diff --git a/thys/Hermite_Lindemann/document/root.tex b/thys/Hermite_Lindemann/document/root.tex --- a/thys/Hermite_Lindemann/document/root.tex +++ b/thys/Hermite_Lindemann/document/root.tex @@ -1,63 +1,64 @@ \documentclass[11pt,a4paper]{article} \usepackage{isabelle,isabellesym} +\usepackage[T1]{fontenc} \usepackage{amsfonts, amsmath, amssymb} % this should be the last package used \usepackage{pdfsetup} \usepackage[shortcuts]{extdash} % urls in roman style, theory text in math-similar italics \urlstyle{rm} \isabellestyle{it} \begin{document} \title{The Hermite--Lindemann--Weierstraß Transcendence Theorem} \author{Manuel Eberl} \maketitle \begin{abstract} This article provides a formalisation of the Her\-mite\--Lin\-de\-mann\--Wei\-er\-straß Theorem (also known as simply Her\-mite\--Lin\-de\-mann or Lin\-de\-mann\--Wei\-er\-straß). This theorem is one of the crowning achievements of 19th century number theory. The 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 $\mathbb{Q}$. Like the previous formalisation in Coq by Bernard~\cite{bernard}, I proceeded by formalising Baker's alternative formulation of the theorem~\cite{baker} 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: \[\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0 \quad\quad\text{iff}\quad\quad \forall i.\ \beta_i = 0\] This has a number of immediate corollaries, e.g.: \begin{itemize} \item $e$ and $\pi$ are transcendental \item $e^z$, $\sin z$, $\tan z$, etc.\ are transcendental for algebraic $z\in\mathbb{C}\setminus\{0\}$ \item $\ln z$ is transcendental for algebraic $z\in\mathbb{C}\setminus\{0, 1\}$ \end{itemize} \end{abstract} \newpage \tableofcontents \newpage \parindent 0pt\parskip 0.5ex \input{session} \nocite{baker} \nocite{redheffer_steinberg} \nocite{bernard} \bibliographystyle{abbrv} \bibliography{root} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: diff --git a/thys/Mereology/document/root.tex b/thys/Mereology/document/root.tex --- a/thys/Mereology/document/root.tex +++ b/thys/Mereology/document/root.tex @@ -1,61 +1,62 @@ \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 \, \, \, \, \, \, %\, \, \, \, \, %\, \, \ %\usepackage{eurosym} %for \ %\usepackage[only,bigsqcap]{stmaryrd} %for \ %\usepackage{eufrak} %for \ ... \, \ ... \ (also included in amssymb) %\usepackage{textcomp} %for \, \, \, \, \, %\ % 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{Mereology} \author{Ben Blumson} \maketitle \abstract{We use Isabelle/HOL to verify elementary theorems and alternative axiomatizations of classical extensional mereology.} \tableofcontents % sane default for proof documents \parindent 0pt\parskip 0.5ex % generated text of all theories \input{session} % optional bibliography \bibliographystyle{apalike} \bibliography{root} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: diff --git a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/document/root.tex b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/document/root.tex --- a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/document/root.tex +++ b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/document/root.tex @@ -1,66 +1,67 @@ \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 \, \, \, \, \, \, %\, \, \, \, \, %\, \, \ %\usepackage{eurosym} %for \ %\usepackage[only,bigsqcap]{stmaryrd} %for \ %\usepackage{eufrak} %for \ ... \, \ ... \ (also included in amssymb) %\usepackage{textcomp} %for \, \, \, \, \, %\ % 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{Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation\footnote{Supported by FWF (Austrian Science Fund) project Y757 and by project MTM2017-88804-P (Spanish Ministry of Science and Innovation).}} %\title{Modulo arithmetic-based algorithms for lattice basis reduction and for computing the Hermite normal form} \author{Ralph Bottesch \and Jose Divas\'on \and Ren\'e Thiemann} \maketitle \begin{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. \end{abstract} \tableofcontents % sane default for proof documents \parindent 0pt\parskip 0.5ex % generated text of all theories \input{session} % optional bibliography %\bibliographystyle{abbrv} %\bibliography{root} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: diff --git a/thys/Projective_Measurements/document/root.tex b/thys/Projective_Measurements/document/root.tex --- a/thys/Projective_Measurements/document/root.tex +++ b/thys/Projective_Measurements/document/root.tex @@ -1,59 +1,60 @@ \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 \, \, \, \, \, \, %\, \, \, \, \, %\, \, \ %\usepackage{eurosym} %for \ %\usepackage[only,bigsqcap]{stmaryrd} %for \ %\usepackage{eufrak} %for \ ... \, \ ... \ (also included in amssymb) %\usepackage{textcomp} %for \, \, \, \, \, %\ % 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{Projective-Measurements} \author{Mnacho} \maketitle \tableofcontents % sane default for proof documents \parindent 0pt\parskip 0.5ex % generated text of all theories \input{session} % optional bibliography %\bibliographystyle{abbrv} %\bibliography{root} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: diff --git a/thys/Sunflowers/document/root.tex b/thys/Sunflowers/document/root.tex --- a/thys/Sunflowers/document/root.tex +++ b/thys/Sunflowers/document/root.tex @@ -1,36 +1,37 @@ \documentclass[11pt,a4paper]{article} +\usepackage[T1]{fontenc} \usepackage{isabelle,isabellesym} \usepackage{amsmath} \usepackage{amssymb} % 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{The Sunflower Lemma of Erdős and Rado} \author{René Thiemann} \maketitle \begin{abstract} We formally define sunflowers and provide a formalization of the sunflower lemma of Erdős and Rado: whenever a set of size-$k$-sets has a larger cardinality than $(r - 1)^k \cdot k!$, then it contains a sunflower of cardinality $r$. \end{abstract} % include generated text of all theories \input{session} % optional bibliography %\addcontentsline{toc}{section}{Bibliography} %\nocite{*} \bibliographystyle{plain} \bibliography{root} \end{document}