diff --git a/NEWS b/NEWS --- a/NEWS +++ b/NEWS @@ -1,16516 +1,16523 @@ Isabelle NEWS -- history of user-relevant changes ================================================= (Note: Isabelle/jEdit shows a tree-view of the NEWS file in Sidekick.) New in this Isabelle version ---------------------------- *** General *** * Timeouts for Isabelle/ML tools are subject to system option "timeout_scale" --- this already used for the overall session build process before, and allows to adapt to slow machines. The underlying Timeout.apply in Isabelle/ML treats an original timeout specification 0 as no timeout; before it meant immediate timeout. Rare INCOMPATIBILITY in boundary cases. * Remote provers from SystemOnTPTP (notably for Sledgehammer) are now managed via Isabelle/Scala instead of perl; the dependency on libwww-perl has been eliminated (notably on Linux). Rare INCOMPATIBILITY: HTTP proxy configuration now works via JVM properties https://docs.oracle.com/en/java/javase/11/docs/api/java.base/java/net/doc-files/net-properties.html * More symbol definitions for the Z Notation (Isabelle fonts and LaTeX). See also the group "Z Notation" in the Symbols dockable of Isabelle/jEdit. *** Isabelle/jEdit Prover IDE *** * More robust 'proof' outline for method "induct": support nested cases. * Support for built-in font substitution of jEdit text area. * The main plugin for Isabelle/jEdit can be deactivated and reactivated as documented --- was broken at least since Isabelle2018. *** Document preparation *** * More predefined symbols: \ \ (package "stmaryrd"), \ \ (LaTeX package "pifont"). * High-quality blackboard-bold symbols from font "txmia" (LaTeX package "pxfonts"): \\\\\\\\\\\\\\\\\\\\\\\\\\. * Document antiquotations for ML text have been refined: "def" and "ref" variants support index entries, e.g. @{ML} (no entry) vs. @{ML_def} (bold entry) vs. @{ML_ref} (regular entry); @{ML_type} supports explicit type arguments for constructors (only relevant for index), e.g. @{ML_type \'a list\} vs. @{ML_type 'a \list\}; @{ML_op} has been renamed to @{ML_infix}. Minor INCOMPATIBILITY concerning name and syntax. * Option "document_logo" determines if an instance of the Isabelle logo should be created in the document output directory. The given string specifies the name of the logo variant, while "_" (underscore) refers to the unnamed variant. The output file name is always "isabelle_logo.pdf". * Option "document_preprocessor" specifies the name of an executable that is run within the document output directory, after preparing the document sources and before the actual build process. This allows to apply adhoc patches, without requiring a separate "build" script. * Option "document_build" determines the document build engine, as defined in Isabelle/Scala (as system service). The subsequent engines are provided by the Isabelle distribution: - "lualatex" (default): use ISABELLE_LUALATEX for a standard LaTeX build with optional ISABELLE_BIBTEX and ISABELLE_MAKEINDEX - "pdflatex": as above, but use ISABELLE_PDFLATEX (legacy mode for special LaTeX styles) - "build": delegate to the executable "./build pdf" The presence of a "build" command within the document output directory explicitly requires document_build=build. Minor INCOMPATIBILITY, need to adjust session ROOT options. * The command-line tool "isabelle latex" has been discontinued, INCOMPATIBILITY for old document build scripts. - Former "isabelle latex -o sty" has become obsolete: Isabelle .sty files are automatically generated within the document output directory. - Former "isabelle latex -o pdf" should be replaced by "$ISABELLE_LUALATEX root" or "$ISABELLE_PDFLATEX root" (without quotes), according to the intended LaTeX engine. - Former "isabelle latex -o bbl" should be replaced by "$ISABELLE_BIBTEX root" (without quotes). - Former "isabelle latex -o idx" should be replaced by "$ISABELLE_MAKEINDEX root" (without quotes). * Option "document_bibliography" explicitly enables the use of bibtex; the default is to check the presence of root.bib, but it could have a different name. * Improved LaTeX typesetting of \...\ using \guilsinglleft ... \guilsinglright. INCOMPATIBILITY, need to use \usepackage[T1]{fontenc} (which is now also the default in "isabelle mkroot"). * Simplified typesetting of \...\ using \guillemotleft ... \guillemotright from \usepackage[T1]{fontenc} --- \usepackage{babel} is no longer required. *** Pure *** * "global_interpretation" is applicable in instantiation and overloading targets and in any nested target built on top of a target supporting "global_interpretation". *** HOL *** * Theory "HOL-Library.Multiset": dedicated predicate "multiset" is gone, use explict expression instead. Minor INCOMPATIBILITY. * Theory "HOL-Library.Multiset": consolidated abbreviations Mempty, Melem, not_Melem to empty_mset, member_mset, not_member_mset respectively. Minor INCOMPATIBILITY. * Theory "HOL-Library.Multiset": consolidated operation and fact names: inf_subset_mset ~> inter_mset sup_subset_mset ~> union_mset multiset_inter_def ~> inter_mset_def sup_subset_mset_def ~> union_mset_def multiset_inter_count ~> count_inter_mset sup_subset_mset_count ~> count_union_mset * Theory "HOL-Library.Multiset": syntax precendence for membership operations has been adjusted to match the corresponding precendences on sets. Rare INCOMPATIBILITY. * Theory "HOL-Library.Cardinality": code generator setup based on the type classes finite_UNIV and card_UNIV has been moved to "HOL-Library.Code_Cardinality", to avoid incompatibilities with other code setups for sets in AFP/Containers. Applications relying on this code setup should import "HOL-Library.Code_Cardinality". Minor INCOMPATIBILITY. * Session "HOL-Analysis" and "HOL-Probability": indexed products of discrete distributions, negative binomial distribution, Hoeffding's inequality, Chernoff bounds, Cauchy–Schwarz inequality for nn_integral, and some more small lemmas. Some theorems that were stated awkwardly before were corrected. Minor INCOMPATIBILITY. * Session "HOL-Analysis": the complex Arg function has been identified with the function "arg" of Complex_Main, renaming arg->Arg also in the names of arg_bounded. Minor INCOMPATIBILITY. * Theorems "antisym" and "eq_iff" in class "order" have been renamed to "order.antisym" and "order.eq_iff", to coexist locally with "antisym" and "eq_iff" from locale "ordering". INCOMPATIBILITY: significant potential for change can be avoided if interpretations of type class "order" are replaced or augmented by interpretations of locale "ordering". * Theorem "swap_def" now is always qualified as "Fun.swap_def". Minor INCOMPATIBILITY; note that for most applications less elementary lemmas exists. * Theory "HOL-Library.Permutation" has been renamed to the more specific "HOL-Library.List_Permutation". Note that most notions from that theory are already present in theory "HOL-Combinatorics.Permutations". INCOMPATIBILITY. * Dedicated session "HOL-Combinatorics". INCOMPATIBILITY: theories "Permutations", "List_Permutation" (formerly "Permutation"), "Stirling", "Multiset_Permutations", "Perm" have been moved there from session HOL-Library. * Theory "HOL-Combinatorics.Transposition" provides elementary swap operation "transpose". * Lemma "permutes_induct" has been given stronger hypotheses and named premises. INCOMPATIBILITY. * Combinator "Fun.swap" resolved into a mere input abbreviation in separate theory "Transposition" in HOL-Combinatorics. INCOMPATIBILITY. * Infix syntax for bit operations AND, OR, XOR is now organized in bundle bit_operations_syntax. INCOMPATIBILITY. * Bit operations set_bit, unset_bit and flip_bit are now class operations. INCOMPATIBILITY. * Theory Bit_Operations is now part of HOL-Main. Minor INCOMPATIBILITY. +* Simplified class hierarchy for bit operations: bit operations reside +in classes (semi)ring_bit_operations, class semiring_bit_shifts is +gone. + * Abbreviation "max_word" has been moved to session Word_Lib in the AFP, as also have constants "shiftl1", "shiftr1", "sshiftr1", "bshiftr1", "setBit", "clearBit". See there further the changelog in theory Guide. INCOMPATIBILITY. +* Reorganized classes and locales for boolean algebras. +INCOMPATIBILITY. + * New simp rules: less_exp, min.absorb1, min.absorb2, min.absorb3, min.absorb4, max.absorb1, max.absorb2, max.absorb3, max.absorb4. Minor INCOMPATIBILITY. * Sledgehammer: - Removed legacy "lam_lifting" (synonym for "lifting") from option "lam_trans". Minor INCOMPATIBILITY. - Renamed "hide_lams" to "opaque_lifting" in option "lam_trans". Minor INCOMPATIBILITY. - Added "opaque_combs" to option "lam_trans": lambda expressions are rewritten using combinators, but the combinators are kept opaque, i.e. without definitions. * Metis: - Renamed option "hide_lams" to "opaque_lifting". Minor INCOMPATIBILITY. *** ML *** * ML antiquotations \<^try>\expr\ and \<^can>\expr\ operate directly on the given ML expression, in contrast to functions "try" and "can" that modify application of a function. * ML antiquotations for conditional ML text: \<^if_linux>\...\ \<^if_macos>\...\ \<^if_windows>\...\ \<^if_unix>\...\ * External bash processes are always managed by Isabelle/Scala, in contrast to Isabelle2021 where this was only done for macOS on Apple Silicon. The main Isabelle/ML interface is Isabelle_System.bash_process with result type Process_Result.T (resembling class Process_Result in Scala); derived operations Isabelle_System.bash and Isabelle_System.bash_output provide similar functionality as before. Rare INCOMPATIBILITY due to subtle semantic differences: - Processes invoked from Isabelle/ML actually run in the context of the Java VM of Isabelle/Scala. The settings environment and current working directory are usually the same on both sides, but there can be subtle corner cases (e.g. unexpected uses of "cd" or "putenv" in ML). - Output via stdout and stderr is line-oriented: Unix vs. Windows line-endings are normalized towards Unix; presence or absence of a final newline is irrelevant. The original lines are available as Process_Result.out_lines/err_lines; the concatenated versions Process_Result.out/err *omit* a trailing newline (using Library.trim_line, which was occasional seen in applications before, but is no longer necessary). - Output needs to be plain text encoded in UTF-8: Isabelle/Scala recodes it temporarily as UTF-16. This works for well-formed Unicode text, but not for arbitrary byte strings. In such cases, the bash script should write tempory files, managed by Isabelle/ML operations like Isabelle_System.with_tmp_file to create a file name and File.read to retrieve its content. - Just like any other Scala function invoked from ML, Isabelle_System.bash_process requires a proper PIDE session context. This could be a regular batch session (e.g. "isabelle build"), a PIDE editor session (e.g. "isabelle jedit"), or headless PIDE (e.g. "isabelle dump" or "isabelle server"). Note that old "isabelle console" or raw "isabelle process" don't have that. New Process_Result.timing works as in Isabelle/Scala, based on direct measurements of the bash_process wrapper in C: elapsed time is always available, CPU time is only available on Linux and macOS, GC time is unavailable. * Likewise, the following Isabelle/ML system operations are run in the context of Isabelle/Scala: - Isabelle_System.make_directory - Isabelle_System.copy_dir - Isabelle_System.copy_file - Isabelle_System.copy_base_file - Isabelle_System.rm_tree - Isabelle_System.download *** System *** * Each Isabelle component may specify a Scala/Java jar module declaratively via etc/build.props (file names are relative to the component directory). E.g. see $ISABELLE_HOME/etc/build.props with further explanations in the "system" manual. * Command-line tool "isabelle scala_build" allows to invoke the build process of all Scala/Java modules explicitly. Normally this is done implicitly on demand, e.g. for "isabelle scala" or "isabelle jedit". * Command-line tool "isabelle scala_project" is has been improved in various ways: - sources from all components with etc/build.props are included, - sources of for the jEdit text editor and the Isabelle/jEdit plugins (jedit_base and jedit_main) are included by default, - more sources may be given on the command-line, - options -f and -D make the tool more convenient. * Isabelle/jEdit is now composed more conventionally from the original jEdit text editor in $JEDIT_HOME (with minor patches), plus two Isabelle plugins that are produced in $JEDIT_SETTINGS/jars on demand. The main isabelle.jedit module is now part of Isabelle/Scala (as one big $ISABELLE_SCALA_JAR). * ML profiling has been updated and reactivated, after some degration in Isabelle2021: - "isabelle build -o threads=1 -o profiling=..." works properly within the PIDE session context; - "isabelle profiling_report" now uses the session build database (like "isabelle log"); - output uses non-intrusive tracing messages, instead of warnings. * System option "system_log" specifies an optional log file for internal messages produced by Output.system_message in Isabelle/ML; the value "true" refers to console progress of the build job. This works for "isabelle build" or any derivative of it. * System options of type string may be set to "true" using the short notation of type bool. E.g. "isabelle build -o system_log". * System option "document=true" is an alias for "document=pdf" and thus can be used in the short form. E.g. "isabelle build -o document". * Command-line tool "isabelle version" supports repository archives (without full .hg directory). More options. * Obsolete settings variable ISABELLE_PLATFORM32 has been discontinued. Note that only Windows supports old 32 bit executables, via settings variable ISABELLE_WINDOWS_PLATFORM32. Everything else should be ISABELLE_PLATFORM64 (generic Posix) or ISABELLE_WINDOWS_PLATFORM64 (native Windows) or ISABELLE_APPLE_PLATFORM64 (Apple Silicon). New in Isabelle2021 (February 2021) ----------------------------------- *** General *** * On macOS, the IsabelleXYZ.app directory layout now follows the other platforms, without indirection via Contents/Resources/. INCOMPATIBILITY, use e.g. IsabelleXYZ.app/bin/isabelle instead of former IsabelleXYZ.app/Isabelle/bin/isabelle or IsabelleXYZ.app/Isabelle/Contents/Resources/IsabelleXYZ/bin/isabelle. * HTML presentation uses rich markup produced by Isabelle/PIDE, resulting in more colors and links. * HTML presentation includes auxiliary files (e.g. ML) for each theory. * Proof method "subst" is confined to the original subgoal range: its included distinct_subgoals_tac no longer affects unrelated subgoals. Rare INCOMPATIBILITY. * Theory_Data extend operation is obsolete and needs to be the identity function; merge should be conservative and not reset to the empty value. Subtle INCOMPATIBILITY and change of semantics (due to Theory.join_theory from Isabelle2020). Special extend/merge behaviour at the begin of a new theory can be achieved via Theory.at_begin. *** Isabelle/jEdit Prover IDE *** * Improved GUI look-and-feel: the portable and scalable "FlatLaf Light" is used by default on all platforms (appearance similar to IntelliJ IDEA). * Improved markup for theory header imports: hyperlinks for theory files work without formal checking of content. * The prover process can download auxiliary files (e.g. 'ML_file') for theories with remote URL. This requires the external "curl" program. * Action "isabelle.goto-entity" (shortcut CS+d) jumps to the definition of the formal entity at the caret position. * The visual feedback on caret entity focus is normally restricted to definitions within the visible text area. The keyboard modifier "CS" overrides this: then all defining and referencing positions are shown. See also option "jedit_focus_modifier". * The jEdit status line includes widgets both for JVM and ML heap usage. Ongoing ML ongoing garbage collection is shown as "ML cleanup". * The Monitor dockable provides buttons to request a full garbage collection and sharing of live data on the ML heap. It also includes information about the Java Runtime system. * PIDE support for session ROOTS: markup for directories. * Update to jedit-5.6.0, the latest release. This version works properly on macOS by default, without the special MacOSX plugin. * Action "full-screen-mode" (shortcut F11 or S+F11) has been modified for better approximate window size on macOS and Linux/X11. * Improved GUI support for macOS 11.1 Big Sur: native fullscreen mode, but non-native look-and-feel (FlatLaf). * Hyperlinks to various file-formats (.pdf, .png, etc.) open an external viewer, instead of re-using the jEdit text editor. * IDE support for Naproche-SAD: Proof Checking of Natural Mathematical Documents. See also $NAPROCHE_HOME/examples for files with .ftl or .ftl.tex extension. The corresponding Naproche-SAD server process can be disabled by setting the system option naproche_server=false and restarting the Isabelle application. *** Document preparation *** * Keyword 'document_theories' within ROOT specifies theories from other sessions that should be included in the generated document source directory. This does not affect the generated session.tex: \input{...} needs to be used separately. * The standard LaTeX engine is now lualatex, according to settings variable ISABELLE_PDFLATEX. This is mostly upwards compatible with old pdflatex, but text encoding needs to conform strictly to utf8. Rare INCOMPATIBILITY. * Discontinued obsolete DVI format and ISABELLE_LATEX settings variable: document output is always PDF. * Antiquotation @{tool} refers to Isabelle command-line tools, with completion and formal reference to the source (external script or internal Scala function). * Antiquotation @{bash_function} refers to GNU bash functions that are checked within the Isabelle settings environment. * Antiquotations @{scala}, @{scala_object}, @{scala_type}, @{scala_method} refer to checked Isabelle/Scala entities. *** Pure *** * Session Pure-Examples contains notable examples for Isabelle/Pure (former entries of HOL-Isar_Examples). * Named contexts (locale and class specifications, locale and class context blocks) allow bundle mixins for the surface context. This allows syntax notations to be organized within bundles conveniently. See theory "HOL-ex.Specifications_with_bundle_mixins" for examples and the isar-ref manual for syntax descriptions. * Definitions in locales produce rule which can be added as congruence rule to protect foundational terms during simplification. * Consolidated terminology and function signatures for nested targets: - Local_Theory.begin_nested replaces Local_Theory.open_target - Local_Theory.end_nested replaces Local_Theory.close_target - Combination of Local_Theory.begin_nested and Local_Theory.end_nested(_result) replaces Local_Theory.subtarget(_result) INCOMPATIBILITY. * Local_Theory.init replaces Generic_Target.init. Minor INCOMPATIBILITY. *** HOL *** * Session HOL-Examples contains notable examples for Isabelle/HOL (former entries of HOL-Isar_Examples, HOL-ex etc.). * An updated version of the veriT solver is now included as Isabelle component. It can be used in the "smt" proof method via "smt (verit)" or via "declare [[smt_solver = verit]]" in the context; see also session HOL-Word-SMT_Examples. * Zipperposition 2.0 is now included as Isabelle component for experimentation, e.g. in "sledgehammer [prover = zipperposition]". * Sledgehammer: - support veriT in proof preplay - take adventage of more cores in proof preplay * Updated the Metis prover underlying the "metis" proof method to version 2.4 (release 20180810). The new version fixes one soundness defect and two incompleteness defects. Very slight INCOMPATIBILITY. * Nitpick/Kodkod may be invoked directly within the running Isabelle/Scala session (instead of an external Java process): this improves reactivity and saves resources. This experimental feature is guarded by system option "kodkod_scala" (default: true in PIDE interaction, false in batch builds). * Simproc "defined_all" and rewrite rule "subst_all" perform more aggressive substitution with variables from assumptions. INCOMPATIBILITY, consider repairing proofs locally like this: supply subst_all [simp del] [[simproc del: defined_all]] * Simproc "datatype_no_proper_subterm" rewrites equalities "lhs = rhs" on datatypes to "False" if either side is a proper subexpression of the other (for any datatype with a reasonable size function). * Syntax for state monad combinators fcomp and scomp is organized in bundle state_combinator_syntax. Minor INCOMPATIBILITY. * Syntax for reflected term syntax is organized in bundle term_syntax, discontinuing previous locale term_syntax. Minor INCOMPATIBILITY. * New constant "power_int" for exponentiation with integer exponent, written as "x powi n". * Added the "at most 1" quantifier, Uniq. * For the natural numbers, "Sup {} = 0". * New constant semiring_char gives the characteristic of any type of class semiring_1, with the convenient notation CHAR('a). For example, CHAR(nat) = CHAR(int) = CHAR(real) = 0, CHAR(17) = 17. * HOL-Computational_Algebra.Polynomial: Definition and basic properties of algebraic integers. * Library theory "Bit_Operations" with generic bit operations. * Library theory "Signed_Division" provides operations for signed division, instantiated for type int. * Theory "Multiset": removed misleading notation \# for sum_mset; replaced with \\<^sub>#. Analogous notation for prod_mset also exists now. * New theory "HOL-Library.Word" takes over material from former session "HOL-Word". INCOMPATIBILITY: need to adjust imports. * Theory "HOL-Library.Word": Type word is restricted to bit strings consisting of at least one bit. INCOMPATIBILITY. * Theory "HOL-Library.Word": Bit operations NOT, AND, OR, XOR are based on generic algebraic bit operations from theory "HOL-Library.Bit_Operations". INCOMPATIBILITY. * Theory "HOL-Library.Word": Most operations on type word are set up for transfer and lifting. INCOMPATIBILITY. * Theory "HOL-Library.Word": Generic type conversions. INCOMPATIBILITY, sometimes additional rewrite rules must be added to applications to get a confluent system again. * Theory "HOL-Library.Word": Uniform polymorphic "mask" operation for both types int and word. INCOMPATIBILITY. * Theory "HOL-Library.Word": Syntax for signed compare operators has been consolidated with syntax of regular compare operators. Minor INCOMPATIBILITY. * Former session "HOL-Word": Various operations dealing with bit values represented as reversed lists of bools are separated into theory Reversed_Bit_Lists in session Word_Lib in the AFP. INCOMPATIBILITY. * Former session "HOL-Word": Theory "Word_Bitwise" has been moved to AFP entry Word_Lib as theory "Bitwise". INCOMPATIBILITY. * Former session "HOL-Word": Compound operation "bin_split" simplifies by default into its components "drop_bit" and "take_bit". INCOMPATIBILITY. * Former session "HOL-Word": Operations lsb, msb and set_bit are separated into theories Least_significant_bit, Most_significant_bit and Generic_set_bit respectively in session Word_Lib in the AFP. INCOMPATIBILITY. * Former session "HOL-Word": Ancient int numeral representation has been factored out in separate theory "Ancient_Numeral" in session Word_Lib in the AFP. INCOMPATIBILITY. * Former session "HOL-Word": Operations "bin_last", "bin_rest", "bin_nth", "bintrunc", "sbintrunc", "norm_sint", "bin_cat" and "max_word" are now mere input abbreviations. Minor INCOMPATIBILITY. * Former session "HOL-Word": Misc ancient material has been factored out into separate theories and moved to session Word_Lib in the AFP. See theory "Guide" there for further information. INCOMPATIBILITY. * Session HOL-TPTP: The "tptp_isabelle" and "tptp_sledgehammer" commands are in working order again, as opposed to outputting "GaveUp" on nearly all problems. * Session "HOL-Hoare": concrete syntax only for Hoare triples, not abstract language constructors. * Session "HOL-Hoare": now provides a total correctness logic as well. *** FOL *** * Added the "at most 1" quantifier, Uniq, as in HOL. * Simproc "defined_all" and rewrite rule "subst_all" have been changed as in HOL. *** ML *** * Antiquotations @{scala_function}, @{scala}, @{scala_thread} refer to registered Isabelle/Scala functions (of type String => String): invocation works via the PIDE protocol. * Path.append is available as overloaded "+" operator, similar to corresponding Isabelle/Scala operation. * ML statistics via an external Poly/ML process: this allows monitoring the runtime system while the ML program sleeps. *** System *** * Isabelle server allows user-defined commands via isabelle_scala_service. * Update/rebuild