</title>
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	<title>Strong Eventual Consistency of the Collaborative Editing Framework WOOT</title>
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	<pubDate>Wed, 25 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/WOOT_Strong_Eventual_Consistency.html</guid>
	<description></description>
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	<item>
	<title>Verified Approximation Algorithms</title>
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	<pubDate>Thu, 16 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Approximation_Algorithms.html</guid>
	<description></description>
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	<item>
	<title>Closest Pair of Points Algorithms</title>
	<link>/entries/Closest_Pair_Points.html</link>
	<pubDate>Mon, 13 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Closest_Pair_Points.html</guid>
	<description></description>
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	<item>
	<title>Skip Lists</title>
	<link>/entries/Skip_Lists.html</link>
	<pubDate>Thu, 09 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Skip_Lists.html</guid>
	<description></description>
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	<item>
	<title>An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</title>
	<link>/entries/Generalized_Counting_Sort.html</link>
	<pubDate>Wed, 04 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generalized_Counting_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2019 -- Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2019.html</link>
	<pubDate>Wed, 16 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Multiway-Join Algorithms</title>
	<link>/entries/Generic_Join.html</link>
	<pubDate>Mon, 16 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Join.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation of an Adaptive State Counting Algorithm</title>
	<link>/entries/Adaptive_State_Counting.html</link>
	<pubDate>Fri, 16 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Adaptive_State_Counting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar Quorum Systems</title>
	<link>/entries/Stellar_Quorums.html</link>
	<pubDate>Thu, 01 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Stellar_Quorums.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</title>
	<link>/entries/MFOTL_Monitor.html</link>
	<pubDate>Thu, 04 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/MFOTL_Monitor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra</title>
	<link>/entries/Prim_Dijkstra_Simple.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prim_Dijkstra_Simple.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Heaps for IMP2</title>
	<link>/entries/IMP2_Binary_Heap.html</link>
	<pubDate>Thu, 13 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2_Binary_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal's Algorithm for Minimum Spanning Forest</title>
	<link>/entries/Kruskal.html</link>
	<pubDate>Thu, 14 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Kruskal.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Primality Testing</title>
	<link>/entries/Probabilistic_Prime_Tests.html</link>
	<pubDate>Mon, 11 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Prime_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inversions of a List</title>
	<link>/entries/List_Inversions.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/List_Inversions.html</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2 – Simple Program Verification in Isabelle/HOL</title>
	<link>/entries/IMP2.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Imperative Programs using Auto2</title>
	<link>/entries/Auto2_Imperative_HOL.html</link>
	<pubDate>Fri, 21 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Binary Search Trees</title>
	<link>/entries/Randomised_BSTs.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
	<link>/entries/Signature_Groebner.html</link>
	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Signature_Groebner.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Incremental Simplex Algorithm with Unsatisfiable Core Generation</title>
	<link>/entries/Simplex.html</link>
	<pubDate>Fri, 24 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Simplex.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal Binary Search Trees</title>
	<link>/entries/Optimal_BST.html</link>
	<pubDate>Sun, 27 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Optimal_BST.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden Markov Models</title>
	<link>/entries/Hidden_Markov_Models.html</link>
	<pubDate>Fri, 25 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hidden_Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monadification, Memoization and Dynamic Programming</title>
	<link>/entries/Monad_Memo_DP.html</link>
	<pubDate>Tue, 22 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Memo_DP.html</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets: Sequential Specifications for Replicated Datatypes</title>
	<link>/entries/OpSets.html</link>
	<pubDate>Thu, 10 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/OpSets.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2018 - Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2018.html</link>
	<pubDate>Fri, 27 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2018.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Terms</title>
	<link>/entries/First_Order_Terms.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/First_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified LLL algorithm</title>
	<link>/entries/LLL_Basis_Reduction.html</link>
	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Median-of-Medians Selection Algorithm</title>
	<link>/entries/Median_Of_Medians_Selection.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Of_Medians_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>The string search algorithm by Knuth, Morris and Pratt</title>
	<link>/entries/Knuth_Morris_Pratt.html</link>
	<pubDate>Mon, 18 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Morris_Pratt.html</guid>
	<description></description>
	</item>

	<item>
	<title>The IMAP CmRDT</title>
	<link>/entries/IMAP-CRDT.html</link>
	<pubDate>Thu, 09 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/IMAP-CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Homogeneous Linear Diophantine Equations</title>
	<link>/entries/Diophantine_Eqns_Lin_Hom.html</link>
	<pubDate>Sat, 14 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Diophantine_Eqns_Lin_Hom.html</guid>
	<description></description>
	</item>

	<item>
	<title>A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</title>
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	<pubDate>Fri, 07 Jul 2017 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Formalizing Push-Relabel Algorithms</title>
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	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>The Floyd-Warshall Algorithm for Shortest Paths</title>
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	<pubDate>Mon, 08 May 2017 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
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	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Expected Shape of Random Binary Search Trees</title>
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	<pubDate>Tue, 04 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Random_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower bound on comparison-based sorting algorithms</title>
	<link>/entries/Comparison_Sort_Lower_Bound.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Comparison_Sort_Lower_Bound.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
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	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
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	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher–Yates shuffle</title>
	<link>/entries/Fisher_Yates.html</link>
	<pubDate>Fri, 30 Sep 2016 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Formalizing the Edmonds-Karp Algorithm</title>
	<link>/entries/EdmondsKarp_Maxflow.html</link>
	<pubDate>Fri, 12 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/EdmondsKarp_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Framework for Verifying Depth-First Search Algorithms</title>
	<link>/entries/DFS_Framework.html</link>
	<pubDate>Tue, 05 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/DFS_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases Theory</title>
	<link>/entries/Groebner_Bases.html</link>
	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Bases.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formalisation of the Cocke-Younger-Kasami algorithm</title>
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	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Algorithms for Reduced Ordered Binary Decision Diagrams</title>
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	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
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	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
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	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/List_Update.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Tortoise and Hare Algorithm</title>
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	<pubDate>Wed, 18 Nov 2015 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Hermite Normal Form</title>
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	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

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	<item>
	<title>Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO</title>
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	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Consensus Refined</title>
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	<pubDate>Wed, 18 Mar 2015 00:00:00 +0000</pubDate>

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	<item>
	<title>Echelon Form</title>
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	<description></description>
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	<item>
	<title>QR Decomposition</title>
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	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Verification of the UpDown Scheme</title>
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	<pubDate>Wed, 28 Jan 2015 00:00:00 +0000</pubDate>

	<guid>/entries/UpDown_Scheme.html</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative Insertion Sort</title>
	<link>/entries/Imperative_Insertion_Sort.html</link>
	<pubDate>Thu, 25 Sep 2014 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Algorithm and Its Applications</title>
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	<description></description>
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	<item>
	<title>Boolean Expression Checkers</title>
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	<pubDate>Sun, 08 Jun 2014 00:00:00 +0000</pubDate>

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	<item>
	<title>Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</title>
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	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gabow_SCC.html</guid>
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	<item>
	<title>Transitive closure according to Roy-Floyd-Warshall</title>
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	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

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	<item>
	<title>Verification of Selection and Heap Sort Using Locales</title>
	<link>/entries/Selection_Heap_Sort.html</link>
	<pubDate>Tue, 11 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Selection_Heap_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</title>
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	<pubDate>Fri, 27 Jul 2012 00:00:00 +0000</pubDate>

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	<item>
	<title>Abortable Linearizable Modules</title>
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	<pubDate>Thu, 01 Mar 2012 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Executable Transitive Closures</title>
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	<pubDate>Wed, 29 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure-II.html</guid>
	<description></description>
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	<item>
	<title>Dijkstra's Shortest Path Algorithm</title>
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	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Efficient Mergesort</title>
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	<pubDate>Wed, 09 Nov 2011 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
	<link>/entries/Gauss-Jordan-Elim-Fun.html</link>
	<pubDate>Fri, 19 Aug 2011 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Executable Transitive Closures of Finite Relations</title>
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	<pubDate>Mon, 14 Mar 2011 00:00:00 +0000</pubDate>

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	<title>Executable Multivariate Polynomials</title>
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	<pubDate>Tue, 10 Aug 2010 00:00:00 +0000</pubDate>

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	<title>Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</title>
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	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Formal Verification of Modern SAT Solvers</title>
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	<pubDate>Wed, 23 Jul 2008 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Much Ado About Two</title>
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	<pubDate>Tue, 06 Nov 2007 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Instances of Schneider's generalized protocol of clock synchronization</title>
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	<pubDate>Wed, 15 Mar 2006 00:00:00 +0000</pubDate>

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	<pubDate>Wed, 12 Oct 2005 00:00:00 +0000</pubDate>

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	<pubDate>Fri, 24 Jun 2005 00:00:00 +0000</pubDate>

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	<description></description>
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	<title>Proving the Correctness of Disk Paxos</title>
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	<pubDate>Wed, 22 Jun 2005 00:00:00 +0000</pubDate>

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	<h5><a class="title" href="../../entries/Quick_Sort_Cost.html">The number of comparisons in QuickSort</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
	</span>
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	<h5><a class="title" href="../../entries/Quick_Sort_Cost.html">The number of comparisons in QuickSort</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
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	<h5><a class="title" href="../../entries/Comparison_Sort_Lower_Bound.html">Lower bound on comparison-based sorting algorithms</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 15
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	<h5><a class="title" href="../../entries/Elliptic_Curves_Group_Law.html">The Group Law for Elliptic Curves</a></h5> <br>by <a href="../../authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Feb 28
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	<h5><a class="title" href="../../entries/Differential_Dynamic_Logic.html">Differential Dynamic Logic</a></h5> <br>by <a href="../../authors/bohrer">Rose Bohrer</a></div>
	<span class="date">
	Feb 13
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	<h5><a class="title" href="../../entries/Key_Agreement_Strong_Adversaries.html">Refining Authenticated Key Agreement with Strong Adversaries</a></h5> <br>by <a href="../../authors/lallemand">Joseph Lallemand</a> and <a href="../../authors/sprenger">Christoph Sprenger</a></div>
	<span class="date">
	Jan 31
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	<h5><a class="title" href="../../entries/Minimal_SSA.html">Minimal Static Single Assignment Form</a></h5> <br>by <a href="../../authors/wagner">Max Wagner</a> and <a href="../../authors/lohner">Denis Lohner</a></div>
	<span class="date">
	Jan 17
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	<h5><a class="title" href="../../entries/UPF_Firewall.html">Formal Network Models and Their Application to Firewall Policies</a></h5> <br>by <a href="../../authors/brucker">Achim D. Brucker</a>, <a href="../../authors/bruegger">Lukas Brügger</a> and <a href="../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Jan 08
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	<h5><a class="title" href="../../entries/UPF_Firewall.html">Formal Network Models and Their Application to Firewall Policies</a></h5> <br>by <a href="../../authors/brucker">Achim D. Brucker</a>, <a href="../../authors/bruegger">Lukas Brügger</a> and <a href="../../authors/wolff">Burkhart Wolff</a></div>
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	Jan 08
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	<h5><a class="title" href="../../entries/Password_Authentication_Protocol.html">Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Jan 03
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	<h2 class="head">2016</h2><article class="entry">
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	<h5><a class="title" href="../../entries/Concurrent_Ref_Alg.html">Concurrent Refinement Algebra and Rely Quotients</a></h5> <br>by <a href="../../authors/fell">Julian Fell</a>, <a href="../../authors/hayes">Ian J. Hayes</a> and <a href="../../authors/velykis">Andrius Velykis</a></div>
	<span class="date">
	Dec 30
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	<h5><a class="title" href="../../entries/Complx.html">COMPLX: A Verification Framework for Concurrent Imperative Programs</a></h5> <br>by <a href="../../authors/amani">Sidney Amani</a>, <a href="../../authors/andronick">June Andronick</a>, <a href="../../authors/bortin">Maksym Bortin</a>, <a href="../../authors/lewis">Corey Lewis</a>, <a href="../../authors/rizkallah">Christine Rizkallah</a> and <a href="../../authors/tuongj">Joseph Tuong</a></div>
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	Nov 29
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	<h5><a class="title" href="../../entries/Complx.html">COMPLX: A Verification Framework for Concurrent Imperative Programs</a></h5> <br>by <a href="../../authors/amani">Sidney Amani</a>, <a href="../../authors/andronick">June Andronick</a>, <a href="../../authors/bortin">Maksym Bortin</a>, <a href="../../authors/lewis">Corey Lewis</a>, <a href="../../authors/rizkallah">Christine Rizkallah</a> and <a href="../../authors/tuongj">Joseph Tuong</a></div>
	<span class="date">
	Nov 29
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	<h5><a class="title" href="../../entries/Abs_Int_ITP2012.html">Abstract Interpretation of Annotated Commands</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Nov 23
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	<h5><a class="title" href="../../entries/Separata.html">Separata: Isabelle tactics for Separation Algebra</a></h5> <br>by <a href="../../authors/hou">Zhe Hou</a>, <a href="../../authors/sanan">David Sanan</a>, <a href="../../authors/tiu">Alwen Tiu</a>, <a href="../../authors/gore">Rajeev Gore</a> and <a href="../../authors/clouston">Ranald Clouston</a></div>
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	Nov 16
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	<h5><a class="title" href="../../entries/Deep_Learning.html">Expressiveness of Deep Learning</a></h5> <br>by <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Nov 10
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	<h5><a class="title" href="../../entries/Modal_Logics_for_NTS.html">Modal Logics for Nominal Transition Systems</a></h5> <br>by <a href="../../authors/weber">Tjark Weber</a>, <a href="../../authors/eriksson">Lars-Henrik Eriksson</a>, <a href="../../authors/parrow">Joachim Parrow</a>, <a href="../../authors/borgstroem">Johannes Borgström</a> and <a href="../../authors/gutkovas">Ramunas Gutkovas</a></div>
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	<h5><a class="title" href="../../entries/LOFT.html">LOFT — Verified Migration of Linux Firewalls to SDN</a></h5> <br>by <a href="../../authors/michaelis">Julius Michaelis</a> and <a href="../../authors/diekmann">Cornelius Diekmann</a></div>
	<span class="date">
	Oct 21
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	<h5><a class="title" href="../../entries/SPARCv8.html">A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</a></h5> <br>by <a href="../../authors/hou">Zhe Hou</a>, <a href="../../authors/sanan">David Sanan</a>, <a href="../../authors/tiu">Alwen Tiu</a> and <a href="../../authors/liuy">Yang Liu</a></div>
	<span class="date">
	Oct 19
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	<h5><a class="title" href="../../entries/SPARCv8.html">A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</a></h5> <br>by <a href="../../authors/hou">Zhe Hou</a>, <a href="../../authors/sanan">David Sanan</a>, <a href="../../authors/tiu">Alwen Tiu</a> and <a href="../../authors/liuy">Yang Liu</a></div>
	<span class="date">
	Oct 19
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	<h5><a class="title" href="../../entries/Fisher_Yates.html">Fisher–Yates shuffle</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 30
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	<h5><a class="title" href="../../entries/Iptables_Semantics.html">Iptables Semantics</a></h5> <br>by <a href="../../authors/diekmann">Cornelius Diekmann</a> and <a href="../../authors/hupel">Lars Hupel</a></div>
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	Sep 09
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	<h5><a class="title" href="../../entries/Routing.html">Routing</a></h5> <br>by <a href="../../authors/michaelis">Julius Michaelis</a> and <a href="../../authors/diekmann">Cornelius Diekmann</a></div>
	<span class="date">
	Aug 31
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	<h5><a class="title" href="../../entries/Simple_Firewall.html">Simple Firewall</a></h5> <br>by <a href="../../authors/diekmann">Cornelius Diekmann</a>, <a href="../../authors/michaelis">Julius Michaelis</a> and <a href="../../authors/haslbeck">Max W. Haslbeck</a></div>
	<span class="date">
	Aug 24
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	<h5><a class="title" href="../../entries/InfPathElimination.html">Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths</a></h5> <br>by <a href="../../authors/aissat">Romain Aissat</a>, <a href="../../authors/voisin">Frederic Voisin</a> and <a href="../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Aug 18
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	<h5><a class="title" href="../../entries/EdmondsKarp_Maxflow.html">Formalizing the Edmonds-Karp Algorithm</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/sefidgar">S. Reza Sefidgar</a></div>
	<span class="date">
	Aug 12
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	<h5><a class="title" href="../../entries/Refine_Imperative_HOL.html">The Imperative Refinement Framework</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Aug 08
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	<h5><a class="title" href="../../entries/Refine_Imperative_HOL.html">The Imperative Refinement Framework</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Aug 08
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	<h5><a class="title" href="../../entries/Pairing_Heap.html">Pairing Heap</a></h5> <br>by <a href="../../authors/brinkop">Hauke Brinkop</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jul 14
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	<div class="item-text">
	<h5><a class="title" href="../../entries/DFS_Framework.html">A Framework for Verifying Depth-First Search Algorithms</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/neumann">René Neumann</a></div>
	<span class="date">
	Jul 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/IP_Addresses.html">IP Addresses</a></h5> <br>by <a href="../../authors/diekmann">Cornelius Diekmann</a>, <a href="../../authors/michaelis">Julius Michaelis</a> and <a href="../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Jun 28
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Dependent_SIFUM_Refinement.html">Compositional Security-Preserving Refinement for Concurrent Imperative Programs</a></h5> <br>by <a href="../../authors/murray">Toby Murray</a>, <a href="../../authors/sison">Robert Sison</a>, <a href="../../authors/pierzchalski">Edward Pierzchalski</a> and <a href="../../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jun 28
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Dependent_SIFUM_Type_Systems.html">A Dependent Security Type System for Concurrent Imperative Programs</a></h5> <br>by <a href="../../authors/murray">Toby Murray</a>, <a href="../../authors/sison">Robert Sison</a>, <a href="../../authors/pierzchalski">Edward Pierzchalski</a> and <a href="../../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jun 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Dependent_SIFUM_Type_Systems.html">A Dependent Security Type System for Concurrent Imperative Programs</a></h5> <br>by <a href="../../authors/murray">Toby Murray</a>, <a href="../../authors/sison">Robert Sison</a>, <a href="../../authors/pierzchalski">Edward Pierzchalski</a> and <a href="../../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jun 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Noninterference_Concurrent_Composition.html">Conservation of CSP Noninterference Security under Concurrent Composition</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Jun 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Noninterference_Concurrent_Composition.html">Conservation of CSP Noninterference Security under Concurrent Composition</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Jun 13
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Word_Lib.html">Finite Machine Word Library</a></h5> <br>by <a href="../../authors/beeren">Joel Beeren</a>, <a href="../../authors/fernandez">Matthew Fernandez</a>, <a href="../../authors/gao">Xin Gao</a>, <a href="../../authors/klein">Gerwin Klein</a>, <a href="../../authors/kolanski">Rafal Kolanski</a>, <a href="../../authors/lim">Japheth Lim</a>, <a href="../../authors/lewis">Corey Lewis</a>, <a href="../../authors/matichuk">Daniel Matichuk</a> and <a href="../../authors/sewell">Thomas Sewell</a></div>
	<span class="date">
	Jun 09
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Posix-Lexing.html">POSIX Lexing with Derivatives of Regular Expressions</a></h5> <br>by <a href="../../authors/ausaf">Fahad Ausaf</a>, <a href="../../authors/dyckhoff">Roy Dyckhoff</a> and <a href="../../authors/urban">Christian Urban</a></div>
	<span class="date">
	May 24
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	<h5><a class="title" href="../../entries/FLP.html">A Constructive Proof for FLP</a></h5> <br>by <a href="../../authors/bisping">Benjamin Bisping</a>, <a href="../../authors/brodmann">Paul-David Brodmann</a>, <a href="../../authors/jungnickel">Tim Jungnickel</a>, <a href="../../authors/rickmann">Christina Rickmann</a>, <a href="../../authors/seidler">Henning Seidler</a>, <a href="../../authors/stueber">Anke Stüber</a>, <a href="../../authors/weidner">Arno Wilhelm-Weidner</a>, <a href="../../authors/peters">Kirstin Peters</a> and <a href="../../authors/nestmann">Uwe Nestmann</a></div>
	<span class="date">
	May 18
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Groebner_Bases.html">Gröbner Bases Theory</a></h5> <br>by <a href="../../authors/immler">Fabian Immler</a> and <a href="../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/ROBDD.html">Algorithms for Reduced Ordered Binary Decision Diagrams</a></h5> <br>by <a href="../../authors/michaelis">Julius Michaelis</a>, <a href="../../authors/haslbeck">Max W. Haslbeck</a>, <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/ROBDD.html">Algorithms for Reduced Ordered Binary Decision Diagrams</a></h5> <br>by <a href="../../authors/michaelis">Julius Michaelis</a>, <a href="../../authors/haslbeck">Max W. Haslbeck</a>, <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/CYK.html">A formalisation of the Cocke-Younger-Kasami algorithm</a></h5> <br>by <a href="../../authors/bortin">Maksym Bortin</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/CYK.html">A formalisation of the Cocke-Younger-Kasami algorithm</a></h5> <br>by <a href="../../authors/bortin">Maksym Bortin</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Noninterference_Sequential_Composition.html">Conservation of CSP Noninterference Security under Sequential Composition</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Noninterference_Sequential_Composition.html">Conservation of CSP Noninterference Security under Sequential Composition</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../authors/guttmann">Walter Guttmann</a>, <a href="../../authors/hoefner">Peter Höfner</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Apr 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../authors/guttmann">Walter Guttmann</a>, <a href="../../authors/hoefner">Peter Höfner</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Apr 12
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Timed_Automata.html">Timed Automata</a></h5> <br>by <a href="../../authors/wimmer">Simon Wimmer</a></div>
	<span class="date">
	Mar 08
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	</article>
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	<div class="item-text">
	<h5><a class="title" href="../../entries/LTL.html">Linear Temporal Logic</a></h5> <br>by <a href="../../authors/sickert">Salomon Sickert</a></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/List_Update.html">Analysis of List Update Algorithms</a></h5> <br>by <a href="../../authors/haslbeckm">Maximilian P. L. Haslbeck</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 17
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/List_Update.html">Analysis of List Update Algorithms</a></h5> <br>by <a href="../../authors/haslbeckm">Maximilian P. L. Haslbeck</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 17
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Formal_SSA.html">Verified Construction of Static Single Assignment Form</a></h5> <br>by <a href="../../authors/ullrich">Sebastian Ullrich</a> and <a href="../../authors/lohner">Denis Lohner</a></div>
	<span class="date">
	Feb 05
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Matrix_Tensor.html">Tensor Product of Matrices</a></h5> <br>by <a href="../../authors/prathamesh">T.V.H. Prathamesh</a></div>
	<span class="date">
	Jan 18
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	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Applicative_Lifting.html">Applicative Lifting</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../authors/schneider">Joshua Schneider</a></div>
	<span class="date">
	Dec 22
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/TortoiseHare.html">The Tortoise and Hare Algorithm</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Nov 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Isabelle_Meta_Model.html">A Meta-Model for the Isabelle API</a></h5> <br>by <a href="../../authors/tuong">Frédéric Tuong</a> and <a href="../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/LTL_to_DRA.html">Converting Linear Temporal Logic to Deterministic (Generalized) Rabin Automata</a></h5> <br>by <a href="../../authors/sickert">Salomon Sickert</a></div>
	<span class="date">
	Sep 04
	</span>
	</article>
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	<div class="item-text">
	<h5><a class="title" href="../../entries/Noninterference_Inductive_Unwinding.html">The Inductive Unwinding Theorem for CSP Noninterference Security</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Aug 18
	</span>
	</article>
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	<h5><a class="title" href="../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Featherweight_OCL.html">Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5</a></h5> <br>by <a href="../../authors/brucker">Achim D. Brucker</a>, <a href="../../authors/tuong">Frédéric Tuong</a> and <a href="../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/CryptoBasedCompositionalProperties.html">Compositional Properties of Crypto-Based Components</a></h5> <br>by <a href="../../authors/spichkova">Maria Spichkova</a></div>
	<span class="date">
	Jan 11
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Tail_Recursive_Functions.html">A General Method for the Proof of Theorems on Tail-recursive Functions</a></h5> <br>by <a href="../../authors/noce">Pasquale Noce</a></div>
	<span class="date">
	Dec 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Coinductive_Languages.html">A Codatatype of Formal Languages</a></h5> <br>by <a href="../../authors/traytel">Dmitriy Traytel</a></div>
	<span class="date">
	Nov 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FocusStreamsCaseStudies.html">Stream Processing Components: Isabelle/HOL Formalisation and Case Studies</a></h5> <br>by <a href="../../authors/spichkova">Maria Spichkova</a></div>
	<span class="date">
	Nov 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Automatic_Refinement.html">Automatic Data Refinement</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Native_Word.html">Native Word</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Sep 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/IEEE_Floating_Point.html">A Formal Model of IEEE Floating Point Arithmetic</a></h5> <br>by <a href="../../authors/yu">Lei Yu</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Containers.html">Light-weight Containers</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Launchbury.html">The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</a></h5> <br>by <a href="../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jan 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Launchbury.html">The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</a></h5> <br>by <a href="../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jan 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Ribbon_Proofs.html">Ribbon Proofs</a></h5> <br>by <a href="../../authors/wickerson">John Wickerson</a></div>
	<span class="date">
	Jan 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Separation_Logic_Imperative_HOL.html">A Separation Logic Framework for Imperative HOL</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/meis">Rene Meis</a></div>
	<span class="date">
	Nov 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Possibilistic_Noninterference.html">Possibilistic Noninterference</a></h5> <br>by <a href="../../authors/popescu">Andrei Popescu</a> and <a href="../../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Possibilistic_Noninterference.html">Possibilistic Noninterference</a></h5> <br>by <a href="../../authors/popescu">Andrei Popescu</a> and <a href="../../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Sep 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Datatype_Order_Generator.html">Generating linear orders for datatypes</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Aug 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Heard_Of.html">Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</a></h5> <br>by <a href="../../authors/debrat">Henri Debrat</a> and <a href="../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/PCF.html">Logical Relations for PCF</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Jul 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Tycon.html">Type Constructor Classes and Monad Transformers</a></h5> <br>by <a href="../../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Jun 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Pi_Calculus.html">The pi-calculus in nominal logic</a></h5> <br>by <a href="../../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Psi_Calculi.html">Psi-calculi in Isabelle</a></h5> <br>by <a href="../../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/CCS.html">CCS in nominal logic</a></h5> <br>by <a href="../../authors/bengtson">Jesper Bengtson</a></div>
	<span class="date">
	May 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Circus.html">Isabelle/Circus</a></h5> <br>by <a href="../../authors/feliachi">Abderrahmane Feliachi</a>, <a href="../../authors/wolff">Burkhart Wolff</a> and <a href="../../authors/gaudel">Marie-Claude Gaudel</a></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Circus.html">Isabelle/Circus</a></h5> <br>by <a href="../../authors/feliachi">Abderrahmane Feliachi</a>, <a href="../../authors/wolff">Burkhart Wolff</a> and <a href="../../authors/gaudel">Marie-Claude Gaudel</a></div>
	<span class="date">
	May 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Separation_Algebra.html">Separation Algebra</a></h5> <br>by <a href="../../authors/klein">Gerwin Klein</a>, <a href="../../authors/kolanski">Rafal Kolanski</a> and <a href="../../authors/boyton">Andrew Boyton</a></div>
	<span class="date">
	May 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Stuttering_Equivalence.html">Stuttering Equivalence</a></h5> <br>by <a href="../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	May 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Inductive_Confidentiality.html">Inductive Study of Confidentiality</a></h5> <br>by <a href="../../authors/bella">Giampaolo Bella</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Abortable_Linearizable_Modules.html">Abortable Linearizable Modules</a></h5> <br>by <a href="../../authors/guerraoui">Rachid Guerraoui</a>, <a href="../../authors/kuncak">Viktor Kuncak</a> and <a href="../../authors/losa">Giuliano Losa</a></div>
	<span class="date">
	Mar 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Transitive-Closure-II.html">Executable Transitive Closures</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Refine_Monadic.html">Refinement for Monadic Programs</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Dijkstra_Shortest_Path.html">Dijkstra's Shortest Path Algorithm</a></h5> <br>by <a href="../../authors/nordhoff">Benedikt Nordhoff</a> and <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Markov_Models.html">Markov Models</a></h5> <br>by <a href="../../authors/hoelzl">Johannes Hölzl</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/TLA.html">A Definitional Encoding of TLA* in Isabelle/HOL</a></h5> <br>by <a href="../../authors/grov">Gudmund Grov</a> and <a href="../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Efficient-Mergesort.html">Efficient Mergesort</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/MonoBoolTranAlgebra.html">Algebra of Monotonic Boolean Transformers</a></h5> <br>by <a href="../../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Myhill-Nerode.html">The Myhill-Nerode Theorem Based on Regular Expressions</a></h5> <br>by <a href="../../authors/wu">Chunhan Wu</a>, <a href="../../authors/zhangx">Xingyuan Zhang</a> and <a href="../../authors/urban">Christian Urban</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KBPs.html">Knowledge-based programs</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	May 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Transitive-Closure.html">Executable Transitive Closures of Finite Relations</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a> and <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/List-Infinite.html">Infinite Lists</a></h5> <br>by <a href="../../authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/AutoFocus-Stream.html">AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</a></h5> <br>by <a href="../../authors/trachtenherz">David Trachtenherz</a></div>
	<span class="date">
	Feb 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/LightweightJava.html">Lightweight Java</a></h5> <br>by <a href="../../authors/strnisa">Rok Strniša</a> and <a href="../../authors/parkinson">Matthew Parkinson</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/RIPEMD-160-SPARK.html">RIPEMD-160</a></h5> <br>by <a href="../../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jan 10
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Shivers-CFA.html">Shivers' Control Flow Analysis</a></h5> <br>by <a href="../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Nov 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Binomial-Queues.html">Functional Binomial Queues</a></h5> <br>by <a href="../../authors/neumann">René Neumann</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Finger-Trees.html">Finger Trees</a></h5> <br>by <a href="../../authors/nordhoff">Benedikt Nordhoff</a>, <a href="../../authors/koerner">Stefan Körner</a> and <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Binomial-Heaps.html">Binomial Heaps and Skew Binomial Heaps</a></h5> <br>by <a href="../../authors/meis">Rene Meis</a>, <a href="../../authors/nielsen">Finn Nielsen</a> and <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Lam-ml-Normalization.html">Strong Normalization of Moggis's Computational Metalanguage</a></h5> <br>by <a href="../../authors/doczkal">Christian Doczkal</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a>, <a href="../../authors/thiemann">René Thiemann</a>, <a href="../../authors/maletzky">Alexander Maletzky</a>, <a href="../../authors/immler">Fabian Immler</a>, <a href="../../authors/haftmann">Florian Haftmann</a>, <a href="../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Statecharts.html">Formalizing Statecharts using Hierarchical Automata</a></h5> <br>by <a href="../../authors/helke">Steffen Helke</a> and <a href="../../authors/kammueller">Florian Kammüller</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Matrix.html">Executable Matrix Operations on Matrices of Arbitrary Dimensions</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a> and <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/GraphMarkingIBP.html">Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</a></h5> <br>by <a href="../../authors/preoteasa">Viorel Preoteasa</a> and <a href="../../authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/DataRefinementIBP.html">Semantics and Data Refinement of Invariant Based Programs</a></h5> <br>by <a href="../../authors/preoteasa">Viorel Preoteasa</a> and <a href="../../authors/back">Ralph-Johan Back</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Regular-Sets.html">Regular Sets and Expressions</a></h5> <br>by <a href="../../authors/krauss">Alexander Krauss</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Locally-Nameless-Sigma.html">Locally Nameless Sigma Calculus</a></h5> <br>by <a href="../../authors/henrio">Ludovic Henrio</a>, <a href="../../authors/kammueller">Florian Kammüller</a>, <a href="../../authors/lutz">Bianca Lutz</a> and <a href="../../authors/sudhof">Henry Sudhof</a></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/InformationFlowSlicing_Inter.html">Inter-Procedural Information Flow Noninterference via Slicing</a></h5> <br>by <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/InformationFlowSlicing.html">Information Flow Noninterference via Slicing</a></h5> <br>by <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/List-Index.html">List Index</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Coinductive.html">Coinductive</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Presburger-Automata.html">Formalizing the Logic-Automaton Connection</a></h5> <br>by <a href="../../authors/berghofer">Stefan Berghofer</a> and <a href="../../authors/reiter">Markus Reiter</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Tree-Automata.html">Tree Automata</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Collections.html">Collections Framework</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/HRB-Slicing.html">Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</a></h5> <br>by <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Nov 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/WorkerWrapper.html">The Worker/Wrapper Transformation</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Oct 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FinFun.html">Code Generation for Functions as Data</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Stream-Fusion.html">Stream Fusion</a></h5> <br>by <a href="../../authors/huffman">Brian Huffman</a></div>
	<span class="date">
	Apr 29
	</span>
	</article>