external provers on currently supported OS platforms, notably CVC4 1.8, E prover 2.5, SPASS 3.8ds, CSDP 6.1.1. * The command-line tool "isabelle log" prints prover messages from the build database of the given session, following the the order of theory sources, instead of erratic parallel evaluation. Consequently, the session log file is restricted to system messages of the overall build process, and thus becomes more informative. * Discontinued obsolete isabelle display tool, and DVI_VIEWER settings variable. * The command-line tool "isabelle logo" only outputs PDF; obsolete EPS (for DVI documents) has been discontinued. Former option -n has been turned into -o with explicit file name. Minor INCOMPATIBILITY. * The command-line tool "isabelle components" supports new options -u and -x to manage $ISABELLE_HOME_USER/etc/components without manual editing of Isabelle configuration files. * The shell function "isabelle_directory" (within etc/settings of components) augments the list of special directories for persistent symbolic path names. This improves portability of heap images and session databases. It used to be hard-wired for Isabelle + AFP, but other projects may now participate on equal terms. * The command-line tool "isabelle process" now prints output to stdout/stderr separately and incrementally, instead of just one bulk to stdout after termination. Potential INCOMPATIBILITY for external tools. * The command-line tool "isabelle console" now supports interrupts properly (on Linux and macOS). * Batch-builds via "isabelle build" use a PIDE session with special protocol: this allows to invoke Isabelle/Scala operations from Isabelle/ML. Big build jobs (e.g. AFP) require extra heap space for the java process, e.g. like this in $ISABELLE_HOME_USER/etc/settings: ISABELLE_TOOL_JAVA_OPTIONS="$ISABELLE_TOOL_JAVA_OPTIONS -Xmx8g" This includes full PIDE markup, if option "build_pide_reports" is enabled. * The command-line tool "isabelle build" provides option -P DIR to produce PDF/HTML presentation in the specified directory; -P: refers to the standard directory according to ISABELLE_BROWSER_INFO / ISABELLE_BROWSER_INFO_SYSTEM settings. Generated PDF documents are taken from the build database -- from this or earlier builds with option document=pdf. * The command-line tool "isabelle document" generates theory documents on the spot, using the underlying session build database (exported LaTeX sources or existing PDF files). INCOMPATIBILITY, the former "isabelle document" tool was rather different and has been discontinued. * The command-line tool "isabelle sessions" explores the structure of Isabelle sessions and prints result names in topological order (on stdout). * The Isabelle/Scala "Progress" interface changed slightly and "No_Progress" has been discontinued. INCOMPATIBILITY, use "new Progress" instead. * General support for Isabelle/Scala system services, configured via the shell function "isabelle_scala_service" in etc/settings (e.g. of an Isabelle component); see implementations of class Isabelle_System.Service in Isabelle/Scala. This supersedes former "isabelle_scala_tools" and "isabelle_file_format": minor INCOMPATIBILITY. * The syntax of theory load commands (for auxiliary files) is now specified in Isabelle/Scala, as instance of class isabelle.Command_Span.Load_Command registered via isabelle_scala_service in etc/settings. This allows more flexible schemes than just a list of file extensions. Minor INCOMPATIBILITY, e.g. see theory HOL-SPARK.SPARK_Setup to emulate the old behaviour. * JVM system property "isabelle.laf" has been discontinued; the default Swing look-and-feel is ""FlatLaf Light". * Isabelle/Phabricator supports Ubuntu 20.04 LTS. * Isabelle/Phabricator setup has been updated to follow ongoing development: libphutil has been discontinued. Minor INCOMPATIBILITY: existing server installations should remove libphutil from /usr/local/bin/isabelle-phabricator-upgrade and each installation root directory (e.g. /var/www/phabricator-vcs/libphutil). * Experimental support for arm64-linux platform. The reference platform is Raspberry Pi 4 with 8 GB RAM running Pi OS (64 bit). * Support for Apple Silicon, using mostly x86_64-darwin runtime translation via Rosetta 2 (e.g. Poly/ML and external provers), but also some native arm64-darwin executables (e.g. Java). New in Isabelle2020 (April 2020) -------------------------------- *** General *** * Session ROOT files need to specify explicit 'directories' for import of theory files. Directories cannot be shared by different sessions. (Recall that import of theories from other sessions works via session-qualified theory names, together with suitable 'sessions' declarations in the ROOT.) * Internal derivations record dependencies on oracles and other theorems accurately, including the implicit type-class reasoning wrt. proven class relations and type arities. In particular, the formal tagging with "Pure.skip_proofs" of results stemming from "instance ... sorry" is now propagated properly to theorems depending on such type instances. * Command 'sorry' (oracle "Pure.skip_proofs") is more precise about the actual proposition that is assumed in the goal and proof context. This requires at least Proofterm.proofs = 1 to show up in theorem dependencies. * Command 'thm_oracles' prints all oracles used in given theorems, covering the full graph of transitive dependencies. * Command 'thm_deps' prints immediate theorem dependencies of the given facts. The former graph visualization has been discontinued, because it was hardly usable. * Refined treatment of proof terms, including type-class proofs for minor object-logics (FOL, FOLP, Sequents). * The inference kernel is now confined to one main module: structure Thm, without the former circular dependency on structure Axclass. * Mixfix annotations may use "' " (single quote followed by space) to separate delimiters (as documented in the isar-ref manual), without requiring an auxiliary empty block. A literal single quote needs to be escaped properly. Minor INCOMPATIBILITY. *** Isar *** * The proof method combinator (subproofs m) applies the method expression m consecutively to each subgoal, constructing individual subproofs internally. This impacts the internal construction of proof terms: it makes a cascade of let-expressions within the derivation tree and may thus improve scalability. * Attribute "trace_locales" activates tracing of locale instances during roundup. It replaces the diagnostic command 'print_dependencies', which has been discontinued. *** Isabelle/jEdit Prover IDE *** * Prover IDE startup is now much faster, because theory dependencies are no longer explored in advance. The overall session structure with its declarations of 'directories' is sufficient to locate theory files. Thus the "session focus" of option "isabelle jedit -S" has become obsolete (likewise for "isabelle vscode_server -S"). Existing option "-R" is both sufficient and more convenient to start editing a particular session. * Actions isabelle.tooltip (CS+b) and isabelle.message (CS+m) display tooltip message popups, corresponding to mouse hovering with/without the CONTROL/COMMAND key pressed. * The following actions allow to navigate errors within the current document snapshot: isabelle.first-error (CS+a) isabelle.last-error (CS+z) isabelle.next-error (CS+n) isabelle.prev-error (CS+p) * Support more brackets: \ \ (intended for implicit argument syntax). * Action isabelle.jconsole (menu item Plugins / Isabelle / Java/VM Monitor) applies the jconsole tool on the running Isabelle/jEdit process. This allows to monitor resource usage etc. * More adequate default font sizes for Linux on HD / UHD displays: automatic font scaling is usually absent on Linux, in contrast to Windows and macOS. * The default value for the jEdit property "view.antiAlias" (menu item Utilities / Global Options / Text Area / Anti Aliased smooth text) is now "subpixel HRGB", instead of former "standard". Especially on Linux this often leads to faster text rendering, but can also cause problems with odd color shades. An alternative is to switch back to "standard" here, and set the following Java system property: isabelle jedit -Dsun.java2d.opengl=true This can be made persistent via JEDIT_JAVA_OPTIONS in $ISABELLE_HOME_USER/etc/settings. For the "Isabelle2020" desktop application there is a corresponding options file in the same directory. *** Isabelle/VSCode Prover IDE *** * Update of State and Preview panels to use new WebviewPanel API of VSCode. *** HOL *** * Improvements of the 'lift_bnf' command: - Add support for quotient types. - Generate transfer rules for the lifted map/set/rel/pred constants (theorems "._transfer_raw"). * Term_XML.Encode/Decode.term uses compact representation of Const "typargs" from the given declaration environment. This also makes more sense for translations to lambda-calculi with explicit polymorphism. INCOMPATIBILITY, use Term_XML.Encode/Decode.term_raw in special applications. * ASCII membership syntax concerning big operators for infimum and supremum has been discontinued. INCOMPATIBILITY. * Removed multiplicativity assumption from class "normalization_semidom". Introduced various new intermediate classes with the multiplicativity assumption; many theorem statements (especially involving GCD/LCM) had to be adapted. This allows for a more natural instantiation of the algebraic typeclasses for e.g. Gaussian integers. INCOMPATIBILITY. * Clear distinction between types for bits (False / True) and Z2 (0 / 1): theory HOL-Library.Bit has been renamed accordingly. INCOMPATIBILITY. * Dynamic facts "algebra_split_simps" and "field_split_simps" correspond to algebra_simps and field_simps but contain more aggressive rules potentially splitting goals; algebra_split_simps roughly replaces sign_simps and field_split_simps can be used instead of divide_simps. INCOMPATIBILITY. * Theory HOL.Complete_Lattices: renamed Inf_Sup -> Inf_eq_Sup and Sup_Inf -> Sup_eq_Inf * Theory HOL-Library.Monad_Syntax: infix operation "bind" (\) associates to the left now as is customary. * Theory HOL-Library.Ramsey: full finite Ramsey's theorem with multiple colours and arbitrary exponents. * Session HOL-Proofs: build faster thanks to better treatment of proof terms in Isabelle/Pure. * Session HOL-Word: bitwise NOT-operator has proper prefix syntax. Minor INCOMPATIBILITY. * Session HOL-Analysis: proof method "metric" implements a decision procedure for simple linear statements in metric spaces. * Session HOL-Complex_Analysis has been split off from HOL-Analysis. *** ML *** * Theory construction may be forked internally, the operation Theory.join_theory recovers a single result theory. See also the example in theory "HOL-ex.Join_Theory". * Antiquotation @{oracle_name} inlines a formally checked oracle name. * Minimal support for a soft-type system within the Isabelle logical framework (module Soft_Type_System). * Former Variable.auto_fixes has been replaced by slightly more general Proof_Context.augment: it is subject to an optional soft-type system of the underlying object-logic. Minor INCOMPATIBILITY. * More scalable Export.export using XML.tree to avoid premature string allocations, with convenient shortcut XML.blob. Minor INCOMPATIBILITY. * Prover IDE support for the underlying Poly/ML compiler (not the basis library). Open $ML_SOURCES/ROOT.ML in Isabelle/jEdit to browse the implementation with full markup. *** System *** * Standard rendering for more Isabelle symbols: \ \ \ \ * The command-line tool "isabelle scala_project" creates a Gradle project configuration for Isabelle/Scala/jEdit, to support Scala IDEs such as IntelliJ IDEA. * The command-line tool "isabelle phabricator_setup" facilitates self-hosting of the Phabricator software-development platform, with support for Git, Mercurial, Subversion repositories. This helps to avoid monoculture and to escape the gravity of centralized version control by Github and/or Bitbucket. For further documentation, see chapter "Phabricator server administration" in the "system" manual. A notable example installation is https://isabelle-dev.sketis.net/. * The command-line tool "isabelle hg_setup" simplifies the setup of Mercurial repositories, with hosting via Phabricator or SSH file server access. * The command-line tool "isabelle imports" has been discontinued: strict checking of session directories enforces session-qualified theory names in applications -- users are responsible to specify session ROOT entries properly. * The command-line tool "isabelle dump" and its underlying Isabelle/Scala module isabelle.Dump has become more scalable, by splitting sessions and supporting a base logic image. Minor INCOMPATIBILITY in options and parameters. * The command-line tool "isabelle build_docker" has been slightly improved: it is now properly documented in the "system" manual. * Isabelle/Scala support for the Linux platform (Ubuntu): packages, users, system services. * Isabelle/Scala support for proof terms (with full type/term information) in module isabelle.Term. * Isabelle/Scala: more scalable output of YXML files, e.g. relevant for "isabelle dump". * Theory export via Isabelle/Scala has been reworked. The former "fact" name space is now split into individual "thm" items: names are potentially indexed, such as "foo" for singleton facts, or "bar(1)", "bar(2)", "bar(3)" for multi-facts. Theorem dependencies are now exported as well: this spans an overall dependency graph of internal inferences; it might help to reconstruct the formal structure of theory libraries. See also the module isabelle.Export_Theory in Isabelle/Scala. * Theory export of structured specifications, based on internal declarations of Spec_Rules by packages like 'definition', 'inductive', 'primrec', 'function'. * Old settings variables ISABELLE_PLATFORM and ISABELLE_WINDOWS_PLATFORM have been discontinued -- deprecated since Isabelle2018. * More complete x86_64 platform support on macOS, notably Catalina where old x86 has been discontinued. * Update to GHC stack 2.1.3 with stackage lts-13.19/ghc-8.6.4. * Update to OCaml Opam 2.0.6 (using ocaml 4.05.0 as before). New in Isabelle2019 (June 2019) ------------------------------- *** General *** * The font collection "Isabelle DejaVu" is systematically derived from the existing "DejaVu" fonts, with variants "Sans Mono", "Sans", "Serif" and styles "Normal", "Bold", "Italic/Oblique", "Bold-Italic/Oblique". The DejaVu base fonts are retricted to well-defined Unicode ranges and augmented by special Isabelle symbols, taken from the former "IsabelleText" font (which is no longer provided separately). The line metrics and overall rendering quality is closer to original DejaVu. INCOMPATIBILITY with display configuration expecting the old "IsabelleText" font: use e.g. "Isabelle DejaVu Sans Mono" instead. * The Isabelle fonts render "\" properly as superscript "-1". * Old-style inner comments (* ... *) within the term language are no longer supported (legacy feature in Isabelle2018). * Old-style {* verbatim *} tokens are explicitly marked as legacy feature and will be removed soon. Use \cartouche\ syntax instead, e.g. via "isabelle update_cartouches -t" (available since Isabelle2015). * Infix operators that begin or end with a "*" are now parenthesized without additional spaces, e.g. "(*)" instead of "( * )". Minor INCOMPATIBILITY. * Mixfix annotations may use cartouches instead of old-style double quotes, e.g. (infixl \+\ 60). The command-line tool "isabelle update -u mixfix_cartouches" allows to update existing theory sources automatically. * ML setup commands (e.g. 'setup', 'method_setup', 'parse_translation') need to provide a closed expression -- without trailing semicolon. Minor INCOMPATIBILITY. * Commands 'generate_file', 'export_generated_files', and 'compile_generated_files' support a stateless (PIDE-conformant) model for generated sources and compiled binaries of other languages. The compilation process is managed in Isabelle/ML, and results exported to the session database for further use (e.g. with "isabelle export" or "isabelle build -e"). *** Isabelle/jEdit Prover IDE *** * Fonts for the text area, gutter, GUI elements etc. use the "Isabelle DejaVu" collection by default, which provides uniform rendering quality with the usual Isabelle symbols. Line spacing no longer needs to be adjusted: properties for the old IsabelleText font had "Global Options / Text Area / Extra vertical line spacing (in pixels): -2", it now defaults to 1, but 0 works as well. * The jEdit File Browser is more prominent in the default GUI layout of Isabelle/jEdit: various virtual file-systems provide access to Isabelle resources, notably via "favorites:" (or "Edit Favorites"). * Further markup and rendering for "plain text" (e.g. informal prose) and "raw text" (e.g. verbatim sources). This improves the visual appearance of formal comments inside the term language, or in general for repeated alternation of formal and informal text. * Action "isabelle-export-browser" points the File Browser to the theory exports of the current buffer, based on the "isabelle-export:" virtual file-system. The directory view needs to be reloaded manually to follow ongoing document processing. * Action "isabelle-session-browser" points the File Browser to session information, based on the "isabelle-session:" virtual file-system. Its entries are structured according to chapter / session names, the open operation is redirected to the session ROOT file. * Support for user-defined file-formats via class isabelle.File_Format in Isabelle/Scala (e.g. see isabelle.Bibtex.File_Format), configured via the shell function "isabelle_file_format" in etc/settings (e.g. of an Isabelle component). * System option "jedit_text_overview" allows to disable the text overview column. * Command-line options "-s" and "-u" of "isabelle jedit" override the default for system option "system_heaps" that determines the heap storage directory for "isabelle build". Option "-n" is now clearly separated from option "-s". * The Isabelle/jEdit desktop application uses the same options as "isabelle jedit" for its internal "isabelle build" process: the implicit option "-o system_heaps" (or "-s") has been discontinued. This reduces the potential for surprise wrt. command-line tools. * The official download of the Isabelle/jEdit application already contains heap images for Isabelle/HOL within its main directory: thus the first encounter becomes faster and more robust (e.g. when run from a read-only directory). * Isabelle DejaVu fonts are available with hinting by default, which is relevant for low-resolution displays. This may be disabled via system option "isabelle_fonts_hinted = false" in $ISABELLE_HOME_USER/etc/preferences -- it occasionally yields better results. * OpenJDK 11 has quite different font rendering, with better glyph shapes and improved sub-pixel anti-aliasing. In some situations results might be *worse* than Oracle Java 8, though -- a proper HiDPI / UHD display is recommended. * OpenJDK 11 supports GTK version 2.2 and 3 (according to system property jdk.gtk.version). The factory default is version 3, but ISABELLE_JAVA_SYSTEM_OPTIONS includes "-Djdk.gtk.version=2.2" to make this more conservative (as in Java 8). Depending on the GTK theme configuration, "-Djdk.gtk.version=3" might work better or worse. *** Document preparation *** * Document markers are formal comments of the form \<^marker>\marker_body\ that are stripped from document output: the effect is to modify the semantic presentation context or to emit markup to the PIDE document. Some predefined markers are taken from the Dublin Core Metadata Initiative, e.g. \<^marker>\contributor arg\ or \<^marker>\license arg\ and produce PIDE markup that can be retrieved from the document database. * Old-style command tags %name are re-interpreted as markers with proof-scope \<^marker>\tag (proof) name\ and produce LaTeX environments as before. Potential INCOMPATIBILITY: multiple markers are composed in canonical order, resulting in a reversed list of tags in the presentation context. * Marker \<^marker>\tag name\ does not apply to the proof of a top-level goal statement by default (e.g. 'theorem', 'lemma'). This is a subtle change of semantics wrt. old-style %name. * In Isabelle/jEdit, the string "\tag" may be completed to a "\<^marker>\tag \" template. * Document antiquotation option "cartouche" indicates if the output should be delimited as cartouche; this takes precedence over the analogous option "quotes". * Many document antiquotations are internally categorized as "embedded" and expect one cartouche argument, which is typically used with the \<^control>\cartouche\ notation (e.g. \<^term>\\x y. x\). The cartouche delimiters are stripped in output of the source (antiquotation option "source"), but it is possible to enforce delimiters via option "source_cartouche", e.g. @{term [source_cartouche] \\x y. x\}. *** Isar *** * Implicit cases goal1, goal2, goal3, etc. have been discontinued (legacy feature since Isabelle2016). * More robust treatment of structural errors: begin/end blocks take precedence over goal/proof. This is particularly relevant for the headless PIDE session and server. * Command keywords of kind thy_decl / thy_goal may be more specifically fit into the traditional document model of "definition-statement-proof" via thy_defn / thy_stmt / thy_goal_defn / thy_goal_stmt. *** HOL *** * Command 'export_code' produces output as logical files within the theory context, as well as formal session exports that can be materialized via command-line tools "isabelle export" or "isabelle build -e" (with 'export_files' in the session ROOT). Isabelle/jEdit also provides a virtual file-system "isabelle-export:" that can be explored in the regular file-browser. A 'file_prefix' argument allows to specify an explicit name prefix for the target file (SML, OCaml, Scala) or directory (Haskell); the default is "export" with a consecutive number within each theory. * Command 'export_code': the 'file' argument is now legacy and will be removed soon: writing to the physical file-system is not well-defined in a reactive/parallel application like Isabelle. The empty 'file' argument has been discontinued already: it is superseded by the file-browser in Isabelle/jEdit on "isabelle-export:". Minor INCOMPATIBILITY. * Command 'code_reflect' no longer supports the 'file' argument: it has been superseded by 'file_prefix' for stateless file management as in 'export_code'. Minor INCOMPATIBILITY. * Code generation for OCaml: proper strings are used for literals. Minor INCOMPATIBILITY. * Code generation for OCaml: Zarith supersedes Nums as library for proper integer arithmetic. The library is located via standard invocations of "ocamlfind" (via ISABELLE_OCAMLFIND settings variable). The environment provided by "isabelle ocaml_setup" already contains this tool and the required packages. Minor INCOMPATIBILITY. * Code generation for Haskell: code includes for Haskell must contain proper module frame, nothing is added magically any longer. INCOMPATIBILITY. * Code generation: slightly more conventional syntax for 'code_stmts' antiquotation. Minor INCOMPATIBILITY. * Theory List: the precedence of the list_update operator has changed: "f a [n := x]" now needs to be written "(f a)[n := x]". * The functions \, \, \, \ (not the corresponding binding operators) now have the same precedence as any other prefix function symbol. Minor INCOMPATIBILITY. * Simplified syntax setup for big operators under image. In rare situations, type conversions are not inserted implicitly any longer and need to be given explicitly. Auxiliary abbreviations INFIMUM, SUPREMUM, UNION, INTER should now rarely occur in output and are just retained as migration auxiliary. Abbreviations MINIMUM and MAXIMUM are gone INCOMPATIBILITY. * The simplifier uses image_cong_simp as a congruence rule. The historic and not really well-formed congruence rules INF_cong*, SUP_cong*, are not used by default any longer. INCOMPATIBILITY; consider using declare image_cong_simp [cong del] in extreme situations. * INF_image and SUP_image are no default simp rules any longer. INCOMPATIBILITY, prefer image_comp as simp rule if needed. * Strong congruence rules (with =simp=> in the premises) for constant f are now uniformly called f_cong_simp, in accordance with congruence rules produced for mappers by the datatype package. INCOMPATIBILITY. * Retired lemma card_Union_image; use the simpler card_UN_disjoint instead. INCOMPATIBILITY. * Facts sum_mset.commute and prod_mset.commute have been renamed to sum_mset.swap and prod_mset.swap, similarly to sum.swap and prod.swap. INCOMPATIBILITY. * ML structure Inductive: slightly more conventional naming schema. Minor INCOMPATIBILITY. * ML: Various _global variants of specification tools have been removed. Minor INCOMPATIBILITY, prefer combinators Named_Target.theory_map[_result] to lift specifications to the global theory level. * Theory HOL-Library.Simps_Case_Conv: 'case_of_simps' now supports overlapping and non-exhaustive patterns and handles arbitrarily nested patterns. It uses on the same algorithm as HOL-Library.Code_Lazy, which assumes sequential left-to-right pattern matching. The generated equation no longer tuples the arguments on the right-hand side. INCOMPATIBILITY. * Theory HOL-Library.Multiset: the \# operator now has the same precedence as any other prefix function symbol. * Theory HOL-Library.Cardinal_Notations has been discontinued in favor of the bundle cardinal_syntax (available in theory Main). Minor INCOMPATIBILITY. * Session HOL-Library and HOL-Number_Theory: Exponentiation by squaring, used for computing powers in class "monoid_mult" and modular exponentiation. * Session HOL-Computational_Algebra: Formal Laurent series and overhaul of Formal power series. * Session HOL-Number_Theory: More material on residue rings in Carmichael's function, primitive roots, more properties for "ord". * Session HOL-Analysis: Better organization and much more material at the level of abstract topological spaces. * Session HOL-Algebra: Free abelian groups, etc., ported from HOL Light; algebraic closure of a field by de Vilhena and Baillon. * Session HOL-Homology has been added. It is a port of HOL Light's homology library, with new proofs of "invariance of domain" and related results. * Session HOL-SPARK: .prv files are no longer written to the file-system, but exported to the session database. Results may be retrieved via "isabelle build -e HOL-SPARK-Examples" on the command-line. * Sledgehammer: - The URL for SystemOnTPTP, which is used by remote provers, has been updated. - The machine-learning-based filter MaSh has been optimized to take less time (in most cases). * SMT: reconstruction is now possible using the SMT solver veriT. * Session HOL-Word: * New theory More_Word as comprehensive entrance point. * Merged type class bitss into type class bits. INCOMPATIBILITY. *** ML *** * Command 'generate_file' allows to produce sources for other languages, with antiquotations in the Isabelle context (only the control-cartouche form). The default "cartouche" antiquotation evaluates an ML expression of type string and inlines the result as a string literal of the target language. For example, this works for Haskell as follows: generate_file "Pure.hs" = \ module Isabelle.Pure where allConst, impConst, eqConst :: String allConst = \\<^const_name>\Pure.all\\ impConst = \\<^const_name>\Pure.imp\\ eqConst = \\<^const_name>\Pure.eq\\ \ See also commands 'export_generated_files' and 'compile_generated_files' to use the results. * ML evaluation (notably via command 'ML' or 'ML_file') is subject to option ML_environment to select a named environment, such as "Isabelle" for Isabelle/ML, or "SML" for official Standard ML. * ML antiquotation @{master_dir} refers to the master directory of the underlying theory, i.e. the directory of the theory file. * ML antiquotation @{verbatim} inlines its argument as string literal, preserving newlines literally. The short form \<^verbatim>\abc\ is particularly useful. * Local_Theory.reset is no longer available in user space. Regular definitional packages should use balanced blocks of Local_Theory.open_target versus Local_Theory.close_target instead, or the Local_Theory.subtarget(_result) combinator. Rare INCOMPATIBILITY. * Original PolyML.pointerEq is retained as a convenience for tools that don't use Isabelle/ML (where this is called "pointer_eq"). *** System *** * Update to OpenJDK 11: the current long-term support version of Java. * Update to Poly/ML 5.8 allows to use the native x86_64 platform without the full overhead of 64-bit values everywhere. This special x86_64_32 mode provides up to 16GB ML heap, while program code and stacks are allocated elsewhere. Thus approx. 5 times more memory is available for applications compared to old x86 mode (which is no longer used by Isabelle). The switch to the x86_64 CPU architecture also avoids compatibility problems with Linux and macOS, where 32-bit applications are gradually phased out. * System option "checkpoint" has been discontinued: obsolete thanks to improved memory management in Poly/ML. * System option "system_heaps" determines where to store the session image of "isabelle build" (and other tools using that internally). Former option "-s" is superseded by option "-o system_heaps". INCOMPATIBILITY in command-line syntax. * Session directory $ISABELLE_HOME/src/Tools/Haskell provides some source modules for Isabelle tools implemented in Haskell, notably for Isabelle/PIDE. * The command-line tool "isabelle build -e" retrieves theory exports from the session build database, using 'export_files' in session ROOT entries. * The command-line tool "isabelle update" uses Isabelle/PIDE in batch-mode to update theory sources based on semantic markup produced in Isabelle/ML. Actual updates depend on system options that may be enabled via "-u OPT" (for "update_OPT"), see also $ISABELLE_HOME/etc/options section "Theory update". Theory sessions are specified as in "isabelle dump". * The command-line tool "isabelle update -u control_cartouches" changes antiquotations into control-symbol format (where possible): @{NAME} becomes \<^NAME> and @{NAME ARG} becomes \<^NAME>\ARG\. * Support for Isabelle command-line tools defined in Isabelle/Scala. Instances of class Isabelle_Scala_Tools may be configured via the shell function "isabelle_scala_tools" in etc/settings (e.g. of an Isabelle component). * Isabelle Server command "use_theories" supports "nodes_status_delay" for continuous output of node status information. The time interval is specified in seconds; a negative value means it is disabled (default). * Isabelle Server command "use_theories" terminates more robustly in the presence of structurally broken sources: full consolidation of theories is no longer required. * OCaml tools and libraries are now accesed via ISABELLE_OCAMLFIND, which needs to point to a suitable version of "ocamlfind" (e.g. via OPAM, see below). INCOMPATIBILITY: settings variables ISABELLE_OCAML and ISABELLE_OCAMLC are no longer supported. * Support for managed installations of Glasgow Haskell Compiler and OCaml via the following command-line tools: isabelle ghc_setup isabelle ghc_stack isabelle ocaml_setup isabelle ocaml_opam The global installation state is determined by the following settings (and corresponding directory contents): ISABELLE_STACK_ROOT ISABELLE_STACK_RESOLVER ISABELLE_GHC_VERSION ISABELLE_OPAM_ROOT ISABELLE_OCAML_VERSION After setup, the following Isabelle settings are automatically redirected (overriding existing user settings): ISABELLE_GHC ISABELLE_OCAMLFIND The old meaning of these settings as locally installed executables may be recovered by purging the directories ISABELLE_STACK_ROOT / ISABELLE_OPAM_ROOT, or by resetting these variables in $ISABELLE_HOME_USER/etc/settings. New in Isabelle2018 (August 2018) --------------------------------- *** General *** * Session-qualified theory names are mandatory: it is no longer possible to refer to unqualified theories from the parent session. INCOMPATIBILITY for old developments that have not been updated to Isabelle2017 yet (using the "isabelle imports" tool). * Only the most fundamental theory names are global, usually the entry points to major logic sessions: Pure, Main, Complex_Main, HOLCF, IFOL, FOL, ZF, ZFC etc. INCOMPATIBILITY, need to use qualified names for formerly global "HOL-Probability.Probability" and "HOL-SPARK.SPARK". * Global facts need to be closed: no free variables and no hypotheses. Rare INCOMPATIBILITY. * Facts stemming from locale interpretation are subject to lazy evaluation for improved performance. Rare INCOMPATIBILITY: errors stemming from interpretation morphisms might be deferred and thus difficult to locate; enable system option "strict_facts" temporarily to avoid this. * Marginal comments need to be written exclusively in the new-style form "\ \text\", old ASCII variants like "-- {* ... *}" are no longer supported. INCOMPATIBILITY, use the command-line tool "isabelle update_comments" to update existing theory files. * Old-style inner comments (* ... *) within the term language are legacy and will be discontinued soon: use formal comments "\ \...\" or "\<^cancel>\...\" instead. * The "op " syntax for infix operators has been replaced by "()". If begins or ends with a "*", there needs to be a space between the "*" and the corresponding parenthesis. INCOMPATIBILITY, use the command-line tool "isabelle update_op" to convert theory and ML files to the new syntax. Because it is based on regular expression matching, the result may need a bit of manual postprocessing. Invoking "isabelle update_op" converts all files in the current directory (recursively). In case you want to exclude conversion of ML files (because the tool frequently also converts ML's "op" syntax), use option "-m". * Theory header 'abbrevs' specifications need to be separated by 'and'. INCOMPATIBILITY. * Command 'external_file' declares the formal dependency on the given file name, such that the Isabelle build process knows about it, but without specific Prover IDE management. * Session ROOT entries no longer allow specification of 'files'. Rare INCOMPATIBILITY, use command 'external_file' within a proper theory context. * Session root directories may be specified multiple times: each accessible ROOT file is processed only once. This facilitates specification of $ISABELLE_HOME_USER/ROOTS or command-line options like -d or -D for "isabelle build" and "isabelle jedit". Example: isabelle build -D '~~/src/ZF' * The command 'display_drafts' has been discontinued. INCOMPATIBILITY, use action "isabelle.draft" (or "print") in Isabelle/jEdit instead. * In HTML output, the Isabelle symbol "\" is rendered as explicit Unicode hyphen U+2010, to avoid unclear meaning of the old "soft hyphen" U+00AD. Rare INCOMPATIBILITY, e.g. copy-paste of historic Isabelle HTML output. *** Isabelle/jEdit Prover IDE *** * The command-line tool "isabelle jedit" provides more flexible options for session management: - option -R builds an auxiliary logic image with all theories from other sessions that are not already present in its parent - option -S is like -R, with a focus on the selected session and its descendants (this reduces startup time for big projects like AFP) - option -A specifies an alternative ancestor session for options -R and -S - option -i includes additional sessions into the name-space of theories Examples: isabelle jedit -R HOL-Number_Theory isabelle jedit -R HOL-Number_Theory -A HOL isabelle jedit -d '$AFP' -S Formal_SSA -A HOL isabelle jedit -d '$AFP' -S Formal_SSA -A HOL-Analysis isabelle jedit -d '$AFP' -S Formal_SSA -A HOL-Analysis -i CryptHOL * PIDE markup for session ROOT files: allows to complete session names, follow links to theories and document files etc. * Completion supports theory header imports, using theory base name. E.g. "Prob" may be completed to "HOL-Probability.Probability". * Named control symbols (without special Unicode rendering) are shown as bold-italic keyword. This is particularly useful for the short form of antiquotations with control symbol: \<^name>\argument\. The action "isabelle.antiquoted_cartouche" turns an antiquotation with 0 or 1 arguments into this format. * Completion provides templates for named symbols with arguments, e.g. "\ \ARGUMENT\" or "\<^emph>\ARGUMENT\". * Slightly more parallel checking, notably for high priority print functions (e.g. State output). * The view title is set dynamically, according to the Isabelle distribution and the logic session name. The user can override this via set-view-title (stored persistently in $JEDIT_SETTINGS/perspective.xml). * System options "spell_checker_include" and "spell_checker_exclude" supersede former "spell_checker_elements" to determine regions of text that are subject to spell-checking. Minor INCOMPATIBILITY. * Action "isabelle.preview" is able to present more file formats, notably bibtex database files and ML files. * Action "isabelle.draft" is similar to "isabelle.preview", but shows a plain-text document draft. Both are available via the menu "Plugins / Isabelle". * When loading text files, the Isabelle symbols encoding UTF-8-Isabelle is only used if there is no conflict with existing Unicode sequences in the file. Otherwise, the fallback encoding is plain UTF-8 and Isabelle symbols remain in literal \ form. This avoids accidental loss of Unicode content when saving the file. * Bibtex database files (.bib) are semantically checked. * Update to jedit-5.5.0, the latest release. *** Isabelle/VSCode Prover IDE *** * HTML preview of theories and other file-formats similar to Isabelle/jEdit. * Command-line tool "isabelle vscode_server" accepts the same options -A, -R, -S, -i for session selection as "isabelle jedit". This is relevant for isabelle.args configuration settings in VSCode. The former option -A (explore all known session files) has been discontinued: it is enabled by default, unless option -S is used to focus on a particular spot in the session structure. INCOMPATIBILITY. *** Document preparation *** * Formal comments work uniformly in outer syntax, inner syntax (term language), Isabelle/ML and some other embedded languages of Isabelle. See also "Document comments" in the isar-ref manual. The following forms are supported: - marginal text comment: \ \\\ - canceled source: \<^cancel>\\\ - raw LaTeX: \<^latex>\\\ * Outside of the inner theory body, the default presentation context is theory Pure. Thus elementary antiquotations may be used in markup commands (e.g. 'chapter', 'section', 'text') and formal comments. * System option "document_tags" specifies alternative command tags. This is occasionally useful to control the global visibility of commands via session options (e.g. in ROOT). * Document markup commands ('section', 'text' etc.) are implicitly tagged as "document" and visible by default. This avoids the application of option "document_tags" to these commands. * Isabelle names are mangled into LaTeX macro names to allow the full identifier syntax with underscore, prime, digits. This is relevant for antiquotations in control symbol notation, e.g. \<^const_name> becomes \isactrlconstUNDERSCOREname. * Document preparation with skip_proofs option now preserves the content more accurately: only terminal proof steps ('by' etc.) are skipped. * Document antiquotation @{theory name} requires the long session-qualified theory name: this is what users reading the text normally need to import. * Document antiquotation @{session name} checks and prints the given session name verbatim. * Document antiquotation @{cite} now checks the given Bibtex entries against the Bibtex database files -- only in batch-mode session builds. * Command-line tool "isabelle document" has been re-implemented in Isabelle/Scala, with simplified arguments and explicit errors from the latex and bibtex process. Minor INCOMPATIBILITY. * Session ROOT entry: empty 'document_files' means there is no document for this session. There is no need to specify options [document = false] anymore. *** Isar *** * Command 'interpret' no longer exposes resulting theorems as literal facts, notably for the \prop\ notation or the "fact" proof method. This improves modularity of proofs and scalability of locale interpretation. Rare INCOMPATIBILITY, need to refer to explicitly named facts instead (e.g. use 'find_theorems' or 'try' to figure this out). * The old 'def' command has been discontinued (legacy since Isbelle2016-1). INCOMPATIBILITY, use 'define' instead -- usually with object-logic equality or equivalence. *** Pure *** * The inner syntax category "sort" now includes notation "_" for the dummy sort: it is effectively ignored in type-inference. * Rewrites clauses (keyword 'rewrites') were moved into the locale expression syntax, where they are part of locale instances. In interpretation commands rewrites clauses now need to occur before 'for' and 'defines'. Rare INCOMPATIBILITY; definitions immediately subject to rewriting may need to be pulled up into the surrounding theory. * For 'rewrites' clauses, if activating a locale instance fails, fall back to reading the clause first. This helps avoid qualification of locale instances where the qualifier's sole purpose is avoiding duplicate constant declarations. * Proof method "simp" now supports a new modifier "flip:" followed by a list of theorems. Each of these theorems is removed from the simpset (without warning if it is not there) and the symmetric version of the theorem (i.e. lhs and rhs exchanged) is added to the simpset. For "auto" and friends the modifier is "simp flip:". *** HOL *** * Sledgehammer: bundled version of "vampire" (for non-commercial users) helps to avoid fragility of "remote_vampire" service. * Clarified relationship of characters, strings and code generation: - Type "char" is now a proper datatype of 8-bit values. - Conversions "nat_of_char" and "char_of_nat" are gone; use more general conversions "of_char" and "char_of" with suitable type constraints instead. - The zero character is just written "CHR 0x00", not "0" any longer. - Type "String.literal" (for code generation) is now isomorphic to lists of 7-bit (ASCII) values; concrete values can be written as "STR ''...''" for sequences of printable characters and "STR 0x..." for one single ASCII code point given as hexadecimal numeral. - Type "String.literal" supports concatenation "... + ..." for all standard target languages. - Theory HOL-Library.Code_Char is gone; study the explanations concerning "String.literal" in the tutorial on code generation to get an idea how target-language string literals can be converted to HOL string values and vice versa. - Session Imperative-HOL: operation "raise" directly takes a value of type "String.literal" as argument, not type "string". INCOMPATIBILITY. * Code generation: Code generation takes an explicit option "case_insensitive" to accomodate case-insensitive file systems. * Abstract bit operations as part of Main: push_bit, take_bit, drop_bit. * New, more general, axiomatization of complete_distrib_lattice. The former axioms: "sup x (Inf X) = Inf (sup x ` X)" and "inf x (Sup X) = Sup (inf x ` X)" are replaced by: "Inf (Sup ` A) <= Sup (Inf ` {f ` A | f . (! Y \ A . f Y \ Y)})" The instantiations of sets and functions as complete_distrib_lattice are moved to Hilbert_Choice.thy because their proofs need the Hilbert choice operator. The dual of this property is also proved in theory HOL.Hilbert_Choice. * New syntax for the minimum/maximum of a function over a finite set: MIN x\A. B and even MIN x. B (only useful for finite types), also MAX. * Clarifed theorem names: Min.antimono ~> Min.subset_imp Max.antimono ~> Max.subset_imp Minor INCOMPATIBILITY. * SMT module: - The 'smt_oracle' option is now necessary when using the 'smt' method with a solver other than Z3. INCOMPATIBILITY. - The encoding to first-order logic is now more complete in the presence of higher-order quantifiers. An 'smt_explicit_application' option has been added to control this. INCOMPATIBILITY. * Facts sum.commute(_restrict) and prod.commute(_restrict) renamed to sum.swap(_restrict) and prod.swap(_restrict), to avoid name clashes on interpretation of abstract locales. INCOMPATIBILITY. * Predicate coprime is now a real definition, not a mere abbreviation. INCOMPATIBILITY. * Predicate pairwise_coprime abolished, use "pairwise coprime" instead. INCOMPATIBILITY. * The relator rel_filter on filters has been strengthened to its canonical categorical definition with better properties. INCOMPATIBILITY. * Generalized linear algebra involving linear, span, dependent, dim from type class real_vector to locales module and vector_space. Renamed: span_inc ~> span_superset span_superset ~> span_base span_eq ~> span_eq_iff INCOMPATIBILITY. * Class linordered_semiring_1 covers zero_less_one also, ruling out pathologic instances. Minor INCOMPATIBILITY. * Theory HOL.List: functions "sorted_wrt" and "sorted" now compare every element in a list to all following elements, not just the next one. * Theory HOL.List syntax: - filter-syntax "[x <- xs. P]" is no longer output syntax, but only input syntax - list comprehension syntax now supports tuple patterns in "pat <- xs" * Theory Map: "empty" must now be qualified as "Map.empty". * Removed nat-int transfer machinery. Rare INCOMPATIBILITY. * Fact mod_mult_self4 (on nat) renamed to Suc_mod_mult_self3, to avoid clash with fact mod_mult_self4 (on more generic semirings). INCOMPATIBILITY. * Eliminated some theorem aliasses: even_times_iff ~> even_mult_iff mod_2_not_eq_zero_eq_one_nat ~> not_mod_2_eq_0_eq_1 even_of_nat ~> even_int_iff INCOMPATIBILITY. * Eliminated some theorem duplicate variations: - dvd_eq_mod_eq_0_numeral can be replaced by dvd_eq_mod_eq_0 - mod_Suc_eq_Suc_mod can be replaced by mod_Suc - mod_Suc_eq_Suc_mod [symmetrict] can be replaced by mod_simps - mod_eq_0_iff can be replaced by mod_eq_0_iff_dvd and dvd_def - the witness of mod_eqD can be given directly as "_ div _" INCOMPATIBILITY. * Classical setup: Assumption "m mod d = 0" (for m d :: nat) is no longer aggresively destroyed to "\q. m = d * q". INCOMPATIBILITY, adding "elim!: dvd" to classical proof methods in most situations restores broken proofs. * Theory HOL-Library.Conditional_Parametricity provides command 'parametric_constant' for proving parametricity of non-recursive definitions. For constants that are not fully parametric the command will infer conditions on relations (e.g., bi_unique, bi_total, or type class conditions such as "respects 0") sufficient for parametricity. See theory HOL-ex.Conditional_Parametricity_Examples for some examples. * Theory HOL-Library.Code_Lazy provides a new preprocessor for the code generator to generate code for algebraic types with lazy evaluation semantics even in call-by-value target languages. See the theories HOL-ex.Code_Lazy_Demo and HOL-Codegenerator_Test.Code_Lazy_Test for some examples. * Theory HOL-Library.Landau_Symbols has been moved here from AFP. * Theory HOL-Library.Old_Datatype no longer provides the legacy command 'old_datatype'. INCOMPATIBILITY. * Theory HOL-Computational_Algebra.Polynomial_Factorial does not provide instances of rat, real, complex as factorial rings etc. Import HOL-Computational_Algebra.Field_as_Ring explicitly in case of need. INCOMPATIBILITY. * Session HOL-Algebra: renamed (^) to [^] to avoid conflict with new infix/prefix notation. * Session HOL-Algebra: revamped with much new material. The set of isomorphisms between two groups is now denoted iso rather than iso_set. INCOMPATIBILITY. * Session HOL-Analysis: the Arg function now respects the same interval as Ln, namely (-pi,pi]; the old Arg function has been renamed Arg2pi. INCOMPATIBILITY. * Session HOL-Analysis: the functions zorder, zer_poly, porder and pol_poly have been redefined. All related lemmas have been reworked. INCOMPATIBILITY. * Session HOL-Analysis: infinite products, Moebius functions, the Riemann mapping theorem, the Vitali covering theorem, change-of-variables results for integration and measures. * Session HOL-Real_Asymp: proof method "real_asymp" proves asymptotics or real-valued functions (limits, "Big-O", etc.) automatically. See also ~~/src/HOL/Real_Asymp/Manual for some documentation. * Session HOL-Types_To_Sets: more tool support (unoverload_type combines internalize_sorts and unoverload) and larger experimental application (type based linear algebra transferred to linear algebra on subspaces). *** ML *** * Operation Export.export emits theory exports (arbitrary blobs), which are stored persistently in the session build database. * Command 'ML_export' exports ML toplevel bindings to the global bootstrap environment of the ML process. This allows ML evaluation without a formal theory context, e.g. in command-line tools like "isabelle process". *** System *** * Mac OS X 10.10 Yosemite is now the baseline version; Mavericks is no longer supported. * Linux and Windows/Cygwin is for x86_64 only, old 32bit platform support has been discontinued. * Java runtime is for x86_64 only. Corresponding Isabelle settings have been renamed to ISABELLE_TOOL_JAVA_OPTIONS and JEDIT_JAVA_OPTIONS, instead of former 32/64 variants. INCOMPATIBILITY. * Old settings ISABELLE_PLATFORM and ISABELLE_WINDOWS_PLATFORM should