	<h2 class="head">2008</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/BytecodeLogicJmlTypes.html">A Bytecode Logic for JML and Types</a></h5> <br>by <a href="../../authors/beringer">Lennart Beringer</a> and <a href="../../authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Dec 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/SIFPL.html">Secure information flow and program logics</a></h5> <br>by <a href="../../authors/beringer">Lennart Beringer</a> and <a href="../../authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/SIFPL.html">Secure information flow and program logics</a></h5> <br>by <a href="../../authors/beringer">Lennart Beringer</a> and <a href="../../authors/hofmann">Martin Hofmann</a></div>
	<span class="date">
	Nov 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Huffman.html">The Textbook Proof of Huffman's Algorithm</a></h5> <br>by <a href="../../authors/blanchette">Jasmin Christian Blanchette</a></div>
	<span class="date">
	Oct 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Slicing.html">Towards Certified Slicing</a></h5> <br>by <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/VolpanoSmith.html">A Correctness Proof for the Volpano/Smith Security Typing System</a></h5> <br>by <a href="../../authors/snelting">Gregor Snelting</a> and <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/VolpanoSmith.html">A Correctness Proof for the Volpano/Smith Security Typing System</a></h5> <br>by <a href="../../authors/snelting">Gregor Snelting</a> and <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/SATSolverVerification.html">Formal Verification of Modern SAT Solvers</a></h5> <br>by <a href="../../authors/maricf">Filip Marić</a></div>
	<span class="date">
	Jul 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/BDD.html">BDD Normalisation</a></h5> <br>by <a href="../../authors/ortner">Veronika Ortner</a> and <a href="../../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Simpl.html">A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</a></h5> <br>by <a href="../../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Simpl.html">A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment</a></h5> <br>by <a href="../../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Feb 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/NormByEval.html">Normalization by Evaluation</a></h5> <br>by <a href="../../authors/aehlig">Klaus Aehlig</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Feb 18
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Program-Conflict-Analysis.html">Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a> and <a href="../../authors/olm">Markus Müller-Olm</a></div>
	<span class="date">
	Dec 14
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/JinjaThreads.html">Jinja with Threads</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Dec 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/MuchAdoAboutTwo.html">Much Ado About Two</a></h5> <br>by <a href="../../authors/boehme">Sascha Böhme</a></div>
	<span class="date">
	Nov 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/POPLmark-deBruijn.html">POPLmark Challenge Via de Bruijn Indices</a></h5> <br>by <a href="../../authors/berghofer">Stefan Berghofer</a></div>
	<span class="date">
	Aug 02
	</span>
	</article>


	<h2 class="head">2006</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/HotelKeyCards.html">Hotel Key Card System</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Abstract-Hoare-Logics.html">Abstract Hoare Logics</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/CoreC++.html">CoreC++</a></h5> <br>by <a href="../../authors/wasserrab">Daniel Wasserrab</a></div>
	<span class="date">
	May 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FeatherweightJava.html">A Theory of Featherweight Java in Isabelle/HOL</a></h5> <br>by <a href="../../authors/fosterj">J. Nathan Foster</a> and <a href="../../authors/vytiniotis">Dimitrios Vytiniotis</a></div>
	<span class="date">
	Mar 31
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/ClockSynchInst.html">Instances of Schneider's generalized protocol of clock synchronization</a></h5> <br>by <a href="../../authors/barsotti">Damián Barsotti</a></div>
	<span class="date">
	Mar 15
	</span>
	</article>


	<h2 class="head">2005</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FFT.html">Fast Fourier Transform</a></h5> <br>by <a href="../../authors/ballarin">Clemens Ballarin</a></div>
	<span class="date">
	Oct 12
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/GenClock.html">Formalization of a Generalized Protocol for Clock Synchronization</a></h5> <br>by <a href="../../authors/tiu">Alwen Tiu</a></div>
	<span class="date">
	Jun 24
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/DiskPaxos.html">Proving the Correctness of Disk Paxos</a></h5> <br>by <a href="../../authors/jaskelioff">Mauro Jaskelioff</a> and <a href="../../authors/merz">Stephan Merz</a></div>
	<span class="date">
	Jun 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/JiveDataStoreModel.html">Jive Data and Store Model</a></h5> <br>by <a href="../../authors/rauch">Nicole Rauch</a> and <a href="../../authors/schirmer">Norbert Schirmer</a></div>
	<span class="date">
	Jun 20
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	<div class="item-text">
	<h5><a class="title" href="../../entries/Jinja.html">Jinja is not Java</a></h5> <br>by <a href="../../authors/klein">Gerwin Klein</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
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	Jun 01
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	<div class="item-text">
	<h5><a class="title" href="../../entries/RSAPSS.html">SHA1, RSA, PSS and more</a></h5> <br>by <a href="../../authors/lindenberg">Christina Lindenberg</a> and <a href="../../authors/wirt">Kai Wirt</a></div>
	<span class="date">
	May 02
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FileRefinement.html">File Refinement</a></h5> <br>by <a href="../../authors/zee">Karen Zee</a> and <a href="../../authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Dec 09
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Compiling-Exceptions-Correctly.html">Compiling Exceptions Correctly</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jul 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Depth-First-Search.html">Depth First Search</a></h5> <br>by <a href="../../authors/nishihara">Toshiaki Nishihara</a> and <a href="../../authors/minamide">Yasuhiko Minamide</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Lazy-Lists-II.html">Lazy Lists II</a></h5> <br>by <a href="../../authors/friedrich">Stefan Friedrich</a></div>
	<span class="date">
	Apr 26
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/BinarySearchTree.html">Binary Search Trees</a></h5> <br>by <a href="../../authors/kuncak">Viktor Kuncak</a></div>
	<span class="date">
	Apr 05
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Functional-Automata.html">Functional Automata</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Mar 30
	</span>
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	<div class="item-text">
	<h5><a class="title" href="../../entries/MiniML.html">Mini ML</a></h5> <br>by <a href="../../authors/naraschewski">Wolfgang Naraschewski</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
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	Mar 19
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	<h5><a class="title" href="../../entries/AVL-Trees.html">AVL Trees</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a> and <a href="../../authors/pusch">Cornelia Pusch</a></div>
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	Mar 19
	</span>
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	<title>Maximum Segment Sum</title>
	<link>/entries/Maximum_Segment_Sum.html</link>
	<pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Maximum_Segment_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Making Arbitrary Relational Calculus Queries Safe-Range</title>
	<link>/entries/Safe_Range_RC.html</link>
	<pubDate>Wed, 28 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Range_RC.html</guid>
	<description></description>
	</item>

	<item>
	<title>CRYSTALS-Kyber</title>
	<link>/entries/CRYSTALS-Kyber.html</link>
	<pubDate>Thu, 08 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/CRYSTALS-Kyber.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unbounded Separation Logic</title>
	<link>/entries/Separation_Logic_Unbounded.html</link>
	<pubDate>Mon, 05 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Unbounded.html</guid>
	<description></description>
	</item>

	<item>
	<title>Number Theoretic Transform</title>
	<link>/entries/Number_Theoretic_Transform.html</link>
	<pubDate>Thu, 18 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Number_Theoretic_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Correctness of a Set-based Algorithm for Computing Strongly Connected Components of a Graph</title>
	<link>/entries/SCC_Bloemen_Sequential.html</link>
	<pubDate>Wed, 17 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/SCC_Bloemen_Sequential.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Complete Test Strategies for Finite State Machines</title>
	<link>/entries/FSM_Tests.html</link>
	<pubDate>Tue, 09 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FSM_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Complete Test Strategies for Finite State Machines</title>
	<link>/entries/FSM_Tests.html</link>
	<pubDate>Tue, 09 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FSM_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Nano JSON: Working with JSON formatted data in Isabelle/HOL and Isabelle/ML</title>
	<link>/entries/Nano_JSON.html</link>
	<pubDate>Fri, 29 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Nano_JSON.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL</title>
	<link>/entries/Solidity.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Solidity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL</title>
	<link>/entries/Solidity.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Solidity.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reuse-Based Multi-Stage Compiler Verification for Language IMP</title>
	<link>/entries/IMP_Compiler_Reuse.html</link>
	<pubDate>Sun, 10 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler_Reuse.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Time Double-Ended Queue</title>
	<link>/entries/Real_Time_Deque.html</link>
	<pubDate>Thu, 23 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Time_Deque.html</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet: Formalization of a Verification Framework for Secure Data Plane Protocols</title>
	<link>/entries/IsaNet.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IsaNet.html</guid>
	<description></description>
	</item>

	<item>
	<title>IsaNet: Formalization of a Verification Framework for Secure Data Plane Protocols</title>
	<link>/entries/IsaNet.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/IsaNet.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Restricted Definition of the Magic Wand to Soundly Combine Fractions of a Wand</title>
	<link>/entries/Combinable_Wands.html</link>
	<pubDate>Mon, 30 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Combinable_Wands.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Framework for the Sound Automation of Magic Wands</title>
	<link>/entries/Package_logic.html</link>
	<pubDate>Wed, 18 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Package_logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Combinator Library for Prefix-Free Codes</title>
	<link>/entries/Prefix_Free_Code_Combinators.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Prefix_Free_Code_Combinators.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Combinator Library for Prefix-Free Codes</title>
	<link>/entries/Prefix_Free_Code_Combinators.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Prefix_Free_Code_Combinators.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Randomized Approximation Algorithms for Frequency Moments</title>
	<link>/entries/Frequency_Moments.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Frequency_Moments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Randomized Approximation Algorithms for Frequency Moments</title>
	<link>/entries/Frequency_Moments.html</link>
	<pubDate>Fri, 08 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Frequency_Moments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Transition Systems</title>
	<link>/entries/ResiduatedTransitionSystem.html</link>
	<pubDate>Mon, 28 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/ResiduatedTransitionSystem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Transition Systems</title>
	<link>/entries/ResiduatedTransitionSystem.html</link>
	<pubDate>Mon, 28 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/ResiduatedTransitionSystem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Transition Systems</title>
	<link>/entries/ResiduatedTransitionSystem.html</link>
	<pubDate>Mon, 28 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/ResiduatedTransitionSystem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Hash Families</title>
	<link>/entries/Universal_Hash_Families.html</link>
	<pubDate>Sun, 20 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Hash_Families.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Query Evaluation</title>
	<link>/entries/Eval_FO.html</link>
	<pubDate>Tue, 15 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Eval_FO.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Head Monitoring of Metric Dynamic Logic</title>
	<link>/entries/VYDRA_MDL.html</link>
	<pubDate>Sun, 13 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/VYDRA_MDL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Enumeration of Equivalence Relations</title>
	<link>/entries/Equivalence_Relation_Enumeration.html</link>
	<pubDate>Fri, 04 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Equivalence_Relation_Enumeration.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi-Borel Spaces</title>
	<link>/entries/Quasi_Borel_Spaces.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Quasi_Borel_Spaces.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Theory of Rewriting</title>
	<link>/entries/FO_Theory_Rewriting.html</link>
	<pubDate>Wed, 02 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/FO_Theory_Rewriting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Median Method</title>
	<link>/entries/Median_Method.html</link>
	<pubDate>Tue, 25 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Shapley Algorithm</title>
	<link>/entries/Gale_Shapley.html</link>
	<pubDate>Wed, 29 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Gale_Shapley.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Algorithms for Solving Markov Decision Processes</title>
	<link>/entries/MDP-Algorithms.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Tree Relations</title>
	<link>/entries/Regular_Tree_Relations.html</link>
	<pubDate>Wed, 15 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Tree_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>van Emde Boas Trees</title>
	<link>/entries/Van_Emde_Boas_Trees.html</link>
	<pubDate>Tue, 23 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_Emde_Boas_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum and Classical Registers</title>
	<link>/entries/Registers.html</link>
	<pubDate>Thu, 28 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Registers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum and Classical Registers</title>
	<link>/entries/Registers.html</link>
	<pubDate>Thu, 28 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Registers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum and Classical Registers</title>
	<link>/entries/Registers.html</link>
	<pubDate>Thu, 28 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Registers.html</guid>
	<description></description>
	</item>

	<item>
	<title>X86 instruction semantics and basic block symbolic execution</title>
	<link>/entries/X86_Semantics.html</link>
	<pubDate>Wed, 13 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/X86_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>X86 instruction semantics and basic block symbolic execution</title>
	<link>/entries/X86_Semantics.html</link>
	<pubDate>Wed, 13 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/X86_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebras for Iteration, Infinite Executions and Correctness of Sequential Computations</title>
	<link>/entries/Correctness_Algebras.html</link>
	<pubDate>Tue, 12 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Correctness_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Quadratic Virtual Substitution for Real Arithmetic</title>
	<link>/entries/Virtual_Substitution.html</link>
	<pubDate>Sat, 02 Oct 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Virtual_Substitution.html</guid>
	<description></description>
	</item>

	<item>
	<title>A data flow analysis algorithm for computing dominators</title>
	<link>/entries/Dominance_CHK.html</link>
	<pubDate>Sun, 05 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Dominance_CHK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logging-independent Message Anonymity in the Relational Method</title>
	<link>/entries/Logging_Independent_Anonymity.html</link>
	<pubDate>Thu, 26 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Logging_Independent_Anonymity.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoCon: A Confidentiality-Verified Conference Management System</title>
	<link>/entries/CoCon.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoCon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional BD Security</title>
	<link>/entries/BD_Security_Compositional.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BD_Security_Compositional.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMed: A confidentiality-verified social media platform</title>
	<link>/entries/CoSMed.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMed.html</guid>
	<description></description>
	</item>

	<item>
	<title>CoSMeDis: A confidentiality-verified distributed social media platform</title>
	<link>/entries/CoSMeDis.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CoSMeDis.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fresh identifiers</title>
	<link>/entries/Fresh_Identifiers.html</link>
	<pubDate>Mon, 16 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Fresh_Identifiers.html</guid>
	<description></description>
	</item>

	<item>
	<title>MiniSail - A kernel language for the ISA specification language SAIL</title>
	<link>/entries/MiniSail.html</link>
	<pubDate>Fri, 18 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MiniSail.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Shorter Compiler Correctness Proof for Language IMP</title>
	<link>/entries/IMP_Compiler.html</link>
	<pubDate>Fri, 04 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IMP_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorics on Words Basics</title>
	<link>/entries/Combinatorics_Words.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Lemma</title>
	<link>/entries/Combinatorics_Words_Graph_Lemma.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Graph_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lyndon words</title>
	<link>/entries/Combinatorics_Words_Lyndon.html</link>
	<pubDate>Mon, 24 May 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Combinatorics_Words_Lyndon.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regression Test Selection</title>
	<link>/entries/Regression_Test_Selection.html</link>
	<pubDate>Fri, 30 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Regression_Test_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>The BKR Decision Procedure for Univariate Real Arithmetic</title>
	<link>/entries/BenOr_Kozen_Reif.html</link>
	<pubDate>Sat, 24 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BenOr_Kozen_Reif.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Timely Dataflow's Progress Tracking Protocol</title>
	<link>/entries/Progress_Tracking.html</link>
	<pubDate>Tue, 13 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Progress_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Control via Dependency Tracking</title>
	<link>/entries/IFC_Tracking.html</link>
	<pubDate>Thu, 01 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IFC_Tracking.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL: the Communication Modeling Aspect</title>
	<link>/entries/Constructive_Cryptography_CM.html</link>
	<pubDate>Wed, 17 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography_CM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation</title>
	<link>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</link>
	<pubDate>Fri, 12 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_arithmetic_LLL_and_HNF_algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum projective measurements and the CHSH inequality</title>
	<link>/entries/Projective_Measurements.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Measurements.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Imperative Implementation of B-Trees</title>
	<link>/entries/BTree.html</link>
	<pubDate>Wed, 24 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BTree.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Imperative Implementation of B-Trees</title>
	<link>/entries/BTree.html</link>
	<pubDate>Wed, 24 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/BTree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hood-Melville Queue</title>
	<link>/entries/Hood_Melville_Queue.html</link>
	<pubDate>Mon, 18 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hood_Melville_Queue.html</guid>
	<description></description>
	</item>

	<item>
	<title>JinjaDCI: a Java semantics with dynamic class initialization</title>
	<link>/entries/JinjaDCI.html</link>
	<pubDate>Mon, 11 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/JinjaDCI.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Minimum Spanning Tree Algorithms</title>
	<link>/entries/Relational_Minimum_Spanning_Trees.html</link>
	<pubDate>Tue, 08 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Minimum_Spanning_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inline Caching and Unboxing Optimization for Interpreters</title>
	<link>/entries/Interpreter_Optimizations.html</link>
	<pubDate>Mon, 07 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Interpreter_Optimizations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Relational Method with Message Anonymity for the Verification of Cryptographic Protocols</title>
	<link>/entries/Relational_Method.html</link>
	<pubDate>Sat, 05 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Method.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information</title>
	<link>/entries/Isabelle_Marries_Dirac.html</link>
	<pubDate>Sun, 22 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Marries_Dirac.html</guid>
	<description></description>
	</item>

	<item>
	<title>The HOL-CSP Refinement Toolkit</title>
	<link>/entries/CSP_RefTK.html</link>
	<pubDate>Thu, 19 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/CSP_RefTK.html</guid>
	<description></description>
	</item>

	<item>
	<title>The HOL-CSP Refinement Toolkit</title>
	<link>/entries/CSP_RefTK.html</link>
	<pubDate>Thu, 19 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/CSP_RefTK.html</guid>
	<description></description>
	</item>

	<item>
	<title>AI Planning Languages Semantics</title>
	<link>/entries/AI_Planning_Languages_Semantics.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/AI_Planning_Languages_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified SAT-Based AI Planning</title>
	<link>/entries/Verified_SAT_Based_AI_Planning.html</link>
	<pubDate>Thu, 29 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Verified_SAT_Based_AI_Planning.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sound Type System for Physical Quantities, Units, and Measurements</title>
	<link>/entries/Physical_Quantities.html</link>
	<pubDate>Tue, 20 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Physical_Quantities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Map Extras</title>
	<link>/entries/Finite-Map-Extras.html</link>
	<pubDate>Mon, 12 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Finite-Map-Extras.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Safely Composable Document Object Model with Shadow Roots</title>
	<link>/entries/Shadow_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Shadow_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Safely Composable Web Components</title>
	<link>/entries/SC_DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/SC_DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Web Components</title>
	<link>/entries/DOM_Components.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/DOM_Components.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safely Composable DOM</title>
	<link>/entries/Core_SC_DOM.html</link>
	<pubDate>Mon, 28 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Core_SC_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machines.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machines.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inference of Extended Finite State Machines</title>
	<link>/entries/Extended_Finite_State_Machine_Inference.html</link>
	<pubDate>Mon, 07 Sep 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Extended_Finite_State_Machine_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Practical Algebraic Calculus Checker</title>
	<link>/entries/PAC_Checker.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/PAC_Checker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Some classical results in inductive inference of recursive functions</title>
	<link>/entries/Inductive_Inference.html</link>
	<pubDate>Mon, 31 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Disjoint-Set Forests</title>
	<link>/entries/Relational_Disjoint_Set_Forests.html</link>
	<pubDate>Wed, 26 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Disjoint_Set_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Putting the `K' into Bird's derivation of Knuth-Morris-Pratt string matching</title>
	<link>/entries/BirdKMP.html</link>
	<pubDate>Tue, 25 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/BirdKMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of The Chandy--Lamport Distributed Snapshot Algorithm</title>
	<link>/entries/Chandy_Lamport.html</link>
	<pubDate>Tue, 21 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Chandy_Lamport.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles</title>
	<link>/entries/Safe_Distance.html</link>
	<pubDate>Mon, 01 Jun 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Distance.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified algorithm for computing the Smith normal form of a matrix</title>
	<link>/entries/Smith_Normal_Form.html</link>
	<pubDate>Sat, 23 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Smith_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Normalisation Procedure for Linear Temporal Logic: Isabelle/HOL Formalisation</title>
	<link>/entries/LTL_Normal_Form.html</link>
	<pubDate>Fri, 08 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Attack Trees in Isabelle for GDPR compliance of IoT healthcare systems</title>
	<link>/entries/Attack_Trees.html</link>
	<pubDate>Mon, 27 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Attack_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Authenticated Data Structures As Functors</title>
	<link>/entries/ADS_Functor.html</link>
	<pubDate>Thu, 16 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/ADS_Functor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Algorithm for Greedily Computing Associative Aggregations on Sliding Windows</title>
	<link>/entries/Sliding_Window_Algorithm.html</link>
	<pubDate>Fri, 10 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Sliding_Window_Algorithm.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations</title>
	<link>/entries/MFODL_Monitor_Optimized.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/MFODL_Monitor_Optimized.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of an Optimized Monitoring Algorithm for Metric First-Order Dynamic Logic with Aggregations</title>
	<link>/entries/MFODL_Monitor_Optimized.html</link>
	<pubDate>Thu, 09 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/MFODL_Monitor_Optimized.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automated Stateful Protocol Verification</title>
	<link>/entries/Automated_Stateful_Protocol_Verification.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Automated_Stateful_Protocol_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stateful Protocol Composition and Typing</title>
	<link>/entries/Stateful_Protocol_Composition_and_Typing.html</link>
	<pubDate>Wed, 08 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Stateful_Protocol_Composition_and_Typing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Eventual Consistency of the Collaborative Editing Framework WOOT</title>
	<link>/entries/WOOT_Strong_Eventual_Consistency.html</link>
	<pubDate>Wed, 25 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/WOOT_Strong_Eventual_Consistency.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Under-Approximate Relational Logic</title>
	<link>/entries/Relational-Incorrectness-Logic.html</link>
	<pubDate>Thu, 12 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational-Incorrectness-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Under-Approximate Relational Logic</title>
	<link>/entries/Relational-Incorrectness-Logic.html</link>
	<pubDate>Thu, 12 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational-Incorrectness-Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hello World</title>
	<link>/entries/Hello_World.html</link>
	<pubDate>Sat, 07 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Hello_World.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Generic Framework for Verified Compilers</title>
	<link>/entries/VeriComp.html</link>
	<pubDate>Mon, 10 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/VeriComp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Approximation Algorithms</title>
	<link>/entries/Approximation_Algorithms.html</link>
	<pubDate>Thu, 16 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Approximation_Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Closest Pair of Points Algorithms</title>
	<link>/entries/Closest_Pair_Points.html</link>
	<pubDate>Mon, 13 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Closest_Pair_Points.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skip Lists</title>
	<link>/entries/Skip_Lists.html</link>
	<pubDate>Thu, 09 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Skip_Lists.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skip Lists</title>
	<link>/entries/Skip_Lists.html</link>
	<pubDate>Thu, 09 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Skip_Lists.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</title>
	<link>/entries/Generalized_Counting_Sort.html</link>
	<pubDate>Wed, 04 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generalized_Counting_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Efficient Generalization of Counting Sort for Large, possibly Infinite Key Ranges</title>
	<link>/entries/Generalized_Counting_Sort.html</link>
	<pubDate>Wed, 04 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generalized_Counting_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interval Arithmetic on 32-bit Words</title>
	<link>/entries/Interval_Arithmetic_Word32.html</link>
	<pubDate>Wed, 27 Nov 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Interval_Arithmetic_Word32.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/C</title>
	<link>/entries/Isabelle_C.html</link>
	<pubDate>Tue, 22 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_C.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/C</title>
	<link>/entries/Isabelle_C.html</link>
	<pubDate>Tue, 22 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_C.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2019 -- Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2019.html</link>
	<pubDate>Wed, 16 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Sigma Protocols and Commitment Schemes</title>
	<link>/entries/Sigma_Commit_Crypto.html</link>
	<pubDate>Mon, 07 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Sigma_Commit_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clean - An Abstract Imperative Programming Language and its Theory</title>
	<link>/entries/Clean.html</link>
	<pubDate>Fri, 04 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Clean.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clean - An Abstract Imperative Programming Language and its Theory</title>
	<link>/entries/Clean.html</link>
	<pubDate>Fri, 04 Oct 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Clean.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Multiway-Join Algorithms</title>
	<link>/entries/Generic_Join.html</link>
	<pubDate>Mon, 16 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Join.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Multiway-Join Algorithms</title>
	<link>/entries/Generic_Join.html</link>
	<pubDate>Mon, 16 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Join.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation of an Adaptive State Counting Algorithm</title>
	<link>/entries/Adaptive_State_Counting.html</link>
	<pubDate>Fri, 16 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Adaptive_State_Counting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation of an Adaptive State Counting Algorithm</title>
	<link>/entries/Adaptive_State_Counting.html</link>
	<pubDate>Fri, 16 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Adaptive_State_Counting.html</guid>
	<description></description>
	</item>

	<item>
	<title>Communicating Concurrent Kleene Algebra for Distributed Systems Specification</title>
	<link>/entries/C2KA_DistributedSystems.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/C2KA_DistributedSystems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stellar Quorum Systems</title>
	<link>/entries/Stellar_Quorums.html</link>
	<pubDate>Thu, 01 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Stellar_Quorums.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Development of a Polychronous Polytimed Coordination Language</title>
	<link>/entries/TESL_Language.html</link>
	<pubDate>Tue, 30 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/TESL_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Development of a Polychronous Polytimed Coordination Language</title>
	<link>/entries/TESL_Language.html</link>
	<pubDate>Tue, 30 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/TESL_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Development of a Polychronous Polytimed Coordination Language</title>
	<link>/entries/TESL_Language.html</link>
	<pubDate>Tue, 30 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/TESL_Language.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Code Generator from Isabelle/HOL to CakeML</title>
	<link>/entries/CakeML_Codegen.html</link>
	<pubDate>Mon, 08 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML_Codegen.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</title>
	<link>/entries/MFOTL_Monitor.html</link>
	<pubDate>Thu, 04 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/MFOTL_Monitor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of a Monitoring Algorithm for Metric First-Order Temporal Logic</title>
	<link>/entries/MFOTL_Monitor.html</link>
	<pubDate>Thu, 04 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/MFOTL_Monitor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Search Trees</title>
	<link>/entries/Priority_Search_Trees.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Search_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Purely Functional, Simple, and Efficient Implementation of Prim and Dijkstra</title>
	<link>/entries/Prim_Dijkstra_Simple.html</link>
	<pubDate>Tue, 25 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prim_Dijkstra_Simple.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Heaps for IMP2</title>
	<link>/entries/IMP2_Binary_Heap.html</link>
	<pubDate>Thu, 13 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2_Binary_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Heaps for IMP2</title>
	<link>/entries/IMP2_Binary_Heap.html</link>
	<pubDate>Thu, 13 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2_Binary_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Game Logic</title>
	<link>/entries/Differential_Game_Logic.html</link>
	<pubDate>Mon, 03 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Game_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multidimensional Binary Search Trees</title>
	<link>/entries/KD_Tree.html</link>
	<pubDate>Thu, 30 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/KD_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Generic Authenticated Data Structures</title>
	<link>/entries/LambdaAuth.html</link>
	<pubDate>Tue, 14 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaAuth.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Generic Authenticated Data Structures</title>
	<link>/entries/LambdaAuth.html</link>
	<pubDate>Tue, 14 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaAuth.html</guid>
	<description></description>
	</item>

	<item>
	<title>Multi-Party Computation</title>
	<link>/entries/Multi_Party_Computation.html</link>
	<pubDate>Thu, 09 May 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Multi_Party_Computation.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP Version 2.0</title>
	<link>/entries/HOL-CSP.html</link>
	<pubDate>Fri, 26 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/HOL-CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOL-CSP Version 2.0</title>
	<link>/entries/HOL-CSP.html</link>
	<pubDate>Fri, 26 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/HOL-CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Compositional and Unified Translation of LTL into ω-Automata</title>
	<link>/entries/LTL_Master_Theorem.html</link>
	<pubDate>Tue, 16 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_Master_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Theory of Syntax with Bindings</title>
	<link>/entries/Binding_Syntax_Theory.html</link>
	<pubDate>Sat, 06 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Binding_Syntax_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Theory of Syntax with Bindings</title>
	<link>/entries/Binding_Syntax_Theory.html</link>
	<pubDate>Sat, 06 Apr 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Binding_Syntax_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum Hoare Logic</title>
	<link>/entries/QHLProver.html</link>
	<pubDate>Sun, 24 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/QHLProver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum Hoare Logic</title>
	<link>/entries/QHLProver.html</link>
	<pubDate>Sun, 24 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/QHLProver.html</guid>
	<description></description>
	</item>

	<item>
	<title>Safe OCL</title>
	<link>/entries/Safe_OCL.html</link>
	<pubDate>Sat, 09 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kruskal's Algorithm for Minimum Spanning Forest</title>
	<link>/entries/Kruskal.html</link>
	<pubDate>Thu, 14 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Kruskal.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Primality Testing</title>
	<link>/entries/Probabilistic_Prime_Tests.html</link>
	<pubDate>Mon, 11 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Prime_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Universal Turing Machine</title>
	<link>/entries/Universal_Turing_Machine.html</link>
	<pubDate>Fri, 08 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Universal_Turing_Machine.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/UTP: Mechanised Theory Engineering for Unifying Theories of Programming</title>
	<link>/entries/UTP.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/UTP.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inversions of a List</title>
	<link>/entries/List_Inversions.html</link>
	<pubDate>Fri, 01 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/List_Inversions.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Algebra for Higher-Order Terms</title>
	<link>/entries/Higher_Order_Terms.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Higher_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2 – Simple Program Verification in Isabelle/HOL</title>
	<link>/entries/IMP2.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2.html</guid>
	<description></description>
	</item>

	<item>
	<title>IMP2 – Simple Program Verification in Isabelle/HOL</title>
	<link>/entries/IMP2.html</link>
	<pubDate>Tue, 15 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMP2.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Reduction Theorem for Store Buffers</title>
	<link>/entries/Store_Buffer_Reduction.html</link>
	<pubDate>Mon, 07 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Store_Buffer_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of the Document Object Model</title>
	<link>/entries/Core_DOM.html</link>
	<pubDate>Wed, 26 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Core_DOM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Concurrent Revisions</title>
	<link>/entries/Concurrent_Revisions.html</link>
	<pubDate>Tue, 25 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Revisions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Imperative Programs using Auto2</title>
	<link>/entries/Auto2_Imperative_HOL.html</link>
	<pubDate>Fri, 21 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Imperative Programs using Auto2</title>
	<link>/entries/Auto2_Imperative_HOL.html</link>
	<pubDate>Fri, 21 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Auto2_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL</title>
	<link>/entries/Constructive_Cryptography.html</link>
	<pubDate>Mon, 17 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer Semantics</title>
	<link>/entries/Transformer_Semantics.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Transformer_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving generic class instances for datatypes</title>
	<link>/entries/Generic_Deriving.html</link>
	<pubDate>Tue, 06 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Generic_Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Binary Search Trees</title>
	<link>/entries/Randomised_BSTs.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Binary Search Trees</title>
	<link>/entries/Randomised_BSTs.html</link>
	<pubDate>Fri, 19 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Upper Bounding Diameters of State Spaces of Factored Transition Systems</title>
	<link>/entries/Factored_Transition_System_Bounding.html</link>
	<pubDate>Fri, 12 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Factored_Transition_System_Bounding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
	<link>/entries/Signature_Groebner.html</link>
	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Signature_Groebner.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Incremental Simplex Algorithm with Unsatisfiable Core Generation</title>
	<link>/entries/Simplex.html</link>
	<pubDate>Fri, 24 Aug 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Simplex.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Order Reduction</title>
	<link>/entries/Partial_Order_Reduction.html</link>
	<pubDate>Tue, 05 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Order_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal Binary Search Trees</title>
	<link>/entries/Optimal_BST.html</link>
	<pubDate>Sun, 27 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Optimal_BST.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optimal Binary Search Trees</title>
	<link>/entries/Optimal_BST.html</link>
	<pubDate>Sun, 27 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Optimal_BST.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden Markov Models</title>
	<link>/entries/Hidden_Markov_Models.html</link>
	<pubDate>Fri, 25 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hidden_Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Timed Automata</title>
	<link>/entries/Probabilistic_Timed_Automata.html</link>
	<pubDate>Thu, 24 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monadification, Memoization and Dynamic Programming</title>
	<link>/entries/Monad_Memo_DP.html</link>
	<pubDate>Tue, 22 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Memo_DP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monadification, Memoization and Dynamic Programming</title>
	<link>/entries/Monad_Memo_DP.html</link>
	<pubDate>Tue, 22 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Memo_DP.html</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets: Sequential Specifications for Replicated Datatypes</title>
	<link>/entries/OpSets.html</link>
	<pubDate>Thu, 10 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/OpSets.html</guid>
	<description></description>
	</item>

	<item>
	<title>OpSets: Sequential Specifications for Replicated Datatypes</title>
	<link>/entries/OpSets.html</link>
	<pubDate>Thu, 10 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/OpSets.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL Formalization of the Modular Assembly Kit for Security Properties</title>
	<link>/entries/Modular_Assembly_Kit_Security.html</link>
	<pubDate>Mon, 07 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Modular_Assembly_Kit_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>WebAssembly</title>
	<link>/entries/WebAssembly.html</link>
	<pubDate>Sun, 29 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/WebAssembly.html</guid>
	<description></description>
	</item>

	<item>
	<title>VerifyThis 2018 - Polished Isabelle Solutions</title>
	<link>/entries/VerifyThis2018.html</link>
	<pubDate>Fri, 27 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/VerifyThis2018.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded Natural Functors with Covariance and Contravariance</title>
	<link>/entries/BNF_CC.html</link>
	<pubDate>Tue, 24 Apr 2018 00:00:00 +0000</pubDate>

	<guid>/entries/BNF_CC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Weight-Balanced Trees</title>
	<link>/entries/Weight_Balanced_Trees.html</link>
	<pubDate>Tue, 13 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Weight_Balanced_Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>CakeML</title>
	<link>/entries/CakeML.html</link>
	<pubDate>Mon, 12 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/CakeML.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Theory of Architectural Design Patterns</title>
	<link>/entries/Architectural_Design_Patterns.html</link>
	<pubDate>Thu, 01 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Architectural_Design_Patterns.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hoare Logics for Time Bounds</title>
	<link>/entries/Hoare_Time.html</link>
	<pubDate>Mon, 26 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hoare_Time.html</guid>
	<description></description>
	</item>

	<item>
	<title>First-Order Terms</title>
	<link>/entries/First_Order_Terms.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/First_Order_Terms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Treaps</title>
	<link>/entries/Treaps.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Treaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified LLL algorithm</title>
	<link>/entries/LLL_Basis_Reduction.html</link>
	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Median-of-Medians Selection Algorithm</title>
	<link>/entries/Median_Of_Medians_Selection.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Median_Of_Medians_Selection.html</guid>
	<description></description>
	</item>

	<item>
	<title>The string search algorithm by Knuth, Morris and Pratt</title>
	<link>/entries/Knuth_Morris_Pratt.html</link>
	<pubDate>Mon, 18 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Knuth_Morris_Pratt.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic Matrices and the Perron-Frobenius Theorem</title>
	<link>/entries/Stochastic_Matrices.html</link>
	<pubDate>Wed, 22 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stochastic_Matrices.html</guid>
	<description></description>
	</item>

	<item>
	<title>The IMAP CmRDT</title>
	<link>/entries/IMAP-CRDT.html</link>
	<pubDate>Thu, 09 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/IMAP-CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>The IMAP CmRDT</title>
	<link>/entries/IMAP-CRDT.html</link>
	<pubDate>Thu, 09 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/IMAP-CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Büchi Complementation</title>
	<link>/entries/Buchi_Complementation.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buchi_Complementation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transition Systems and Automata</title>
	<link>/entries/Transition_Systems_and_Automata.html</link>
	<pubDate>Thu, 19 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Transition_Systems_and_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Homogeneous Linear Diophantine Equations</title>
	<link>/entries/Diophantine_Eqns_Lin_Hom.html</link>
	<pubDate>Sat, 14 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Diophantine_Eqns_Lin_Hom.html</guid>
	<description></description>
	</item>

	<item>
	<title>Root-Balanced Tree</title>
	<link>/entries/Root_Balanced_Tree.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Root_Balanced_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>The LambdaMu-calculus</title>
	<link>/entries/LambdaMu.html</link>
	<pubDate>Wed, 16 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LambdaMu.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dynamic Architectures</title>
	<link>/entries/DynamicArchitectures.html</link>
	<pubDate>Fri, 28 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/DynamicArchitectures.html</guid>
	<description></description>
	</item>

	<item>
	<title>Declarative Semantics for Functional Languages</title>
	<link>/entries/Decl_Sem_Fun_PL.html</link>
	<pubDate>Fri, 21 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Decl_Sem_Fun_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>HOLCF-Prelude</title>
	<link>/entries/HOLCF-Prelude.html</link>
	<pubDate>Sat, 15 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/HOLCF-Prelude.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Metatheory and Type Inference for a Name-Carrying Simply-Typed Lambda Calculus</title>
	<link>/entries/Name_Carrying_Type_Inference.html</link>
	<pubDate>Sun, 09 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Name_Carrying_Type_Inference.html</guid>
	<description></description>
	</item>

	<item>
	<title>A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</title>
	<link>/entries/CRDT.html</link>
	<pubDate>Fri, 07 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>A framework for establishing Strong Eventual Consistency for Conflict-free Replicated Datatypes</title>
	<link>/entries/CRDT.html</link>
	<pubDate>Fri, 07 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CRDT.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Push-Relabel Algorithms</title>
	<link>/entries/Prpu_Maxflow.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Prpu_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/entries/Optics.html</link>
	<pubDate>Thu, 25 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Optics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Developing Security Protocols by Refinement</title>
	<link>/entries/Security_Protocol_Refinement.html</link>
	<pubDate>Wed, 24 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Security_Protocol_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Floyd-Warshall Algorithm for Shortest Paths</title>
	<link>/entries/Floyd_Warshall.html</link>
	<pubDate>Mon, 08 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/entries/CryptHOL.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CryptHOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/entries/CryptHOL.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CryptHOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Effect polymorphism in higher-order logic</title>
	<link>/entries/Monomorphic_Monad.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monomorphic_Monad.html</guid>
	<description></description>
	</item>

	<item>
	<title>Game-based cryptography in HOL</title>
	<link>/entries/Game_Based_Crypto.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Game_Based_Crypto.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monad normalisation</title>
	<link>/entries/Monad_Normalisation.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Monad_Normalisation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
	<link>/entries/Probabilistic_While.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_While.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
	<link>/entries/Probabilistic_While.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_While.html</guid>
	<description></description>
	</item>

	<item>
	<title>Local Lexing</title>
	<link>/entries/LocalLexing.html</link>
	<pubDate>Fri, 28 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/LocalLexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expected Shape of Random Binary Search Trees</title>
	<link>/entries/Random_BSTs.html</link>
	<pubDate>Tue, 04 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Random_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expected Shape of Random Binary Search Trees</title>
	<link>/entries/Random_BSTs.html</link>
	<pubDate>Tue, 04 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Random_BSTs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lower bound on comparison-based sorting algorithms</title>
	<link>/entries/Comparison_Sort_Lower_Bound.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Comparison_Sort_Lower_Bound.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
	<link>/entries/Quick_Sort_Cost.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>The number of comparisons in QuickSort</title>
	<link>/entries/Quick_Sort_Cost.html</link>
	<pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Quick_Sort_Cost.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Group Law for Elliptic Curves</title>
	<link>/entries/Elliptic_Curves_Group_Law.html</link>
	<pubDate>Tue, 28 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Elliptic_Curves_Group_Law.html</guid>
	<description></description>
	</item>

	<item>
	<title>Differential Dynamic Logic</title>
	<link>/entries/Differential_Dynamic_Logic.html</link>
	<pubDate>Mon, 13 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Differential_Dynamic_Logic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refining Authenticated Key Agreement with Strong Adversaries</title>
	<link>/entries/Key_Agreement_Strong_Adversaries.html</link>
	<pubDate>Tue, 31 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Key_Agreement_Strong_Adversaries.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minimal Static Single Assignment Form</title>
	<link>/entries/Minimal_SSA.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minimal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Network Models and Their Application to Firewall Policies</title>
	<link>/entries/UPF_Firewall.html</link>
	<pubDate>Sun, 08 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/UPF_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Network Models and Their Application to Firewall Policies</title>
	<link>/entries/UPF_Firewall.html</link>
	<pubDate>Sun, 08 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/UPF_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of a Diffie-Hellman Password-based Authentication Protocol by Extending the Inductive Method</title>
	<link>/entries/Password_Authentication_Protocol.html</link>
	<pubDate>Tue, 03 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Password_Authentication_Protocol.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent Refinement Algebra and Rely Quotients</title>
	<link>/entries/Concurrent_Ref_Alg.html</link>
	<pubDate>Fri, 30 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Concurrent_Ref_Alg.html</guid>
	<description></description>
	</item>

	<item>
	<title>COMPLX: A Verification Framework for Concurrent Imperative Programs</title>
	<link>/entries/Complx.html</link>
	<pubDate>Tue, 29 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Complx.html</guid>
	<description></description>
	</item>

	<item>
	<title>COMPLX: A Verification Framework for Concurrent Imperative Programs</title>
	<link>/entries/Complx.html</link>
	<pubDate>Tue, 29 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Complx.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abstract Interpretation of Annotated Commands</title>
	<link>/entries/Abs_Int_ITP2012.html</link>
	<pubDate>Wed, 23 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Abs_Int_ITP2012.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separata: Isabelle tactics for Separation Algebra</title>
	<link>/entries/Separata.html</link>
	<pubDate>Wed, 16 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Separata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expressiveness of Deep Learning</title>
	<link>/entries/Deep_Learning.html</link>
	<pubDate>Thu, 10 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Deep_Learning.html</guid>
	<description></description>
	</item>

	<item>
	<title>Modal Logics for Nominal Transition Systems</title>
	<link>/entries/Modal_Logics_for_NTS.html</link>
	<pubDate>Tue, 25 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Modal_Logics_for_NTS.html</guid>
	<description></description>
	</item>

	<item>
	<title>LOFT — Verified Migration of Linux Firewalls to SDN</title>
	<link>/entries/LOFT.html</link>
	<pubDate>Fri, 21 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LOFT.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</title>
	<link>/entries/SPARCv8.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SPARCv8.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formal model for the SPARCv8 ISA and a proof of non-interference for the LEON3 processor</title>
	<link>/entries/SPARCv8.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SPARCv8.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher–Yates shuffle</title>
	<link>/entries/Fisher_Yates.html</link>
	<pubDate>Fri, 30 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Fisher_Yates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Iptables Semantics</title>
	<link>/entries/Iptables_Semantics.html</link>
	<pubDate>Fri, 09 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Iptables_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Routing</title>
	<link>/entries/Routing.html</link>
	<pubDate>Wed, 31 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Routing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simple Firewall</title>
	<link>/entries/Simple_Firewall.html</link>
	<pubDate>Wed, 24 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Simple_Firewall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infeasible Paths Elimination by Symbolic Execution Techniques: Proof of Correctness and Preservation of Paths</title>
	<link>/entries/InfPathElimination.html</link>
	<pubDate>Thu, 18 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/InfPathElimination.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Edmonds-Karp Algorithm</title>
	<link>/entries/EdmondsKarp_Maxflow.html</link>
	<pubDate>Fri, 12 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/EdmondsKarp_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Imperative Refinement Framework</title>
	<link>/entries/Refine_Imperative_HOL.html</link>
	<pubDate>Mon, 08 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Imperative Refinement Framework</title>
	<link>/entries/Refine_Imperative_HOL.html</link>
	<pubDate>Mon, 08 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pairing Heap</title>
	<link>/entries/Pairing_Heap.html</link>
	<pubDate>Thu, 14 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Pairing_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Framework for Verifying Depth-First Search Algorithms</title>
	<link>/entries/DFS_Framework.html</link>
	<pubDate>Tue, 05 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/DFS_Framework.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Security-Preserving Refinement for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Refinement.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>IP Addresses</title>
	<link>/entries/IP_Addresses.html</link>
	<pubDate>Tue, 28 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/IP_Addresses.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Dependent Security Type System for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Type_Systems.html</link>
	<pubDate>Sat, 25 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Dependent Security Type System for Concurrent Imperative Programs</title>
	<link>/entries/Dependent_SIFUM_Type_Systems.html</link>
	<pubDate>Sat, 25 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Dependent_SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Concurrent Composition</title>
	<link>/entries/Noninterference_Concurrent_Composition.html</link>
	<pubDate>Mon, 13 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Concurrent_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Concurrent Composition</title>
	<link>/entries/Noninterference_Concurrent_Composition.html</link>
	<pubDate>Mon, 13 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Concurrent_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Machine Word Library</title>
	<link>/entries/Word_Lib.html</link>
	<pubDate>Thu, 09 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Word_Lib.html</guid>
	<description></description>
	</item>

	<item>
	<title>POSIX Lexing with Derivatives of Regular Expressions</title>
	<link>/entries/Posix-Lexing.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Posix-Lexing.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Constructive Proof for FLP</title>
	<link>/entries/FLP.html</link>
	<pubDate>Wed, 18 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/FLP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases Theory</title>
	<link>/entries/Groebner_Bases.html</link>
	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Bases.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formalisation of the Cocke-Younger-Kasami algorithm</title>
	<link>/entries/CYK.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/CYK.html</guid>
	<description></description>
	</item>

	<item>
	<title>A formalisation of the Cocke-Younger-Kasami algorithm</title>
	<link>/entries/CYK.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/CYK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algorithms for Reduced Ordered Binary Decision Diagrams</title>
	<link>/entries/ROBDD.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/ROBDD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algorithms for Reduced Ordered Binary Decision Diagrams</title>
	<link>/entries/ROBDD.html</link>
	<pubDate>Wed, 27 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/ROBDD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Sequential Composition</title>
	<link>/entries/Noninterference_Sequential_Composition.html</link>
	<pubDate>Tue, 26 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Sequential_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Conservation of CSP Noninterference Security under Sequential Composition</title>
	<link>/entries/Noninterference_Sequential_Composition.html</link>
	<pubDate>Tue, 26 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Sequential_Composition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebras with Domain</title>
	<link>/entries/KAD.html</link>
	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebras with Domain</title>
	<link>/entries/KAD.html</link>
	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
	</item>

	<item>
	<title>Timed Automata</title>
	<link>/entries/Timed_Automata.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Temporal Logic</title>
	<link>/entries/LTL.html</link>
	<pubDate>Tue, 01 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/LTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
	<link>/entries/List_Update.html</link>
	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/List_Update.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysis of List Update Algorithms</title>
	<link>/entries/List_Update.html</link>
	<pubDate>Wed, 17 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/List_Update.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Construction of Static Single Assignment Form</title>
	<link>/entries/Formal_SSA.html</link>
	<pubDate>Fri, 05 Feb 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_SSA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tensor Product of Matrices</title>
	<link>/entries/Matrix_Tensor.html</link>
	<pubDate>Mon, 18 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix_Tensor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Applicative Lifting</title>
	<link>/entries/Applicative_Lifting.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Applicative_Lifting.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Tortoise and Hare Algorithm</title>
	<link>/entries/TortoiseHare.html</link>
	<pubDate>Wed, 18 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/TortoiseHare.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Meta-Model for the Isabelle API</title>
	<link>/entries/Isabelle_Meta_Model.html</link>
	<pubDate>Wed, 16 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Meta_Model.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear Temporal Logic to Deterministic (Generalized) Rabin Automata</title>
	<link>/entries/LTL_to_DRA.html</link>
	<pubDate>Fri, 04 Sep 2015 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Inductive Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Inductive_Unwinding.html</link>
	<pubDate>Tue, 18 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Inductive_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Analysing and Comparing Encodability Criteria for Process Calculi</title>
	<link>/entries/Encodability_Process_Calculi.html</link>
	<pubDate>Mon, 10 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Encodability_Process_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating Cases from Labeled Subgoals</title>
	<link>/entries/Case_Labeling.html</link>
	<pubDate>Tue, 21 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Case_Labeling.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite Normal Form</title>
	<link>/entries/Hermite.html</link>
	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Reasoning about Lists via List Interleaving</title>
	<link>/entries/List_Interleaving.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/List_Interleaving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generic Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Generic_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Generic_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Generic Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Generic_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Generic_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Ipurge Unwinding Theorem for CSP Noninterference Security</title>
	<link>/entries/Noninterference_Ipurge_Unwinding.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_Ipurge_Unwinding.html</guid>
	<description></description>
	</item>

	<item>
	<title>Parameterized Dynamic Tables</title>
	<link>/entries/Dynamic_Tables.html</link>
	<pubDate>Sun, 07 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Dynamic_Tables.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derivatives of Logical Formulas</title>
	<link>/entries/Formula_Derivatives.html</link>
	<pubDate>Thu, 28 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Formula_Derivatives.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Zoo of Probabilistic Systems</title>
	<link>/entries/Probabilistic_System_Zoo.html</link>
	<pubDate>Wed, 27 May 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_System_Zoo.html</guid>
	<description></description>
	</item>