be phased out due to unclear preference of 32bit vs. 64bit architecture. Explicit GNU bash expressions are now preferred, for example (with quotes): #Posix executables (Unix or Cygwin), with preference for 64bit "${ISABELLE_PLATFORM64:-$ISABELLE_PLATFORM32}" #native Windows or Unix executables, with preference for 64bit "${ISABELLE_WINDOWS_PLATFORM64:-${ISABELLE_WINDOWS_PLATFORM32:-${ISABELLE_PLATFORM64:-$ISABELLE_PLATFORM32}}}" #native Windows (32bit) or Unix executables (preference for 64bit) "${ISABELLE_WINDOWS_PLATFORM32:-${ISABELLE_PLATFORM64:-$ISABELLE_PLATFORM32}}" * Command-line tool "isabelle build" supports new options: - option -B NAME: include session NAME and all descendants - option -S: only observe changes of sources, not heap images - option -f: forces a fresh build * Command-line tool "isabelle build" options -c -x -B refer to descendants wrt. the session parent or import graph. Subtle INCOMPATIBILITY: options -c -x used to refer to the session parent graph only. * Command-line tool "isabelle build" takes "condition" options with the corresponding environment values into account, when determining the up-to-date status of a session. * The command-line tool "dump" dumps information from the cumulative PIDE session database: many sessions may be loaded into a given logic image, results from all loaded theories are written to the output directory. * Command-line tool "isabelle imports -I" also reports actual session imports. This helps to minimize the session dependency graph. * The command-line tool "export" and 'export_files' in session ROOT entries retrieve theory exports from the session build database. * The command-line tools "isabelle server" and "isabelle client" provide access to the Isabelle Server: it supports responsive session management and concurrent use of theories, based on Isabelle/PIDE infrastructure. See also the "system" manual. * The command-line tool "isabelle update_comments" normalizes formal comments in outer syntax as follows: \ \text\ (whith a single space to approximate the appearance in document output). This is more specific than former "isabelle update_cartouches -c": the latter tool option has been discontinued. * The command-line tool "isabelle mkroot" now always produces a document outline: its options have been adapted accordingly. INCOMPATIBILITY. * The command-line tool "isabelle mkroot -I" initializes a Mercurial repository for the generated session files. * Settings ISABELLE_HEAPS + ISABELLE_BROWSER_INFO (or ISABELLE_HEAPS_SYSTEM + ISABELLE_BROWSER_INFO_SYSTEM in "system build mode") determine the directory locations of the main build artefacts -- instead of hard-wired directories in ISABELLE_HOME_USER (or ISABELLE_HOME). * Settings ISABELLE_PATH and ISABELLE_OUTPUT have been discontinued: heap images and session databases are always stored in $ISABELLE_HEAPS/$ML_IDENTIFIER (command-line default) or $ISABELLE_HEAPS_SYSTEM/$ML_IDENTIFIER (main Isabelle application or "isabelle jedit -s" or "isabelle build -s"). * ISABELLE_LATEX and ISABELLE_PDFLATEX now include platform-specific options for improved error reporting. Potential INCOMPATIBILITY with unusual LaTeX installations, may have to adapt these settings. * Update to Poly/ML 5.7.1 with slightly improved performance and PIDE markup for identifier bindings. It now uses The GNU Multiple Precision Arithmetic Library (libgmp) on all platforms, notably Mac OS X with 32/64 bit. New in Isabelle2017 (October 2017) ---------------------------------- *** General *** * Experimental support for Visual Studio Code (VSCode) as alternative Isabelle/PIDE front-end, see also https://marketplace.visualstudio.com/items?itemName=makarius.Isabelle2017 VSCode is a new type of application that continues the concepts of "programmer's editor" and "integrated development environment" towards fully semantic editing and debugging -- in a relatively light-weight manner. Thus it fits nicely on top of the Isabelle/PIDE infrastructure. Technically, VSCode is based on the Electron application framework (Node.js + Chromium browser + V8), which is implemented in JavaScript and TypeScript, while Isabelle/VSCode mainly consists of Isabelle/Scala modules around a Language Server implementation. * Theory names are qualified by the session name that they belong to. This affects imports, but not the theory name space prefix (which is just the theory base name as before). In order to import theories from other sessions, the ROOT file format provides a new 'sessions' keyword. In contrast, a theory that is imported in the old-fashioned manner via an explicit file-system path belongs to the current session, and might cause theory name conflicts later on. Theories that are imported from other sessions are excluded from the current session document. The command-line tool "isabelle imports" helps to update theory imports. * The main theory entry points for some non-HOL sessions have changed, to avoid confusion with the global name "Main" of the session HOL. This leads to the follow renamings: CTT/Main.thy ~> CTT/CTT.thy ZF/Main.thy ~> ZF/ZF.thy ZF/Main_ZF.thy ~> ZF/ZF.thy ZF/Main_ZFC.thy ~> ZF/ZFC.thy ZF/ZF.thy ~> ZF/ZF_Base.thy INCOMPATIBILITY. * Commands 'alias' and 'type_alias' introduce aliases for constants and type constructors, respectively. This allows adhoc changes to name-space accesses within global or local theory contexts, e.g. within a 'bundle'. * Document antiquotations @{prf} and @{full_prf} output proof terms (again) in the same way as commands 'prf' and 'full_prf'. * Computations generated by the code generator can be embedded directly into ML, alongside with @{code} antiquotations, using the following antiquotations: @{computation ... terms: ... datatypes: ...} : ((term -> term) -> 'ml option -> 'a) -> Proof.context -> term -> 'a @{computation_conv ... terms: ... datatypes: ...} : (Proof.context -> 'ml -> conv) -> Proof.context -> conv @{computation_check terms: ... datatypes: ...} : Proof.context -> conv See src/HOL/ex/Computations.thy, src/HOL/Decision_Procs/Commutative_Ring.thy and src/HOL/Decision_Procs/Reflective_Field.thy for examples and the tutorial on code generation. *** Prover IDE -- Isabelle/Scala/jEdit *** * Session-qualified theory imports allow the Prover IDE to process arbitrary theory hierarchies independently of the underlying logic session image (e.g. option "isabelle jedit -l"), but the directory structure needs to be known in advance (e.g. option "isabelle jedit -d" or a line in the file $ISABELLE_HOME_USER/ROOTS). * The PIDE document model maintains file content independently of the status of jEdit editor buffers. Reloading jEdit buffers no longer causes changes of formal document content. Theory dependencies are always resolved internally, without the need for corresponding editor buffers. The system option "jedit_auto_load" has been discontinued: it is effectively always enabled. * The Theories dockable provides a "Purge" button, in order to restrict the document model to theories that are required for open editor buffers. * The Theories dockable indicates the overall status of checking of each entry. When all forked tasks of a theory are finished, the border is painted with thick lines; remaining errors in this situation are represented by a different border color. * Automatic indentation is more careful to avoid redundant spaces in intermediate situations. Keywords are indented after input (via typed characters or completion); see also option "jedit_indent_input". * Action "isabelle.preview" opens an HTML preview of the current theory document in the default web browser. * Command-line invocation "isabelle jedit -R -l LOGIC" opens the ROOT entry of the specified logic session in the editor, while its parent is used for formal checking. * The main Isabelle/jEdit plugin may be restarted manually (using the jEdit Plugin Manager), as long as the "Isabelle Base" plugin remains enabled at all times. * Update to current jedit-5.4.0. *** Pure *** * Deleting the last code equations for a particular function using [code del] results in function with no equations (runtime abort) rather than an unimplemented function (generation time abort). Use explicit [[code drop:]] to enforce the latter. Minor INCOMPATIBILITY. * Proper concept of code declarations in code.ML: - Regular code declarations act only on the global theory level, being ignored with warnings if syntactically malformed. - Explicitly global code declarations yield errors if syntactically malformed. - Default code declarations are silently ignored if syntactically malformed. Minor INCOMPATIBILITY. * Clarified and standardized internal data bookkeeping of code declarations: history of serials allows to track potentially non-monotonous declarations appropriately. Minor INCOMPATIBILITY. *** HOL *** * The Nunchaku model finder is now part of "Main". * SMT module: - A new option, 'smt_nat_as_int', has been added to translate 'nat' to 'int' and benefit from the SMT solver's theory reasoning. It is disabled by default. - The legacy module "src/HOL/Library/Old_SMT.thy" has been removed. - Several small issues have been rectified in the 'smt' command. * (Co)datatype package: The 'size_gen_o_map' lemma is no longer generated for datatypes with type class annotations. As a result, the tactic that derives it no longer fails on nested datatypes. Slight INCOMPATIBILITY. * Command and antiquotation "value" with modified default strategy: terms without free variables are always evaluated using plain evaluation only, with no fallback on normalization by evaluation. Minor INCOMPATIBILITY. * Theories "GCD" and "Binomial" are already included in "Main" (instead of "Complex_Main"). * Constant "surj" is a full input/output abbreviation (again). Minor INCOMPATIBILITY. * Dropped aliasses RangeP, DomainP for Rangep, Domainp respectively. INCOMPATIBILITY. * Renamed ii to imaginary_unit in order to free up ii as a variable name. The syntax \ remains available. INCOMPATIBILITY. * Dropped abbreviations transP, antisymP, single_valuedP; use constants transp, antisymp, single_valuedp instead. INCOMPATIBILITY. * Constant "subseq" in Topological_Spaces has been removed -- it is subsumed by "strict_mono". Some basic lemmas specific to "subseq" have been renamed accordingly, e.g. "subseq_o" -> "strict_mono_o" etc. * Theory List: "sublist" renamed to "nths" in analogy with "nth", and "sublisteq" renamed to "subseq". Minor INCOMPATIBILITY. * Theory List: new generic function "sorted_wrt". * Named theorems mod_simps covers various congruence rules concerning mod, replacing former zmod_simps. INCOMPATIBILITY. * Swapped orientation of congruence rules mod_add_left_eq, mod_add_right_eq, mod_add_eq, mod_mult_left_eq, mod_mult_right_eq, mod_mult_eq, mod_minus_eq, mod_diff_left_eq, mod_diff_right_eq, mod_diff_eq. INCOMPATIBILITY. * Generalized some facts: measure_induct_rule measure_induct zminus_zmod ~> mod_minus_eq zdiff_zmod_left ~> mod_diff_left_eq zdiff_zmod_right ~> mod_diff_right_eq zmod_eq_dvd_iff ~> mod_eq_dvd_iff INCOMPATIBILITY. * Algebraic type class hierarchy of euclidean (semi)rings in HOL: euclidean_(semi)ring, euclidean_(semi)ring_cancel, unique_euclidean_(semi)ring; instantiation requires provision of a euclidean size. * Theory "HOL-Number_Theory.Euclidean_Algorithm" has been reworked: - Euclidean induction is available as rule eucl_induct. - Constants Euclidean_Algorithm.gcd, Euclidean_Algorithm.lcm, Euclidean_Algorithm.Gcd and Euclidean_Algorithm.Lcm allow easy instantiation of euclidean (semi)rings as GCD (semi)rings. - Coefficients obtained by extended euclidean algorithm are available as "bezout_coefficients". INCOMPATIBILITY. * Theory "Number_Theory.Totient" introduces basic notions about Euler's totient function previously hidden as solitary example in theory Residues. Definition changed so that "totient 1 = 1" in agreement with the literature. Minor INCOMPATIBILITY. * New styles in theory "HOL-Library.LaTeXsugar": - "dummy_pats" for printing equations with "_" on the lhs; - "eta_expand" for printing eta-expanded terms. * Theory "HOL-Library.Permutations": theorem bij_swap_ompose_bij has been renamed to bij_swap_compose_bij. INCOMPATIBILITY. * New theory "HOL-Library.Going_To_Filter" providing the "f going_to F" filter for describing points x such that f(x) is in the filter F. * Theory "HOL-Library.Formal_Power_Series": constants X/E/L/F have been renamed to fps_X/fps_exp/fps_ln/fps_hypergeo to avoid polluting the name space. INCOMPATIBILITY. * Theory "HOL-Library.FinFun" has been moved to AFP (again). INCOMPATIBILITY. * Theory "HOL-Library.FuncSet": some old and rarely used ASCII replacement syntax has been removed. INCOMPATIBILITY, standard syntax with symbols should be used instead. The subsequent commands help to reproduce the old forms, e.g. to simplify porting old theories: syntax (ASCII) "_PiE" :: "pttrn \ 'a set \ 'b set \ ('a \ 'b) set" ("(3PIE _:_./ _)" 10) "_Pi" :: "pttrn \ 'a set \ 'b set \ ('a \ 'b) set" ("(3PI _:_./ _)" 10) "_lam" :: "pttrn \ 'a set \ 'a \ 'b \ ('a \ 'b)" ("(3%_:_./ _)" [0,0,3] 3) * Theory "HOL-Library.Multiset": the simprocs on subsets operators of multisets have been renamed: msetless_cancel_numerals ~> msetsubset_cancel msetle_cancel_numerals ~> msetsubset_eq_cancel INCOMPATIBILITY. * Theory "HOL-Library.Pattern_Aliases" provides input and output syntax for pattern aliases as known from Haskell, Scala and ML. * Theory "HOL-Library.Uprod" formalizes the type of unordered pairs. * Session HOL-Analysis: more material involving arcs, paths, covering spaces, innessential maps, retracts, infinite products, simplicial complexes. Baire Category theorem. Major results include the Jordan Curve Theorem and the Great Picard Theorem. * Session HOL-Algebra has been extended by additional lattice theory: the Knaster-Tarski fixed point theorem and Galois Connections. * Sessions HOL-Computational_Algebra and HOL-Number_Theory: new notions of squarefreeness, n-th powers, and prime powers. * Session "HOL-Computional_Algebra" covers many previously scattered theories, notably Euclidean_Algorithm, Factorial_Ring, Formal_Power_Series, Fraction_Field, Fundamental_Theorem_Algebra, Normalized_Fraction, Polynomial_FPS, Polynomial, Primes. Minor INCOMPATIBILITY. *** System *** * Isabelle/Scala: the SQL module supports access to relational databases, either as plain file (SQLite) or full-scale server (PostgreSQL via local port or remote ssh connection). * Results of "isabelle build" are recorded as SQLite database (i.e. "Application File Format" in the sense of https://www.sqlite.org/appfileformat.html). This allows systematic access via operations from module Sessions.Store in Isabelle/Scala. * System option "parallel_proofs" is 1 by default (instead of more aggressive 2). This requires less heap space and avoids burning parallel CPU cycles, while full subproof parallelization is enabled for repeated builds (according to parallel_subproofs_threshold). * System option "record_proofs" allows to change the global Proofterm.proofs variable for a session. Regular values are are 0, 1, 2; a negative value means the current state in the ML heap image remains unchanged. * Isabelle settings variable ISABELLE_SCALA_BUILD_OPTIONS has been renamed to ISABELLE_SCALAC_OPTIONS. Rare INCOMPATIBILITY. * Isabelle settings variables ISABELLE_WINDOWS_PLATFORM, ISABELLE_WINDOWS_PLATFORM32, ISABELLE_WINDOWS_PLATFORM64 indicate the native Windows platform (independently of the Cygwin installation). This is analogous to ISABELLE_PLATFORM, ISABELLE_PLATFORM32, ISABELLE_PLATFORM64. * Command-line tool "isabelle build_docker" builds a Docker image from the Isabelle application bundle for Linux. See also https://hub.docker.com/r/makarius/isabelle * Command-line tool "isabelle vscode_server" provides a Language Server Protocol implementation, e.g. for the Visual Studio Code editor. It serves as example for alternative PIDE front-ends. * Command-line tool "isabelle imports" helps to maintain theory imports wrt. session structure. Examples for the main Isabelle distribution: isabelle imports -I -a isabelle imports -U -a isabelle imports -U -i -a isabelle imports -M -a -d '~~/src/Benchmarks' New in Isabelle2016-1 (December 2016) ------------------------------------- *** General *** * Splitter in proof methods "simp", "auto" and friends: - The syntax "split add" has been discontinued, use plain "split", INCOMPATIBILITY. - For situations with many conditional or case expressions, there is an alternative splitting strategy that can be much faster. It is selected by writing "split!" instead of "split". It applies safe introduction and elimination rules after each split rule. As a result the subgoal may be split into several subgoals. * Command 'bundle' provides a local theory target to define a bundle from the body of specification commands (such as 'declare', 'declaration', 'notation', 'lemmas', 'lemma'). For example: bundle foo begin declare a [simp] declare b [intro] end * Command 'unbundle' is like 'include', but works within a local theory context. Unlike "context includes ... begin", the effect of 'unbundle' on the target context persists, until different declarations are given. * Simplified outer syntax: uniform category "name" includes long identifiers. Former "xname" / "nameref" / "name reference" has been discontinued. * Embedded content (e.g. the inner syntax of types, terms, props) may be delimited uniformly via cartouches. This works better than old-fashioned quotes when sub-languages are nested. * Mixfix annotations support general block properties, with syntax "(\x=a y=b z \\". Notable property names are "indent", "consistent", "unbreakable", "markup". The existing notation "(DIGITS" is equivalent to "(\indent=DIGITS\". The former notation "(00" for unbreakable blocks is superseded by "(\unbreabable\" --- rare INCOMPATIBILITY. * Proof method "blast" is more robust wrt. corner cases of Pure statements without object-logic judgment. * Commands 'prf' and 'full_prf' are somewhat more informative (again): proof terms are reconstructed and cleaned from administrative thm nodes. * Code generator: config option "code_timing" triggers measurements of different phases of code generation. See src/HOL/ex/Code_Timing.thy for examples. * Code generator: implicits in Scala (stemming from type class instances) are generated into companion object of corresponding type class, to resolve some situations where ambiguities may occur. * Solve direct: option "solve_direct_strict_warnings" gives explicit warnings for lemma statements with trivial proofs. *** Prover IDE -- Isabelle/Scala/jEdit *** * More aggressive flushing of machine-generated input, according to system option editor_generated_input_delay (in addition to existing editor_input_delay for regular user edits). This may affect overall PIDE reactivity and CPU usage. * Syntactic indentation according to Isabelle outer syntax. Action "indent-lines" (shortcut C+i) indents the current line according to command keywords and some command substructure. Action "isabelle.newline" (shortcut ENTER) indents the old and the new line according to command keywords only; see also option "jedit_indent_newline". * Semantic indentation for unstructured proof scripts ('apply' etc.) via number of subgoals. This requires information of ongoing document processing and may thus lag behind, when the user is editing too quickly; see also option "jedit_script_indent" and "jedit_script_indent_limit". * Refined folding mode "isabelle" based on Isar syntax: 'next' and 'qed' are treated as delimiters for fold structure; 'begin' and 'end' structure of theory specifications is treated as well. * Command 'proof' provides information about proof outline with cases, e.g. for proof methods "cases", "induct", "goal_cases". * Completion templates for commands involving "begin ... end" blocks, e.g. 'context', 'notepad'. * Sidekick parser "isabelle-context" shows nesting of context blocks according to 'begin' and 'end' structure. * Highlighting of entity def/ref positions wrt. cursor. * Action "isabelle.select-entity" (shortcut CS+ENTER) selects all occurrences of the formal entity at the caret position. This facilitates systematic renaming. * PIDE document markup works across multiple Isar commands, e.g. the results established at the end of a proof are properly identified in the theorem statement. * Cartouche abbreviations work both for " and ` to accomodate typical situations where old ASCII notation may be updated. * Dockable window "Symbols" also provides access to 'abbrevs' from the outer syntax of the current theory buffer. This provides clickable syntax templates, including entries with empty abbrevs name (which are inaccessible via keyboard completion). * IDE support for the Isabelle/Pure bootstrap process, with the following independent stages: src/Pure/ROOT0.ML src/Pure/ROOT.ML src/Pure/Pure.thy src/Pure/ML_Bootstrap.thy The ML ROOT files act like quasi-theories in the context of theory ML_Bootstrap: this allows continuous checking of all loaded ML files. The theory files are presented with a modified header to import Pure from the running Isabelle instance. Results from changed versions of each stage are *not* propagated to the next stage, and isolated from the actual Isabelle/Pure that runs the IDE itself. The sequential dependencies of the above files are only observed for batch build. * Isabelle/ML and Standard ML files are presented in Sidekick with the tree structure of section headings: this special comment format is described in "implementation" chapter 0, e.g. (*** section ***). * Additional abbreviations for syntactic completion may be specified within the theory header as 'abbrevs'. The theory syntax for 'keywords' has been simplified accordingly: optional abbrevs need to go into the new 'abbrevs' section. * Global abbreviations via $ISABELLE_HOME/etc/abbrevs and $ISABELLE_HOME_USER/etc/abbrevs are no longer supported. Minor INCOMPATIBILITY, use 'abbrevs' within theory header instead. * Action "isabelle.keymap-merge" asks the user to resolve pending Isabelle keymap changes that are in conflict with the current jEdit keymap; non-conflicting changes are always applied implicitly. This action is automatically invoked on Isabelle/jEdit startup and thus increases chances that users see new keyboard shortcuts when re-using old keymaps. * ML and document antiquotations for file-systems paths are more uniform and diverse: @{path NAME} -- no file-system check @{file NAME} -- check for plain file @{dir NAME} -- check for directory Minor INCOMPATIBILITY, former uses of @{file} and @{file_unchecked} may have to be changed. *** Document preparation *** * New symbol \, e.g. for temporal operator. * New document and ML antiquotation @{locale} for locales, similar to existing antiquotation @{class}. * Mixfix annotations support delimiters like \<^control>\cartouche\ -- this allows special forms of document output. * Raw LaTeX output now works via \<^latex>\...\ instead of raw control symbol \<^raw:...>. INCOMPATIBILITY, notably for LaTeXsugar.thy and its derivatives. * \<^raw:...> symbols are no longer supported. * Old 'header' command is no longer supported (legacy since Isabelle2015). *** Isar *** * Many specification elements support structured statements with 'if' / 'for' eigen-context, e.g. 'axiomatization', 'abbreviation', 'definition', 'inductive', 'function'. * Toplevel theorem statements support eigen-context notation with 'if' / 'for' (in postfix), which corresponds to 'assumes' / 'fixes' in the traditional long statement form (in prefix). Local premises are called "that" or "assms", respectively. Empty premises are *not* bound in the context: INCOMPATIBILITY. * Command 'define' introduces a local (non-polymorphic) definition, with optional abstraction over local parameters. The syntax resembles 'definition' and 'obtain'. It fits better into the Isar language than old 'def', which is now a legacy feature. * Command 'obtain' supports structured statements with 'if' / 'for' context. * Command '\' is an alias for 'sorry', with different typesetting. E.g. to produce proof holes in examples and documentation. * The defining position of a literal fact \prop\ is maintained more carefully, and made accessible as hyperlink in the Prover IDE. * Commands 'finally' and 'ultimately' used to expose the result as literal fact: this accidental behaviour has been discontinued. Rare INCOMPATIBILITY, use more explicit means to refer to facts in Isar. * Command 'axiomatization' has become more restrictive to correspond better to internal axioms as singleton facts with mandatory name. Minor INCOMPATIBILITY. * Proof methods may refer to the main facts via the dynamic fact "method_facts". This is particularly useful for Eisbach method definitions. * Proof method "use" allows to modify the main facts of a given method expression, e.g. (use facts in simp) (use facts in \simp add: ...\) * The old proof method "default" has been removed (legacy since Isabelle2016). INCOMPATIBILITY, use "standard" instead. *** Pure *** * Pure provides basic versions of proof methods "simp" and "simp_all" that only know about meta-equality (==). Potential INCOMPATIBILITY in theory imports that merge Pure with e.g. Main of Isabelle/HOL: the order is relevant to avoid confusion of Pure.simp vs. HOL.simp. * The command 'unfolding' and proof method "unfold" include a second stage where given equations are passed through the attribute "abs_def" before rewriting. This ensures that definitions are fully expanded, regardless of the actual parameters that are provided. Rare INCOMPATIBILITY in some corner cases: use proof method (simp only:) instead, or declare [[unfold_abs_def = false]] in the proof context. * Type-inference improves sorts of newly introduced type variables for the object-logic, using its base sort (i.e. HOL.type for Isabelle/HOL). Thus terms like "f x" or "\x. P x" without any further syntactic context produce x::'a::type in HOL instead of x::'a::{} in Pure. Rare INCOMPATIBILITY, need to provide explicit type constraints for Pure types where this is really intended. *** HOL *** * New proof method "argo" using the built-in Argo solver based on SMT technology. The method can be used to prove goals of quantifier-free propositional logic, goals based on a combination of quantifier-free propositional logic with equality, and goals based on a combination of quantifier-free propositional logic with linear real arithmetic including min/max/abs. See HOL/ex/Argo_Examples.thy for examples. * The new "nunchaku" command integrates the Nunchaku model finder. The tool is experimental. See ~~/src/HOL/Nunchaku/Nunchaku.thy for details. * Metis: The problem encoding has changed very slightly. This might break existing proofs. INCOMPATIBILITY. * Sledgehammer: - The MaSh relevance filter is now faster than before. - Produce syntactically correct Vampire 4.0 problem files. * (Co)datatype package: - New commands for defining corecursive functions and reasoning about them in "~~/src/HOL/Library/BNF_Corec.thy": 'corec', 'corecursive', 'friend_of_corec', and 'corecursion_upto'; and 'corec_unique' proof method. See 'isabelle doc corec'. - The predicator :: ('a \ bool) \ 'a F \ bool is now a first-class citizen in bounded natural functors. - 'primrec' now allows nested calls through the predicator in addition to the map function. - 'bnf' automatically discharges reflexive proof obligations. - 'bnf' outputs a slightly modified proof obligation expressing rel in terms of map and set (not giving a specification for rel makes this one reflexive). - 'bnf' outputs a new proof obligation expressing pred in terms of set (not giving a specification for pred makes this one reflexive). INCOMPATIBILITY: manual 'bnf' declarations may need adjustment. - Renamed lemmas: rel_prod_apply ~> rel_prod_inject pred_prod_apply ~> pred_prod_inject INCOMPATIBILITY. - The "size" plugin has been made compatible again with locales. - The theorems about "rel" and "set" may have a slightly different (but equivalent) form. INCOMPATIBILITY. * The 'coinductive' command produces a proper coinduction rule for mutual coinductive predicates. This new rule replaces the old rule, which exposed details of the internal fixpoint construction and was hard to use. INCOMPATIBILITY. * New abbreviations for negated existence (but not bounded existence): \x. P x \ \ (\x. P x) \!x. P x \ \ (\!x. P x) * The print mode "HOL" for ASCII syntax of binders "!", "?", "?!", "@" has been removed for output. It is retained for input only, until it is eliminated altogether. * The unique existence quantifier no longer provides 'binder' syntax, but uses syntax translations (as for bounded unique existence). Thus iterated quantification \!x y. P x y with its slightly confusing sequential meaning \!x. \!y. P x y is no longer possible. Instead, pattern abstraction admits simultaneous unique existence \!(x, y). P x y (analogous to existing notation \!(x, y)\A. P x y). Potential INCOMPATIBILITY in rare situations. * Conventional syntax "%(). t" for unit abstractions. Slight syntactic INCOMPATIBILITY. * Renamed constants and corresponding theorems: setsum ~> sum setprod ~> prod listsum ~> sum_list listprod ~> prod_list INCOMPATIBILITY. * Sligthly more standardized theorem names: sgn_times ~> sgn_mult sgn_mult' ~> Real_Vector_Spaces.sgn_mult divide_zero_left ~> div_0 zero_mod_left ~> mod_0 divide_zero ~> div_by_0 divide_1 ~> div_by_1 nonzero_mult_divide_cancel_left ~> nonzero_mult_div_cancel_left div_mult_self1_is_id ~> nonzero_mult_div_cancel_left nonzero_mult_divide_cancel_right ~> nonzero_mult_div_cancel_right div_mult_self2_is_id ~> nonzero_mult_div_cancel_right is_unit_divide_mult_cancel_left ~> is_unit_div_mult_cancel_left is_unit_divide_mult_cancel_right ~> is_unit_div_mult_cancel_right mod_div_equality ~> div_mult_mod_eq mod_div_equality2 ~> mult_div_mod_eq mod_div_equality3 ~> mod_div_mult_eq mod_div_equality4 ~> mod_mult_div_eq minus_div_eq_mod ~> minus_div_mult_eq_mod minus_div_eq_mod2 ~> minus_mult_div_eq_mod minus_mod_eq_div ~> minus_mod_eq_div_mult minus_mod_eq_div2 ~> minus_mod_eq_mult_div div_mod_equality' ~> minus_mod_eq_div_mult [symmetric] mod_div_equality' ~> minus_div_mult_eq_mod [symmetric] zmod_zdiv_equality ~> mult_div_mod_eq [symmetric] zmod_zdiv_equality' ~> minus_div_mult_eq_mod [symmetric] Divides.mult_div_cancel ~> minus_mod_eq_mult_div [symmetric] mult_div_cancel ~> minus_mod_eq_mult_div [symmetric] zmult_div_cancel ~> minus_mod_eq_mult_div [symmetric] div_1 ~> div_by_Suc_0 mod_1 ~> mod_by_Suc_0 INCOMPATIBILITY. * New type class "idom_abs_sgn" specifies algebraic properties of sign and absolute value functions. Type class "sgn_if" has disappeared. Slight INCOMPATIBILITY. * Dedicated syntax LENGTH('a) for length of types. * Characters (type char) are modelled as finite algebraic type corresponding to {0..255}. - Logical representation: * 0 is instantiated to the ASCII zero character. * All other characters are represented as "Char n" with n being a raw numeral expression less than 256. * Expressions of the form "Char n" with n greater than 255 are non-canonical. - Printing and parsing: * Printable characters are printed and parsed as "CHR ''\''" (as before). * The ASCII zero character is printed and parsed as "0". * All other canonical characters are printed as "CHR 0xXX" with XX being the hexadecimal character code. "CHR n" is parsable for every numeral expression n. * Non-canonical characters have no special syntax and are printed as their logical representation. - Explicit conversions from and to the natural numbers are provided as char_of_nat, nat_of_char (as before). - The auxiliary nibble type has been discontinued. INCOMPATIBILITY. * Type class "div" with operation "mod" renamed to type class "modulo" with operation "modulo", analogously to type class "divide". This eliminates the need to qualify any of those names in the presence of infix "mod" syntax. INCOMPATIBILITY. * Statements and proofs of Knaster-Tarski fixpoint combinators lfp/gfp have been clarified. The fixpoint properties are lfp_fixpoint, its symmetric lfp_unfold (as before), and the duals for gfp. Auxiliary items for the proof (lfp_lemma2 etc.) are no longer exported, but can be easily recovered by composition with eq_refl. Minor INCOMPATIBILITY. * Constant "surj" is a mere input abbreviation, to avoid hiding an equation in term output. Minor INCOMPATIBILITY. * Command 'code_reflect' accepts empty constructor lists for datatypes, which renders those abstract effectively. * Command 'export_code' checks given constants for abstraction violations: a small guarantee that given constants specify a safe interface for the generated code. * Code generation for Scala: ambiguous implicts in class diagrams are spelt out explicitly. * Static evaluators (Code_Evaluation.static_* in Isabelle/ML) rely on explicitly provided auxiliary definitions for required type class dictionaries rather than half-working magic. INCOMPATIBILITY, see the tutorial on code generation for details. * Theory Set_Interval: substantial new theorems on indexed sums and products. * Locale bijection establishes convenient default simp rules such as "inv f (f a) = a" for total bijections. * Abstract locales semigroup, abel_semigroup, semilattice, semilattice_neutr, ordering, ordering_top, semilattice_order, semilattice_neutr_order, comm_monoid_set, semilattice_set, semilattice_neutr_set, semilattice_order_set, semilattice_order_neutr_set monoid_list, comm_monoid_list, comm_monoid_list_set, comm_monoid_mset, comm_monoid_fun use boldified syntax uniformly that does not clash with corresponding global syntax. INCOMPATIBILITY. * Former locale lifting_syntax is now a bundle, which is easier to include in a local context or theorem statement, e.g. "context includes lifting_syntax begin ... end". Minor INCOMPATIBILITY. * Some old / obsolete theorems have been renamed / removed, potential INCOMPATIBILITY. nat_less_cases -- removed, use linorder_cases instead inv_image_comp -- removed, use image_inv_f_f instead image_surj_f_inv_f ~> image_f_inv_f * Some theorems about groups and orders have been generalised from groups to semi-groups that are also monoids: le_add_same_cancel1 le_add_same_cancel2 less_add_same_cancel1 less_add_same_cancel2 add_le_same_cancel1 add_le_same_cancel2 add_less_same_cancel1 add_less_same_cancel2 * Some simplifications theorems about rings have been removed, since superseeded by a more general version: less_add_cancel_left_greater_zero ~> less_add_same_cancel1 less_add_cancel_right_greater_zero ~> less_add_same_cancel2 less_eq_add_cancel_left_greater_eq_zero ~> le_add_same_cancel1 less_eq_add_cancel_right_greater_eq_zero ~> le_add_same_cancel2 less_eq_add_cancel_left_less_eq_zero ~> add_le_same_cancel1 less_eq_add_cancel_right_less_eq_zero ~> add_le_same_cancel2 less_add_cancel_left_less_zero ~> add_less_same_cancel1 less_add_cancel_right_less_zero ~> add_less_same_cancel2 INCOMPATIBILITY. * Renamed split_if -> if_split and split_if_asm -> if_split_asm to resemble the f.split naming convention, INCOMPATIBILITY. * Added class topological_monoid. * The following theorems have been renamed: setsum_left_distrib ~> sum_distrib_right setsum_right_distrib ~> sum_distrib_left INCOMPATIBILITY. * Compound constants INFIMUM and SUPREMUM are mere abbreviations now. INCOMPATIBILITY. * "Gcd (f ` A)" and "Lcm (f ` A)" are printed with optional comprehension-like syntax analogously to "Inf (f ` A)" and "Sup (f ` A)". * Class semiring_Lcd merged into semiring_Gcd. INCOMPATIBILITY. * The type class ordered_comm_monoid_add is now called ordered_cancel_comm_monoid_add. A new type class ordered_comm_monoid_add is introduced as the combination of ordered_ab_semigroup_add + comm_monoid_add. INCOMPATIBILITY. * Introduced the type classes canonically_ordered_comm_monoid_add and dioid. * Introduced the type class ordered_ab_semigroup_monoid_add_imp_le. When instantiating linordered_semiring_strict and ordered_ab_group_add, an explicit instantiation of ordered_ab_semigroup_monoid_add_imp_le might be required. INCOMPATIBILITY. * Dropped various legacy fact bindings, whose replacements are often of a more general type also: lcm_left_commute_nat ~> lcm.left_commute lcm_left_commute_int ~> lcm.left_commute gcd_left_commute_nat ~> gcd.left_commute gcd_left_commute_int ~> gcd.left_commute gcd_greatest_iff_nat ~> gcd_greatest_iff gcd_greatest_iff_int ~> gcd_greatest_iff coprime_dvd_mult_nat ~> coprime_dvd_mult coprime_dvd_mult_int ~> coprime_dvd_mult zpower_numeral_even ~> power_numeral_even gcd_mult_cancel_nat ~> gcd_mult_cancel gcd_mult_cancel_int ~> gcd_mult_cancel div_gcd_coprime_nat ~> div_gcd_coprime div_gcd_coprime_int ~> div_gcd_coprime zpower_numeral_odd ~> power_numeral_odd zero_less_int_conv ~> of_nat_0_less_iff gcd_greatest_nat ~> gcd_greatest gcd_greatest_int ~> gcd_greatest coprime_mult_nat ~> coprime_mult coprime_mult_int ~> coprime_mult lcm_commute_nat ~> lcm.commute lcm_commute_int ~> lcm.commute int_less_0_conv ~> of_nat_less_0_iff gcd_commute_nat ~> gcd.commute gcd_commute_int ~> gcd.commute Gcd_insert_nat ~> Gcd_insert Gcd_insert_int ~> Gcd_insert of_int_int_eq ~> of_int_of_nat_eq lcm_least_nat ~> lcm_least lcm_least_int ~> lcm_least lcm_assoc_nat ~> lcm.assoc lcm_assoc_int ~> lcm.assoc int_le_0_conv ~> of_nat_le_0_iff int_eq_0_conv ~> of_nat_eq_0_iff Gcd_empty_nat ~> Gcd_empty Gcd_empty_int ~> Gcd_empty gcd_assoc_nat ~> gcd.assoc gcd_assoc_int ~> gcd.assoc zero_zle_int ~> of_nat_0_le_iff lcm_dvd2_nat ~> dvd_lcm2 lcm_dvd2_int ~> dvd_lcm2 lcm_dvd1_nat ~> dvd_lcm1 lcm_dvd1_int ~> dvd_lcm1 gcd_zero_nat ~> gcd_eq_0_iff gcd_zero_int ~> gcd_eq_0_iff gcd_dvd2_nat ~> gcd_dvd2 gcd_dvd2_int ~> gcd_dvd2 gcd_dvd1_nat ~> gcd_dvd1 gcd_dvd1_int ~> gcd_dvd1 int_numeral ~> of_nat_numeral lcm_ac_nat ~> ac_simps lcm_ac_int ~> ac_simps gcd_ac_nat ~> ac_simps gcd_ac_int ~> ac_simps abs_int_eq ~> abs_of_nat zless_int ~> of_nat_less_iff zdiff_int ~> of_nat_diff zadd_int ~> of_nat_add int_mult ~> of_nat_mult int_Suc ~> of_nat_Suc inj_int ~> inj_of_nat int_1 ~> of_nat_1 int_0 ~> of_nat_0 Lcm_empty_nat ~> Lcm_empty Lcm_empty_int ~> Lcm_empty Lcm_insert_nat ~> Lcm_insert Lcm_insert_int ~> Lcm_insert comp_fun_idem_gcd_nat ~> comp_fun_idem_gcd comp_fun_idem_gcd_int ~> comp_fun_idem_gcd comp_fun_idem_lcm_nat ~> comp_fun_idem_lcm comp_fun_idem_lcm_int ~> comp_fun_idem_lcm Lcm_eq_0 ~> Lcm_eq_0_I Lcm0_iff ~> Lcm_0_iff Lcm_dvd_int ~> Lcm_least divides_mult_nat ~> divides_mult divides_mult_int ~> divides_mult lcm_0_nat ~> lcm_0_right lcm_0_int ~> lcm_0_right lcm_0_left_nat ~> lcm_0_left lcm_0_left_int ~> lcm_0_left dvd_gcd_D1_nat ~> dvd_gcdD1 dvd_gcd_D1_int ~> dvd_gcdD1 dvd_gcd_D2_nat ~> dvd_gcdD2 dvd_gcd_D2_int ~> dvd_gcdD2 coprime_dvd_mult_iff_nat ~> coprime_dvd_mult_iff coprime_dvd_mult_iff_int ~> coprime_dvd_mult_iff realpow_minus_mult ~> power_minus_mult realpow_Suc_le_self ~> power_Suc_le_self dvd_Gcd, dvd_Gcd_nat, dvd_Gcd_int removed in favour of Gcd_greatest INCOMPATIBILITY. * Renamed HOL/Quotient_Examples/FSet.thy to HOL/Quotient_Examples/Quotient_FSet.thy INCOMPATIBILITY. * Session HOL-Library: theory FinFun bundles "finfun_syntax" and "no_finfun_syntax" allow to control optional syntax in local contexts; this supersedes former theory FinFun_Syntax. INCOMPATIBILITY, e.g. use "unbundle finfun_syntax" to imitate import of "~~/src/HOL/Library/FinFun_Syntax". * Session HOL-Library: theory Multiset_Permutations (executably) defines the set of permutations of a given set or multiset, i.e. the set of all lists that contain every element of the carrier (multi-)set exactly once. * Session HOL-Library: multiset membership is now expressed using set_mset rather than count. - Expressions "count M a > 0" and similar simplify to membership by default. - Converting between "count M a = 0" and non-membership happens using equations count_eq_zero_iff and not_in_iff. - Rules count_inI and in_countE obtain facts of the form "count M a = n" from membership. - Rules count_in_diffI and in_diff_countE obtain facts of the form "count M a = n + count N a" from membership on difference sets. INCOMPATIBILITY. * Session HOL-Library: theory LaTeXsugar uses new-style "dummy_pats" for displaying equations in functional programming style --- variables present on the left-hand but not on the righ-hand side are replaced by underscores. * Session HOL-Library: theory Combinator_PER provides combinator to build partial equivalence relations from a predicate and an equivalence relation. * Session HOL-Library: theory Perm provides basic facts about almost everywhere fix bijections. * Session HOL-Library: theory Normalized_Fraction allows viewing an element of a field of fractions as a normalized fraction (i.e. a pair of numerator and denominator such that the two are coprime and the denominator is normalized wrt. unit factors). * Session HOL-NSA has been renamed to HOL-Nonstandard_Analysis. * Session HOL-Multivariate_Analysis has been renamed to HOL-Analysis. * Session HOL-Analysis: measure theory has been moved here from HOL-Probability. When importing HOL-Analysis some theorems need additional name spaces prefixes due to name clashes. INCOMPATIBILITY. * Session HOL-Analysis: more complex analysis including Cauchy's inequality, Liouville theorem, open mapping theorem, maximum modulus principle, Residue theorem, Schwarz Lemma. * Session HOL-Analysis: Theory of polyhedra: faces, extreme points, polytopes, and the Krein–Milman Minkowski theorem. * Session HOL-Analysis: Numerous results ported from the HOL Light libraries: homeomorphisms, continuous function extensions, invariance of domain. * Session HOL-Probability: the type of emeasure and nn_integral was changed from ereal to ennreal, INCOMPATIBILITY. emeasure :: 'a measure \ 'a set \ ennreal nn_integral :: 'a measure \ ('a \ ennreal) \ ennreal * Session HOL-Probability: Code generation and QuickCheck for Probability Mass Functions. * Session HOL-Probability: theory Random_Permutations contains some theory about choosing a permutation of a set uniformly at random and folding over a list in random order. * Session HOL-Probability: theory SPMF formalises discrete subprobability distributions. * Session HOL-Library: the names of multiset theorems have been normalised to distinguish which ordering the theorems are about mset_less_eqI ~> mset_subset_eqI mset_less_insertD ~> mset_subset_insertD mset_less_eq_count ~> mset_subset_eq_count mset_less_diff_self ~> mset_subset_diff_self mset_le_exists_conv ~> mset_subset_eq_exists_conv mset_le_mono_add_right_cancel ~> mset_subset_eq_mono_add_right_cancel mset_le_mono_add_left_cancel ~> mset_subset_eq_mono_add_left_cancel mset_le_mono_add ~> mset_subset_eq_mono_add mset_le_add_left ~> mset_subset_eq_add_left mset_le_add_right ~> mset_subset_eq_add_right mset_le_single ~> mset_subset_eq_single mset_le_multiset_union_diff_commute ~> mset_subset_eq_multiset_union_diff_commute diff_le_self ~> diff_subset_eq_self mset_leD ~> mset_subset_eqD mset_lessD ~> mset_subsetD mset_le_insertD ~> mset_subset_eq_insertD mset_less_of_empty ~> mset_subset_of_empty mset_less_size ~> mset_subset_size wf_less_mset_rel ~> wf_subset_mset_rel count_le_replicate_mset_le ~> count_le_replicate_mset_subset_eq mset_remdups_le ~> mset_remdups_subset_eq ms_lesseq_impl ~> subset_eq_mset_impl Some functions have been renamed: ms_lesseq_impl -> subset_eq_mset_impl * HOL-Library: multisets are now ordered with the multiset ordering #\# ~> \ #\# ~> < le_multiset ~> less_eq_multiset less_multiset ~> le_multiset INCOMPATIBILITY. * Session HOL-Library: the prefix multiset_order has been discontinued: the theorems can be directly accessed. As a consequence, the lemmas "order_multiset" and "linorder_multiset" have been discontinued, and the interpretations "multiset_linorder" and "multiset_wellorder" have been replaced by instantiations. INCOMPATIBILITY. * Session HOL-Library: some theorems about the multiset ordering have been renamed: le_multiset_def ~> less_eq_multiset_def less_multiset_def ~> le_multiset_def less_eq_imp_le_multiset ~> subset_eq_imp_le_multiset mult_less_not_refl ~> mset_le_not_refl mult_less_trans ~> mset_le_trans mult_less_not_sym ~> mset_le_not_sym mult_less_asym ~> mset_le_asym mult_less_irrefl ~> mset_le_irrefl union_less_mono2{,1,2} ~> union_le_mono2{,1,2} le_multiset\<^sub>H\<^sub>O ~> less_eq_multiset\<^sub>H\<^sub>O le_multiset_total ~> less_eq_multiset_total less_multiset_right_total ~> subset_eq_imp_le_multiset le_multiset_empty_left ~> less_eq_multiset_empty_left le_multiset_empty_right ~> less_eq_multiset_empty_right less_multiset_empty_right ~> le_multiset_empty_left less_multiset_empty_left ~> le_multiset_empty_right union_less_diff_plus ~> union_le_diff_plus ex_gt_count_imp_less_multiset ~> ex_gt_count_imp_le_multiset less_multiset_plus_left_nonempty ~> le_multiset_plus_left_nonempty le_multiset_plus_right_nonempty ~> le_multiset_plus_right_nonempty INCOMPATIBILITY. * Session HOL-Library: the lemma mset_map has now the attribute [simp]. INCOMPATIBILITY. * Session HOL-Library: some theorems about multisets have been removed. INCOMPATIBILITY, use the following replacements: le_multiset_plus_plus_left_iff ~> add_less_cancel_right less_multiset_plus_plus_left_iff ~> add_less_cancel_right le_multiset_plus_plus_right_iff ~> add_less_cancel_left less_multiset_plus_plus_right_iff ~> add_less_cancel_left add_eq_self_empty_iff ~> add_cancel_left_right mset_subset_add_bothsides ~> subset_mset.add_less_cancel_right mset_less_add_bothsides ~> subset_mset.add_less_cancel_right mset_le_add_bothsides ~> subset_mset.add_less_cancel_right empty_inter ~> subset_mset.inf_bot_left inter_empty ~> subset_mset.inf_bot_right empty_sup ~> subset_mset.sup_bot_left sup_empty ~> subset_mset.sup_bot_right bdd_below_multiset ~> subset_mset.bdd_above_bot subset_eq_empty ~> subset_mset.le_zero_eq le_empty ~> subset_mset.le_zero_eq mset_subset_empty_nonempty ~> subset_mset.zero_less_iff_neq_zero mset_less_empty_nonempty ~> subset_mset.zero_less_iff_neq_zero * Session HOL-Library: some typeclass constraints about multisets have been reduced from ordered or linordered to preorder. Multisets have the additional typeclasses order_bot, no_top, ordered_ab_semigroup_add_imp_le, ordered_cancel_comm_monoid_add, linordered_cancel_ab_semigroup_add, and ordered_ab_semigroup_monoid_add_imp_le. INCOMPATIBILITY. * Session HOL-Library: there are some new simplification rules about multisets, the multiset ordering, and the subset ordering on multisets. INCOMPATIBILITY. * Session HOL-Library: the subset ordering on multisets has now the interpretations ordered_ab_semigroup_monoid_add_imp_le and bounded_lattice_bot. INCOMPATIBILITY. * Session HOL-Library, theory Multiset: single has been removed in favor of add_mset that roughly corresponds to Set.insert. Some theorems have removed or changed: single_not_empty ~> add_mset_not_empty or empty_not_add_mset fold_mset_insert ~> fold_mset_add_mset image_mset_insert ~> image_mset_add_mset union_single_eq_diff multi_self_add_other_not_self diff_single_eq_union INCOMPATIBILITY. * Session HOL-Library, theory Multiset: some theorems have been changed to use add_mset instead of single: mset_add multi_self_add_other_not_self diff_single_eq_union union_single_eq_diff union_single_eq_member add_eq_conv_diff insert_noteq_member add_eq_conv_ex multi_member_split multiset_add_sub_el_shuffle mset_subset_eq_insertD mset_subset_insertD insert_subset_eq_iff insert_union_subset_iff multi_psub_of_add_self inter_add_left1 inter_add_left2 inter_add_right1 inter_add_right2 sup_union_left1 sup_union_left2 sup_union_right1 sup_union_right2 size_eq_Suc_imp_eq_union multi_nonempty_split mset_insort mset_update mult1I less_add mset_zip_take_Cons_drop_twice rel_mset_Zero msed_map_invL msed_map_invR msed_rel_invL msed_rel_invR le_multiset_right_total multiset_induct multiset_induct2_size multiset_induct2 INCOMPATIBILITY. * Session HOL-Library, theory Multiset: the definitions of some constants have changed to use add_mset instead of adding a single element: image_mset mset replicate_mset mult1 pred_mset rel_mset' mset_insort INCOMPATIBILITY. * Session HOL-Library, theory Multiset: due to the above changes, the attributes of some multiset theorems have been changed: insert_DiffM [] ~> [simp] insert_DiffM2 [simp] ~> [] diff_add_mset_swap [simp] fold_mset_add_mset [simp] diff_diff_add [simp] (for multisets only) diff_cancel [simp] ~> [] count_single [simp] ~> [] set_mset_single [simp] ~> [] size_multiset_single [simp] ~> [] size_single [simp] ~> [] image_mset_single [simp] ~> [] mset_subset_eq_mono_add_right_cancel [simp] ~> [] mset_subset_eq_mono_add_left_cancel [simp] ~> [] fold_mset_single [simp] ~> [] subset_eq_empty [simp] ~> [] empty_sup [simp] ~> [] sup_empty [simp] ~> [] inter_empty [simp] ~> [] empty_inter [simp] ~> [] INCOMPATIBILITY. * Session HOL-Library, theory Multiset: the order of the variables in the second cases of multiset_induct, multiset_induct2_size, multiset_induct2 has been changed (e.g. Add A a ~> Add a A). INCOMPATIBILITY. * Session HOL-Library, theory Multiset: there is now a simplification procedure on multisets. It mimics the behavior of the procedure on natural numbers. INCOMPATIBILITY. * Session HOL-Library, theory Multiset: renamed sums and products of multisets: msetsum ~> sum_mset msetprod ~> prod_mset * Session HOL-Library, theory