	<item>
	<title>Concurrent IMP</title>
	<link>/entries/ConcurrentIMP.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentIMP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relaxing Safely: Verified On-the-Fly Garbage Collection for x86-TSO</title>
	<link>/entries/ConcurrentGC.html</link>
	<pubDate>Mon, 13 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/ConcurrentGC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Trie</title>
	<link>/entries/Trie.html</link>
	<pubDate>Mon, 30 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Trie.html</guid>
	<description></description>
	</item>

	<item>
	<title>Consensus Refined</title>
	<link>/entries/Consensus_Refined.html</link>
	<pubDate>Wed, 18 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Consensus_Refined.html</guid>
	<description></description>
	</item>

	<item>
	<title>Deriving class instances for datatypes</title>
	<link>/entries/Deriving.html</link>
	<pubDate>Wed, 11 Mar 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Deriving.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Safety of Call Arity</title>
	<link>/entries/Call_Arity.html</link>
	<pubDate>Fri, 20 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Call_Arity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Echelon Form</title>
	<link>/entries/Echelon_Form.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Echelon_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>QR Decomposition</title>
	<link>/entries/QR_Decomposition.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/QR_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Automata in Hereditarily Finite Set Theory</title>
	<link>/entries/Finite_Automata_HF.html</link>
	<pubDate>Thu, 05 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Automata_HF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the UpDown Scheme</title>
	<link>/entries/UpDown_Scheme.html</link>
	<pubDate>Wed, 28 Jan 2015 00:00:00 +0000</pubDate>

	<guid>/entries/UpDown_Scheme.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Unified Policy Framework (UPF)</title>
	<link>/entries/UPF.html</link>
	<pubDate>Fri, 28 Nov 2014 00:00:00 +0000</pubDate>

	<guid>/entries/UPF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Loop freedom of the (untimed) AODV routing protocol</title>
	<link>/entries/AODV.html</link>
	<pubDate>Thu, 23 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AODV.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting Definition Option</title>
	<link>/entries/Lifting_Definition_Option.html</link>
	<pubDate>Mon, 13 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_Definition_Option.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion in HOL with Code Generation</title>
	<link>/entries/Stream_Fusion_Code.html</link>
	<pubDate>Fri, 10 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Stream_Fusion_Code.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Compiler for Probability Density Functions</title>
	<link>/entries/Density_Compiler.html</link>
	<pubDate>Thu, 09 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Density_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalization of Refinement Calculus for Reactive Systems</title>
	<link>/entries/RefinementReactive.html</link>
	<pubDate>Wed, 08 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/RefinementReactive.html</guid>
	<description></description>
	</item>

	<item>
	<title>Certification Monads</title>
	<link>/entries/Certification_Monads.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Certification_Monads.html</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/entries/XML.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/XML.html</guid>
	<description></description>
	</item>

	<item>
	<title>XML</title>
	<link>/entries/XML.html</link>
	<pubDate>Fri, 03 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/XML.html</guid>
	<description></description>
	</item>

	<item>
	<title>Imperative Insertion Sort</title>
	<link>/entries/Imperative_Insertion_Sort.html</link>
	<pubDate>Thu, 25 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Imperative_Insertion_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Priority Queues Based on Braun Trees</title>
	<link>/entries/Priority_Queue_Braun.html</link>
	<pubDate>Thu, 04 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Priority_Queue_Braun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Algorithm and Its Applications</title>
	<link>/entries/Gauss_Jordan.html</link>
	<pubDate>Wed, 03 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Jordan.html</guid>
	<description></description>
	</item>

	<item>
	<title>Skew Heap</title>
	<link>/entries/Skew_Heap.html</link>
	<pubDate>Wed, 13 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Skew_Heap.html</guid>
	<description></description>
	</item>

	<item>
	<title>Splay Tree</title>
	<link>/entries/Splay_Tree.html</link>
	<pubDate>Tue, 12 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Splay_Tree.html</guid>
	<description></description>
	</item>

	<item>
	<title>Haskell's Show Class in Isabelle/HOL</title>
	<link>/entries/Show.html</link>
	<pubDate>Tue, 29 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Show.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Specification of a Generic Separation Kernel</title>
	<link>/entries/CISC-Kernel.html</link>
	<pubDate>Fri, 18 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CISC-Kernel.html</guid>
	<description></description>
	</item>

	<item>
	<title>pGCL for Isabelle</title>
	<link>/entries/pGCL.html</link>
	<pubDate>Sun, 13 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/pGCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amortized Complexity Verified</title>
	<link>/entries/Amortized_Complexity.html</link>
	<pubDate>Mon, 07 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Amortized_Complexity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Network Security Policy Verification</title>
	<link>/entries/Network_Security_Policy_Verification.html</link>
	<pubDate>Fri, 04 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Network_Security_Policy_Verification.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pop-Refinement</title>
	<link>/entries/Pop_Refinement.html</link>
	<pubDate>Thu, 03 Jul 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Pop_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Decision Procedures for MSO on Words Based on Derivatives of Regular Expressions</title>
	<link>/entries/MSO_Regex_Equivalence.html</link>
	<pubDate>Thu, 12 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/MSO_Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Boolean Expression Checkers</title>
	<link>/entries/Boolean_Expression_Checkers.html</link>
	<pubDate>Sun, 08 Jun 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Boolean_Expression_Checkers.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Fully Verified Executable LTL Model Checker</title>
	<link>/entries/CAVA_LTL_Modelchecker.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_LTL_Modelchecker.html</guid>
	<description></description>
	</item>

	<item>
	<title>Converting Linear-Time Temporal Logic to Generalized Büchi Automata</title>
	<link>/entries/LTL_to_GBA.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/LTL_to_GBA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Promela Formalization</title>
	<link>/entries/Promela.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Promela.html</guid>
	<description></description>
	</item>

	<item>
	<title>The CAVA Automata Library</title>
	<link>/entries/CAVA_Automata.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CAVA_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</title>
	<link>/entries/Gabow_SCC.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gabow_SCC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Noninterference Security in Communicating Sequential Processes</title>
	<link>/entries/Noninterference_CSP.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Noninterference_CSP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transitive closure according to Roy-Floyd-Warshall</title>
	<link>/entries/Roy_Floyd_Warshall.html</link>
	<pubDate>Fri, 23 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Roy_Floyd_Warshall.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Algebras</title>
	<link>/entries/Regular_Algebras.html</link>
	<pubDate>Wed, 21 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalisation and Analysis of Component Dependencies</title>
	<link>/entries/ComponentDependencies.html</link>
	<pubDate>Mon, 28 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/ComponentDependencies.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Assumptions and Guarantees for Compositional Noninterference</title>
	<link>/entries/SIFUM_Type_Systems.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Assumptions and Guarantees for Compositional Noninterference</title>
	<link>/entries/SIFUM_Type_Systems.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/SIFUM_Type_Systems.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Declassification with WHAT-and-WHERE-Security</title>
	<link>/entries/WHATandWHERE_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/WHATandWHERE_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Declassification with WHAT-and-WHERE-Security</title>
	<link>/entries/WHATandWHERE_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/WHATandWHERE_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Strong Security</title>
	<link>/entries/Strong_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Strong_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formalization of Strong Security</title>
	<link>/entries/Strong_Security.html</link>
	<pubDate>Wed, 23 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Strong_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bounded-Deducibility Security</title>
	<link>/entries/Bounded_Deducibility_Security.html</link>
	<pubDate>Tue, 22 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Bounded_Deducibility_Security.html</guid>
	<description></description>
	</item>

	<item>
	<title>A shallow embedding of HyperCTL*</title>
	<link>/entries/HyperCTL.html</link>
	<pubDate>Wed, 16 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/HyperCTL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Syntax and semantics of a GPU kernel programming language</title>
	<link>/entries/GPU_Kernel_PL.html</link>
	<pubDate>Thu, 03 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/GPU_Kernel_PL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Noninterference</title>
	<link>/entries/Probabilistic_Noninterference.html</link>
	<pubDate>Tue, 11 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mechanization of the Algebra for Wireless Networks (AWN)</title>
	<link>/entries/AWN.html</link>
	<pubDate>Sat, 08 Mar 2014 00:00:00 +0000</pubDate>

	<guid>/entries/AWN.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mutually Recursive Partial Functions</title>
	<link>/entries/Partial_Function_MR.html</link>
	<pubDate>Tue, 18 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Partial_Function_MR.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of Selection and Heap Sort Using Locales</title>
	<link>/entries/Selection_Heap_Sort.html</link>
	<pubDate>Tue, 11 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Selection_Heap_Sort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Unified Decision Procedures for Regular Expression Equivalence</title>
	<link>/entries/Regex_Equivalence.html</link>
	<pubDate>Thu, 30 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regex_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
	<link>/entries/KAT_and_DRA.html</link>
	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/KAT_and_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
	<link>/entries/KAT_and_DRA.html</link>
	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/KAT_and_DRA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Featherweight OCL: A Proposal for a Machine-Checked Formal Semantics for OCL 2.5</title>
	<link>/entries/Featherweight_OCL.html</link>
	<pubDate>Thu, 16 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Featherweight_OCL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Compositional Properties of Crypto-Based Components</title>
	<link>/entries/CryptoBasedCompositionalProperties.html</link>
	<pubDate>Sat, 11 Jan 2014 00:00:00 +0000</pubDate>

	<guid>/entries/CryptoBasedCompositionalProperties.html</guid>
	<description></description>
	</item>

	<item>
	<title>A General Method for the Proof of Theorems on Tail-recursive Functions</title>
	<link>/entries/Tail_Recursive_Functions.html</link>
	<pubDate>Sun, 01 Dec 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Tail_Recursive_Functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Codatatype of Formal Languages</title>
	<link>/entries/Coinductive_Languages.html</link>
	<pubDate>Fri, 15 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive_Languages.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Processing Components: Isabelle/HOL Formalisation and Case Studies</title>
	<link>/entries/FocusStreamsCaseStudies.html</link>
	<pubDate>Thu, 14 Nov 2013 00:00:00 +0000</pubDate>

	<guid>/entries/FocusStreamsCaseStudies.html</guid>
	<description></description>
	</item>

	<item>
	<title>Automatic Data Refinement</title>
	<link>/entries/Automatic_Refinement.html</link>
	<pubDate>Wed, 02 Oct 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Automatic_Refinement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Native Word</title>
	<link>/entries/Native_Word.html</link>
	<pubDate>Tue, 17 Sep 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Native_Word.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Model of IEEE Floating Point Arithmetic</title>
	<link>/entries/IEEE_Floating_Point.html</link>
	<pubDate>Sat, 27 Jul 2013 00:00:00 +0000</pubDate>

	<guid>/entries/IEEE_Floating_Point.html</guid>
	<description></description>
	</item>

	<item>
	<title>Light-weight Containers</title>
	<link>/entries/Containers.html</link>
	<pubDate>Mon, 15 Apr 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Containers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</title>
	<link>/entries/Launchbury.html</link>
	<pubDate>Thu, 31 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Launchbury.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</title>
	<link>/entries/Launchbury.html</link>
	<pubDate>Thu, 31 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Launchbury.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ribbon Proofs</title>
	<link>/entries/Ribbon_Proofs.html</link>
	<pubDate>Sat, 19 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Ribbon_Proofs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra</title>
	<link>/entries/Kleene_Algebra.html</link>
	<pubDate>Tue, 15 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Kleene_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebra</title>
	<link>/entries/Kleene_Algebra.html</link>
	<pubDate>Tue, 15 Jan 2013 00:00:00 +0000</pubDate>

	<guid>/entries/Kleene_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Separation Logic Framework for Imperative HOL</title>
	<link>/entries/Separation_Logic_Imperative_HOL.html</link>
	<pubDate>Wed, 14 Nov 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Logic_Imperative_HOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic Noninterference</title>
	<link>/entries/Possibilistic_Noninterference.html</link>
	<pubDate>Mon, 10 Sep 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Possibilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Possibilistic Noninterference</title>
	<link>/entries/Possibilistic_Noninterference.html</link>
	<pubDate>Mon, 10 Sep 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Possibilistic_Noninterference.html</guid>
	<description></description>
	</item>

	<item>
	<title>Generating linear orders for datatypes</title>
	<link>/entries/Datatype_Order_Generator.html</link>
	<pubDate>Tue, 07 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Datatype_Order_Generator.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model</title>
	<link>/entries/Heard_Of.html</link>
	<pubDate>Fri, 27 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Heard_Of.html</guid>
	<description></description>
	</item>

	<item>
	<title>Logical Relations for PCF</title>
	<link>/entries/PCF.html</link>
	<pubDate>Sun, 01 Jul 2012 00:00:00 +0000</pubDate>

	<guid>/entries/PCF.html</guid>
	<description></description>
	</item>

	<item>
	<title>Type Constructor Classes and Monad Transformers</title>
	<link>/entries/Tycon.html</link>
	<pubDate>Tue, 26 Jun 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Tycon.html</guid>
	<description></description>
	</item>

	<item>
	<title>CCS in nominal logic</title>
	<link>/entries/CCS.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/CCS.html</guid>
	<description></description>
	</item>

	<item>
	<title>Psi-calculi in Isabelle</title>
	<link>/entries/Psi_Calculi.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Psi_Calculi.html</guid>
	<description></description>
	</item>

	<item>
	<title>The pi-calculus in nominal logic</title>
	<link>/entries/Pi_Calculus.html</link>
	<pubDate>Tue, 29 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Circus</title>
	<link>/entries/Circus.html</link>
	<pubDate>Sun, 27 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Circus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle/Circus</title>
	<link>/entries/Circus.html</link>
	<pubDate>Sun, 27 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Circus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Separation Algebra</title>
	<link>/entries/Separation_Algebra.html</link>
	<pubDate>Fri, 11 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Separation_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stuttering Equivalence</title>
	<link>/entries/Stuttering_Equivalence.html</link>
	<pubDate>Mon, 07 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Stuttering_Equivalence.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inductive Study of Confidentiality</title>
	<link>/entries/Inductive_Confidentiality.html</link>
	<pubDate>Wed, 02 May 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Inductive_Confidentiality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Abortable Linearizable Modules</title>
	<link>/entries/Abortable_Linearizable_Modules.html</link>
	<pubDate>Thu, 01 Mar 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Abortable_Linearizable_Modules.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures</title>
	<link>/entries/Transitive-Closure-II.html</link>
	<pubDate>Wed, 29 Feb 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure-II.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dijkstra's Shortest Path Algorithm</title>
	<link>/entries/Dijkstra_Shortest_Path.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Dijkstra_Shortest_Path.html</guid>
	<description></description>
	</item>

	<item>
	<title>Refinement for Monadic Programs</title>
	<link>/entries/Refine_Monadic.html</link>
	<pubDate>Mon, 30 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Refine_Monadic.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Models</title>
	<link>/entries/Markov_Models.html</link>
	<pubDate>Tue, 03 Jan 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Definitional Encoding of TLA* in Isabelle/HOL</title>
	<link>/entries/TLA.html</link>
	<pubDate>Sat, 19 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/TLA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Efficient Mergesort</title>
	<link>/entries/Efficient-Mergesort.html</link>
	<pubDate>Wed, 09 Nov 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Efficient-Mergesort.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebra of Monotonic Boolean Transformers</title>
	<link>/entries/MonoBoolTranAlgebra.html</link>
	<pubDate>Thu, 22 Sep 2011 00:00:00 +0000</pubDate>

	<guid>/entries/MonoBoolTranAlgebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Myhill-Nerode Theorem Based on Regular Expressions</title>
	<link>/entries/Myhill-Nerode.html</link>
	<pubDate>Fri, 26 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Myhill-Nerode.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
	<link>/entries/Gauss-Jordan-Elim-Fun.html</link>
	<pubDate>Fri, 19 Aug 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss-Jordan-Elim-Fun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knowledge-based programs</title>
	<link>/entries/KBPs.html</link>
	<pubDate>Tue, 17 May 2011 00:00:00 +0000</pubDate>

	<guid>/entries/KBPs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Transitive Closures of Finite Relations</title>
	<link>/entries/Transitive-Closure.html</link>
	<pubDate>Mon, 14 Mar 2011 00:00:00 +0000</pubDate>

	<guid>/entries/Transitive-Closure.html</guid>
	<description></description>
	</item>

	<item>
	<title>AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics</title>
	<link>/entries/AutoFocus-Stream.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/AutoFocus-Stream.html</guid>
	<description></description>
	</item>

	<item>
	<title>Infinite Lists</title>
	<link>/entries/List-Infinite.html</link>
	<pubDate>Wed, 23 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/List-Infinite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lightweight Java</title>
	<link>/entries/LightweightJava.html</link>
	<pubDate>Mon, 07 Feb 2011 00:00:00 +0000</pubDate>

	<guid>/entries/LightweightJava.html</guid>
	<description></description>
	</item>

	<item>
	<title>RIPEMD-160</title>
	<link>/entries/RIPEMD-160-SPARK.html</link>
	<pubDate>Mon, 10 Jan 2011 00:00:00 +0000</pubDate>

	<guid>/entries/RIPEMD-160-SPARK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Shivers' Control Flow Analysis</title>
	<link>/entries/Shivers-CFA.html</link>
	<pubDate>Tue, 16 Nov 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Shivers-CFA.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binomial Heaps and Skew Binomial Heaps</title>
	<link>/entries/Binomial-Heaps.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Heaps.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finger Trees</title>
	<link>/entries/Finger-Trees.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Finger-Trees.html</guid>
	<description></description>
	</item>

	<item>
	<title>Functional Binomial Queues</title>
	<link>/entries/Binomial-Queues.html</link>
	<pubDate>Thu, 28 Oct 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Binomial-Queues.html</guid>
	<description></description>
	</item>

	<item>
	<title>Strong Normalization of Moggis's Computational Metalanguage</title>
	<link>/entries/Lam-ml-Normalization.html</link>
	<pubDate>Sun, 29 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Lam-ml-Normalization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Multivariate Polynomials</title>
	<link>/entries/Polynomials.html</link>
	<pubDate>Tue, 10 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Statecharts using Hierarchical Automata</title>
	<link>/entries/Statecharts.html</link>
	<pubDate>Sun, 08 Aug 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Statecharts.html</guid>
	<description></description>
	</item>

	<item>
	<title>Executable Matrix Operations on Matrices of Arbitrary Dimensions</title>
	<link>/entries/Matrix.html</link>
	<pubDate>Thu, 17 Jun 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix.html</guid>
	<description></description>
	</item>

	<item>
	<title>Semantics and Data Refinement of Invariant Based Programs</title>
	<link>/entries/DataRefinementIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/DataRefinementIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement</title>
	<link>/entries/GraphMarkingIBP.html</link>
	<pubDate>Fri, 28 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/GraphMarkingIBP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Sets and Expressions</title>
	<link>/entries/Regular-Sets.html</link>
	<pubDate>Wed, 12 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Regular-Sets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Locally Nameless Sigma Calculus</title>
	<link>/entries/Locally-Nameless-Sigma.html</link>
	<pubDate>Fri, 30 Apr 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Locally-Nameless-Sigma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Inter-Procedural Information Flow Noninterference via Slicing</title>
	<link>/entries/InformationFlowSlicing_Inter.html</link>
	<pubDate>Tue, 23 Mar 2010 00:00:00 +0000</pubDate>

	<guid>/entries/InformationFlowSlicing_Inter.html</guid>
	<description></description>
	</item>

	<item>
	<title>List Index</title>
	<link>/entries/List-Index.html</link>
	<pubDate>Sat, 20 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/List-Index.html</guid>
	<description></description>
	</item>

	<item>
	<title>Coinductive</title>
	<link>/entries/Coinductive.html</link>
	<pubDate>Fri, 12 Feb 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Coinductive.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing the Logic-Automaton Connection</title>
	<link>/entries/Presburger-Automata.html</link>
	<pubDate>Thu, 03 Dec 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Presburger-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Collections Framework</title>
	<link>/entries/Collections.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Collections.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Automata</title>
	<link>/entries/Tree-Automata.html</link>
	<pubDate>Wed, 25 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Tree-Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer</title>
	<link>/entries/HRB-Slicing.html</link>
	<pubDate>Fri, 13 Nov 2009 00:00:00 +0000</pubDate>

	<guid>/entries/HRB-Slicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Worker/Wrapper Transformation</title>
	<link>/entries/WorkerWrapper.html</link>
	<pubDate>Fri, 30 Oct 2009 00:00:00 +0000</pubDate>

	<guid>/entries/WorkerWrapper.html</guid>
	<description></description>
	</item>

	<item>
	<title>Code Generation for Functions as Data</title>
	<link>/entries/FinFun.html</link>
	<pubDate>Wed, 06 May 2009 00:00:00 +0000</pubDate>

	<guid>/entries/FinFun.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stream Fusion</title>
	<link>/entries/Stream-Fusion.html</link>
	<pubDate>Wed, 29 Apr 2009 00:00:00 +0000</pubDate>

	<guid>/entries/Stream-Fusion.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Bytecode Logic for JML and Types</title>
	<link>/entries/BytecodeLogicJmlTypes.html</link>
	<pubDate>Fri, 12 Dec 2008 00:00:00 +0000</pubDate>

	<guid>/entries/BytecodeLogicJmlTypes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secure information flow and program logics</title>
	<link>/entries/SIFPL.html</link>
	<pubDate>Mon, 10 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SIFPL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Secure information flow and program logics</title>
	<link>/entries/SIFPL.html</link>
	<pubDate>Mon, 10 Nov 2008 00:00:00 +0000</pubDate>

	<guid>/entries/SIFPL.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Textbook Proof of Huffman's Algorithm</title>
	<link>/entries/Huffman.html</link>
	<pubDate>Wed, 15 Oct 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Huffman.html</guid>
	<description></description>
	</item>

	<item>
	<title>Towards Certified Slicing</title>
	<link>/entries/Slicing.html</link>
	<pubDate>Tue, 16 Sep 2008 00:00:00 +0000</pubDate>

	<guid>/entries/Slicing.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Correctness Proof for the Volpano/Smith Security Typing System</title>
	<link>/entries/VolpanoSmith.html</link>
	<pubDate>Tue, 02 Sep 2008 00:00:00 +0000</pubDate>

	<guid>/entries/VolpanoSmith.html</guid>
	<description></description>
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	<h1 >
	<span class='first'>C</span>omputer <span class='first'>S</span>cience/<span class='first'>S</span>emantics and <span class='first'>R</span>easoning</h1>
	<div>


	</div>
	-</header><div><h2>Subject Classification</h2><p>ACM: <a href="https://dl.acm.org/topic/ccs2012/10003752.10010124">Theory of computation~Semantics and reasoning</a></p><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=68Q55">Computer science / Theory of computing / Semantics in the theory of computing</a></p><h2 class="head">2022</h2><article class="entry">
	+</header><div><h2>Subject Classification</h2><p>ACM: <a href="https://dl.acm.org/topic/ccs2012/10003752.10010124">Theory of computation~Semantics and reasoning</a></p><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=68Q55">Computer science / Theory of computing / Semantics in the theory of computing</a></p><h2 class="head">2023</h2><article class="entry">
	+ <div class="item-text">
	+ <h5><a class="title" href="../../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a> and <a href="../../../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+
	+
	+<h2 class="head">2022</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/CHERI-C_Memory_Model.html">A Formal CHERI-C Memory Model</a></h5> <br>by <a href="../../../authors/park">Seung Hoon Park</a></div>
	<span class="date">
	Nov 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Solidity.html">Isabelle/Solidity: A deep Embedding of Solidity in Isabelle/HOL</a></h5> <br>by <a href="../../../authors/marmsoler">Diego Marmsoler</a> and <a href="../../../authors/brucker">Achim D. Brucker</a></div>
	<span class="date">
	Jul 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Quasi_Borel_Spaces.html">Quasi-Borel Spaces</a></h5> <br>by <a href="../../../authors/hirata">Michikazu Hirata</a>, <a href="../../../authors/minamide">Yasuhiko Minamide</a> and <a href="../../../authors/sato">Tetsuya Sato</a></div>
	<span class="date">
	Feb 03
	</span>
	</article>


	<h2 class="head">2021</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Registers.html">Quantum and Classical Registers</a></h5> <br>by <a href="../../../authors/unruh">Dominique Unruh</a></div>
	<span class="date">
	Oct 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/X86_Semantics.html">X86 instruction semantics and basic block symbolic execution</a></h5> <br>by <a href="../../../authors/verbeek">Freek Verbeek</a>, <a href="../../../authors/bharadwaj">Abhijith Bharadwaj</a>, <a href="../../../authors/bockenek">Joshua Bockenek</a>, <a href="../../../authors/roessle">Ian Roessle</a>, <a href="../../../authors/weerwag">Timmy Weerwag</a> and <a href="../../../authors/ravindran">Binoy Ravindran</a></div>
	<span class="date">
	Oct 13
	</span>
	</article>


	<h2 class="head">2020</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/CSP_RefTK.html">The HOL-CSP Refinement Toolkit</a></h5> <br>by <a href="../../../authors/taha">Safouan Taha</a>, <a href="../../../authors/wolff">Burkhart Wolff</a> and <a href="../../../authors/ye">Lina Ye</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>


	<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Isabelle_C.html">Isabelle/C</a></h5> <br>by <a href="../../../authors/tuong">Frédéric Tuong</a> and <a href="../../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Oct 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Clean.html">Clean - An Abstract Imperative Programming Language and its Theory</a></h5> <br>by <a href="../../../authors/tuong">Frédéric Tuong</a> and <a href="../../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Oct 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/TESL_Language.html">A Formal Development of a Polychronous Polytimed Coordination Language</a></h5> <br>by <a href="../../../authors/van">Hai Nguyen Van</a>, <a href="../../../authors/boulanger">Frédéric Boulanger</a> and <a href="../../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Jul 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/HOL-CSP.html">HOL-CSP Version 2.0</a></h5> <br>by <a href="../../../authors/taha">Safouan Taha</a>, <a href="../../../authors/ye">Lina Ye</a> and <a href="../../../authors/wolff">Burkhart Wolff</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/QHLProver.html">Quantum Hoare Logic</a></h5> <br>by <a href="../../../authors/liu">Junyi Liu</a>, <a href="../../../authors/zhan">Bohua Zhan</a>, <a href="../../../authors/wang">Shuling Wang</a>, <a href="../../../authors/ying">Shenggang Ying</a>, <a href="../../../authors/liut">Tao Liu</a>, <a href="../../../authors/liy">Yangjia Li</a>, <a href="../../../authors/yingm">Mingsheng Ying</a> and <a href="../../../authors/zhann">Naijun Zhan</a></div>
	<span class="date">
	Mar 24
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Transformer_Semantics.html">Transformer Semantics</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Refine_Imperative_HOL.html">The Imperative Refinement Framework</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Launchbury.html">The Correctness of Launchbury's Natural Semantics for Lazy Evaluation</a></h5> <br>by <a href="../../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jan 31
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Refine_Monadic.html">Refinement for Monadic Programs</a></h5> <br>by <a href="../../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	Jan 30
	</span>
	</article>