Multiset: the notation for intersection and union of multisets have been changed: #\ ~> \# #\ ~> \# INCOMPATIBILITY. * Session HOL-Library, theory Multiset: the lemma one_step_implies_mult_aux on multisets has been removed, use one_step_implies_mult instead. INCOMPATIBILITY. * Session HOL-Library: theory Complete_Partial_Order2 provides reasoning support for monotonicity and continuity in chain-complete partial orders and about admissibility conditions for fixpoint inductions. * Session HOL-Library: theory Library/Polynomial contains also derivation of polynomials (formerly in Library/Poly_Deriv) but not gcd/lcm on polynomials over fields. This has been moved to a separate theory Library/Polynomial_GCD_euclidean.thy, to pave way for a possible future different type class instantiation for polynomials over factorial rings. INCOMPATIBILITY. * Session HOL-Library: theory Sublist provides function "prefixes" with the following renaming prefixeq -> prefix prefix -> strict_prefix suffixeq -> suffix suffix -> strict_suffix Added theory of longest common prefixes. * Session HOL-Number_Theory: algebraic foundation for primes: Generalisation of predicate "prime" and introduction of predicates "prime_elem", "irreducible", a "prime_factorization" function, and the "factorial_ring" typeclass with instance proofs for nat, int, poly. Some theorems now have different names, most notably "prime_def" is now "prime_nat_iff". INCOMPATIBILITY. * Session Old_Number_Theory has been removed, after porting remaining theories. * Session HOL-Types_To_Sets provides an experimental extension of Higher-Order Logic to allow translation of types to sets. *** ML *** * Integer.gcd and Integer.lcm use efficient operations from the Poly/ML library (notably for big integers). Subtle change of semantics: Integer.gcd and Integer.lcm both normalize the sign, results are never negative. This coincides with the definitions in HOL/GCD.thy. INCOMPATIBILITY. * Structure Rat for rational numbers is now an integral part of Isabelle/ML, with special notation @int/nat or @int for numerals (an abbreviation for antiquotation @{Pure.rat argument}) and ML pretty printing. Standard operations on type Rat.rat are provided via ad-hoc overloading of + - * / < <= > >= ~ abs. INCOMPATIBILITY, need to use + instead of +/ etc. Moreover, exception Rat.DIVZERO has been superseded by General.Div. * ML antiquotation @{path} is superseded by @{file}, which ensures that the argument is a plain file. Minor INCOMPATIBILITY. * Antiquotation @{make_string} is available during Pure bootstrap -- with approximative output quality. * Low-level ML system structures (like PolyML and RunCall) are no longer exposed to Isabelle/ML user-space. Potential INCOMPATIBILITY. * The ML function "ML" provides easy access to run-time compilation. This is particularly useful for conditional compilation, without requiring separate files. * Option ML_exception_debugger controls detailed exception trace via the Poly/ML debugger. Relevant ML modules need to be compiled beforehand with ML_file_debug, or with ML_file and option ML_debugger enabled. Note debugger information requires consirable time and space: main Isabelle/HOL with full debugger support may need ML_system_64. * Local_Theory.restore has been renamed to Local_Theory.reset to emphasize its disruptive impact on the cumulative context, notably the scope of 'private' or 'qualified' names. Note that Local_Theory.reset is only appropriate when targets are managed, e.g. starting from a global theory and returning to it. Regular definitional packages should use balanced blocks of Local_Theory.open_target versus Local_Theory.close_target instead. Rare INCOMPATIBILITY. * Structure TimeLimit (originally from the SML/NJ library) has been replaced by structure Timeout, with slightly different signature. INCOMPATIBILITY. * Discontinued cd and pwd operations, which are not well-defined in a multi-threaded environment. Note that files are usually located relatively to the master directory of a theory (see also File.full_path). Potential INCOMPATIBILITY. * Binding.empty_atts supersedes Thm.empty_binding and Attrib.empty_binding. Minor INCOMPATIBILITY. *** System *** * SML/NJ and old versions of Poly/ML are no longer supported. * Poly/ML heaps now follow the hierarchy of sessions, and thus require much less disk space. * The Isabelle ML process is now managed directly by Isabelle/Scala, and shell scripts merely provide optional command-line access. In particular: . Scala module ML_Process to connect to the raw ML process, with interaction via stdin/stdout/stderr or in batch mode; . command-line tool "isabelle console" as interactive wrapper; . command-line tool "isabelle process" as batch mode wrapper. * The executable "isabelle_process" has been discontinued. Tools and prover front-ends should use ML_Process or Isabelle_Process in Isabelle/Scala. INCOMPATIBILITY. * New command-line tool "isabelle process" supports ML evaluation of literal expressions (option -e) or files (option -f) in the context of a given heap image. Errors lead to premature exit of the ML process with return code 1. * The command-line tool "isabelle build" supports option -N for cyclic shuffling of NUMA CPU nodes. This may help performance tuning on Linux servers with separate CPU/memory modules. * System option "threads" (for the size of the Isabelle/ML thread farm) is also passed to the underlying ML runtime system as --gcthreads, unless there is already a default provided via ML_OPTIONS settings. * System option "checkpoint" helps to fine-tune the global heap space management of isabelle build. This is relevant for big sessions that may exhaust the small 32-bit address space of the ML process (which is used by default). * System option "profiling" specifies the mode for global ML profiling in "isabelle build". Possible values are "time", "allocations". The command-line tool "isabelle profiling_report" helps to digest the resulting log files. * System option "ML_process_policy" specifies an optional command prefix for the underlying ML process, e.g. to control CPU affinity on multiprocessor systems. The "isabelle jedit" tool allows to override the implicit default via option -p. * Command-line tool "isabelle console" provides option -r to help to bootstrapping Isabelle/Pure interactively. * Command-line tool "isabelle yxml" has been discontinued. INCOMPATIBILITY, use operations from the modules "XML" and "YXML" in Isabelle/ML or Isabelle/Scala. * Many Isabelle tools that require a Java runtime system refer to the settings ISABELLE_TOOL_JAVA_OPTIONS32 / ISABELLE_TOOL_JAVA_OPTIONS64, depending on the underlying platform. The settings for "isabelle build" ISABELLE_BUILD_JAVA_OPTIONS32 / ISABELLE_BUILD_JAVA_OPTIONS64 have been discontinued. Potential INCOMPATIBILITY. * The Isabelle system environment always ensures that the main executables are found within the shell search $PATH: "isabelle" and "isabelle_scala_script". * Isabelle tools may consist of .scala files: the Scala compiler is invoked on the spot. The source needs to define some object that extends Isabelle_Tool.Body. * File.bash_string, File.bash_path etc. represent Isabelle/ML and Isabelle/Scala strings authentically within GNU bash. This is useful to produce robust shell scripts under program control, without worrying about spaces or special characters. Note that user output works via Path.print (ML) or Path.toString (Scala). INCOMPATIBILITY, the old (and less versatile) operations File.shell_quote, File.shell_path etc. have been discontinued. * The isabelle_java executable allows to run a Java process within the name space of Java and Scala components that are bundled with Isabelle, but without the Isabelle settings environment. * Isabelle/Scala: the SSH module supports ssh and sftp connections, for remote command-execution and file-system access. This resembles operations from module File and Isabelle_System to some extent. Note that Path specifications need to be resolved remotely via ssh.remote_path instead of File.standard_path: the implicit process environment is different, Isabelle settings are not available remotely. * Isabelle/Scala: the Mercurial module supports repositories via the regular hg command-line interface. The repositroy clone and working directory may reside on a local or remote file-system (via ssh connection). New in Isabelle2016 (February 2016) ----------------------------------- *** General *** * Eisbach is now based on Pure instead of HOL. Objects-logics may import either the theory ~~/src/HOL/Eisbach/Eisbach (for HOL etc.) or ~~/src/HOL/Eisbach/Eisbach_Old_Appl_Syntax (for FOL, ZF etc.). Note that the HOL-Eisbach session located in ~~/src/HOL/Eisbach/ contains further examples that do require HOL. * Better resource usage on all platforms (Linux, Windows, Mac OS X) for both Isabelle/ML and Isabelle/Scala. Slightly reduced heap space usage. * Former "xsymbols" syntax with Isabelle symbols is used by default, without any special print mode. Important ASCII replacement syntax remains available under print mode "ASCII", but less important syntax has been removed (see below). * Support for more arrow symbols, with rendering in LaTeX and Isabelle fonts: \ \ \ \ \ \. * Special notation \ for the first implicit 'structure' in the context has been discontinued. Rare INCOMPATIBILITY, use explicit structure name instead, notably in indexed notation with block-subscript (e.g. \\<^bsub>A\<^esub>). * The glyph for \ in the IsabelleText font now corresponds better to its counterpart \ as quantifier-like symbol. A small diamond is available as \; the old symbol \ loses this rendering and any special meaning. * Syntax for formal comments "-- text" now also supports the symbolic form "\ text". Command-line tool "isabelle update_cartouches -c" helps to update old sources. * Toplevel theorem statements have been simplified as follows: theorems ~> lemmas schematic_lemma ~> schematic_goal schematic_theorem ~> schematic_goal schematic_corollary ~> schematic_goal Command-line tool "isabelle update_theorems" updates theory sources accordingly. * Toplevel theorem statement 'proposition' is another alias for 'theorem'. * The old 'defs' command has been removed (legacy since Isabelle2014). INCOMPATIBILITY, use regular 'definition' instead. Overloaded and/or deferred definitions require a surrounding 'overloading' block. *** Prover IDE -- Isabelle/Scala/jEdit *** * IDE support for the source-level debugger of Poly/ML, to work with Isabelle/ML and official Standard ML. Option "ML_debugger" and commands 'ML_file_debug', 'ML_file_no_debug', 'SML_file_debug', 'SML_file_no_debug' control compilation of sources with or without debugging information. The Debugger panel allows to set breakpoints (via context menu), step through stopped threads, evaluate local ML expressions etc. At least one Debugger view needs to be active to have any effect on the running ML program. * The State panel manages explicit proof state output, with dynamic auto-update according to cursor movement. Alternatively, the jEdit action "isabelle.update-state" (shortcut S+ENTER) triggers manual update. * The Output panel no longer shows proof state output by default, to avoid GUI overcrowding. INCOMPATIBILITY, use the State panel instead or enable option "editor_output_state". * The text overview column (status of errors, warnings etc.) is updated asynchronously, leading to much better editor reactivity. Moreover, the full document node content is taken into account. The width of the column is scaled according to the main text area font, for improved visibility. * The main text area no longer changes its color hue in outdated situations. The text overview column takes over the role to indicate unfinished edits in the PIDE pipeline. This avoids flashing text display due to ad-hoc updates by auxiliary GUI components, such as the State panel. * Slightly improved scheduling for urgent print tasks (e.g. command state output, interactive queries) wrt. long-running background tasks. * Completion of symbols via prefix of \ or \<^name> or \name is always possible, independently of the language context. It is never implicit: a popup will show up unconditionally. * Additional abbreviations for syntactic completion may be specified in $ISABELLE_HOME/etc/abbrevs and $ISABELLE_HOME_USER/etc/abbrevs, with support for simple templates using ASCII 007 (bell) as placeholder. * Symbols \, \, \, \, \, \, \, \ no longer provide abbreviations for completion like "+o", "*o", ".o" etc. -- due to conflicts with other ASCII syntax. INCOMPATIBILITY, use plain backslash-completion or define suitable abbreviations in $ISABELLE_HOME_USER/etc/abbrevs. * Action "isabelle-emph" (with keyboard shortcut C+e LEFT) controls emphasized text style; the effect is visible in document output, not in the editor. * Action "isabelle-reset" now uses keyboard shortcut C+e BACK_SPACE, instead of former C+e LEFT. * The command-line tool "isabelle jedit" and the isabelle.Main application wrapper treat the default $USER_HOME/Scratch.thy more uniformly, and allow the dummy file argument ":" to open an empty buffer instead. * New command-line tool "isabelle jedit_client" allows to connect to an already running Isabelle/jEdit process. This achieves the effect of single-instance applications seen on common GUI desktops. * The default look-and-feel for Linux is the traditional "Metal", which works better with GUI scaling for very high-resolution displays (e.g. 4K). Moreover, it is generally more robust than "Nimbus". * Update to jedit-5.3.0, with improved GUI scaling and support of high-resolution displays (e.g. 4K). * The main Isabelle executable is managed as single-instance Desktop application uniformly on all platforms: Linux, Windows, Mac OS X. *** Document preparation *** * Commands 'paragraph' and 'subparagraph' provide additional section headings. Thus there are 6 levels of standard headings, as in HTML. * Command 'text_raw' has been clarified: input text is processed as in 'text' (with antiquotations and control symbols). The key difference is the lack of the surrounding isabelle markup environment in output. * Text is structured in paragraphs and nested lists, using notation that is similar to Markdown. The control symbols for list items are as follows: \<^item> itemize \<^enum> enumerate \<^descr> description * There is a new short form for antiquotations with a single argument that is a cartouche: \<^name>\...\ is equivalent to @{name \...\} and \...\ without control symbol is equivalent to @{cartouche \...\}. \<^name> without following cartouche is equivalent to @{name}. The standard Isabelle fonts provide glyphs to render important control symbols, e.g. "\<^verbatim>", "\<^emph>", "\<^bold>". * Antiquotations @{noindent}, @{smallskip}, @{medskip}, @{bigskip} with corresponding control symbols \<^noindent>, \<^smallskip>, \<^medskip>, \<^bigskip> specify spacing formally, using standard LaTeX macros of the same names. * Antiquotation @{cartouche} in Isabelle/Pure is the same as @{text}. Consequently, \...\ without any decoration prints literal quasi-formal text. Command-line tool "isabelle update_cartouches -t" helps to update old sources, by approximative patching of the content of string and cartouche tokens seen in theory sources. * The @{text} antiquotation now ignores the antiquotation option "source". The given text content is output unconditionally, without any surrounding quotes etc. Subtle INCOMPATIBILITY, put quotes into the argument where they are really intended, e.g. @{text \"foo"\}. Initial or terminal spaces are ignored. * Antiquotations @{emph} and @{bold} output LaTeX source recursively, adding appropriate text style markup. These may be used in the short form \<^emph>\...\ and \<^bold>\...\. * Document antiquotation @{footnote} outputs LaTeX source recursively, marked as \footnote{}. This may be used in the short form \<^footnote>\...\. * Antiquotation @{verbatim [display]} supports option "indent". * Antiquotation @{theory_text} prints uninterpreted theory source text (Isar outer syntax with command keywords etc.). This may be used in the short form \<^theory_text>\...\. @{theory_text [display]} supports option "indent". * Antiquotation @{doc ENTRY} provides a reference to the given documentation, with a hyperlink in the Prover IDE. * Antiquotations @{command}, @{method}, @{attribute} print checked entities of the Isar language. * HTML presentation uses the standard IsabelleText font and Unicode rendering of Isabelle symbols like Isabelle/Scala/jEdit. The former print mode "HTML" loses its special meaning. *** Isar *** * Local goals ('have', 'show', 'hence', 'thus') allow structured rule statements like fixes/assumes/shows in theorem specifications, but the notation is postfix with keywords 'if' (or 'when') and 'for'. For example: have result: "C x y" if "A x" and "B y" for x :: 'a and y :: 'a The local assumptions are bound to the name "that". The result is exported from context of the statement as usual. The above roughly corresponds to a raw proof block like this: { fix x :: 'a and y :: 'a assume that: "A x" "B y" have "C x y" } note result = this The keyword 'when' may be used instead of 'if', to indicate 'presume' instead of 'assume' above. * Assumptions ('assume', 'presume') allow structured rule statements using 'if' and 'for', similar to 'have' etc. above. For example: assume result: "C x y" if "A x" and "B y" for x :: 'a and y :: 'a This assumes "\x y::'a. A x \ B y \ C x y" and produces a general result as usual: "A ?x \ B ?y \ C ?x ?y". Vacuous quantification in assumptions is omitted, i.e. a for-context only effects propositions according to actual use of variables. For example: assume "A x" and "B y" for x and y is equivalent to: assume "\x. A x" and "\y. B y" * The meaning of 'show' with Pure rule statements has changed: premises are treated in the sense of 'assume', instead of 'presume'. This means, a goal like "\x. A x \ B x \ C x" can be solved completely as follows: show "\x. A x \ B x \ C x" or: show "C x" if "A x" "B x" for x Rare INCOMPATIBILITY, the old behaviour may be recovered as follows: show "C x" when "A x" "B x" for x * New command 'consider' states rules for generalized elimination and case splitting. This is like a toplevel statement "theorem obtains" used within a proof body; or like a multi-branch 'obtain' without activation of the local context elements yet. * Proof method "cases" allows to specify the rule as first entry of chained facts. This is particularly useful with 'consider': consider (a) A | (b) B | (c) C then have something proof cases case a then show ?thesis next case b then show ?thesis next case c then show ?thesis qed * Command 'case' allows fact name and attribute specification like this: case a: (c xs) case a [attributes]: (c xs) Facts that are introduced by invoking the case context are uniformly qualified by "a"; the same name is used for the cumulative fact. The old form "case (c xs) [attributes]" is no longer supported. Rare INCOMPATIBILITY, need to adapt uses of case facts in exotic situations, and always put attributes in front. * The standard proof method of commands 'proof' and '..' is now called "standard" to make semantically clear what it is; the old name "default" is still available as legacy for some time. Documentation now explains '..' more accurately as "by standard" instead of "by rule". * Nesting of Isar goal structure has been clarified: the context after the initial backwards refinement is retained for the whole proof, within all its context sections (as indicated via 'next'). This is e.g. relevant for 'using', 'including', 'supply': have "A \ A" if a: A for A supply [simp] = a proof show A by simp next show A by simp qed * Command 'obtain' binds term abbreviations (via 'is' patterns) in the proof body as well, abstracted over relevant parameters. * Improved type-inference for theorem statement 'obtains': separate parameter scope for of each clause. * Term abbreviations via 'is' patterns also work for schematic statements: result is abstracted over unknowns. * Command 'subgoal' allows to impose some structure on backward refinements, to avoid proof scripts degenerating into long of 'apply' sequences. Further explanations and examples are given in the isar-ref manual. * Command 'supply' supports fact definitions during goal refinement ('apply' scripts). * Proof method "goal_cases" turns the current subgoals into cases within the context; the conclusion is bound to variable ?case in each case. For example: lemma "\x. A x \ B x \ C x" and "\y z. U y \ V z \ W y z" proof goal_cases case (1 x) then show ?case using \A x\ \B x\ sorry next case (2 y z) then show ?case using \U y\ \V z\ sorry qed lemma "\x. A x \ B x \ C x" and "\y z. U y \ V z \ W y z" proof goal_cases case prems: 1 then show ?case using prems sorry next case prems: 2 then show ?case using prems sorry qed * The undocumented feature of implicit cases goal1, goal2, goal3, etc. is marked as legacy, and will be removed eventually. The proof method "goals" achieves a similar effect within regular Isar; often it can be done more adequately by other means (e.g. 