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	<title>Transformer Semantics</title>
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	<article class="entry">
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	<h5><a class="title" href="../../../entries/Jacobson_Basic_Algebra.html">A Case Study in Basic Algebra</a></h5> <br>by <a href="../../../authors/ballarin">Clemens Ballarin</a></div>
	<span class="date">
	Aug 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Linear_Programming.html">Linear Programming</a></h5> <br>by <a href="../../../authors/parsert">Julian Parsert</a> and <a href="../../../authors/kaliszyk">Cezary Kaliszyk</a></div>
	<span class="date">
	Aug 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/C2KA_DistributedSystems.html">Communicating Concurrent Kleene Algebra for Distributed Systems Specification</a></h5> <br>by <a href="../../../authors/buyse">Maxime Buyse</a> and <a href="../../../authors/jaskolka">Jason Jaskolka</a></div>
	<span class="date">
	Aug 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Linear_Inequalities.html">Linear Inequalities</a></h5> <br>by <a href="../../../authors/bottesch">Ralph Bottesch</a>, <a href="../../../authors/reynaud">Alban Reynaud</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jun 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Nullstellensatz.html">Hilbert's Nullstellensatz</a></h5> <br>by <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	Jun 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Groebner_Macaulay.html">Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds</a></h5> <br>by <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	Jun 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Farkas.html">Farkas' Lemma and Motzkin's Transposition Theorem</a></h5> <br>by <a href="../../../authors/bottesch">Ralph Bottesch</a>, <a href="../../../authors/haslbeck">Max W. Haslbeck</a> and <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jan 17
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Transformer_Semantics.html">Transformer Semantics</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Quantales.html">Quantales</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Symmetric_Polynomials.html">Symmetric Polynomials</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Signature_Groebner.html">Signature-Based Gröbner Basis Algorithms</a></h5> <br>by <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Aggregation_Algebras.html">Aggregation Algebras</a></h5> <br>by <a href="../../../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Sep 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Octonions.html">Octonions</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	<span class="date">
	Sep 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Quaternions.html">Quaternions</a></h5> <br>by <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Sep 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Localization_Ring.html">The Localization of a Commutative Ring</a></h5> <br>by <a href="../../../authors/bordg">Anthony Bordg</a></div>
	<span class="date">
	Jun 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/LLL_Factorization.html">A verified factorization algorithm for integer polynomials with polynomial complexity</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a>, <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/LLL_Basis_Reduction.html">A verified LLL algorithm</a></h5> <br>by <a href="../../../authors/bottesch">Ralph Bottesch</a>, <a href="../../../authors/divason">Jose Divasón</a>, <a href="../../../authors/haslbeck">Max W. Haslbeck</a>, <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Feb 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Taylor_Models.html">Taylor Models</a></h5> <br>by <a href="../../../authors/traut">Christoph Traut</a> and <a href="../../../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2017</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Mason_Stothers.html">The Mason–Stothers Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Dirichlet_L.html">Dirichlet L-Functions and Dirichlet's Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stochastic_Matrices.html">Stochastic Matrices and the Perron-Frobenius Theorem</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Orbit_Stabiliser.html">Orbit-Stabiliser Theorem with Application to Rotational Symmetries</a></h5> <br>by <a href="../../../authors/raedle">Jonas Rädle</a></div>
	<span class="date">
	Aug 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stone_Kleene_Relation_Algebras.html">Stone-Kleene Relation Algebras</a></h5> <br>by <a href="../../../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Jul 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/PSemigroupsConvolution.html">Partial Semigroups and Convolution Algebras</a></h5> <br>by <a href="../../../authors/dongol">Brijesh Dongol</a>, <a href="../../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../../authors/hayes">Ian J. Hayes</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Optics.html">Optics</a></h5> <br>by <a href="../../../authors/fosters">Simon Foster</a>, <a href="../../../authors/laursen">Christian Pardillo-Laursen</a> and <a href="../../../authors/zeyda">Frank Zeyda</a></div>
	<span class="date">
	May 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Subresultants.html">Subresultants</a></h5> <br>by <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Apr 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stone_Relation_Algebras.html">Stone Relation Algebras</a></h5> <br>by <a href="../../../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Berlekamp_Zassenhaus.html">The Factorization Algorithm of Berlekamp and Zassenhaus</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a>, <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Buildings.html">Chamber Complexes, Coxeter Systems, and Buildings</a></h5> <br>by <a href="../../../authors/sylvestre">Jeremy Sylvestre</a></div>
	<span class="date">
	Jul 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Algebraic_VCs.html">Program Construction and Verification Components Based on Kleene Algebra</a></h5> <br>by <a href="../../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 18
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Perron_Frobenius.html">Perron-Frobenius Theorem for Spectral Radius Analysis</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a>, <a href="../../../authors/kuncar">Ondřej Kunčar</a>, <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	May 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Groebner_Bases.html">Gröbner Bases Theory</a></h5> <br>by <a href="../../../authors/immler">Fabian Immler</a> and <a href="../../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	May 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../../authors/guttmann">Walter Guttmann</a>, <a href="../../../authors/hoefner">Peter Höfner</a>, <a href="../../../authors/struth">Georg Struth</a> and <a href="../../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Apr 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Polynomial_Interpolation.html">Polynomial Interpolation</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Jan 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Polynomial_Factorization.html">Polynomial Factorization</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Jan 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Matrix_Tensor.html">Tensor Product of Matrices</a></h5> <br>by <a href="../../../authors/prathamesh">T.V.H. Prathamesh</a></div>
	<span class="date">
	Jan 18
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Algebraic_Numbers.html">Algebraic Numbers in Isabelle/HOL</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a>, <a href="../../../authors/yamada">Akihisa Yamada</a> and <a href="../../../authors/joosten">Sebastiaan J. C. Joosten</a></div>
	<span class="date">
	Dec 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Jordan_Normal_Form.html">Matrices, Jordan Normal Forms, and Spectral Radius Theory</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a> and <a href="../../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Aug 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Rep_Fin_Groups.html">Representations of Finite Groups</a></h5> <br>by <a href="../../../authors/sylvestre">Jeremy Sylvestre</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Hermite.html">Hermite Normal Form</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Multirelations.html">Binary Multirelations</a></h5> <br>by <a href="../../../authors/furusawa">Hitoshi Furusawa</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Residuated_Lattices.html">Residuated Lattices</a></h5> <br>by <a href="../../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/QR_Decomposition.html">QR Decomposition</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Echelon_Form.html">Echelon Form</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Cayley_Hamilton.html">The Cayley-Hamilton Theorem</a></h5> <br>by <a href="../../../authors/adelsberger">Stephan Adelsberger</a>, <a href="../../../authors/hetzl">Stefan Hetzl</a> and <a href="../../../authors/pollak">Florian Pollak</a></div>
	<span class="date">
	Sep 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Jordan_Hoelder.html">The Jordan-Hölder Theorem</a></h5> <br>by <a href="../../../authors/raumer">Jakob von Raumer</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/VectorSpace.html">Vector Spaces</a></h5> <br>by <a href="../../../authors/lee">Holden Lee</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Regular_Algebras.html">Regular Algebras</a></h5> <br>by <a href="../../../authors/fosters">Simon Foster</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	May 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Secondary_Sylow.html">Secondary Sylow Theorems</a></h5> <br>by <a href="../../../authors/raumer">Jakob von Raumer</a></div>
	<span class="date">
	Jan 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Relation_Algebra.html">Relation Algebra</a></h5> <br>by <a href="../../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../../authors/fosters">Simon Foster</a>, <a href="../../../authors/struth">Georg Struth</a> and <a href="../../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Rank_Nullity_Theorem.html">Rank-Nullity Theorem in Linear Algebra</a></h5> <br>by <a href="../../../authors/divason">Jose Divasón</a> and <a href="../../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../../authors/struth">Georg Struth</a> and <a href="../../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Impossible_Geometry.html">Proving the Impossibility of Trisecting an Angle and Doubling the Cube</a></h5> <br>by <a href="../../../authors/romanos">Ralph Romanos</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 05
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/PseudoHoops.html">Pseudo Hoops</a></h5> <br>by <a href="../../../authors/georgescu">George Georgescu</a>, <a href="../../../authors/leustean">Laurentiu Leustean</a> and <a href="../../../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>by <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a>, <a href="../../../authors/thiemann">René Thiemann</a>, <a href="../../../authors/maletzky">Alexander Maletzky</a>, <a href="../../../authors/immler">Fabian Immler</a>, <a href="../../../authors/haftmann">Florian Haftmann</a>, <a href="../../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Free-Groups.html">Free Groups</a></h5> <br>by <a href="../../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Robbins-Conjecture.html">A Complete Proof of the Robbins Conjecture</a></h5> <br>by <a href="../../../authors/doty">Matthew Doty</a></div>
	<span class="date">
	May 22
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/CofGroups.html">An Example of a Cofinitary Group in Isabelle/HOL</a></h5> <br>by <a href="../../../authors/kastermans">Bart Kastermans</a></div>
	<span class="date">
	Aug 04
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Valuation.html">Fundamental Properties of Valuation Theory and Hensel's Lemma</a></h5> <br>by <a href="../../../authors/kobayashi">Hidetsune Kobayashi</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Group-Ring-Module.html">Groups, Rings and Modules</a></h5> <br>by <a href="../../../authors/kobayashi">Hidetsune Kobayashi</a>, <a href="../../../authors/chen">L. Chen</a> and <a href="../../../authors/murao">H. Murao</a></div>
	<span class="date">
	May 18
	</span>
	</article>




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	<title>Inner Structure, Determinism and Modal Algebra of Multirelations</title>
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	<title>Birkhoff's Representation Theorem For Finite Distributive Lattices</title>
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	<title>p-adic Fields and p-adic Semialgebraic Sets</title>
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	<title>Simultaneous diagonalization of pairwise commuting Hermitian matrices</title>
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	<title>Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</title>
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	<title>Duality of Linear Programming</title>
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	<title>Interpolation Polynomials (in HOL-Algebra)</title>
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	<title>Hyperdual Numbers and Forward Differentiation</title>
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	<title>Factorization of Polynomials with Algebraic Coefficients</title>
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	<title>Finitely Generated Abelian Groups</title>
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	<item>
	<title>Grothendieck's Schemes in Algebraic Geometry</title>
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	<pubDate>Mon, 29 Mar 2021 00:00:00 +0000</pubDate>

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	<title>Formal Puiseux Series</title>
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	<item>
	<title>A verified algorithm for computing the Smith normal form of a matrix</title>
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	<pubDate>Sat, 23 May 2020 00:00:00 +0000</pubDate>

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	<item>
	<title>Power Sum Polynomials</title>
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	<title>Matrices for ODEs</title>
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	<pubDate>Sun, 19 Apr 2020 00:00:00 +0000</pubDate>

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	<title>A Hierarchy of Algebras for Boolean Subsets</title>
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	<pubDate>Fri, 31 Jan 2020 00:00:00 +0000</pubDate>

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	<title>Verification Components for Hybrid Systems</title>
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	<item>
	<title>A Case Study in Basic Algebra</title>
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	<pubDate>Fri, 30 Aug 2019 00:00:00 +0000</pubDate>

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	<title>Communicating Concurrent Kleene Algebra for Distributed Systems Specification</title>
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	<title>Linear Inequalities</title>
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	<title>Hilbert's Nullstellensatz</title>
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	<title>Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds</title>
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	<pubDate>Sat, 15 Jun 2019 00:00:00 +0000</pubDate>

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	<title>Farkas' Lemma and Motzkin's Transposition Theorem</title>
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	<title>Quantales</title>
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	<item>
	<title>Transformer Semantics</title>
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	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

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	<item>
	<title>Symmetric Polynomials</title>
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	<pubDate>Tue, 25 Sep 2018 00:00:00 +0000</pubDate>

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	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
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	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

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	<item>
	<title>Aggregation Algebras</title>
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	<pubDate>Sat, 15 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Aggregation_Algebras.html</guid>
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	<item>
	<title>Octonions</title>
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	<pubDate>Fri, 14 Sep 2018 00:00:00 +0000</pubDate>

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	<item>
	<title>Quaternions</title>
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	<pubDate>Wed, 05 Sep 2018 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>The Localization of a Commutative Ring</title>
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	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

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	<item>
	<title>A verified factorization algorithm for integer polynomials with polynomial complexity</title>
	<link>/entries/LLL_Factorization.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Factorization.html</guid>
	<description></description>
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	<item>
	<title>A verified LLL algorithm</title>
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	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
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	<item>
	<title>Taylor Models</title>
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	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
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	<item>
	<title>Dirichlet L-Functions and Dirichlet's Theorem</title>
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	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

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	<item>
	<title>The Mason–Stothers Theorem</title>
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	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Stochastic Matrices and the Perron-Frobenius Theorem</title>
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	<pubDate>Wed, 22 Nov 2017 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Orbit-Stabiliser Theorem with Application to Rotational Symmetries</title>
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	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Stone-Kleene Relation Algebras</title>
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	<pubDate>Thu, 06 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Kleene_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Semigroups and Convolution Algebras</title>
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	<pubDate>Tue, 13 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PSemigroupsConvolution.html</guid>
	<description></description>
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	<item>
	<title>Optics</title>
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	<pubDate>Thu, 25 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Optics.html</guid>
	<description></description>
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	<item>
	<title>Subresultants</title>
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	<pubDate>Thu, 06 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Subresultants.html</guid>
	<description></description>
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	<item>
	<title>Stone Relation Algebras</title>
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	<pubDate>Tue, 07 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Relation_Algebras.html</guid>
	<description></description>
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	<item>
	<title>The Factorization Algorithm of Berlekamp and Zassenhaus</title>
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	<pubDate>Fri, 14 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Berlekamp_Zassenhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Chamber Complexes, Coxeter Systems, and Buildings</title>
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	<pubDate>Fri, 01 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Buildings.html</guid>
	<description></description>
	</item>

	<item>
	<title>Program Construction and Verification Components Based on Kleene Algebra</title>
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	<pubDate>Sat, 18 Jun 2016 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Perron-Frobenius Theorem for Spectral Radius Analysis</title>
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	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Perron_Frobenius.html</guid>
	<description></description>
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	<item>
	<title>Gröbner Bases Theory</title>
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	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Kleene Algebras with Domain</title>
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	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
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	<item>
	<title>Polynomial Factorization</title>
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	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Factorization.html</guid>
	<description></description>
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	<item>
	<title>Polynomial Interpolation</title>
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	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Tensor Product of Matrices</title>
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	<pubDate>Mon, 18 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix_Tensor.html</guid>
	<description></description>
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	<item>
	<title>Algebraic Numbers in Isabelle/HOL</title>
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	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_Numbers.html</guid>
	<description></description>
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	<item>
	<title>Matrices, Jordan Normal Forms, and Spectral Radius Theory</title>
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	<pubDate>Fri, 21 Aug 2015 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Representations of Finite Groups</title>
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	<pubDate>Wed, 12 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Rep_Fin_Groups.html</guid>
	<description></description>
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	<item>
	<title>Hermite Normal Form</title>
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	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Multirelations</title>
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	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Residuated Lattices</title>
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	<pubDate>Wed, 15 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Residuated_Lattices.html</guid>
	<description></description>
	</item>

	<item>
	<title>Echelon Form</title>
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	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Echelon_Form.html</guid>
	<description></description>
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	<item>
	<title>QR Decomposition</title>
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	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

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	<description></description>
	</item>

	<item>
	<title>The Cayley-Hamilton Theorem</title>
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	<pubDate>Mon, 15 Sep 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>The Jordan-Hölder Theorem</title>
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	<pubDate>Tue, 09 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Hoelder.html</guid>
	<description></description>
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	<item>
	<title>Vector Spaces</title>
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	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Regular Algebras</title>
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	<pubDate>Wed, 21 May 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Secondary Sylow Theorems</title>
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	<pubDate>Tue, 28 Jan 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Relation Algebra</title>
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	<pubDate>Sat, 25 Jan 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
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	<pubDate>Thu, 23 Jan 2014 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Rank-Nullity Theorem in Linear Algebra</title>
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	<pubDate>Wed, 16 Jan 2013 00:00:00 +0000</pubDate>

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	<item>
	<title>Kleene Algebra</title>
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	<description></description>
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	<item>
	<title>Proving the Impossibility of Trisecting an Angle and Doubling the Cube</title>
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	<pubDate>Sun, 05 Aug 2012 00:00:00 +0000</pubDate>

	<guid>/entries/Impossible_Geometry.html</guid>
	<description></description>
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	<item>
	<title>Pseudo Hoops</title>
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	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
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	<item>
	<title>Executable Multivariate Polynomials</title>
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	<item>
	<title>Free Groups</title>
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	<item>
	<title>A Complete Proof of the Robbins Conjecture</title>
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	<pubDate>Sat, 22 May 2010 00:00:00 +0000</pubDate>

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	<item>
	<title>An Example of a Cofinitary Group in Isabelle/HOL</title>
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	<pubDate>Tue, 04 Aug 2009 00:00:00 +0000</pubDate>

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	<description></description>
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	<item>
	<title>Fundamental Properties of Valuation Theory and Hensel's Lemma</title>
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	<pubDate>Wed, 08 Aug 2007 00:00:00 +0000</pubDate>

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	<item>
	<title>Groups, Rings and Modules</title>
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	<pubDate>Tue, 18 May 2004 00:00:00 +0000</pubDate>

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	+++ b/web/topics/mathematics/combinatorics/index.html
	@@ -1,373 +1,380 @@
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	<h1 >
	<span class='first'>M</span>athematics/<span class='first'>C</span>ombinatorics</h1>
	<div>


	</div>
	</header><div><h2>Subject Classification</h2><p>ACM: <a href="https://dl.acm.org/topic/ccs2012/10002950.10003624.10003625">Mathematics of computing~Combinatorics</a></p><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=05-XX">Combinatorics</a></p><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../../authors/leek">Kevin Lee</a>, <a href="../../../authors/yez">Zhengkun Ye</a> and <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="../../../entries/Gray_Codes.html">Gray Codes for Arbitrary Numeral Systems</a></h5> <br>by <a href="../../../authors/spitz">Maximilian Spitz</a></div>
	<span class="date">
	Jul 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Tree_Enumeration.html">Tree Enumeration</a></h5> <br>by <a href="../../../authors/cremer">Nils Cremer</a></div>
	<span class="date">
	May 09
	</span>
	</article>


	<h2 class="head">2022</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Sauer_Shelah_Lemma.html">Sauer-Shelah Lemma</a></h5> <br>by <a href="../../../authors/keskin">Ata Keskin</a></div>
	<span class="date">
	Nov 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Kneser_Cauchy_Davenport.html">Kneser's Theorem and the Cauchy–Davenport Theorem</a></h5> <br>by <a href="../../../authors/baksys">Mantas Bakšys</a> and <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	<span class="date">
	Nov 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Balog_Szemeredi_Gowers.html">The Balog–Szemerédi–Gowers Theorem</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a>, <a href="../../../authors/baksys">Mantas Bakšys</a> and <a href="../../../authors/edmonds">Chelsea Edmonds</a></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Combinatorial_Enumeration_Algorithms.html">Combinatorial Enumeration Algorithms</a></h5> <br>by <a href="../../../authors/hofmeier">Paul Hofmeier</a> and <a href="../../../authors/karayel">Emin Karayel</a></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Hales_Jewett.html">The Hales–Jewett Theorem</a></h5> <br>by <a href="../../../authors/sulejmani">Ujkan Sulejmani</a>, <a href="../../../authors/eberl">Manuel Eberl</a> and <a href="../../../authors/kreuzer">Katharina Kreuzer</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Khovanskii_Theorem.html">Khovanskii's Theorem</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Sep 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Pluennecke_Ruzsa_Inequality.html">The Plünnecke-Ruzsa Inequality</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	May 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Clique_and_Monotone_Circuits.html">Clique is not solvable by monotone circuits of polynomial size</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	May 08
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Fishers_Inequality.html">Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</a></h5> <br>by <a href="../../../authors/edmonds">Chelsea Edmonds</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Apr 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Equivalence_Relation_Enumeration.html">Enumeration of Equivalence Relations</a></h5> <br>by <a href="../../../authors/karayel">Emin Karayel</a></div>
	<span class="date">
	Feb 04
	</span>
	</article>


	<h2 class="head">2021</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Roth_Arithmetic_Progressions.html">Roth's Theorem on Arithmetic Progressions</a></h5> <br>by <a href="../../../authors/edmonds">Chelsea Edmonds</a>, <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Dec 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Szemeredi_Regularity.html">Szemerédi's Regularity Lemma</a></h5> <br>by <a href="../../../authors/edmonds">Chelsea Edmonds</a>, <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Design_Theory.html">Combinatorial Design Theory</a></h5> <br>by <a href="../../../authors/edmonds">Chelsea Edmonds</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Van_der_Waerden.html">Van der Waerden's Theorem</a></h5> <br>by <a href="../../../authors/kreuzer">Katharina Kreuzer</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jun 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Sunflowers.html">The Sunflower Lemma of Erdős and Rado</a></h5> <br>by <a href="../../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 25
	</span>
	</article>


	<h2 class="head">2020</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Delta_System_Lemma.html">Cofinality and the Delta System Lemma</a></h5> <br>by <a href="../../../authors/terraf">Pedro Sánchez Terraf</a></div>
	<span class="date">
	Dec 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Ordinal_Partitions.html">Ordinal Partitions</a></h5> <br>by <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Nash_Williams.html">The Nash-Williams Partition Theorem</a></h5> <br>by <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	May 16
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Matroids.html">Matroids</a></h5> <br>by <a href="../../../authors/keinholz">Jonas Keinholz</a></div>
	<span class="date">
	Nov 16
	</span>
	</article>


	<h2 class="head">2017</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Falling_Factorial_Sum.html">The Falling Factorial of a Sum</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Dec 22
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Twelvefold_Way.html">The Twelvefold Way</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Dec 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Card_Multisets.html">Cardinality of Multisets</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Jun 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Catalan_Numbers.html">Catalan Numbers</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jun 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Card_Equiv_Relations.html">Cardinality of Equivalence Relations</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	May 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Bell_Numbers_Spivey.html">Spivey's Generalized Recurrence for Bell Numbers</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	May 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Card_Number_Partitions.html">Cardinality of Number Partitions</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Jan 14
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Card_Partitions.html">Cardinality of Set Partitions</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Dec 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Latin_Square.html">Latin Square</a></h5> <br>by <a href="../../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Dec 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Euler_Partition.html">Euler's Partition Theorem</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Derangements.html">Derangements Formula</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Jun 27
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Discrete_Summation.html">Discrete Summation</a></h5> <br>by <a href="../../../authors/haftmann">Florian Haftmann</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Open_Induction.html">Open Induction</a></h5> <br>by <a href="../../../authors/ogawa">Mizuhito Ogawa</a> and <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Bondy.html">Bondy's Theorem</a></h5> <br>by <a href="../../../authors/avigad">Jeremy Avigad</a> and <a href="../../../authors/hetzl">Stefan Hetzl</a></div>
	<span class="date">
	Oct 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Well_Quasi_Orders.html">Well-Quasi-Orders</a></h5> <br>by <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Marriage.html">Hall's Marriage Theorem</a></h5> <br>by <a href="../../../authors/jiangd">Dongchen Jiang</a> and <a href="../../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Dec 17
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Ramsey-Infinite.html">Ramsey's theorem, infinitary version</a></h5> <br>by <a href="../../../authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>