'consider'). * The vacuous fact "TERM x" may be established "by fact" or as `TERM x` as well, not just "by this" or "." as before. * Method "sleep" succeeds after a real-time delay (in seconds). This is occasionally useful for demonstration and testing purposes. *** Pure *** * Qualifiers in locale expressions default to mandatory ('!') regardless of the command. Previously, for 'locale' and 'sublocale' the default was optional ('?'). The old synatx '!' has been discontinued. INCOMPATIBILITY, remove '!' and add '?' as required. * Keyword 'rewrites' identifies rewrite morphisms in interpretation commands. Previously, the keyword was 'where'. INCOMPATIBILITY. * More gentle suppression of syntax along locale morphisms while printing terms. Previously 'abbreviation' and 'notation' declarations would be suppressed for morphisms except term identity. Now 'abbreviation' is also kept for morphims that only change the involved parameters, and only 'notation' is suppressed. This can be of great help when working with complex locale hierarchies, because proof states are displayed much more succinctly. It also means that only notation needs to be redeclared if desired, as illustrated by this example: locale struct = fixes composition :: "'a => 'a => 'a" (infixl "\" 65) begin definition derived (infixl "\" 65) where ... end locale morphism = left: struct composition + right: struct composition' for composition (infix "\" 65) and composition' (infix "\''" 65) begin notation right.derived ("\''") end * Command 'global_interpretation' issues interpretations into global theories, with optional rewrite definitions following keyword 'defines'. * Command 'sublocale' accepts optional rewrite definitions after keyword 'defines'. * Command 'permanent_interpretation' has been discontinued. Use 'global_interpretation' or 'sublocale' instead. INCOMPATIBILITY. * Command 'print_definitions' prints dependencies of definitional specifications. This functionality used to be part of 'print_theory'. * Configuration option rule_insts_schematic has been discontinued (intermediate legacy feature in Isabelle2015). INCOMPATIBILITY. * Abbreviations in type classes now carry proper sort constraint. Rare INCOMPATIBILITY in situations where the previous misbehaviour has been exploited. * Refinement of user-space type system in type classes: pseudo-local operations behave more similar to abbreviations. Potential INCOMPATIBILITY in exotic situations. *** HOL *** * The 'typedef' command has been upgraded from a partially checked "axiomatization", to a full definitional specification that takes the global collection of overloaded constant / type definitions into account. Type definitions with open dependencies on overloaded definitions need to be specified as "typedef (overloaded)". This provides extra robustness in theory construction. Rare INCOMPATIBILITY. * Qualification of various formal entities in the libraries is done more uniformly via "context begin qualified definition ... end" instead of old-style "hide_const (open) ...". Consequently, both the defined constant and its defining fact become qualified, e.g. Option.is_none and Option.is_none_def. Occasional INCOMPATIBILITY in applications. * Some old and rarely used ASCII replacement syntax has been removed. INCOMPATIBILITY, standard syntax with symbols should be used instead. The subsequent commands help to reproduce the old forms, e.g. to simplify porting old theories: notation iff (infixr "<->" 25) notation Times (infixr "<*>" 80) type_notation Map.map (infixr "~=>" 0) notation Map.map_comp (infixl "o'_m" 55) type_notation FinFun.finfun ("(_ =>f /_)" [22, 21] 21) notation FuncSet.funcset (infixr "->" 60) notation FuncSet.extensional_funcset (infixr "->\<^sub>E" 60) notation Omega_Words_Fun.conc (infixr "conc" 65) notation Preorder.equiv ("op ~~") and Preorder.equiv ("(_/ ~~ _)" [51, 51] 50) notation (in topological_space) tendsto (infixr "--->" 55) notation (in topological_space) LIMSEQ ("((_)/ ----> (_))" [60, 60] 60) notation LIM ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) notation NSA.approx (infixl "@=" 50) notation NSLIMSEQ ("((_)/ ----NS> (_))" [60, 60] 60) notation NSLIM ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60) * The alternative notation "\" for type and sort constraints has been removed: in LaTeX document output it looks the same as "::". INCOMPATIBILITY, use plain "::" instead. * Commands 'inductive' and 'inductive_set' work better when names for intro rules are omitted: the "cases" and "induct" rules no longer declare empty case_names, but no case_names at all. This allows to use numbered cases in proofs, without requiring method "goal_cases". * Inductive definitions ('inductive', 'coinductive', etc.) expose low-level facts of the internal construction only if the option "inductive_internals" is enabled. This refers to the internal predicate definition and its monotonicity result. Rare INCOMPATIBILITY. * Recursive function definitions ('fun', 'function', 'partial_function') expose low-level facts of the internal construction only if the option "function_internals" is enabled. Its internal inductive definition is also subject to "inductive_internals". Rare INCOMPATIBILITY. * BNF datatypes ('datatype', 'codatatype', etc.) expose low-level facts of the internal construction only if the option "bnf_internals" is enabled. This supersedes the former option "bnf_note_all". Rare INCOMPATIBILITY. * Combinator to represent case distinction on products is named "case_prod", uniformly, discontinuing any input aliasses. Very popular theorem aliasses have been retained. Consolidated facts: PairE ~> prod.exhaust Pair_eq ~> prod.inject pair_collapse ~> prod.collapse Pair_fst_snd_eq ~> prod_eq_iff split_twice ~> prod.case_distrib split_weak_cong ~> prod.case_cong_weak split_split ~> prod.split split_split_asm ~> prod.split_asm splitI ~> case_prodI splitD ~> case_prodD splitI2 ~> case_prodI2 splitI2' ~> case_prodI2' splitE ~> case_prodE splitE' ~> case_prodE' split_pair ~> case_prod_Pair split_eta ~> case_prod_eta split_comp ~> case_prod_comp mem_splitI ~> mem_case_prodI mem_splitI2 ~> mem_case_prodI2 mem_splitE ~> mem_case_prodE The_split ~> The_case_prod cond_split_eta ~> cond_case_prod_eta Collect_split_in_rel_leE ~> Collect_case_prod_in_rel_leE Collect_split_in_rel_leI ~> Collect_case_prod_in_rel_leI in_rel_Collect_split_eq ~> in_rel_Collect_case_prod_eq Collect_split_Grp_eqD ~> Collect_case_prod_Grp_eqD Collect_split_Grp_inD ~> Collect_case_prod_Grp_in Domain_Collect_split ~> Domain_Collect_case_prod Image_Collect_split ~> Image_Collect_case_prod Range_Collect_split ~> Range_Collect_case_prod Eps_split ~> Eps_case_prod Eps_split_eq ~> Eps_case_prod_eq split_rsp ~> case_prod_rsp curry_split ~> curry_case_prod split_curry ~> case_prod_curry Changes in structure HOLogic: split_const ~> case_prod_const mk_split ~> mk_case_prod mk_psplits ~> mk_ptupleabs strip_psplits ~> strip_ptupleabs INCOMPATIBILITY. * The coercions to type 'real' have been reorganised. The function 'real' is no longer overloaded, but has type 'nat => real' and abbreviates of_nat for that type. Also 'real_of_int :: int => real' abbreviates of_int for that type. Other overloaded instances of 'real' have been replaced by 'real_of_ereal' and 'real_of_float'. Consolidated facts (among others): real_of_nat_le_iff -> of_nat_le_iff real_of_nat_numeral of_nat_numeral real_of_int_zero of_int_0 real_of_nat_zero of_nat_0 real_of_one of_int_1 real_of_int_add of_int_add real_of_nat_add of_nat_add real_of_int_diff of_int_diff real_of_nat_diff of_nat_diff floor_subtract floor_diff_of_int real_of_int_inject of_int_eq_iff real_of_int_gt_zero_cancel_iff of_int_0_less_iff real_of_int_ge_zero_cancel_iff of_int_0_le_iff real_of_nat_ge_zero of_nat_0_le_iff real_of_int_ceiling_ge le_of_int_ceiling ceiling_less_eq ceiling_less_iff ceiling_le_eq ceiling_le_iff less_floor_eq less_floor_iff floor_less_eq floor_less_iff floor_divide_eq_div floor_divide_of_int_eq real_of_int_zero_cancel of_nat_eq_0_iff ceiling_real_of_int ceiling_of_int INCOMPATIBILITY. * Theory Map: lemma map_of_is_SomeD was a clone of map_of_SomeD and has been removed. INCOMPATIBILITY. * Quickcheck setup for finite sets. * Discontinued simp_legacy_precond. Potential INCOMPATIBILITY. * Sledgehammer: - The MaSh relevance filter has been sped up. - Proof reconstruction has been improved, to minimize the incidence of cases where Sledgehammer gives a proof that does not work. - Auto Sledgehammer now minimizes and preplays the results. - Handle Vampire 4.0 proof output without raising exception. - Eliminated "MASH" environment variable. Use the "MaSh" option in Isabelle/jEdit instead. INCOMPATIBILITY. - Eliminated obsolete "blocking" option and related subcommands. * Nitpick: - Fixed soundness bug in translation of "finite" predicate. - Fixed soundness bug in "destroy_constrs" optimization. - Fixed soundness bug in translation of "rat" type. - Removed "check_potential" and "check_genuine" options. - Eliminated obsolete "blocking" option. * (Co)datatype package: - New commands "lift_bnf" and "copy_bnf" for lifting (copying) a BNF structure on the raw type to an abstract type defined using typedef. - Always generate "case_transfer" theorem. - For mutual types, generate slightly stronger "rel_induct", "rel_coinduct", and "coinduct" theorems. INCOMPATIBILITY. - Allow discriminators and selectors with the same name as the type being defined. - Avoid various internal name clashes (e.g., 'datatype f = f'). * Transfer: new methods for interactive debugging of 'transfer' and 'transfer_prover': 'transfer_start', 'transfer_step', 'transfer_end', 'transfer_prover_start' and 'transfer_prover_end'. * New diagnostic command print_record for displaying record definitions. * Division on integers is bootstrapped directly from division on naturals and uses generic numeral algorithm for computations. Slight INCOMPATIBILITY, simproc numeral_divmod replaces and generalizes former simprocs binary_int_div and binary_int_mod * Tightened specification of class semiring_no_zero_divisors. Minor INCOMPATIBILITY. * Class algebraic_semidom introduces common algebraic notions of integral (semi)domains, particularly units. Although logically subsumed by fields, is is not a super class of these in order not to burden fields with notions that are trivial there. * Class normalization_semidom specifies canonical representants for equivalence classes of associated elements in an integral (semi)domain. This formalizes associated elements as well. * Abstract specification of gcd/lcm operations in classes semiring_gcd, semiring_Gcd, semiring_Lcd. Minor INCOMPATIBILITY: facts gcd_nat.commute and gcd_int.commute are subsumed by gcd.commute, as well as gcd_nat.assoc and gcd_int.assoc by gcd.assoc. * Former constants Fields.divide (_ / _) and Divides.div (_ div _) are logically unified to Rings.divide in syntactic type class Rings.divide, with infix syntax (_ div _). Infix syntax (_ / _) for field division is added later as abbreviation in class Fields.inverse. INCOMPATIBILITY, instantiations must refer to Rings.divide rather than the former separate constants, hence infix syntax (_ / _) is usually not available during instantiation. * New cancellation simprocs for boolean algebras to cancel complementary terms for sup and inf. For example, "sup x (sup y (- x))" simplifies to "top". INCOMPATIBILITY. * Class uniform_space introduces uniform spaces btw topological spaces and metric spaces. Minor INCOMPATIBILITY: open__def needs to be introduced in the form of an uniformity. Some constants are more general now, it may be necessary to add type class constraints. open_real_def \ open_dist open_complex_def \ open_dist * Library/Monad_Syntax: notation uses symbols \ and \. INCOMPATIBILITY. * Library/Multiset: - Renamed multiset inclusion operators: < ~> <# > ~> ># <= ~> <=# >= ~> >=# \ ~> \# \ ~> \# INCOMPATIBILITY. - Added multiset inclusion operator syntax: \# \# \# \# - "'a multiset" is no longer an instance of the "order", "ordered_ab_semigroup_add_imp_le", "ordered_cancel_comm_monoid_diff", "semilattice_inf", and "semilattice_sup" type classes. The theorems previously provided by these type classes (directly or indirectly) are now available through the "subset_mset" interpretation (e.g. add_mono ~> subset_mset.add_mono). INCOMPATIBILITY. - Renamed conversions: multiset_of ~> mset multiset_of_set ~> mset_set set_of ~> set_mset INCOMPATIBILITY - Renamed lemmas: mset_le_def ~> subseteq_mset_def mset_less_def ~> subset_mset_def less_eq_multiset.rep_eq ~> subseteq_mset_def INCOMPATIBILITY - Removed lemmas generated by lift_definition: less_eq_multiset.abs_eq, less_eq_multiset.rsp, less_eq_multiset.transfer, less_eq_multiset_def INCOMPATIBILITY * Library/Omega_Words_Fun: Infinite words modeled as functions nat \ 'a. * Library/Bourbaki_Witt_Fixpoint: Added formalisation of the Bourbaki-Witt fixpoint theorem for increasing functions in chain-complete partial orders. * Library/Old_Recdef: discontinued obsolete 'defer_recdef' command. Minor INCOMPATIBILITY, use 'function' instead. * Library/Periodic_Fun: a locale that provides convenient lemmas for periodic functions. * Library/Formal_Power_Series: proper definition of division (with remainder) for formal power series; instances for Euclidean Ring and GCD. * HOL-Imperative_HOL: obsolete theory Legacy_Mrec has been removed. * HOL-Statespace: command 'statespace' uses mandatory qualifier for import of parent, as for general 'locale' expressions. INCOMPATIBILITY, remove '!' and add '?' as required. * HOL-Decision_Procs: The "approximation" method works with "powr" (exponentiation on real numbers) again. * HOL-Multivariate_Analysis: theory Cauchy_Integral_Thm with Contour integrals (= complex path integrals), Cauchy's integral theorem, winding numbers and Cauchy's integral formula, Liouville theorem, Fundamental Theorem of Algebra. Ported from HOL Light. * HOL-Multivariate_Analysis: topological concepts such as connected components, homotopic paths and the inside or outside of a set. * HOL-Multivariate_Analysis: radius of convergence of power series and various summability tests; Harmonic numbers and the Euler–Mascheroni constant; the Generalised Binomial Theorem; the complex and real Gamma/log-Gamma/Digamma/ Polygamma functions and their most important properties. * HOL-Probability: The central limit theorem based on Levy's uniqueness and continuity theorems, weak convergence, and characterisitc functions. * HOL-Data_Structures: new and growing session of standard data structures. *** ML *** * The following combinators for low-level profiling of the ML runtime system are available: profile_time (*CPU time*) profile_time_thread (*CPU time on this thread*) profile_allocations (*overall heap allocations*) * Antiquotation @{undefined} or \<^undefined> inlines (raise Match). * Antiquotation @{method NAME} inlines the (checked) name of the given Isar proof method. * Pretty printing of Poly/ML compiler output in Isabelle has been improved: proper treatment of break offsets and blocks with consistent breaks. * The auxiliary module Pure/display.ML has been eliminated. Its elementary thm print operations are now in Pure/more_thm.ML and thus called Thm.pretty_thm, Thm.string_of_thm etc. INCOMPATIBILITY. * Simproc programming interfaces have been simplified: Simplifier.make_simproc and Simplifier.define_simproc supersede various forms of Simplifier.mk_simproc, Simplifier.simproc_global etc. Note that term patterns for the left-hand sides are specified with implicitly fixed variables, like top-level theorem statements. INCOMPATIBILITY. * Instantiation rules have been re-organized as follows: Thm.instantiate (*low-level instantiation with named arguments*) Thm.instantiate' (*version with positional arguments*) Drule.infer_instantiate (*instantiation with type inference*) Drule.infer_instantiate' (*version with positional arguments*) The LHS only requires variable specifications, instead of full terms. Old cterm_instantiate is superseded by infer_instantiate. INCOMPATIBILITY, need to re-adjust some ML names and types accordingly. * Old tactic shorthands atac, rtac, etac, dtac, ftac have been discontinued. INCOMPATIBILITY, use regular assume_tac, resolve_tac etc. instead (with proper context). * Thm.instantiate (and derivatives) no longer require the LHS of the instantiation to be certified: plain variables are given directly. * Subgoal.SUBPROOF and Subgoal.FOCUS combinators use anonymous quasi-bound variables (like the Simplifier), instead of accidentally named local fixes. This has the potential to improve stability of proof tools, but can also cause INCOMPATIBILITY for tools that don't observe the proof context discipline. * Isar proof methods are based on a slightly more general type context_tactic, which allows to change the proof context dynamically (e.g. to update cases) and indicate explicit Seq.Error results. Former METHOD_CASES is superseded by CONTEXT_METHOD; further combinators are provided in src/Pure/Isar/method.ML for convenience. INCOMPATIBILITY. *** System *** * Command-line tool "isabelle console" enables print mode "ASCII". * Command-line tool "isabelle update_then" expands old Isar command conflations: hence ~> then have thus ~> then show This syntax is more orthogonal and improves readability and maintainability of proofs. * Global session timeout is multiplied by timeout_scale factor. This allows to adjust large-scale tests (e.g. AFP) to overall hardware performance. * Property values in etc/symbols may contain spaces, if written with the replacement character "␣" (Unicode point 0x2324). For example: \ code: 0x0022c6 group: operator font: Deja␣Vu␣Sans␣Mono * Java runtime environment for x86_64-windows allows to use larger heap space. * Java runtime options are determined separately for 32bit vs. 64bit platforms as follows. - Isabelle desktop application: platform-specific files that are associated with the main app bundle - isabelle jedit: settings JEDIT_JAVA_SYSTEM_OPTIONS JEDIT_JAVA_OPTIONS32 vs. JEDIT_JAVA_OPTIONS64 - isabelle build: settings ISABELLE_BUILD_JAVA_OPTIONS32 vs. ISABELLE_BUILD_JAVA_OPTIONS64 * Bash shell function "jvmpath" has been renamed to "platform_path": it is relevant both for Poly/ML and JVM processes. * Poly/ML default platform architecture may be changed from 32bit to 64bit via system option ML_system_64. A system restart (and rebuild) is required after change. * Poly/ML 5.6 runs natively on x86-windows and x86_64-windows, which both allow larger heap space than former x86-cygwin. * Heap images are 10-15% smaller due to less wasteful persistent theory content (using ML type theory_id instead of theory); New in Isabelle2015 (May 2015) ------------------------------ *** General *** * Local theory specification commands may have a 'private' or 'qualified' modifier to restrict name space accesses to the local scope, as provided by some "context begin ... end" block. For example: context begin private definition ... private lemma ... qualified definition ... qualified lemma ... lemma ... theorem ... end * Command 'experiment' opens an anonymous locale context with private naming policy. * Command 'notepad' requires proper nesting of begin/end and its proof structure in the body: 'oops' is no longer supported here. Minor INCOMPATIBILITY, use 'sorry' instead. * Command 'named_theorems' declares a dynamic fact within the context, together with an attribute to maintain the content incrementally. This supersedes functor Named_Thms in Isabelle/ML, but with a subtle change of semantics due to external visual order vs. internal reverse order. * 'find_theorems': search patterns which are abstractions are schematically expanded before search. Search results match the naive expectation more closely, particularly wrt. abbreviations. INCOMPATIBILITY. * Commands 'method_setup' and 'attribute_setup' now work within a local theory context. * Outer syntax commands are managed authentically within the theory context, without implicit global state. Potential for accidental INCOMPATIBILITY, make sure that required theories are really imported. * Historical command-line terminator ";" is no longer accepted (and already used differently in Isar). Minor INCOMPATIBILITY, use "isabelle update_semicolons" to remove obsolete semicolons from old theory sources. * Structural composition of proof methods (meth1; meth2) in Isar corresponds to (tac1 THEN_ALL_NEW tac2) in ML. * The Eisbach proof method language allows to define new proof methods by combining existing ones with their usual syntax. The "match" proof method provides basic fact/term matching in addition to premise/conclusion matching through Subgoal.focus, and binds fact names from matches as well as term patterns within matches. The Isabelle documentation provides an entry "eisbach" for the Eisbach User Manual. Sources and various examples are in ~~/src/HOL/Eisbach/. *** Prover IDE -- Isabelle/Scala/jEdit *** * Improved folding mode "isabelle" based on Isar syntax. Alternatively, the "sidekick" mode may be used for document structure. * Extended bracket matching based on Isar language structure. System option jedit_structure_limit determines maximum number of lines to scan in the buffer. * Support for BibTeX files: context menu, context-sensitive token marker, SideKick parser. * Document antiquotation @{cite} provides formal markup, which is interpreted semi-formally based on .bib files that happen to be open in the editor (hyperlinks, completion etc.). * Less waste of vertical space via negative line spacing (see Global Options / Text Area). * Improved graphview panel with optional output of PNG or PDF, for display of 'thy_deps', 'class_deps' etc. * The commands 'thy_deps' and 'class_deps' allow optional bounds to restrict the visualized hierarchy. * Improved scheduling for asynchronous print commands (e.g. provers managed by the Sledgehammer panel) wrt. ongoing document processing. *** Document preparation *** * Document markup commands 'chapter', 'section', 'subsection', 'subsubsection', 'text', 'txt', 'text_raw' work uniformly in any context, even before the initial 'theory' command. Obsolete proof commands 'sect', 'subsect', 'subsubsect', 'txt_raw' have been discontinued, use 'section', 'subsection', 'subsubsection', 'text_raw' instead. The old 'header' command is still retained for some time, but should be replaced by 'chapter', 'section' etc. (using "isabelle update_header"). Minor INCOMPATIBILITY. * Official support for "tt" style variants, via \isatt{...} or \begin{isabellett}...\end{isabellett}. The somewhat fragile \verb or verbatim environment of LaTeX is no longer used. This allows @{ML} etc. as argument to other macros (such as footnotes). * Document antiquotation @{verbatim} prints ASCII text literally in "tt" style. * Discontinued obsolete option "document_graph": session_graph.pdf is produced unconditionally for HTML browser_info and PDF-LaTeX document. * Diagnostic commands and document markup commands within a proof do not affect the command tag for output. Thus commands like 'thm' are subject to proof document structure, and no longer "stick out" accidentally. Commands 'text' and 'txt' merely differ in the LaTeX style, not their tags. Potential INCOMPATIBILITY in exotic situations. * System option "pretty_margin" is superseded by "thy_output_margin", which is also accessible via document antiquotation option "margin". Only the margin for document output may be changed, but not the global pretty printing: that is 76 for plain console output, and adapted dynamically in GUI front-ends. Implementations of document antiquotations need to observe the margin explicitly according to Thy_Output.string_of_margin. Minor INCOMPATIBILITY. * Specification of 'document_files' in the session ROOT file is mandatory for document preparation. The legacy mode with implicit copying of the document/ directory is no longer supported. Minor INCOMPATIBILITY. *** Pure *** * Proof methods with explicit instantiation ("rule_tac", "subgoal_tac" etc.) allow an optional context of local variables ('for' declaration): these variables become schematic in the instantiated theorem; this behaviour is analogous to 'for' in attributes "where" and "of". Configuration option rule_insts_schematic (default false) controls use of schematic variables outside the context. Minor INCOMPATIBILITY, declare rule_insts_schematic = true temporarily and update to use local variable declarations or dummy patterns instead. * Explicit instantiation via attributes "where", "of", and proof methods "rule_tac" with derivatives like "subgoal_tac" etc. admit dummy patterns ("_") that stand for anonymous local variables. * Generated schematic variables in standard format of exported facts are incremented to avoid material in the proof context. Rare INCOMPATIBILITY, explicit instantiation sometimes needs to refer to different index. * Lexical separation of signed and unsigned numerals: categories "num" and "float" are unsigned. INCOMPATIBILITY: subtle change in precedence of numeral signs, particularly in expressions involving infix syntax like "(- 1) ^ n". * Old inner token category "xnum" has been discontinued. Potential INCOMPATIBILITY for exotic syntax: may use mixfix grammar with "num" token category instead. *** HOL *** * New (co)datatype package: - The 'datatype_new' command has been renamed 'datatype'. The old command of that name is now called 'old_datatype' and is provided by "~~/src/HOL/Library/Old_Datatype.thy". See 'isabelle doc datatypes' for information on porting. INCOMPATIBILITY. - Renamed theorems: disc_corec ~> corec_disc disc_corec_iff ~> corec_disc_iff disc_exclude ~> distinct_disc disc_exhaust ~> exhaust_disc disc_map_iff ~> map_disc_iff sel_corec ~> corec_sel sel_exhaust ~> exhaust_sel sel_map ~> map_sel sel_set ~> set_sel sel_split ~> split_sel sel_split_asm ~> split_sel_asm strong_coinduct ~> coinduct_strong weak_case_cong ~> case_cong_weak INCOMPATIBILITY. - The "no_code" option to "free_constructors", "datatype_new", and "codatatype" has been renamed "plugins del: code". INCOMPATIBILITY. - The rules "set_empty" have been removed. They are easy consequences of other set rules "by auto". INCOMPATIBILITY. - The rule "set_cases" is now registered with the "[cases set]" attribute. This can influence the behavior of the "cases" proof method when more than one case rule is applicable (e.g., an assumption is of the form "w : set ws" and the method "cases w" is invoked). The solution is to specify the case rule explicitly (e.g. "cases w rule: widget.exhaust"). INCOMPATIBILITY. - Renamed theories: BNF_Comp ~> BNF_Composition BNF_FP_Base ~> BNF_Fixpoint_Base BNF_GFP ~> BNF_Greatest_Fixpoint BNF_LFP ~> BNF_Least_Fixpoint BNF_Constructions_on_Wellorders ~> BNF_Wellorder_Constructions Cardinals/Constructions_on_Wellorders ~> Cardinals/Wellorder_Constructions INCOMPATIBILITY. - Lifting and Transfer setup for basic HOL types sum and prod (also option) is now performed by the BNF package. Theories Lifting_Sum, Lifting_Product and Lifting_Option from Main became obsolete and were removed. Changed definitions of the relators rel_prod and rel_sum (using inductive). INCOMPATIBILITY: use rel_prod.simps and rel_sum.simps instead of rel_prod_def and rel_sum_def. Minor INCOMPATIBILITY: (rarely used by name) transfer theorem names changed (e.g. map_prod_transfer ~> prod.map_transfer). - Parametricity theorems for map functions, relators, set functions, constructors, case combinators, discriminators, selectors and (co)recursors are automatically proved and registered as transfer rules. * Old datatype package: - The old 'datatype' command has been renamed 'old_datatype', and 'rep_datatype' has been renamed 'old_rep_datatype'. They are provided by "~~/src/HOL/Library/Old_Datatype.thy". See 'isabelle doc datatypes' for information on porting. INCOMPATIBILITY. - Renamed theorems: weak_case_cong ~> case_cong_weak INCOMPATIBILITY. - Renamed theory: ~~/src/HOL/Datatype.thy ~> ~~/src/HOL/Library/Old_Datatype.thy INCOMPATIBILITY. * Nitpick: - Fixed soundness bug related to the strict and non-strict subset operations. * Sledgehammer: - CVC4 is now included with Isabelle instead