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	<title>Szemerédi's Regularity Lemma</title>
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	Aug 04
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	<h5><a class="title" href="../../entries/Arith_Prog_Rel_Primes.html">Arithmetic progressions and relative primes</a></h5> <br>by <a href="../../authors/caballero">José Manuel Rodríguez Caballero</a></div>
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	<h5><a class="title" href="../../entries/Complex_Geometry.html">Complex Geometry</a></h5> <br>by <a href="../../authors/maricf">Filip Marić</a> and <a href="../../authors/simic">Danijela Simić</a></div>
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	<h5><a class="title" href="../../entries/Gauss_Sums.html">Gauss Sums and the Pólya–Vinogradov Inequality</a></h5> <br>by <a href="../../authors/raya">Rodrigo Raya</a> and <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Hybrid_Systems_VCs.html">Verification Components for Hybrid Systems</a></h5> <br>by <a href="../../authors/munive">Jonathan Julian Huerta y Munive</a></div>
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	<h5><a class="title" href="../../entries/Quantales.html">Quantales</a></h5> <br>by <a href="../../authors/struth">Georg Struth</a></div>
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	<h5><a class="title" href="../../entries/Hidden_Markov_Models.html">Hidden Markov Models</a></h5> <br>by <a href="../../authors/wimmer">Simon Wimmer</a></div>
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	May 25
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	May 24
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	<h5><a class="title" href="../../entries/Irrationality_J_Hancl.html">Irrational Rapidly Convergent Series</a></h5> <br>by <a href="../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../authors/li">Wenda Li</a></div>
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	<h5><a class="title" href="../../entries/AxiomaticCategoryTheory.html">Axiom Systems for Category Theory in Free Logic</a></h5> <br>by <a href="../../authors/benzmueller">Christoph Benzmüller</a> and <a href="../../authors/scott">Dana Scott</a></div>
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	<h5><a class="title" href="../../entries/Fishburn_Impossibility.html">The Incompatibility of Fishburn-Strategyproofness and Pareto-Efficiency</a></h5> <br>by <a href="../../authors/brandt">Felix Brandt</a>, <a href="../../authors/eberl">Manuel Eberl</a>, <a href="../../authors/saile">Christian Saile</a> and <a href="../../authors/stricker">Christian Stricker</a></div>
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	<h5><a class="title" href="../../entries/LLL_Factorization.html">A verified factorization algorithm for integer polynomials with polynomial complexity</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a>, <a href="../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
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	<h5><a class="title" href="../../entries/Gromov_Hyperbolicity.html">Gromov Hyperbolicity</a></h5> <br>by <a href="../../authors/gouezel">Sebastien Gouezel</a></div>
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	<h5><a class="title" href="../../entries/Green.html">An Isabelle/HOL formalisation of Green's Theorem</a></h5> <br>by <a href="../../authors/abdulaziz">Mohammad Abdulaziz</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
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	<h5><a class="title" href="../../entries/Taylor_Models.html">Taylor Models</a></h5> <br>by <a href="../../authors/traut">Christoph Traut</a> and <a href="../../authors/immler">Fabian Immler</a></div>
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	<h5><a class="title" href="../../entries/Taylor_Models.html">Taylor Models</a></h5> <br>by <a href="../../authors/traut">Christoph Traut</a> and <a href="../../authors/immler">Fabian Immler</a></div>
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	<h5><a class="title" href="../../entries/Falling_Factorial_Sum.html">The Falling Factorial of a Sum</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
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	Dec 22
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	<h5><a class="title" href="../../entries/Mason_Stothers.html">The Mason–Stothers Theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Dec 21
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	<h5><a class="title" href="../../entries/Dirichlet_L.html">Dirichlet L-Functions and Dirichlet's Theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Dec 21
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	<h5><a class="title" href="../../entries/Dirichlet_L.html">Dirichlet L-Functions and Dirichlet's Theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Dec 21
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	<h5><a class="title" href="../../entries/Stochastic_Matrices.html">Stochastic Matrices and the Perron-Frobenius Theorem</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a></div>
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	Nov 22
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	<h5><a class="title" href="../../entries/Kuratowski_Closure_Complement.html">The Kuratowski Closure-Complement Theorem</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a> and <a href="../../authors/gioiosa">Gianpaolo Gioiosa</a></div>
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	Oct 26
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	<h5><a class="title" href="../../entries/Winding_Number_Eval.html">Evaluate Winding Numbers through Cauchy Indices</a></h5> <br>by <a href="../../authors/li">Wenda Li</a></div>
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	<h5><a class="title" href="../../entries/Count_Complex_Roots.html">Count the Number of Complex Roots</a></h5> <br>by <a href="../../authors/li">Wenda Li</a></div>
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	<h5><a class="title" href="../../entries/Diophantine_Eqns_Lin_Hom.html">Homogeneous Linear Diophantine Equations</a></h5> <br>by <a href="../../authors/messner">Florian Messner</a>, <a href="../../authors/parsert">Julian Parsert</a>, <a href="../../authors/schoepf">Jonas Schöpf</a> and <a href="../../authors/sternagel">Christian Sternagel</a></div>
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	<h5><a class="title" href="../../entries/Zeta_Function.html">The Hurwitz and Riemann ζ Functions</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Oct 12
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	<h5><a class="title" href="../../entries/Zeta_Function.html">The Hurwitz and Riemann ζ Functions</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Linear_Recurrences.html">Linear Recurrences</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Dirichlet_Series.html">Dirichlet Series</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/First_Welfare_Theorem.html">Microeconomics and the First Welfare Theorem</a></h5> <br>by <a href="../../authors/parsert">Julian Parsert</a> and <a href="../../authors/kaliszyk">Cezary Kaliszyk</a></div>
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	<h5><a class="title" href="../../entries/Minkowskis_Theorem.html">Minkowski's Theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Stone_Kleene_Relation_Algebras.html">Stone-Kleene Relation Algebras</a></h5> <br>by <a href="../../authors/guttmann">Walter Guttmann</a></div>
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	<h5><a class="title" href="../../entries/PSemigroupsConvolution.html">Partial Semigroups and Convolution Algebras</a></h5> <br>by <a href="../../authors/dongol">Brijesh Dongol</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../authors/hayes">Ian J. Hayes</a> and <a href="../../authors/struth">Georg Struth</a></div>
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	<h5><a class="title" href="../../entries/Buffons_Needle.html">Buffon's Needle Problem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Buffons_Needle.html">Buffon's Needle Problem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Probabilistic_While.html">Probabilistic while loop</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
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	<h5><a class="title" href="../../entries/CryptHOL.html">CryptHOL</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
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	<h5><a class="title" href="../../entries/MonoidalCategory.html">Monoidal Categories</a></h5> <br>by <a href="../../authors/stark">Eugene W. Stark</a></div>
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	Apr 06
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	<h5><a class="title" href="../../entries/Euler_MacLaurin.html">The Euler–MacLaurin Formula</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Mar 10
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	<h5><a class="title" href="../../entries/Menger.html">Menger's Theorem</a></h5> <br>by <a href="../../authors/dittmann">Christoph Dittmann</a></div>
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	Feb 26
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	<h5><a class="title" href="../../entries/Stone_Relation_Algebras.html">Stone Relation Algebras</a></h5> <br>by <a href="../../authors/guttmann">Walter Guttmann</a></div>
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	Feb 07
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	<h5><a class="title" href="../../entries/Bernoulli.html">Bernoulli Numbers</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a> and <a href="../../authors/eberl">Manuel Eberl</a></div>
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	<h5><a class="title" href="../../entries/Bertrands_Postulate.html">Bertrand's postulate</a></h5> <br>by <a href="../../authors/biendarra">Julian Biendarra</a> and <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Jan 12
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	<h5><a class="title" href="../../entries/E_Transcendental.html">The Transcendence of e</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
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	Jan 12
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	<h5><a class="title" href="../../entries/Twelvefold_Way.html">The Twelvefold Way</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
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	Dec 29
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	<h5><a class="title" href="../../entries/Deep_Learning.html">Expressiveness of Deep Learning</a></h5> <br>by <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
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	Nov 10
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	<h5><a class="title" href="../../entries/Stable_Matching.html">Stable Matching</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
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	Oct 24
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	<h5><a class="title" href="../../entries/Source_Coding_Theorem.html">Source Coding Theorem</a></h5> <br>by <a href="../../authors/hibon">Quentin Hibon</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
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	Oct 19
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	<h5><a class="title" href="../../entries/Berlekamp_Zassenhaus.html">The Factorization Algorithm of Berlekamp and Zassenhaus</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a>, <a href="../../authors/joosten">Sebastiaan J. C. Joosten</a>, <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
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	<h5><a class="title" href="../../entries/Ptolemys_Theorem.html">Ptolemy's Theorem</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
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	May 24
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	<h5><a class="title" href="../../entries/Perron_Frobenius.html">Perron-Frobenius Theorem for Spectral Radius Analysis</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a>, <a href="../../authors/kuncar">Ondřej Kunčar</a>, <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	May 20
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	<h5><a class="title" href="../../entries/MFMC_Countable.html">A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks</a></h5> <br>by <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	May 09
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/Randomised_Social_Choice.html">Randomised Social Choice Theory</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	May 05
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/SDS_Impossibility.html">The Incompatibility of SD-Efficiency and SD-Strategy-Proofness</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	May 04
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	<h5><a class="title" href="../../entries/Bell_Numbers_Spivey.html">Spivey's Generalized Recurrence for Bell Numbers</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	May 04
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Groebner_Bases.html">Gröbner Bases Theory</a></h5> <br>by <a href="../../authors/immler">Fabian Immler</a> and <a href="../../authors/maletzky">Alexander Maletzky</a></div>
	<span class="date">
	May 02
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	<h5><a class="title" href="../../entries/No_FTL_observers.html">No Faster-Than-Light Observers</a></h5> <br>by <a href="../../authors/stannett">Mike Stannett</a> and <a href="../../authors/nemeti">István Németi</a></div>
	<span class="date">
	Apr 28
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	<h5><a class="title" href="../../entries/KAD.html">Kleene Algebras with Domain</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a>, <a href="../../authors/guttmann">Walter Guttmann</a>, <a href="../../authors/hoefner">Peter Höfner</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Apr 12
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	<h5><a class="title" href="../../entries/Cartan_FP.html">The Cartan Fixed Point Theorems</a></h5> <br>by <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Mar 08
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	<h5><a class="title" href="../../entries/Polynomial_Interpolation.html">Polynomial Interpolation</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Jan 29
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	<h5><a class="title" href="../../entries/Polynomial_Factorization.html">Polynomial Factorization</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Jan 29
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	<h5><a class="title" href="../../entries/Knot_Theory.html">Knot Theory</a></h5> <br>by <a href="../../authors/prathamesh">T.V.H. Prathamesh</a></div>
	<span class="date">
	Jan 20
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	<h5><a class="title" href="../../entries/Matrix_Tensor.html">Tensor Product of Matrices</a></h5> <br>by <a href="../../authors/prathamesh">T.V.H. Prathamesh</a></div>
	<span class="date">
	Jan 18
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	<h5><a class="title" href="../../entries/Card_Number_Partitions.html">Cardinality of Number Partitions</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Jan 14
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	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Prime_Harmonic_Series.html">The Divergence of the Prime Harmonic Series</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
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	<h5><a class="title" href="../../entries/Liouville_Numbers.html">Liouville numbers</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
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	<article class="entry">
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	<h5><a class="title" href="../../entries/Liouville_Numbers.html">Liouville numbers</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
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	<h5><a class="title" href="../../entries/Descartes_Sign_Rule.html">Descartes' Rule of Signs</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
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	<h5><a class="title" href="../../entries/Triangle.html">Basic Geometric Properties of Triangles</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
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	<div class="item-text">
	<h5><a class="title" href="../../entries/Stern_Brocot.html">The Stern-Brocot Tree</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a> and <a href="../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Dec 22
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	<h5><a class="title" href="../../entries/Algebraic_Numbers.html">Algebraic Numbers in Isabelle/HOL</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a>, <a href="../../authors/yamada">Akihisa Yamada</a> and <a href="../../authors/joosten">Sebastiaan J. C. Joosten</a></div>
	<span class="date">
	Dec 22
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Card_Partitions.html">Cardinality of Set Partitions</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Dec 12
	</span>
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	<h5><a class="title" href="../../entries/Latin_Square.html">Latin Square</a></h5> <br>by <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Dec 02
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	<h5><a class="title" href="../../entries/Ergodic_Theory.html">Ergodic Theory</a></h5> <br>by <a href="../../authors/gouezel">Sebastien Gouezel</a></div>
	<span class="date">
	Dec 01
	</span>
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	<div class="item-text">
	<h5><a class="title" href="../../entries/Euler_Partition.html">Euler's Partition Theorem</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Nov 19
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Planarity_Certificates.html">Planarity Certificates</a></h5> <br>by <a href="../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Nov 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Parity_Game.html">Positional Determinacy of Parity Games</a></h5> <br>by <a href="../../authors/dittmann">Christoph Dittmann</a></div>
	<span class="date">
	Nov 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Parity_Game.html">Positional Determinacy of Parity Games</a></h5> <br>by <a href="../../authors/dittmann">Christoph Dittmann</a></div>
	<span class="date">
	Nov 02
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Jordan_Normal_Form.html">Matrices, Jordan Normal Forms, and Spectral Radius Theory</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a> and <a href="../../authors/yamada">Akihisa Yamada</a></div>
	<span class="date">
	Aug 21
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/Rep_Fin_Groups.html">Representations of Finite Groups</a></h5> <br>by <a href="../../authors/sylvestre">Jeremy Sylvestre</a></div>
	<span class="date">
	Aug 12
	</span>
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Akra_Bazzi.html">The Akra-Bazzi theorem and the Master theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jul 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Landau_Symbols.html">Landau Symbols</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jul 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Hermite.html">Hermite Normal Form</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a> and <a href="../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jul 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Derangements.html">Derangements Formula</a></h5> <br>by <a href="../../authors/bulwahn">Lukas Bulwahn</a></div>
	<span class="date">
	Jun 27
	</span>
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	<h5><a class="title" href="../../entries/Multirelations.html">Binary Multirelations</a></h5> <br>by <a href="../../authors/furusawa">Hitoshi Furusawa</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jun 11
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Vickrey_Clarke_Groves.html">VCG - Combinatorial Vickrey-Clarke-Groves Auctions</a></h5> <br>by <a href="../../authors/caminati">Marco B. Caminati</a>, <a href="../../authors/kerber">Manfred Kerber</a>, <a href="../../authors/lange">Christoph Lange</a> and <a href="../../authors/rowat">Colin Rowat</a></div>
	<span class="date">
	Apr 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Residuated_Lattices.html">Residuated Lattices</a></h5> <br>by <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Apr 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/QR_Decomposition.html">QR Decomposition</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a> and <a href="../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Echelon_Form.html">Echelon Form</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a> and <a href="../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Feb 12
	</span>
	</article>


	<h2 class="head">2014</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Density_Compiler.html">A Verified Compiler for Probability Density Functions</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a>, <a href="../../authors/hoelzl">Johannes Hölzl</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Oct 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Sturm_Tarski.html">The Sturm–Tarski Theorem</a></h5> <br>by <a href="../../authors/li">Wenda Li</a></div>
	<span class="date">
	Sep 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Cayley_Hamilton.html">The Cayley-Hamilton Theorem</a></h5> <br>by <a href="../../authors/adelsberger">Stephan Adelsberger</a>, <a href="../../authors/hetzl">Stefan Hetzl</a> and <a href="../../authors/pollak">Florian Pollak</a></div>
	<span class="date">
	Sep 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Jordan_Hoelder.html">The Jordan-Hölder Theorem</a></h5> <br>by <a href="../../authors/raumer">Jakob von Raumer</a></div>
	<span class="date">
	Sep 09
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/VectorSpace.html">Vector Spaces</a></h5> <br>by <a href="../../authors/lee">Holden Lee</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
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	<h5><a class="title" href="../../entries/Special_Function_Bounds.html">Real-Valued Special Functions: Upper and Lower Bounds</a></h5> <br>by <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 29
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/Gabow_SCC.html">Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</a></h5> <br>by <a href="../../authors/lammich">Peter Lammich</a></div>
	<span class="date">
	May 28
	</span>
	</article>
	<article class="entry">
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	<h5><a class="title" href="../../entries/Regular_Algebras.html">Regular Algebras</a></h5> <br>by <a href="../../authors/fosters">Simon Foster</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	May 21
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	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Discrete_Summation.html">Discrete Summation</a></h5> <br>by <a href="../../authors/haftmann">Florian Haftmann</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Random_Graph_Subgraph_Threshold.html">Properties of Random Graphs -- Subgraph Containment</a></h5> <br>by <a href="../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Feb 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Random_Graph_Subgraph_Threshold.html">Properties of Random Graphs -- Subgraph Containment</a></h5> <br>by <a href="../../authors/hupel">Lars Hupel</a></div>
	<span class="date">
	Feb 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Affine_Arithmetic.html">Affine Arithmetic</a></h5> <br>by <a href="../../authors/immler">Fabian Immler</a></div>
	<span class="date">
	Feb 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Real_Impl.html">Implementing field extensions of the form Q[sqrt(b)]</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Secondary_Sylow.html">Secondary Sylow Theorems</a></h5> <br>by <a href="../../authors/raumer">Jakob von Raumer</a></div>
	<span class="date">
	Jan 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Relation_Algebra.html">Relation Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/fosters">Simon Foster</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 25
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/KAT_and_DRA.html">Kleene Algebra with Tests and Demonic Refinement Algebras</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/gomes">Victor B. F. Gomes</a> and <a href="../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Jan 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Sturm_Sequences.html">Sturm's Theorem</a></h5> <br>by <a href="../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 11
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Pratt_Certificate.html">Pratt's Primality Certificates</a></h5> <br>by <a href="../../authors/wimmer">Simon Wimmer</a> and <a href="../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Lehmer.html">Lehmer's Theorem</a></h5> <br>by <a href="../../authors/wimmer">Simon Wimmer</a> and <a href="../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Koenigsberg_Friendship.html">The Königsberg Bridge Problem and the Friendship Theorem</a></h5> <br>by <a href="../../authors/li">Wenda Li</a></div>
	<span class="date">
	Jul 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/ShortestPath.html">An Axiomatic Characterization of the Single-Source Shortest Path Problem</a></h5> <br>by <a href="../../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Graph_Theory.html">Graph Theory</a></h5> <br>by <a href="../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Apr 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Rank_Nullity_Theorem.html">Rank-Nullity Theorem in Linear Algebra</a></h5> <br>by <a href="../../authors/divason">Jose Divasón</a> and <a href="../../authors/aransay">Jesús Aransay</a></div>
	<span class="date">
	Jan 16
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Kleene_Algebra.html">Kleene Algebra</a></h5> <br>by <a href="../../authors/armstrong">Alasdair Armstrong</a>, <a href="../../authors/struth">Georg Struth</a> and <a href="../../authors/weber">Tjark Weber</a></div>
	<span class="date">
	Jan 15
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Sqrt_Babylonian.html">Computing N-th Roots using the Babylonian Method</a></h5> <br>by <a href="../../authors/thiemann">René Thiemann</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2012</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Open_Induction.html">Open Induction</a></h5> <br>by <a href="../../authors/ogawa">Mizuhito Ogawa</a> and <a href="../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Nov 02
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Tarskis_Geometry.html">The independence of Tarski's Euclidean axiom</a></h5> <br>by <a href="../../authors/makarios">T. J. M. Makarios</a></div>
	<span class="date">
	Oct 30
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Bondy.html">Bondy's Theorem</a></h5> <br>by <a href="../../authors/avigad">Jeremy Avigad</a> and <a href="../../authors/hetzl">Stefan Hetzl</a></div>
	<span class="date">
	Oct 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Impossible_Geometry.html">Proving the Impossibility of Trisecting an Angle and Doubling the Cube</a></h5> <br>by <a href="../../authors/romanos">Ralph Romanos</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Impossible_Geometry.html">Proving the Impossibility of Trisecting an Angle and Doubling the Cube</a></h5> <br>by <a href="../../authors/romanos">Ralph Romanos</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Aug 05
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Ordinary_Differential_Equations.html">Ordinary Differential Equations</a></h5> <br>by <a href="../../authors/immler">Fabian Immler</a> and <a href="../../authors/hoelzl">Johannes Hölzl</a></div>
	<span class="date">
	Apr 26
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Well_Quasi_Orders.html">Well-Quasi-Orders</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Apr 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Girth_Chromatic.html">A Probabilistic Proof of the Girth-Chromatic Number Theorem</a></h5> <br>by <a href="../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Feb 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Markov_Models.html">Markov Models</a></h5> <br>by <a href="../../authors/hoelzl">Johannes Hölzl</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Jan 03
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/PseudoHoops.html">Pseudo Hoops</a></h5> <br>by <a href="../../authors/georgescu">George Georgescu</a>, <a href="../../authors/leustean">Laurentiu Leustean</a> and <a href="../../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/LatticeProperties.html">Lattice Properties</a></h5> <br>by <a href="../../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Gauss-Jordan-Elim-Fun.html">Gauss-Jordan Elimination for Matrices Represented as Functions</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Max-Card-Matching.html">Maximum Cardinality Matching</a></h5> <br>by <a href="../../authors/rizkallah">Christine Rizkallah</a></div>
	<span class="date">
	Jul 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/General-Triangle.html">The General Triangle Is Unique</a></h5> <br>by <a href="../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Apr 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Lower_Semicontinuous.html">Lower Semicontinuous Functions</a></h5> <br>by <a href="../../authors/grechuk">Bogdan Grechuk</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2010</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Marriage.html">Hall's Marriage Theorem</a></h5> <br>by <a href="../../authors/jiangd">Dongchen Jiang</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Dec 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a>, <a href="../../authors/thiemann">René Thiemann</a>, <a href="../../authors/maletzky">Alexander Maletzky</a>, <a href="../../authors/immler">Fabian Immler</a>, <a href="../../authors/haftmann">Florian Haftmann</a>, <a href="../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Polynomials.html">Executable Multivariate Polynomials</a></h5> <br>by <a href="../../authors/sternagel">Christian Sternagel</a>, <a href="../../authors/thiemann">René Thiemann</a>, <a href="../../authors/maletzky">Alexander Maletzky</a>, <a href="../../authors/immler">Fabian Immler</a>, <a href="../../authors/haftmann">Florian Haftmann</a>, <a href="../../authors/lochbihler">Andreas Lochbihler</a> and <a href="../../authors/bentkamp">Alexander Bentkamp</a></div>
	<span class="date">
	Aug 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Free-Groups.html">Free Groups</a></h5> <br>by <a href="../../authors/breitner">Joachim Breitner</a></div>
	<span class="date">
	Jun 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Category2.html">Category Theory</a></h5> <br>by <a href="../../authors/katovsky">Alexander Katovsky</a></div>
	<span class="date">
	Jun 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Robbins-Conjecture.html">A Complete Proof of the Robbins Conjecture</a></h5> <br>by <a href="../../authors/doty">Matthew Doty</a></div>
	<span class="date">
	May 22
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Perfect-Number-Thm.html">Perfect Number Theorem</a></h5> <br>by <a href="../../authors/ijbema">Mark Ijbema</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/CofGroups.html">An Example of a Cofinitary Group in Isabelle/HOL</a></h5> <br>by <a href="../../authors/kastermans">Bart Kastermans</a></div>
	<span class="date">
	Aug 04
	</span>
	</article>


	<h2 class="head">2008</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/SenSocialChoice.html">Some classical results in Social Choice Theory</a></h5> <br>by <a href="../../authors/gammie">Peter Gammie</a></div>
	<span class="date">
	Nov 09
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FunWithTilings.html">Fun With Tilings</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a> and <a href="../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Nov 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/ArrowImpossibilityGS.html">Arrow and Gibbard-Satterthwaite</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Sep 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/FunWithFunctions.html">Fun With Functions</a></h5> <br>by <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	Aug 26
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/SumSquares.html">Sums of Two and Four Squares</a></h5> <br>by <a href="../../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Fermat3_4.html">Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</a></h5> <br>by <a href="../../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Valuation.html">Fundamental Properties of Valuation Theory and Hensel's Lemma</a></h5> <br>by <a href="../../authors/kobayashi">Hidetsune Kobayashi</a></div>
	<span class="date">
	Aug 08
	</span>
	</article>


	<h2 class="head">2006</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Flyspeck-Tame.html">Flyspeck I: Tame Graphs</a></h5> <br>by <a href="../../authors/bauer">Gertrud Bauer</a> and <a href="../../authors/nipkow">Tobias Nipkow</a></div>
	<span class="date">
	May 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Cauchy.html">Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality</a></h5> <br>by <a href="../../authors/porter">Benjamin Porter</a></div>
	<span class="date">
	Mar 14
	</span>
	</article>


	<h2 class="head">2005</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Category.html">Category Theory to Yoneda's Lemma</a></h5> <br>by <a href="../../authors/keefe">Greg O’Keefe</a></div>
	<span class="date">
	Apr 21
	</span>
	</article>


	<h2 class="head">2004</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Integration.html">Integration theory and random variables</a></h5> <br>by <a href="../../authors/richter">Stefan Richter</a></div>
	<span class="date">
	Nov 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Ramsey-Infinite.html">Ramsey's theorem, infinitary version</a></h5> <br>by <a href="../../authors/ridge">Tom Ridge</a></div>
	<span class="date">
	Sep 20
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../entries/Group-Ring-Module.html">Groups, Rings and Modules</a></h5> <br>by <a href="../../authors/kobayashi">Hidetsune Kobayashi</a>, <a href="../../authors/chen">L. Chen</a> and <a href="../../authors/murao">H. Murao</a></div>
	<span class="date">
	May 18
	</span>
	</article>
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	<h5><a class="title" href="../../entries/Topology.html">Topology</a></h5> <br>by <a href="../../authors/friedrich">Stefan Friedrich</a></div>
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	Apr 26
	</span>
	</article>




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	<link>/entries/Khovanskii_Theorem.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Khovanskii_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hales–Jewett Theorem</title>
	<link>/entries/Hales_Jewett.html</link>
	<pubDate>Fri, 02 Sep 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Hales_Jewett.html</guid>
	<description></description>
	</item>

	<item>
	<title>From THE BOOK: Two Squares via Involutions</title>
	<link>/entries/Involutions2Squares.html</link>
	<pubDate>Mon, 15 Aug 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Involutions2Squares.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simultaneous diagonalization of pairwise commuting Hermitian matrices</title>
	<link>/entries/Commuting_Hermitian.html</link>
	<pubDate>Mon, 18 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Commuting_Hermitian.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pólya’s Proof of the Weighted Arithmetic–Geometric Mean Inequality</title>
	<link>/entries/Weighted_Arithmetic_Geometric_Mean.html</link>
	<pubDate>Mon, 11 Jul 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Weighted_Arithmetic_Geometric_Mean.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finite Fields</title>
	<link>/entries/Finite_Fields.html</link>
	<pubDate>Wed, 08 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Finite_Fields.html</guid>
	<description></description>
	</item>

	<item>
	<title>Diophantine Equations and the DPRM Theorem</title>
	<link>/entries/DPRM_Theorem.html</link>
	<pubDate>Mon, 06 Jun 2022 00:00:00 +0000</pubDate>

	<guid>/entries/DPRM_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Plünnecke-Ruzsa Inequality</title>
	<link>/entries/Pluennecke_Ruzsa_Inequality.html</link>
	<pubDate>Thu, 26 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Pluennecke_Ruzsa_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Clique is not solvable by monotone circuits of polynomial size</title>
	<link>/entries/Clique_and_Monotone_Circuits.html</link>
	<pubDate>Sun, 08 May 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Clique_and_Monotone_Circuits.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</title>
	<link>/entries/Fishers_Inequality.html</link>
	<pubDate>Thu, 21 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Fishers_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fisher's Inequality: Linear Algebraic Proof Techniques for Combinatorics</title>
	<link>/entries/Fishers_Inequality.html</link>
	<pubDate>Thu, 21 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Fishers_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Digit Expansions</title>
	<link>/entries/Digit_Expansions.html</link>
	<pubDate>Wed, 20 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Digit_Expansions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sophomore's Dream</title>
	<link>/entries/Sophomores_Dream.html</link>
	<pubDate>Sun, 10 Apr 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Sophomores_Dream.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructing the Reals as Dedekind Cuts of Rationals</title>
	<link>/entries/Dedekind_Real.html</link>
	<pubDate>Thu, 24 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Dedekind_Real.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent</title>
	<link>/entries/Cotangent_PFD_Formula.html</link>
	<pubDate>Tue, 15 Mar 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Cotangent_PFD_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>Wetzel's Problem and the Continuum Hypothesis</title>
	<link>/entries/Wetzels_Problem.html</link>
	<pubDate>Fri, 18 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Wetzels_Problem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Enumeration of Equivalence Relations</title>
	<link>/entries/Equivalence_Relation_Enumeration.html</link>
	<pubDate>Fri, 04 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Equivalence_Relation_Enumeration.html</guid>
	<description></description>
	</item>

	<item>
	<title>Duality of Linear Programming</title>
	<link>/entries/LP_Duality.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/LP_Duality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quasi-Borel Spaces</title>
	<link>/entries/Quasi_Borel_Spaces.html</link>
	<pubDate>Thu, 03 Feb 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Quasi_Borel_Spaces.html</guid>
	<description></description>
	</item>

	<item>
	<title>Young's Inequality for Increasing Functions</title>
	<link>/entries/Youngs_Inequality.html</link>
	<pubDate>Mon, 31 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Youngs_Inequality.html</guid>
	<description></description>
	</item>

	<item>
	<title>Interpolation Polynomials (in HOL-Algebra)</title>
	<link>/entries/Interpolation_Polynomials_HOL_Algebra.html</link>
	<pubDate>Sat, 29 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Interpolation_Polynomials_HOL_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Actuarial Mathematics</title>
	<link>/entries/Actuarial_Mathematics.html</link>
	<pubDate>Sun, 23 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Actuarial_Mathematics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational numbers from THE BOOK</title>
	<link>/entries/Irrationals_From_THEBOOK.html</link>
	<pubDate>Sat, 08 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationals_From_THEBOOK.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knight's Tour Revisited Revisited</title>
	<link>/entries/Knights_Tour.html</link>
	<pubDate>Tue, 04 Jan 2022 00:00:00 +0000</pubDate>

	<guid>/entries/Knights_Tour.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual Numbers and Forward Differentiation</title>
	<link>/entries/Hyperdual.html</link>
	<pubDate>Fri, 31 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hyperdual.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hyperdual Numbers and Forward Differentiation</title>
	<link>/entries/Hyperdual.html</link>
	<pubDate>Fri, 31 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hyperdual.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Shapley Algorithm</title>
	<link>/entries/Gale_Shapley.html</link>
	<pubDate>Wed, 29 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Gale_Shapley.html</guid>
	<description></description>
	</item>

	<item>
	<title>Roth's Theorem on Arithmetic Progressions</title>
	<link>/entries/Roth_Arithmetic_Progressions.html</link>
	<pubDate>Tue, 28 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Roth_Arithmetic_Progressions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Roth's Theorem on Arithmetic Progressions</title>
	<link>/entries/Roth_Arithmetic_Progressions.html</link>
	<pubDate>Tue, 28 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Roth_Arithmetic_Progressions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Markov Decision Processes with Rewards</title>
	<link>/entries/MDP-Rewards.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Rewards.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Algorithms for Solving Markov Decision Processes</title>
	<link>/entries/MDP-Algorithms.html</link>
	<pubDate>Thu, 16 Dec 2021 00:00:00 +0000</pubDate>

	<guid>/entries/MDP-Algorithms.html</guid>
	<description></description>
	</item>

	<item>
	<title>Simplicial Complexes and Boolean functions</title>
	<link>/entries/Simplicial_complexes_and_boolean_functions.html</link>
	<pubDate>Mon, 29 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Simplicial_complexes_and_boolean_functions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Foundation of geometry in planes, and some complements: Excluding the parallel axioms</title>
	<link>/entries/Foundation_of_geometry.html</link>
	<pubDate>Mon, 22 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Foundation_of_geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hahn and Jordan Decomposition Theorems</title>
	<link>/entries/Hahn_Jordan_Decomposition.html</link>
	<pubDate>Fri, 19 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hahn_Jordan_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Factorization of Polynomials with Algebraic Coefficients</title>
	<link>/entries/Factor_Algebraic_Polynomial.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Factor_Algebraic_Polynomial.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real Exponents as the Limits of Sequences of Rational Exponents</title>
	<link>/entries/Real_Power.html</link>
	<pubDate>Mon, 08 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Real_Power.html</guid>
	<description></description>
	</item>

	<item>
	<title>Szemerédi's Regularity Lemma</title>
	<link>/entries/Szemeredi_Regularity.html</link>
	<pubDate>Fri, 05 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Szemeredi_Regularity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Szemerédi's Regularity Lemma</title>
	<link>/entries/Szemeredi_Regularity.html</link>
	<pubDate>Fri, 05 Nov 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Szemeredi_Regularity.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Bounded Operators</title>
	<link>/entries/Complex_Bounded_Operators.html</link>
	<pubDate>Sat, 18 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Bounded_Operators.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL I: Foundations: Design Patterns, Set Theory, Digraphs, Semicategories</title>
	<link>/entries/CZH_Foundations.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Foundations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL II: Elementary Theory of 1-Categories</title>
	<link>/entries/CZH_Elementary_Categories.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Elementary_Categories.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory for ZFC in HOL III: Universal Constructions</title>
	<link>/entries/CZH_Universal_Constructions.html</link>
	<pubDate>Mon, 06 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/CZH_Universal_Constructions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Solving Cubic and Quartic Equations</title>
	<link>/entries/Cubic_Quartic_Equations.html</link>
	<pubDate>Fri, 03 Sep 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Cubic_Quartic_Equations.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Theorem of Three Circles</title>
	<link>/entries/Three_Circles.html</link>
	<pubDate>Sat, 21 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Three_Circles.html</guid>
	<description></description>
	</item>

	<item>
	<title>Combinatorial Design Theory</title>
	<link>/entries/Design_Theory.html</link>
	<pubDate>Fri, 13 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Design_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Forests</title>
	<link>/entries/Relational_Forests.html</link>
	<pubDate>Tue, 03 Aug 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Forests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz' Independent Axioms for Minkowski Spacetime</title>
	<link>/entries/Schutz_Spacetime.html</link>
	<pubDate>Tue, 27 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Schutz_Spacetime.html</guid>
	<description></description>
	</item>

	<item>
	<title>Schutz' Independent Axioms for Minkowski Spacetime</title>
	<link>/entries/Schutz_Spacetime.html</link>
	<pubDate>Tue, 27 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Schutz_Spacetime.html</guid>
	<description></description>
	</item>

	<item>
	<title>Finitely Generated Abelian Groups</title>
	<link>/entries/Finitely_Generated_Abelian_Groups.html</link>
	<pubDate>Wed, 07 Jul 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Finitely_Generated_Abelian_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Van der Waerden's Theorem</title>
	<link>/entries/Van_der_Waerden.html</link>
	<pubDate>Tue, 22 Jun 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Van_der_Waerden.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lifting the Exponent</title>
	<link>/entries/Lifting_the_Exponent.html</link>
	<pubDate>Tue, 27 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Lifting_the_Exponent.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gale-Stewart Games</title>
	<link>/entries/GaleStewart_Games.html</link>
	<pubDate>Fri, 23 Apr 2021 00:00:00 +0000</pubDate>

	<guid>/entries/GaleStewart_Games.html</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck's Schemes in Algebraic Geometry</title>
	<link>/entries/Grothendieck_Schemes.html</link>
	<pubDate>Mon, 29 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Grothendieck_Schemes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Grothendieck's Schemes in Algebraic Geometry</title>
	<link>/entries/Grothendieck_Schemes.html</link>
	<pubDate>Mon, 29 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Grothendieck_Schemes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hensel's Lemma for the p-adic Integers</title>
	<link>/entries/Padic_Ints.html</link>
	<pubDate>Tue, 23 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Padic_Ints.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL: the Communication Modeling Aspect</title>
	<link>/entries/Constructive_Cryptography_CM.html</link>
	<pubDate>Wed, 17 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography_CM.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantum projective measurements and the CHSH inequality</title>
	<link>/entries/Projective_Measurements.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Measurements.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hermite–Lindemann–Weierstraß Transcendence Theorem</title>
	<link>/entries/Hermite_Lindemann.html</link>
	<pubDate>Wed, 03 Mar 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite_Lindemann.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sunflower Lemma of Erdős and Rado</title>
	<link>/entries/Sunflowers.html</link>
	<pubDate>Thu, 25 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Sunflowers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formal Puiseux Series</title>
	<link>/entries/Formal_Puiseux_Series.html</link>
	<pubDate>Wed, 17 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Formal_Puiseux_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Laws of Large Numbers</title>
	<link>/entries/Laws_of_Large_Numbers.html</link>
	<pubDate>Wed, 10 Feb 2021 00:00:00 +0000</pubDate>

	<guid>/entries/Laws_of_Large_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid</title>
	<link>/entries/IsaGeoCoq.html</link>
	<pubDate>Sun, 31 Jan 2021 00:00:00 +0000</pubDate>

	<guid>/entries/IsaGeoCoq.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cofinality and the Delta System Lemma</title>
	<link>/entries/Delta_System_Lemma.html</link>
	<pubDate>Sun, 27 Dec 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Delta_System_Lemma.html</guid>
	<description></description>
	</item>

	<item>
	<title>Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information</title>
	<link>/entries/Isabelle_Marries_Dirac.html</link>
	<pubDate>Sun, 22 Nov 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Isabelle_Marries_Dirac.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Sound Type System for Physical Quantities, Units, and Measurements</title>
	<link>/entries/Physical_Quantities.html</link>
	<pubDate>Tue, 20 Oct 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Physical_Quantities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Amicable Numbers</title>
	<link>/entries/Amicable_Numbers.html</link>
	<pubDate>Tue, 04 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Amicable_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ordinal Partitions</title>
	<link>/entries/Ordinal_Partitions.html</link>
	<pubDate>Mon, 03 Aug 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Ordinal_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Relational Characterisations of Paths</title>
	<link>/entries/Relational_Paths.html</link>
	<pubDate>Mon, 13 Jul 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Relational_Paths.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formally Verified Checker of the Safe Distance Traffic Rules for Autonomous Vehicles</title>
	<link>/entries/Safe_Distance.html</link>
	<pubDate>Mon, 01 Jun 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Safe_Distance.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified algorithm for computing the Smith normal form of a matrix</title>
	<link>/entries/Smith_Normal_Form.html</link>
	<pubDate>Sat, 23 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Smith_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Nash-Williams Partition Theorem</title>
	<link>/entries/Nash_Williams.html</link>
	<pubDate>Sat, 16 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Nash_Williams.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality Criteria for Series by Erdős and Straus</title>
	<link>/entries/Irrational_Series_Erdos_Straus.html</link>
	<pubDate>Tue, 12 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Irrational_Series_Erdos_Straus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrationality Criteria for Series by Erdős and Straus</title>
	<link>/entries/Irrational_Series_Erdos_Straus.html</link>
	<pubDate>Tue, 12 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Irrational_Series_Erdos_Straus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Banach-Steinhaus Theorem</title>
	<link>/entries/Banach_Steinhaus.html</link>
	<pubDate>Sat, 02 May 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Banach_Steinhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gaussian Integers</title>
	<link>/entries/Gaussian_Integers.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Gaussian_Integers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Power Sum Polynomials</title>
	<link>/entries/Power_Sum_Polynomials.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Power_Sum_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Lambert W Function on the Reals</title>
	<link>/entries/Lambert_W.html</link>
	<pubDate>Fri, 24 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lambert_W.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices for ODEs</title>
	<link>/entries/Matrices_for_ODEs.html</link>
	<pubDate>Sun, 19 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Matrices_for_ODEs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices for ODEs</title>
	<link>/entries/Matrices_for_ODEs.html</link>
	<pubDate>Sun, 19 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Matrices_for_ODEs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lucas's Theorem</title>
	<link>/entries/Lucas_Theorem.html</link>
	<pubDate>Tue, 07 Apr 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Lucas_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Furstenberg's topology and his proof of the infinitude of primes</title>
	<link>/entries/Furstenberg_Topology.html</link>
	<pubDate>Sun, 22 Mar 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Furstenberg_Topology.html</guid>
	<description></description>
	</item>

	<item>
	<title>Arithmetic progressions and relative primes</title>
	<link>/entries/Arith_Prog_Rel_Primes.html</link>
	<pubDate>Sat, 01 Feb 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Arith_Prog_Rel_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Hierarchy of Algebras for Boolean Subsets</title>
	<link>/entries/Subset_Boolean_Algebras.html</link>
	<pubDate>Fri, 31 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Subset_Boolean_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Mersenne primes and the Lucas–Lehmer test</title>
	<link>/entries/Mersenne_Primes.html</link>
	<pubDate>Fri, 17 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Mersenne_Primes.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bicategories</title>
	<link>/entries/Bicategory.html</link>
	<pubDate>Mon, 06 Jan 2020 00:00:00 +0000</pubDate>

	<guid>/entries/Bicategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Irrationality of ζ(3)</title>
	<link>/entries/Zeta_3_Irrational.html</link>
	<pubDate>Fri, 27 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_3_Irrational.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Poincaré-Bendixson Theorem</title>
	<link>/entries/Poincare_Bendixson.html</link>
	<pubDate>Wed, 18 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Bendixson.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complex Geometry</title>
	<link>/entries/Complex_Geometry.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complex_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>Poincaré Disc Model</title>
	<link>/entries/Poincare_Disc.html</link>
	<pubDate>Mon, 16 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Poincare_Disc.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gauss Sums and the Pólya–Vinogradov Inequality</title>
	<link>/entries/Gauss_Sums.html</link>
	<pubDate>Tue, 10 Dec 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Gauss_Sums.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification Components for Hybrid Systems</title>
	<link>/entries/Hybrid_Systems_VCs.html</link>
	<pubDate>Tue, 10 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Systems_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verification Components for Hybrid Systems</title>
	<link>/entries/Hybrid_Systems_VCs.html</link>
	<pubDate>Tue, 10 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Hybrid_Systems_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Fourier Series</title>
	<link>/entries/Fourier.html</link>
	<pubDate>Fri, 06 Sep 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Case Study in Basic Algebra</title>
	<link>/entries/Jacobson_Basic_Algebra.html</link>
	<pubDate>Fri, 30 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Jacobson_Basic_Algebra.html</guid>
	<description></description>
	</item>

	<item>
	<title>Laplace Transform</title>
	<link>/entries/Laplace_Transform.html</link>
	<pubDate>Wed, 14 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Laplace_Transform.html</guid>
	<description></description>
	</item>

	<item>
	<title>Communicating Concurrent Kleene Algebra for Distributed Systems Specification</title>
	<link>/entries/C2KA_DistributedSystems.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/C2KA_DistributedSystems.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Programming</title>
	<link>/entries/Linear_Programming.html</link>
	<pubDate>Tue, 06 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Programming.html</guid>
	<description></description>
	</item>

	<item>
	<title>Selected Problems from the International Mathematical Olympiad 2019</title>
	<link>/entries/IMO2019.html</link>
	<pubDate>Mon, 05 Aug 2019 00:00:00 +0000</pubDate>

	<guid>/entries/IMO2019.html</guid>
	<description></description>
	</item>

	<item>
	<title>Order Extension and Szpilrajn's Extension Theorem</title>
	<link>/entries/Szpilrajn.html</link>
	<pubDate>Sat, 27 Jul 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Szpilrajn.html</guid>
	<description></description>
	</item>

	<item>
	<title>Complete Non-Orders and Fixed Points</title>
	<link>/entries/Complete_Non_Orders.html</link>
	<pubDate>Thu, 27 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Complete_Non_Orders.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Inequalities</title>
	<link>/entries/Linear_Inequalities.html</link>
	<pubDate>Fri, 21 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Inequalities.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hilbert's Nullstellensatz</title>
	<link>/entries/Nullstellensatz.html</link>
	<pubDate>Sun, 16 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Nullstellensatz.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hilbert's Nullstellensatz</title>
	<link>/entries/Nullstellensatz.html</link>
	<pubDate>Sun, 16 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Nullstellensatz.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases, Macaulay Matrices and Dubé's Degree Bounds</title>
	<link>/entries/Groebner_Macaulay.html</link>
	<pubDate>Sat, 15 Jun 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Macaulay.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of Certain Infinite Series</title>
	<link>/entries/Transcendence_Series_Hancl_Rucki.html</link>
	<pubDate>Wed, 27 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Transcendence_Series_Hancl_Rucki.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of Certain Infinite Series</title>
	<link>/entries/Transcendence_Series_Hancl_Rucki.html</link>
	<pubDate>Wed, 27 Mar 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Transcendence_Series_Hancl_Rucki.html</guid>
	<description></description>
	</item>

	<item>
	<title>Elementary Facts About the Distribution of Primes</title>
	<link>/entries/Prime_Distribution_Elementary.html</link>
	<pubDate>Thu, 21 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Distribution_Elementary.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Primality Testing</title>
	<link>/entries/Probabilistic_Prime_Tests.html</link>
	<pubDate>Mon, 11 Feb 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Prime_Tests.html</guid>
	<description></description>
	</item>

	<item>
	<title>Farkas' Lemma and Motzkin's Transposition Theorem</title>
	<link>/entries/Farkas.html</link>
	<pubDate>Thu, 17 Jan 2019 00:00:00 +0000</pubDate>

	<guid>/entries/Farkas.html</guid>
	<description></description>
	</item>

	<item>
	<title>Constructive Cryptography in HOL</title>
	<link>/entries/Constructive_Cryptography.html</link>
	<pubDate>Mon, 17 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Constructive_Cryptography.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Orderings and Lattices</title>
	<link>/entries/Order_Lattice_Props.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Order_Lattice_Props.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quantales</title>
	<link>/entries/Quantales.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quantales.html</guid>
	<description></description>
	</item>

	<item>
	<title>Transformer Semantics</title>
	<link>/entries/Transformer_Semantics.html</link>
	<pubDate>Tue, 11 Dec 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Transformer_Semantics.html</guid>
	<description></description>
	</item>

	<item>
	<title>Graph Saturation</title>
	<link>/entries/Graph_Saturation.html</link>
	<pubDate>Fri, 23 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Graph_Saturation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matroids</title>
	<link>/entries/Matroids.html</link>
	<pubDate>Fri, 16 Nov 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Matroids.html</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth Manifolds</title>
	<link>/entries/Smooth_Manifolds.html</link>
	<pubDate>Mon, 22 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Smooth_Manifolds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Smooth Manifolds</title>
	<link>/entries/Smooth_Manifolds.html</link>
	<pubDate>Mon, 22 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Smooth_Manifolds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Upper Bounding Diameters of State Spaces of Factored Transition Systems</title>
	<link>/entries/Factored_Transition_System_Bounding.html</link>
	<pubDate>Fri, 12 Oct 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Factored_Transition_System_Bounding.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of π</title>
	<link>/entries/Pi_Transcendental.html</link>
	<pubDate>Fri, 28 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pi_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>Symmetric Polynomials</title>
	<link>/entries/Symmetric_Polynomials.html</link>
	<pubDate>Tue, 25 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Symmetric_Polynomials.html</guid>
	<description></description>
	</item>

	<item>
	<title>Signature-Based Gröbner Basis Algorithms</title>
	<link>/entries/Signature_Groebner.html</link>
	<pubDate>Thu, 20 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Signature_Groebner.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Prime Number Theorem</title>
	<link>/entries/Prime_Number_Theorem.html</link>
	<pubDate>Wed, 19 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Number_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Aggregation Algebras</title>
	<link>/entries/Aggregation_Algebras.html</link>
	<pubDate>Sat, 15 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Aggregation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/entries/Octonions.html</link>
	<pubDate>Fri, 14 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Octonions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Octonions</title>
	<link>/entries/Octonions.html</link>
	<pubDate>Fri, 14 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Octonions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/entries/Quaternions.html</link>
	<pubDate>Wed, 05 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quaternions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Quaternions</title>
	<link>/entries/Quaternions.html</link>
	<pubDate>Wed, 05 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Quaternions.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Budan–Fourier Theorem and Counting Real Roots with Multiplicity</title>
	<link>/entries/Budan_Fourier.html</link>
	<pubDate>Sun, 02 Sep 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Budan_Fourier.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pricing in discrete financial models</title>
	<link>/entries/DiscretePricing.html</link>
	<pubDate>Mon, 16 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/DiscretePricing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pricing in discrete financial models</title>
	<link>/entries/DiscretePricing.html</link>
	<pubDate>Mon, 16 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/DiscretePricing.html</guid>
	<description></description>
	</item>

	<item>
	<title>Von-Neumann-Morgenstern Utility Theorem</title>
	<link>/entries/Neumann_Morgenstern_Utility.html</link>
	<pubDate>Wed, 04 Jul 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Neumann_Morgenstern_Utility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Pell's Equation</title>
	<link>/entries/Pell.html</link>
	<pubDate>Sat, 23 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Pell.html</guid>
	<description></description>
	</item>

	<item>
	<title>Projective Geometry</title>
	<link>/entries/Projective_Geometry.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Projective_Geometry.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Localization of a Commutative Ring</title>
	<link>/entries/Localization_Ring.html</link>
	<pubDate>Thu, 14 Jun 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Localization_Ring.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hidden Markov Models</title>
	<link>/entries/Hidden_Markov_Models.html</link>
	<pubDate>Fri, 25 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Hidden_Markov_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic Timed Automata</title>
	<link>/entries/Probabilistic_Timed_Automata.html</link>
	<pubDate>Thu, 24 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_Timed_Automata.html</guid>
	<description></description>
	</item>

	<item>
	<title>Axiom Systems for Category Theory in Free Logic</title>
	<link>/entries/AxiomaticCategoryTheory.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/AxiomaticCategoryTheory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational Rapidly Convergent Series</title>
	<link>/entries/Irrationality_J_Hancl.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationality_J_Hancl.html</guid>
	<description></description>
	</item>

	<item>
	<title>Irrational Rapidly Convergent Series</title>
	<link>/entries/Irrationality_J_Hancl.html</link>
	<pubDate>Wed, 23 May 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Irrationality_J_Hancl.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of Fishburn-Strategyproofness and Pareto-Efficiency</title>
	<link>/entries/Fishburn_Impossibility.html</link>
	<pubDate>Thu, 22 Mar 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Fishburn_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified factorization algorithm for integer polynomials with polynomial complexity</title>
	<link>/entries/LLL_Factorization.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Error Function</title>
	<link>/entries/Error_Function.html</link>
	<pubDate>Tue, 06 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Error_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>A verified LLL algorithm</title>
	<link>/entries/LLL_Basis_Reduction.html</link>
	<pubDate>Fri, 02 Feb 2018 00:00:00 +0000</pubDate>

	<guid>/entries/LLL_Basis_Reduction.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gromov Hyperbolicity</title>
	<link>/entries/Gromov_Hyperbolicity.html</link>
	<pubDate>Tue, 16 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Gromov_Hyperbolicity.html</guid>
	<description></description>
	</item>

	<item>
	<title>An Isabelle/HOL formalisation of Green's Theorem</title>
	<link>/entries/Green.html</link>
	<pubDate>Thu, 11 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Green.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>Taylor Models</title>
	<link>/entries/Taylor_Models.html</link>
	<pubDate>Mon, 08 Jan 2018 00:00:00 +0000</pubDate>

	<guid>/entries/Taylor_Models.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Falling Factorial of a Sum</title>
	<link>/entries/Falling_Factorial_Sum.html</link>
	<pubDate>Fri, 22 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Falling_Factorial_Sum.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet L-Functions and Dirichlet's Theorem</title>
	<link>/entries/Dirichlet_L.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_L.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet L-Functions and Dirichlet's Theorem</title>
	<link>/entries/Dirichlet_L.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_L.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Mason–Stothers Theorem</title>
	<link>/entries/Mason_Stothers.html</link>
	<pubDate>Thu, 21 Dec 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Mason_Stothers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stochastic Matrices and the Perron-Frobenius Theorem</title>
	<link>/entries/Stochastic_Matrices.html</link>
	<pubDate>Wed, 22 Nov 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stochastic_Matrices.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Kuratowski Closure-Complement Theorem</title>
	<link>/entries/Kuratowski_Closure_Complement.html</link>
	<pubDate>Thu, 26 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Kuratowski_Closure_Complement.html</guid>
	<description></description>
	</item>

	<item>
	<title>Count the Number of Complex Roots</title>
	<link>/entries/Count_Complex_Roots.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Count_Complex_Roots.html</guid>
	<description></description>
	</item>

	<item>
	<title>Evaluate Winding Numbers through Cauchy Indices</title>
	<link>/entries/Winding_Number_Eval.html</link>
	<pubDate>Tue, 17 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Winding_Number_Eval.html</guid>
	<description></description>
	</item>

	<item>
	<title>Homogeneous Linear Diophantine Equations</title>
	<link>/entries/Diophantine_Eqns_Lin_Hom.html</link>
	<pubDate>Sat, 14 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Diophantine_Eqns_Lin_Hom.html</guid>
	<description></description>
	</item>

	<item>
	<title>Dirichlet Series</title>
	<link>/entries/Dirichlet_Series.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Dirichlet_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Linear Recurrences</title>
	<link>/entries/Linear_Recurrences.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Linear_Recurrences.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hurwitz and Riemann ζ Functions</title>
	<link>/entries/Zeta_Function.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Hurwitz and Riemann ζ Functions</title>
	<link>/entries/Zeta_Function.html</link>
	<pubDate>Thu, 12 Oct 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Zeta_Function.html</guid>
	<description></description>
	</item>

	<item>
	<title>Microeconomics and the First Welfare Theorem</title>
	<link>/entries/First_Welfare_Theorem.html</link>
	<pubDate>Fri, 01 Sep 2017 00:00:00 +0000</pubDate>

	<guid>/entries/First_Welfare_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Orbit-Stabiliser Theorem with Application to Rotational Symmetries</title>
	<link>/entries/Orbit_Stabiliser.html</link>
	<pubDate>Sun, 20 Aug 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Orbit_Stabiliser.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stewart's Theorem and Apollonius' Theorem</title>
	<link>/entries/Stewart_Apollonius.html</link>
	<pubDate>Mon, 31 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stewart_Apollonius.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowski's Theorem</title>
	<link>/entries/Minkowskis_Theorem.html</link>
	<pubDate>Thu, 13 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minkowskis_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Minkowski's Theorem</title>
	<link>/entries/Minkowskis_Theorem.html</link>
	<pubDate>Thu, 13 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Minkowskis_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone-Kleene Relation Algebras</title>
	<link>/entries/Stone_Kleene_Relation_Algebras.html</link>
	<pubDate>Thu, 06 Jul 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Kleene_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Partial Semigroups and Convolution Algebras</title>
	<link>/entries/PSemigroupsConvolution.html</link>
	<pubDate>Tue, 13 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/PSemigroupsConvolution.html</guid>
	<description></description>
	</item>

	<item>
	<title>Buffon's Needle Problem</title>
	<link>/entries/Buffons_Needle.html</link>
	<pubDate>Tue, 06 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buffons_Needle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Buffon's Needle Problem</title>
	<link>/entries/Buffons_Needle.html</link>
	<pubDate>Tue, 06 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Buffons_Needle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Flow Networks and the Min-Cut-Max-Flow Theorem</title>
	<link>/entries/Flow_Networks.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Flow_Networks.html</guid>
	<description></description>
	</item>

	<item>
	<title>Formalizing Push-Relabel Algorithms</title>
	<link>/entries/Prpu_Maxflow.html</link>
	<pubDate>Thu, 01 Jun 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Prpu_Maxflow.html</guid>
	<description></description>
	</item>

	<item>
	<title>Optics</title>
	<link>/entries/Optics.html</link>
	<pubDate>Thu, 25 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Optics.html</guid>
	<description></description>
	</item>

	<item>
	<title>CryptHOL</title>
	<link>/entries/CryptHOL.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/CryptHOL.html</guid>
	<description></description>
	</item>

	<item>
	<title>Probabilistic while loop</title>
	<link>/entries/Probabilistic_While.html</link>
	<pubDate>Fri, 05 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Probabilistic_While.html</guid>
	<description></description>
	</item>

	<item>
	<title>Monoidal Categories</title>
	<link>/entries/MonoidalCategory.html</link>
	<pubDate>Thu, 04 May 2017 00:00:00 +0000</pubDate>

	<guid>/entries/MonoidalCategory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Subresultants</title>
	<link>/entries/Subresultants.html</link>
	<pubDate>Thu, 06 Apr 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Subresultants.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Euler–MacLaurin Formula</title>
	<link>/entries/Euler_MacLaurin.html</link>
	<pubDate>Fri, 10 Mar 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_MacLaurin.html</guid>
	<description></description>
	</item>

	<item>
	<title>Menger's Theorem</title>
	<link>/entries/Menger.html</link>
	<pubDate>Sun, 26 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Menger.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Relation Algebras</title>
	<link>/entries/Stone_Relation_Algebras.html</link>
	<pubDate>Tue, 07 Feb 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Relation_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli Numbers</title>
	<link>/entries/Bernoulli.html</link>
	<pubDate>Tue, 24 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bernoulli.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bernoulli Numbers</title>
	<link>/entries/Bernoulli.html</link>
	<pubDate>Tue, 24 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bernoulli.html</guid>
	<description></description>
	</item>

	<item>
	<title>Bertrand's postulate</title>
	<link>/entries/Bertrands_Postulate.html</link>
	<pubDate>Tue, 17 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/Bertrands_Postulate.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of e</title>
	<link>/entries/E_Transcendental.html</link>
	<pubDate>Thu, 12 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/E_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Transcendence of e</title>
	<link>/entries/E_Transcendental.html</link>
	<pubDate>Thu, 12 Jan 2017 00:00:00 +0000</pubDate>

	<guid>/entries/E_Transcendental.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Twelvefold Way</title>
	<link>/entries/Twelvefold_Way.html</link>
	<pubDate>Thu, 29 Dec 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Twelvefold_Way.html</guid>
	<description></description>
	</item>

	<item>
	<title>Expressiveness of Deep Learning</title>
	<link>/entries/Deep_Learning.html</link>
	<pubDate>Thu, 10 Nov 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Deep_Learning.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stable Matching</title>
	<link>/entries/Stable_Matching.html</link>
	<pubDate>Mon, 24 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stable_Matching.html</guid>
	<description></description>
	</item>

	<item>
	<title>Source Coding Theorem</title>
	<link>/entries/Source_Coding_Theorem.html</link>
	<pubDate>Wed, 19 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Source_Coding_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Factorization Algorithm of Berlekamp and Zassenhaus</title>
	<link>/entries/Berlekamp_Zassenhaus.html</link>
	<pubDate>Fri, 14 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Berlekamp_Zassenhaus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Intersecting Chords Theorem</title>
	<link>/entries/Chord_Segments.html</link>
	<pubDate>Tue, 11 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Chord_Segments.html</guid>
	<description></description>
	</item>

	<item>
	<title>Lp spaces</title>
	<link>/entries/Lp.html</link>
	<pubDate>Wed, 05 Oct 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Lp.html</guid>
	<description></description>
	</item>

	<item>
	<title>Allen's Interval Calculus</title>
	<link>/entries/Allen_Calculus.html</link>
	<pubDate>Thu, 29 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Allen_Calculus.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stone Algebras</title>
	<link>/entries/Stone_Algebras.html</link>
	<pubDate>Tue, 06 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stone_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Stirling's formula</title>
	<link>/entries/Stirling_Formula.html</link>
	<pubDate>Thu, 01 Sep 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Stirling_Formula.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ptolemy's Theorem</title>
	<link>/entries/Ptolemys_Theorem.html</link>
	<pubDate>Sun, 07 Aug 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Ptolemys_Theorem.html</guid>
	<description></description>
	</item>

	<item>
	<title>Chamber Complexes, Coxeter Systems, and Buildings</title>
	<link>/entries/Buildings.html</link>
	<pubDate>Fri, 01 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Buildings.html</guid>
	<description></description>
	</item>

	<item>
	<title>Chamber Complexes, Coxeter Systems, and Buildings</title>
	<link>/entries/Buildings.html</link>
	<pubDate>Fri, 01 Jul 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Buildings.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Multisets</title>
	<link>/entries/Card_Multisets.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Multisets.html</guid>
	<description></description>
	</item>

	<item>
	<title>Category Theory with Adjunctions and Limits</title>
	<link>/entries/Category3.html</link>
	<pubDate>Sun, 26 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Category3.html</guid>
	<description></description>
	</item>

	<item>
	<title>Catalan Numbers</title>
	<link>/entries/Catalan_Numbers.html</link>
	<pubDate>Tue, 21 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Catalan_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Program Construction and Verification Components Based on Kleene Algebra</title>
	<link>/entries/Algebraic_VCs.html</link>
	<pubDate>Sat, 18 Jun 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_VCs.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tree Decomposition</title>
	<link>/entries/Tree_Decomposition.html</link>
	<pubDate>Tue, 31 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Tree_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Equivalence Relations</title>
	<link>/entries/Card_Equiv_Relations.html</link>
	<pubDate>Tue, 24 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Equiv_Relations.html</guid>
	<description></description>
	</item>

	<item>
	<title>Perron-Frobenius Theorem for Spectral Radius Analysis</title>
	<link>/entries/Perron_Frobenius.html</link>
	<pubDate>Fri, 20 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Perron_Frobenius.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks</title>
	<link>/entries/MFMC_Countable.html</link>
	<pubDate>Mon, 09 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/MFMC_Countable.html</guid>
	<description></description>
	</item>

	<item>
	<title>Randomised Social Choice Theory</title>
	<link>/entries/Randomised_Social_Choice.html</link>
	<pubDate>Thu, 05 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Randomised_Social_Choice.html</guid>
	<description></description>
	</item>

	<item>
	<title>Spivey's Generalized Recurrence for Bell Numbers</title>
	<link>/entries/Bell_Numbers_Spivey.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Bell_Numbers_Spivey.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Incompatibility of SD-Efficiency and SD-Strategy-Proofness</title>
	<link>/entries/SDS_Impossibility.html</link>
	<pubDate>Wed, 04 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/SDS_Impossibility.html</guid>
	<description></description>
	</item>

	<item>
	<title>Gröbner Bases Theory</title>
	<link>/entries/Groebner_Bases.html</link>
	<pubDate>Mon, 02 May 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Groebner_Bases.html</guid>
	<description></description>
	</item>

	<item>
	<title>No Faster-Than-Light Observers</title>
	<link>/entries/No_FTL_observers.html</link>
	<pubDate>Thu, 28 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/No_FTL_observers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Kleene Algebras with Domain</title>
	<link>/entries/KAD.html</link>
	<pubDate>Tue, 12 Apr 2016 00:00:00 +0000</pubDate>

	<guid>/entries/KAD.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cartan Fixed Point Theorems</title>
	<link>/entries/Cartan_FP.html</link>
	<pubDate>Tue, 08 Mar 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Cartan_FP.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Factorization</title>
	<link>/entries/Polynomial_Factorization.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Factorization.html</guid>
	<description></description>
	</item>

	<item>
	<title>Polynomial Interpolation</title>
	<link>/entries/Polynomial_Interpolation.html</link>
	<pubDate>Fri, 29 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Polynomial_Interpolation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Knot Theory</title>
	<link>/entries/Knot_Theory.html</link>
	<pubDate>Wed, 20 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Knot_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Tensor Product of Matrices</title>
	<link>/entries/Matrix_Tensor.html</link>
	<pubDate>Mon, 18 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Matrix_Tensor.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Number Partitions</title>
	<link>/entries/Card_Number_Partitions.html</link>
	<pubDate>Thu, 14 Jan 2016 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Number_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Basic Geometric Properties of Triangles</title>
	<link>/entries/Triangle.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Triangle.html</guid>
	<description></description>
	</item>

	<item>
	<title>Descartes' Rule of Signs</title>
	<link>/entries/Descartes_Sign_Rule.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Descartes_Sign_Rule.html</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville numbers</title>
	<link>/entries/Liouville_Numbers.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Liouville_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>Liouville numbers</title>
	<link>/entries/Liouville_Numbers.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Liouville_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Divergence of the Prime Harmonic Series</title>
	<link>/entries/Prime_Harmonic_Series.html</link>
	<pubDate>Mon, 28 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Prime_Harmonic_Series.html</guid>
	<description></description>
	</item>

	<item>
	<title>Algebraic Numbers in Isabelle/HOL</title>
	<link>/entries/Algebraic_Numbers.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Algebraic_Numbers.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Stern-Brocot Tree</title>
	<link>/entries/Stern_Brocot.html</link>
	<pubDate>Tue, 22 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Stern_Brocot.html</guid>
	<description></description>
	</item>

	<item>
	<title>Cardinality of Set Partitions</title>
	<link>/entries/Card_Partitions.html</link>
	<pubDate>Sat, 12 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Card_Partitions.html</guid>
	<description></description>
	</item>

	<item>
	<title>Latin Square</title>
	<link>/entries/Latin_Square.html</link>
	<pubDate>Wed, 02 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Latin_Square.html</guid>
	<description></description>
	</item>

	<item>
	<title>Ergodic Theory</title>
	<link>/entries/Ergodic_Theory.html</link>
	<pubDate>Tue, 01 Dec 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Ergodic_Theory.html</guid>
	<description></description>
	</item>

	<item>
	<title>Euler's Partition Theorem</title>
	<link>/entries/Euler_Partition.html</link>
	<pubDate>Thu, 19 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Euler_Partition.html</guid>
	<description></description>
	</item>

	<item>
	<title>Planarity Certificates</title>
	<link>/entries/Planarity_Certificates.html</link>
	<pubDate>Wed, 11 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Planarity_Certificates.html</guid>
	<description></description>
	</item>

	<item>
	<title>Positional Determinacy of Parity Games</title>
	<link>/entries/Parity_Game.html</link>
	<pubDate>Mon, 02 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Parity_Game.html</guid>
	<description></description>
	</item>

	<item>
	<title>Positional Determinacy of Parity Games</title>
	<link>/entries/Parity_Game.html</link>
	<pubDate>Mon, 02 Nov 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Parity_Game.html</guid>
	<description></description>
	</item>

	<item>
	<title>Matrices, Jordan Normal Forms, and Spectral Radius Theory</title>
	<link>/entries/Jordan_Normal_Form.html</link>
	<pubDate>Fri, 21 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Normal_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>Representations of Finite Groups</title>
	<link>/entries/Rep_Fin_Groups.html</link>
	<pubDate>Wed, 12 Aug 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Rep_Fin_Groups.html</guid>
	<description></description>
	</item>

	<item>
	<title>Landau Symbols</title>
	<link>/entries/Landau_Symbols.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Landau_Symbols.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Akra-Bazzi theorem and the Master theorem</title>
	<link>/entries/Akra_Bazzi.html</link>
	<pubDate>Tue, 14 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Akra_Bazzi.html</guid>
	<description></description>
	</item>

	<item>
	<title>Hermite Normal Form</title>
	<link>/entries/Hermite.html</link>
	<pubDate>Tue, 07 Jul 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Hermite.html</guid>
	<description></description>
	</item>

	<item>
	<title>Derangements Formula</title>
	<link>/entries/Derangements.html</link>
	<pubDate>Sat, 27 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Derangements.html</guid>
	<description></description>
	</item>

	<item>
	<title>Binary Multirelations</title>
	<link>/entries/Multirelations.html</link>
	<pubDate>Thu, 11 Jun 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Multirelations.html</guid>
	<description></description>
	</item>

	<item>
	<title>VCG - Combinatorial Vickrey-Clarke-Groves Auctions</title>
	<link>/entries/Vickrey_Clarke_Groves.html</link>
	<pubDate>Thu, 30 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Vickrey_Clarke_Groves.html</guid>
	<description></description>
	</item>

	<item>
	<title>Residuated Lattices</title>
	<link>/entries/Residuated_Lattices.html</link>
	<pubDate>Wed, 15 Apr 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Residuated_Lattices.html</guid>
	<description></description>
	</item>

	<item>
	<title>Echelon Form</title>
	<link>/entries/Echelon_Form.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/Echelon_Form.html</guid>
	<description></description>
	</item>

	<item>
	<title>QR Decomposition</title>
	<link>/entries/QR_Decomposition.html</link>
	<pubDate>Thu, 12 Feb 2015 00:00:00 +0000</pubDate>

	<guid>/entries/QR_Decomposition.html</guid>
	<description></description>
	</item>

	<item>
	<title>A Verified Compiler for Probability Density Functions</title>
	<link>/entries/Density_Compiler.html</link>
	<pubDate>Thu, 09 Oct 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Density_Compiler.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Sturm–Tarski Theorem</title>
	<link>/entries/Sturm_Tarski.html</link>
	<pubDate>Fri, 19 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Sturm_Tarski.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Cayley-Hamilton Theorem</title>
	<link>/entries/Cayley_Hamilton.html</link>
	<pubDate>Mon, 15 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Cayley_Hamilton.html</guid>
	<description></description>
	</item>

	<item>
	<title>The Jordan-Hölder Theorem</title>
	<link>/entries/Jordan_Hoelder.html</link>
	<pubDate>Tue, 09 Sep 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Jordan_Hoelder.html</guid>
	<description></description>
	</item>

	<item>
	<title>Real-Valued Special Functions: Upper and Lower Bounds</title>
	<link>/entries/Special_Function_Bounds.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Special_Function_Bounds.html</guid>
	<description></description>
	</item>

	<item>
	<title>Vector Spaces</title>
	<link>/entries/VectorSpace.html</link>
	<pubDate>Fri, 29 Aug 2014 00:00:00 +0000</pubDate>

	<guid>/entries/VectorSpace.html</guid>
	<description></description>
	</item>

	<item>
	<title>Verified Efficient Implementation of Gabow's Strongly Connected Components Algorithm</title>
	<link>/entries/Gabow_SCC.html</link>
	<pubDate>Wed, 28 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Gabow_SCC.html</guid>
	<description></description>
	</item>

	<item>
	<title>Regular Algebras</title>
	<link>/entries/Regular_Algebras.html</link>
	<pubDate>Wed, 21 May 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Regular_Algebras.html</guid>
	<description></description>
	</item>

	<item>
	<title>Discrete Summation</title>
	<link>/entries/Discrete_Summation.html</link>
	<pubDate>Sun, 13 Apr 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Discrete_Summation.html</guid>
	<description></description>
	</item>

	<item>
	<title>Properties of Random Graphs -- Subgraph Containment</title>
	<link>/entries/Random_Graph_Subgraph_Threshold.html</link>
	<pubDate>Thu, 13 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Random_Graph_Subgraph_Threshold.html</guid>
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	<item>
	<title>Properties of Random Graphs -- Subgraph Containment</title>
	<link>/entries/Random_Graph_Subgraph_Threshold.html</link>
	<pubDate>Thu, 13 Feb 2014 00:00:00 +0000</pubDate>

	<guid>/entries/Random_Graph_Subgraph_Threshold.html</guid>
	<description></description>
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	<item>
	<title>Affine Arithmetic</title>
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	<pubDate>Fri, 07 Feb 2014 00:00:00 +0000</pubDate>

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	<title>Implementing field extensions of the form Q[sqrt(b)]</title>
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	<pubDate>Thu, 06 Feb 2014 00:00:00 +0000</pubDate>

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	<item>
	<title>Secondary Sylow Theorems</title>
	<link>/entries/Secondary_Sylow.html</link>
	<pubDate>Tue, 28 Jan 2014 00:00:00 +0000</pubDate>

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	<item>
	<title>Relation Algebra</title>
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	<title>Kleene Algebra with Tests and Demonic Refinement Algebras</title>
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	<title>Sturm's Theorem</title>
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	<title>Lehmer's Theorem</title>
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	<title>Pratt's Primality Certificates</title>
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	<title>The Königsberg Bridge Problem and the Friendship Theorem</title>
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	<item>
	<title>An Axiomatic Characterization of the Single-Source Shortest Path Problem</title>
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	<item>
	<title>Graph Theory</title>
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	<title>Rank-Nullity Theorem in Linear Algebra</title>
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	<title>Kleene Algebra</title>
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	<title>Computing N-th Roots using the Babylonian Method</title>
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	<title>Open Induction</title>
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	<title>The independence of Tarski's Euclidean axiom</title>
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	<title>Bondy's Theorem</title>
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	<title>Proving the Impossibility of Trisecting an Angle and Doubling the Cube</title>
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	<title>Proving the Impossibility of Trisecting an Angle and Doubling the Cube</title>
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	<title>A Probabilistic Proof of the Girth-Chromatic Number Theorem</title>
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	<title>Lattice Properties</title>
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	<title>Pseudo Hoops</title>
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	<item>
	<title>Gauss-Jordan Elimination for Matrices Represented as Functions</title>
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	<pubDate>Fri, 19 Aug 2011 00:00:00 +0000</pubDate>

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	<item>
	<title>Maximum Cardinality Matching</title>
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	<pubDate>Thu, 21 Jul 2011 00:00:00 +0000</pubDate>

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	<item>
	<title>The General Triangle Is Unique</title>
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	<item>
	<title>Lower Semicontinuous Functions</title>
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	<pubDate>Sat, 08 Jan 2011 00:00:00 +0000</pubDate>

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	<item>
	<title>Hall's Marriage Theorem</title>
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	<pubDate>Fri, 17 Dec 2010 00:00:00 +0000</pubDate>

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	<title>Executable Multivariate Polynomials</title>
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	<title>Executable Multivariate Polynomials</title>
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	<title>Free Groups</title>
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	<pubDate>Thu, 24 Jun 2010 00:00:00 +0000</pubDate>

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	<item>
	<title>Category Theory</title>
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	<pubDate>Sun, 20 Jun 2010 00:00:00 +0000</pubDate>

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	<item>
	<title>A Complete Proof of the Robbins Conjecture</title>
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	<pubDate>Sat, 22 May 2010 00:00:00 +0000</pubDate>

	<guid>/entries/Robbins-Conjecture.html</guid>
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	<item>
	<title>Perfect Number Theorem</title>
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	<pubDate>Sun, 22 Nov 2009 00:00:00 +0000</pubDate>

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	<item>
	<title>An Example of a Cofinitary Group in Isabelle/HOL</title>
	<link>/entries/CofGroups.html</link>
	<pubDate>Tue, 04 Aug 2009 00:00:00 +0000</pubDate>

	<guid>/entries/CofGroups.html</guid>
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	<item>
	<title>Some classical results in Social Choice Theory</title>
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	<pubDate>Sun, 09 Nov 2008 00:00:00 +0000</pubDate>

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	<item>
	<title>Fun With Tilings</title>
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	<pubDate>Fri, 07 Nov 2008 00:00:00 +0000</pubDate>

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	<title>Arrow and Gibbard-Satterthwaite</title>
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	<title>Fun With Functions</title>
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	<title>Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</title>
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	<pubDate>Sun, 12 Aug 2007 00:00:00 +0000</pubDate>

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	<title>Fundamental Properties of Valuation Theory and Hensel's Lemma</title>
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	<pubDate>Wed, 08 Aug 2007 00:00:00 +0000</pubDate>

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	<title>Flyspeck I: Tame Graphs</title>
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	<pubDate>Mon, 22 May 2006 00:00:00 +0000</pubDate>

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	<title>Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality</title>
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	<h1 >
	<span class='first'>M</span>athematics/<span class='first'>N</span>umber <span class='first'>T</span>heory</h1>
	<div>


	</div>
	</header><div><h2>Subject Classification</h2><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=11-XX">Number theory</a></p><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../../../entries/Polygonal_Number_Theorem.html">Polygonal Number Theorem</a></h5> <br>by <a href="../../../authors/leek">Kevin Lee</a>, <a href="../../../authors/yez">Zhengkun Ye</a> and <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
	+ <span class="date">
	+ Aug 10
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="../../../entries/Zeckendorf.html">Zeckendorf’s Theorem</a></h5> <br>by <a href="../../../authors/dalvit">Christian Dalvit</a></div>
	<span class="date">
	Jun 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Three_Squares.html">Three Squares Theorem</a></h5> <br>by <a href="../../../authors/danilkin">Anton Danilkin</a> and <a href="../../../authors/chevalier">Loïc Chevalier</a></div>
	<span class="date">
	May 03
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/DigitsInBase.html">Positional Notation for Natural Numbers in an Arbitrary Base</a></h5> <br>by <a href="../../../authors/staats">Charles Staats</a></div>
	<span class="date">
	Apr 03
	</span>
	</article>


	<h2 class="head">2022</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Padic_Field.html">p-adic Fields and p-adic Semialgebraic Sets</a></h5> <br>by <a href="../../../authors/crighton">Aaron Crighton</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Involutions2Squares.html">From THE BOOK: Two Squares via Involutions</a></h5> <br>by <a href="../../../authors/bortin">Maksym Bortin</a></div>
	<span class="date">
	Aug 15
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/DPRM_Theorem.html">Diophantine Equations and the DPRM Theorem</a></h5> <br>by <a href="../../../authors/bayer">Jonas Bayer</a>, <a href="../../../authors/david">Marco David</a>, <a href="../../../authors/stock">Benedikt Stock</a>, <a href="../../../authors/pal">Abhik Pal</a>, <a href="../../../authors/matiyasevich">Yuri Matiyasevich</a> and <a href="../../../authors/schleicher">Dierk Schleicher</a></div>
	<span class="date">
	Jun 06
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Digit_Expansions.html">Digit Expansions</a></h5> <br>by <a href="../../../authors/bayer">Jonas Bayer</a>, <a href="../../../authors/david">Marco David</a>, <a href="../../../authors/pal">Abhik Pal</a> and <a href="../../../authors/stock">Benedikt Stock</a></div>
	<span class="date">
	Apr 20
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	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Irrationals_From_THEBOOK.html">Irrational numbers from THE BOOK</a></h5> <br>by <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Jan 08
	</span>
	</article>


	<h2 class="head">2021</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Lifting_the_Exponent.html">Lifting the Exponent</a></h5> <br>by <a href="../../../authors/kadzioka">Maya Kądziołka</a></div>
	<span class="date">
	Apr 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Padic_Ints.html">Hensel's Lemma for the p-adic Integers</a></h5> <br>by <a href="../../../authors/crighton">Aaron Crighton</a></div>
	<span class="date">
	Mar 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Hermite_Lindemann.html">The Hermite–Lindemann–Weierstraß Transcendence Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 03
	</span>
	</article>


	<h2 class="head">2020</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Amicable_Numbers.html">Amicable Numbers</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a></div>
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	Aug 04
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Irrational_Series_Erdos_Straus.html">Irrationality Criteria for Series by Erdős and Straus</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/li">Wenda Li</a></div>
	<span class="date">
	May 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gaussian_Integers.html">Gaussian Integers</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Apr 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Lucas_Theorem.html">Lucas's Theorem</a></h5> <br>by <a href="../../../authors/edmonds">Chelsea Edmonds</a></div>
	<span class="date">
	Apr 07
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Furstenberg_Topology.html">Furstenberg's topology and his proof of the infinitude of primes</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Mar 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Arith_Prog_Rel_Primes.html">Arithmetic progressions and relative primes</a></h5> <br>by <a href="../../../authors/caballero">José Manuel Rodríguez Caballero</a></div>
	<span class="date">
	Feb 01
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Mersenne_Primes.html">Mersenne primes and the Lucas–Lehmer test</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 17
	</span>
	</article>


	<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Zeta_3_Irrational.html">The Irrationality of ζ(3)</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Gauss_Sums.html">Gauss Sums and the Pólya–Vinogradov Inequality</a></h5> <br>by <a href="../../../authors/raya">Rodrigo Raya</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 10
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Transcendence_Series_Hancl_Rucki.html">The Transcendence of Certain Infinite Series</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/li">Wenda Li</a></div>
	<span class="date">
	Mar 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Prime_Distribution_Elementary.html">Elementary Facts About the Distribution of Primes</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Feb 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Probabilistic_Prime_Tests.html">Probabilistic Primality Testing</a></h5> <br>by <a href="../../../authors/stuewe">Daniel Stüwe</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Feb 11
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Pi_Transcendental.html">The Transcendence of π</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Sep 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Prime_Number_Theorem.html">The Prime Number Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a> and <a href="../../../authors/paulson">Lawrence C. Paulson</a></div>
	<span class="date">
	Sep 19
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Pell.html">Pell's Equation</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jun 23
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Irrationality_J_Hancl.html">Irrational Rapidly Convergent Series</a></h5> <br>by <a href="../../../authors/argyraki">Angeliki Koutsoukou-Argyraki</a> and <a href="../../../authors/li">Wenda Li</a></div>
	<span class="date">
	May 23
	</span>
	</article>


	<h2 class="head">2017</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Dirichlet_L.html">Dirichlet L-Functions and Dirichlet's Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 21
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Diophantine_Eqns_Lin_Hom.html">Homogeneous Linear Diophantine Equations</a></h5> <br>by <a href="../../../authors/messner">Florian Messner</a>, <a href="../../../authors/parsert">Julian Parsert</a>, <a href="../../../authors/schoepf">Jonas Schöpf</a> and <a href="../../../authors/sternagel">Christian Sternagel</a></div>
	<span class="date">
	Oct 14
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Zeta_Function.html">The Hurwitz and Riemann ζ Functions</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Dirichlet_Series.html">Dirichlet Series</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Oct 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Minkowskis_Theorem.html">Minkowski's Theorem</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jul 13
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Bernoulli.html">Bernoulli Numbers</a></h5> <br>by <a href="../../../authors/bulwahn">Lukas Bulwahn</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 24
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Bertrands_Postulate.html">Bertrand's postulate</a></h5> <br>by <a href="../../../authors/biendarra">Julian Biendarra</a> and <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 17
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/E_Transcendental.html">The Transcendence of e</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Jan 12
	</span>
	</article>


	<h2 class="head">2015</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Prime_Harmonic_Series.html">The Divergence of the Prime Harmonic Series</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Liouville_Numbers.html">Liouville numbers</a></h5> <br>by <a href="../../../authors/eberl">Manuel Eberl</a></div>
	<span class="date">
	Dec 28
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stern_Brocot.html">The Stern-Brocot Tree</a></h5> <br>by <a href="../../../authors/gammie">Peter Gammie</a> and <a href="../../../authors/lochbihler">Andreas Lochbihler</a></div>
	<span class="date">
	Dec 22
	</span>
	</article>


	<h2 class="head">2013</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Pratt_Certificate.html">Pratt's Primality Certificates</a></h5> <br>by <a href="../../../authors/wimmer">Simon Wimmer</a> and <a href="../../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Lehmer.html">Lehmer's Theorem</a></h5> <br>by <a href="../../../authors/wimmer">Simon Wimmer</a> and <a href="../../../authors/noschinski">Lars Noschinski</a></div>
	<span class="date">
	Jul 22
	</span>
	</article>


	<h2 class="head">2009</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Perfect-Number-Thm.html">Perfect Number Theorem</a></h5> <br>by <a href="../../../authors/ijbema">Mark Ijbema</a></div>
	<span class="date">
	Nov 22
	</span>
	</article>


	<h2 class="head">2007</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/SumSquares.html">Sums of Two and Four Squares</a></h5> <br>by <a href="../../../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Fermat3_4.html">Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples</a></h5> <br>by <a href="../../../authors/oosterhuis">Roelof Oosterhuis</a></div>
	<span class="date">
	Aug 12
	</span>
	</article>




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	<h1 >
	<span class='first'>M</span>athematics/<span class='first'>O</span>rder</h1>
	<div>


	</div>
	</header><div><h2>Subject Classification</h2><p>AMS: <a href="https://mathscinet.ams.org/mathscinet/msc/msc2020.html?t=06-XX">Order, lattices, ordered algebraic structures</a></p><h2 class="head">2023</h2><article class="entry">
	<div class="item-text">
	+ <h5><a class="title" href="../../../entries/Quantales_Converse.html">Modal quantales, involutive quantales, Dedekind Quantales</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a> and <a href="../../../authors/calk">Cameron Calk</a></div>
	+ <span class="date">
	+ Jul 25
	+ </span>
	+</article>
	+<article class="entry">
	+ <div class="item-text">
	<h5><a class="title" href="../../../entries/Directed_Sets.html">Formalizing Results on Directed Sets</a></h5> <br>by <a href="../../../authors/yamada">Akihisa Yamada</a> and <a href="../../../authors/dubut">Jérémy Dubut</a></div>
	<span class="date">
	May 24
	</span>
	</article>


	<h2 class="head">2019</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Szpilrajn.html">Order Extension and Szpilrajn's Extension Theorem</a></h5> <br>by <a href="../../../authors/zeller">Peter Zeller</a> and <a href="../../../authors/stevens">Lukas Stevens</a></div>
	<span class="date">
	Jul 27
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Complete_Non_Orders.html">Complete Non-Orders and Fixed Points</a></h5> <br>by <a href="../../../authors/yamada">Akihisa Yamada</a> and <a href="../../../authors/dubut">Jérémy Dubut</a></div>
	<span class="date">
	Jun 27
	</span>
	</article>


	<h2 class="head">2018</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Order_Lattice_Props.html">Properties of Orderings and Lattices</a></h5> <br>by <a href="../../../authors/struth">Georg Struth</a></div>
	<span class="date">
	Dec 11
	</span>
	</article>


	<h2 class="head">2016</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Allen_Calculus.html">Allen's Interval Calculus</a></h5> <br>by <a href="../../../authors/ghourabi">Fadoua Ghourabi</a></div>
	<span class="date">
	Sep 29
	</span>
	</article>
	<article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/Stone_Algebras.html">Stone Algebras</a></h5> <br>by <a href="../../../authors/guttmann">Walter Guttmann</a></div>
	<span class="date">
	Sep 06
	</span>
	</article>


	<h2 class="head">2011</h2><article class="entry">
	<div class="item-text">
	<h5><a class="title" href="../../../entries/LatticeProperties.html">Lattice Properties</a></h5> <br>by <a href="../../../authors/preoteasa">Viorel Preoteasa</a></div>
	<span class="date">
	Sep 22
	</span>
	</article>




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