of CVC3 and run by default. - Z3 is now always enabled by default, now that it is fully open source. The "z3_non_commercial" option is discontinued. - Minimization is now always enabled by default. Removed sub-command: min - Proof reconstruction, both one-liners and Isar, has been dramatically improved. - Improved support for CVC4 and veriT. * Old and new SMT modules: - The old 'smt' method has been renamed 'old_smt' and moved to 'src/HOL/Library/Old_SMT.thy'. It is provided for compatibility, until applications have been ported to use the new 'smt' method. For the method to work, an older version of Z3 (e.g. Z3 3.2 or 4.0) must be installed, and the environment variable "OLD_Z3_SOLVER" must point to it. INCOMPATIBILITY. - The 'smt2' method has been renamed 'smt'. INCOMPATIBILITY. - New option 'smt_reconstruction_step_timeout' to limit the reconstruction time of Z3 proof steps in the new 'smt' method. - New option 'smt_statistics' to display statistics of the new 'smt' method, especially runtime statistics of Z3 proof reconstruction. * Lifting: command 'lift_definition' allows to execute lifted constants that have as a return type a datatype containing a subtype. This overcomes long-time limitations in the area of code generation and lifting, and avoids tedious workarounds. * Command and antiquotation "value" provide different evaluation slots (again), where the previous strategy (NBE after ML) serves as default. Minor INCOMPATIBILITY. * Add NO_MATCH-simproc, allows to check for syntactic non-equality. * field_simps: Use NO_MATCH-simproc for distribution rules, to avoid non-termination in case of distributing a division. With this change field_simps is in some cases slightly less powerful, if it fails try to add algebra_simps, or use divide_simps. Minor INCOMPATIBILITY. * Separate class no_zero_divisors has been given up in favour of fully algebraic semiring_no_zero_divisors. INCOMPATIBILITY. * Class linordered_semidom really requires no zero divisors. INCOMPATIBILITY. * Classes division_ring, field and linordered_field always demand "inverse 0 = 0". Given up separate classes division_ring_inverse_zero, field_inverse_zero and linordered_field_inverse_zero. INCOMPATIBILITY. * Classes cancel_ab_semigroup_add / cancel_monoid_add specify explicit additive inverse operation. INCOMPATIBILITY. * Complex powers and square roots. The functions "ln" and "powr" are now overloaded for types real and complex, and 0 powr y = 0 by definition. INCOMPATIBILITY: type constraints may be necessary. * The functions "sin" and "cos" are now defined for any type of sort "{real_normed_algebra_1,banach}" type, so in particular on "real" and "complex" uniformly. Minor INCOMPATIBILITY: type constraints may be needed. * New library of properties of the complex transcendental functions sin, cos, tan, exp, Ln, Arctan, Arcsin, Arccos. Ported from HOL Light. * The factorial function, "fact", now has type "nat => 'a" (of a sort that admits numeric types including nat, int, real and complex. INCOMPATIBILITY: an expression such as "fact 3 = 6" may require a type constraint, and the combination "real (fact k)" is likely to be unsatisfactory. If a type conversion is still necessary, then use "of_nat (fact k)" or "real_of_nat (fact k)". * Removed functions "natfloor" and "natceiling", use "nat o floor" and "nat o ceiling" instead. A few of the lemmas have been retained and adapted: in their names "natfloor"/"natceiling" has been replaced by "nat_floor"/"nat_ceiling". * Qualified some duplicated fact names required for boostrapping the type class hierarchy: ab_add_uminus_conv_diff ~> diff_conv_add_uminus field_inverse_zero ~> inverse_zero field_divide_inverse ~> divide_inverse field_inverse ~> left_inverse Minor INCOMPATIBILITY. * Eliminated fact duplicates: mult_less_imp_less_right ~> mult_right_less_imp_less mult_less_imp_less_left ~> mult_left_less_imp_less Minor INCOMPATIBILITY. * Fact consolidation: even_less_0_iff is subsumed by double_add_less_zero_iff_single_add_less_zero (simp by default anyway). * Generalized and consolidated some theorems concerning divsibility: dvd_reduce ~> dvd_add_triv_right_iff dvd_plus_eq_right ~> dvd_add_right_iff dvd_plus_eq_left ~> dvd_add_left_iff Minor INCOMPATIBILITY. * "even" and "odd" are mere abbreviations for "2 dvd _" and "~ 2 dvd _" and part of theory Main. even_def ~> even_iff_mod_2_eq_zero INCOMPATIBILITY. * Lemma name consolidation: divide_Numeral1 ~> divide_numeral_1. Minor INCOMPATIBILITY. * Bootstrap of listsum as special case of abstract product over lists. Fact rename: listsum_def ~> listsum.eq_foldr INCOMPATIBILITY. * Product over lists via constant "listprod". * Theory List: renamed drop_Suc_conv_tl and nth_drop' to Cons_nth_drop_Suc. * New infrastructure for compiling, running, evaluating and testing generated code in target languages in HOL/Library/Code_Test. See HOL/Codegenerator_Test/Code_Test* for examples. * Library/Multiset: - Introduced "replicate_mset" operation. - Introduced alternative characterizations of the multiset ordering in "Library/Multiset_Order". - Renamed multiset ordering: <# ~> #<# <=# ~> #<=# \# ~> #\# \# ~> #\# INCOMPATIBILITY. - Introduced abbreviations for ill-named multiset operations: <#, \# abbreviate < (strict subset) <=#, \#, \# abbreviate <= (subset or equal) INCOMPATIBILITY. - Renamed in_multiset_of ~> in_multiset_in_set Multiset.fold ~> fold_mset Multiset.filter ~> filter_mset INCOMPATIBILITY. - Removed mcard, is equal to size. - Added attributes: image_mset.id [simp] image_mset_id [simp] elem_multiset_of_set [simp, intro] comp_fun_commute_plus_mset [simp] comp_fun_commute.fold_mset_insert [OF comp_fun_commute_plus_mset, simp] in_mset_fold_plus_iff [iff] set_of_Union_mset [simp] in_Union_mset_iff [iff] INCOMPATIBILITY. * Library/Sum_of_Squares: simplified and improved "sos" method. Always use local CSDP executable, which is much faster than the NEOS server. The "sos_cert" functionality is invoked as "sos" with additional argument. Minor INCOMPATIBILITY. * HOL-Decision_Procs: New counterexample generator quickcheck [approximation] for inequalities of transcendental functions. Uses hardware floating point arithmetic to randomly discover potential counterexamples. Counterexamples are certified with the "approximation" method. See HOL/Decision_Procs/ex/Approximation_Quickcheck_Ex.thy for examples. * HOL-Probability: Reworked measurability prover - applies destructor rules repeatedly - removed application splitting (replaced by destructor rule) - added congruence rules to rewrite measure spaces under the sets projection * New proof method "rewrite" (in theory ~~/src/HOL/Library/Rewrite) for single-step rewriting with subterm selection based on patterns. *** ML *** * Subtle change of name space policy: undeclared entries are now considered inaccessible, instead of accessible via the fully-qualified internal name. This mainly affects Name_Space.intern (and derivatives), which may produce an unexpected Long_Name.hidden prefix. Note that contemporary applications use the strict Name_Space.check (and derivatives) instead, which is not affected by the change. Potential INCOMPATIBILITY in rare applications of Name_Space.intern. * Subtle change of error semantics of Toplevel.proof_of: regular user ERROR instead of internal Toplevel.UNDEF. * Basic combinators map, fold, fold_map, split_list, apply are available as parameterized antiquotations, e.g. @{map 4} for lists of quadruples. * Renamed "pairself" to "apply2", in accordance to @{apply 2}. INCOMPATIBILITY. * Former combinators NAMED_CRITICAL and CRITICAL for central critical sections have been discontinued, in favour of the more elementary Multithreading.synchronized and its high-level derivative Synchronized.var (which is usually sufficient in applications). Subtle INCOMPATIBILITY: synchronized access needs to be atomic and cannot be nested. * Synchronized.value (ML) is actually synchronized (as in Scala): subtle change of semantics with minimal potential for INCOMPATIBILITY. * The main operations to certify logical entities are Thm.ctyp_of and Thm.cterm_of with a local context; old-style global theory variants are available as Thm.global_ctyp_of and Thm.global_cterm_of. INCOMPATIBILITY. * Elementary operations in module Thm are no longer pervasive. INCOMPATIBILITY, need to use qualified Thm.prop_of, Thm.cterm_of, Thm.term_of etc. * Proper context for various elementary tactics: assume_tac, resolve_tac, eresolve_tac, dresolve_tac, forward_tac, match_tac, compose_tac, Splitter.split_tac etc. INCOMPATIBILITY. * Tactical PARALLEL_ALLGOALS is the most common way to refer to PARALLEL_GOALS. * Goal.prove_multi is superseded by the fully general Goal.prove_common, which also allows to specify a fork priority. * Antiquotation @{command_spec "COMMAND"} is superseded by @{command_keyword COMMAND} (usually without quotes and with PIDE markup). Minor INCOMPATIBILITY. * Cartouches within ML sources are turned into values of type Input.source (with formal position information). *** System *** * The Isabelle tool "update_cartouches" changes theory files to use cartouches instead of old-style {* verbatim *} or `alt_string` tokens. * The Isabelle tool "build" provides new options -X, -k, -x. * Discontinued old-fashioned "codegen" tool. Code generation can always be externally triggered using an appropriate ROOT file plus a corresponding theory. Parametrization is possible using environment variables, or ML snippets in the most extreme cases. Minor INCOMPATIBILITY. * JVM system property "isabelle.threads" determines size of Scala thread pool, like Isabelle system option "threads" for ML. * JVM system property "isabelle.laf" determines the default Swing look-and-feel, via internal class name or symbolic name as in the jEdit menu Global Options / Appearance. * Support for Proof General and Isar TTY loop has been discontinued. Minor INCOMPATIBILITY, use standard PIDE infrastructure instead. New in Isabelle2014 (August 2014) --------------------------------- *** General *** * Support for official Standard ML within the Isabelle context. Command 'SML_file' reads and evaluates the given Standard ML file. Toplevel bindings are stored within the theory context; the initial environment is restricted to the Standard ML implementation of Poly/ML, without the add-ons of Isabelle/ML. Commands 'SML_import' and 'SML_export' allow to exchange toplevel bindings between the two separate environments. See also ~~/src/Tools/SML/Examples.thy for some examples. * Standard tactics and proof methods such as "clarsimp", "auto" and "safe" now preserve equality hypotheses "x = expr" where x is a free variable. Locale assumptions and chained facts containing "x" continue to be useful. The new method "hypsubst_thin" and the configuration option "hypsubst_thin" (within the attribute name space) restore the previous behavior. INCOMPATIBILITY, especially where induction is done after these methods or when the names of free and bound variables clash. As first approximation, old proofs may be repaired by "using [[hypsubst_thin = true]]" in the critical spot. * More static checking of proof methods, which allows the system to form a closure over the concrete syntax. Method arguments should be processed in the original proof context as far as possible, before operating on the goal state. In any case, the standard discipline for subgoal-addressing needs to be observed: no subgoals or a subgoal number that is out of range produces an empty result sequence, not an exception. Potential INCOMPATIBILITY for non-conformant tactical proof tools. * Lexical syntax (inner and outer) supports text cartouches with arbitrary nesting, and without escapes of quotes etc. The Prover IDE supports input via ` (backquote). * The outer syntax categories "text" (for formal comments and document markup commands) and "altstring" (for literal fact references) allow cartouches as well, in addition to the traditional mix of quotations. * Syntax of document antiquotation @{rail} now uses \ instead of "\\", to avoid the optical illusion of escaped backslash within string token. General renovation of its syntax using text cartouches. Minor INCOMPATIBILITY. * Discontinued legacy_isub_isup, which was a temporary workaround for Isabelle/ML in Isabelle2013-1. The prover process no longer accepts old identifier syntax with \<^isub> or \<^isup>. Potential INCOMPATIBILITY. * Document antiquotation @{url} produces markup for the given URL, which results in an active hyperlink within the text. * Document antiquotation @{file_unchecked} is like @{file}, but does not check existence within the file-system. * Updated and extended manuals: codegen, datatypes, implementation, isar-ref, jedit, system. *** Prover IDE -- Isabelle/Scala/jEdit *** * Improved Document panel: simplified interaction where every single mouse click (re)opens document via desktop environment or as jEdit buffer. * Support for Navigator plugin (with toolbar buttons), with connection to PIDE hyperlinks. * Auxiliary files ('ML_file' etc.) are managed by the Prover IDE. Open text buffers take precedence over copies within the file-system. * Improved support for Isabelle/ML, with jEdit mode "isabelle-ml" for auxiliary ML files. * Improved syntactic and semantic completion mechanism, with simple templates, completion language context, name-space completion, file-name completion, spell-checker completion. * Refined GUI popup for completion: more robust key/mouse event handling and propagation to enclosing text area -- avoid loosing keystrokes with slow / remote graphics displays. * Completion popup supports both ENTER and TAB (default) to select an item, depending on Isabelle options. * Refined insertion of completion items wrt. jEdit text: multiple selections, rectangular selections, rectangular selection as "tall caret". * Integrated spell-checker for document text, comments etc. with completion popup and context-menu. * More general "Query" panel supersedes "Find" panel, with GUI access to commands 'find_theorems' and 'find_consts', as well as print operations for the context. Minor incompatibility in keyboard shortcuts etc.: replace action isabelle-find by isabelle-query. * Search field for all output panels ("Output", "Query", "Info" etc.) to highlight text via regular expression. * Option "jedit_print_mode" (see also "Plugin Options / Isabelle / General") allows to specify additional print modes for the prover process, without requiring old-fashioned command-line invocation of "isabelle jedit -m MODE". * More support for remote files (e.g. http) using standard Java networking operations instead of jEdit virtual file-systems. * Empty editors buffers that are no longer required (e.g.\ via theory imports) are automatically removed from the document model. * Improved monitor panel. * Improved Console/Scala plugin: more uniform scala.Console output, more robust treatment of threads and interrupts. * Improved management of dockable windows: clarified keyboard focus and window placement wrt. main editor view; optional menu item to "Detach" a copy where this makes sense. * New Simplifier Trace panel provides an interactive view of the simplification process, enabled by the "simp_trace_new" attribute within the context. *** Pure *** * Low-level type-class commands 'classes', 'classrel', 'arities' have been discontinued to avoid the danger of non-trivial axiomatization that is not immediately visible. INCOMPATIBILITY, use regular 'instance' command with proof. The required OFCLASS(...) theorem might be postulated via 'axiomatization' beforehand, or the proof finished trivially if the underlying class definition is made vacuous (without any assumptions). See also Isabelle/ML operations Axclass.class_axiomatization, Axclass.classrel_axiomatization, Axclass.arity_axiomatization. * Basic constants of Pure use more conventional names and are always qualified. Rare INCOMPATIBILITY, but with potentially serious consequences, notably for tools in Isabelle/ML. The following renaming needs to be applied: == ~> Pure.eq ==> ~> Pure.imp all ~> Pure.all TYPE ~> Pure.type dummy_pattern ~> Pure.dummy_pattern Systematic porting works by using the following theory setup on a *previous* Isabelle version to introduce the new name accesses for the old constants: setup {* fn thy => thy |> Sign.root_path |> Sign.const_alias (Binding.qualify true "Pure" @{binding eq}) "==" |> Sign.const_alias (Binding.qualify true "Pure" @{binding imp}) "==>" |> Sign.const_alias (Binding.qualify true "Pure" @{binding all}) "all" |> Sign.restore_naming thy *} Thus ML antiquotations like @{const_name Pure.eq} may be used already. Later the application is moved to the current Isabelle version, and the auxiliary aliases are deleted. * Attributes "where" and "of" allow an optional context of local variables ('for' declaration): these variables become schematic in the instantiated theorem. * Obsolete attribute "standard" has been discontinued (legacy since Isabelle2012). Potential INCOMPATIBILITY, use explicit 'for' context where instantiations with schematic variables are intended (for declaration commands like 'lemmas' or attributes like "of"). The following temporary definition may help to port old applications: attribute_setup standard = "Scan.succeed (Thm.rule_attribute (K Drule.export_without_context))" * More thorough check of proof context for goal statements and attributed fact expressions (concerning background theory, declared hyps). Potential INCOMPATIBILITY, tools need to observe standard context discipline. See also Assumption.add_assumes and the more primitive Thm.assume_hyps. * Inner syntax token language allows regular quoted strings "..." (only makes sense in practice, if outer syntax is delimited differently, e.g. via cartouches). * Command 'print_term_bindings' supersedes 'print_binds' for clarity, but the latter is retained some time as Proof General legacy. * Code generator preprocessor: explicit control of simp tracing on a per-constant basis. See attribute "code_preproc". *** HOL *** * Code generator: enforce case of identifiers only for strict target language requirements. INCOMPATIBILITY. * Code generator: explicit proof contexts in many ML interfaces. INCOMPATIBILITY. * Code generator: minimize exported identifiers by default. Minor INCOMPATIBILITY. * Code generation for SML and OCaml: dropped arcane "no_signatures" option. Minor INCOMPATIBILITY. * "declare [[code abort: ...]]" replaces "code_abort ...". INCOMPATIBILITY. * "declare [[code drop: ...]]" drops all code equations associated with the given constants. * Code generations are provided for make, fields, extend and truncate operations on records. * Command and antiquotation "value" are now hardcoded against nbe and ML. Minor INCOMPATIBILITY. * Renamed command 'enriched_type' to 'functor'. INCOMPATIBILITY. * The symbol "\" may be used within char or string literals to represent (Char Nibble0 NibbleA), i.e. ASCII newline. * Qualified String.implode and String.explode. INCOMPATIBILITY. * Simplifier: Enhanced solver of preconditions of rewrite rules can now deal with conjunctions. For help with converting proofs, the old behaviour of the simplifier can be restored like this: declare/using [[simp_legacy_precond]]. This configuration option will disappear again in the future. INCOMPATIBILITY. * Simproc "finite_Collect" is no longer enabled by default, due to spurious crashes and other surprises. Potential INCOMPATIBILITY. * Moved new (co)datatype package and its dependencies from session "HOL-BNF" to "HOL". The commands 'bnf', 'wrap_free_constructors', 'datatype_new', 'codatatype', 'primcorec', 'primcorecursive' are now part of theory "Main". Theory renamings: FunDef.thy ~> Fun_Def.thy (and Fun_Def_Base.thy) Library/Wfrec.thy ~> Wfrec.thy Library/Zorn.thy ~> Zorn.thy Cardinals/Order_Relation.thy ~> Order_Relation.thy Library/Order_Union.thy ~> Cardinals/Order_Union.thy Cardinals/Cardinal_Arithmetic_Base.thy ~> BNF_Cardinal_Arithmetic.thy Cardinals/Cardinal_Order_Relation_Base.thy ~> BNF_Cardinal_Order_Relation.thy Cardinals/Constructions_on_Wellorders_Base.thy ~> BNF_Constructions_on_Wellorders.thy Cardinals/Wellorder_Embedding_Base.thy ~> BNF_Wellorder_Embedding.thy Cardinals/Wellorder_Relation_Base.thy ~> BNF_Wellorder_Relation.thy BNF/Ctr_Sugar.thy ~> Ctr_Sugar.thy BNF/Basic_BNFs.thy ~> Basic_BNFs.thy BNF/BNF_Comp.thy ~> BNF_Comp.thy BNF/BNF_Def.thy ~> BNF_Def.thy BNF/BNF_FP_Base.thy ~> BNF_FP_Base.thy BNF/BNF_GFP.thy ~> BNF_GFP.thy BNF/BNF_LFP.thy ~> BNF_LFP.thy BNF/BNF_Util.thy ~> BNF_Util.thy BNF/Coinduction.thy ~> Coinduction.thy BNF/More_BNFs.thy ~> Library/More_BNFs.thy BNF/Countable_Type.thy ~> Library/Countable_Set_Type.thy BNF/Examples/* ~> BNF_Examples/* New theories: Wellorder_Extension.thy (split from Zorn.thy) Library/Cardinal_Notations.thy Library/BNF_Axomatization.thy BNF_Examples/Misc_Primcorec.thy BNF_Examples/Stream_Processor.thy Discontinued theories: BNF/BNF.thy BNF/Equiv_Relations_More.thy INCOMPATIBILITY. * New (co)datatype package: - Command 'primcorec' is fully implemented. - Command 'datatype_new' generates size functions ("size_xxx" and "size") as required by 'fun'. - BNFs are integrated with the Lifting tool and new-style (co)datatypes with Transfer. - Renamed commands: datatype_new_compat ~> datatype_compat primrec_new ~> primrec wrap_free_constructors ~> free_constructors INCOMPATIBILITY. - The generated constants "xxx_case" and "xxx_rec" have been renamed "case_xxx" and "rec_xxx" (e.g., "prod_case" ~> "case_prod"). INCOMPATIBILITY. - The constant "xxx_(un)fold" and related theorems are no longer generated. Use "xxx_(co)rec" or define "xxx_(un)fold" manually using "prim(co)rec". INCOMPATIBILITY. - No discriminators are generated for nullary constructors by default, eliminating the need for the odd "=:" syntax. INCOMPATIBILITY. - No discriminators or selectors are generated by default by "datatype_new", unless custom names are specified or the new "discs_sels" option is passed. INCOMPATIBILITY. * Old datatype package: - The generated theorems "xxx.cases" and "xxx.recs" have been renamed "xxx.case" and "xxx.rec" (e.g., "sum.cases" -> "sum.case"). INCOMPATIBILITY. - The generated constants "xxx_case", "xxx_rec", and "xxx_size" have been renamed "case_xxx", "rec_xxx", and "size_xxx" (e.g., "prod_case" ~> "case_prod"). INCOMPATIBILITY. * The types "'a list" and "'a option", their set and map functions, their relators, and their selectors are now produced using the new BNF-based datatype package. Renamed constants: Option.set ~> set_option Option.map ~> map_option option_rel ~> rel_option Renamed theorems: set_def ~> set_rec[abs_def] map_def ~> map_rec[abs_def] Option.map_def ~> map_option_case[abs_def] (with "case_option" instead of "rec_option") option.recs ~> option.rec list_all2_def ~> list_all2_iff set.simps ~> set_simps (or the slightly different "list.set") map.simps ~> list.map hd.simps ~> list.sel(1) tl.simps ~> list.sel(2-3) the.simps ~> option.sel INCOMPATIBILITY. * The following map functions and relators have been renamed: sum_map ~> map_sum map_pair ~> map_prod prod_rel ~> rel_prod sum_rel ~> rel_sum fun_rel ~> rel_fun set_rel ~> rel_set filter_rel ~> rel_filter fset_rel ~> rel_fset (in "src/HOL/Library/FSet.thy") cset_rel ~> rel_cset (in "src/HOL/Library/Countable_Set_Type.thy") vset ~> rel_vset (in "src/HOL/Library/Quotient_Set.thy") INCOMPATIBILITY. * Lifting and Transfer: - a type variable as a raw type is supported - stronger reflexivity prover - rep_eq is always generated by lift_definition - setup for Lifting/Transfer is now automated for BNFs + holds for BNFs that do not contain a dead variable + relator_eq, relator_mono, relator_distr, relator_domain, relator_eq_onp, quot_map, transfer rules for bi_unique, bi_total, right_unique, right_total, left_unique, left_total are proved automatically + definition of a predicator is generated automatically + simplification rules for a predicator definition are proved automatically for datatypes - consolidation of the setup of Lifting/Transfer + property that a relator preservers reflexivity is not needed any more Minor INCOMPATIBILITY. + left_total and left_unique rules are now transfer rules (reflexivity_rule attribute not needed anymore) INCOMPATIBILITY. + Domainp does not have to be a separate assumption in relator_domain theorems (=> more natural statement) INCOMPATIBILITY. - registration of code equations is more robust Potential INCOMPATIBILITY. - respectfulness proof obligation is preprocessed to a more readable form Potential INCOMPATIBILITY. - eq_onp is always unfolded in respectfulness proof obligation Potential INCOMPATIBILITY. - unregister lifting setup for Code_Numeral.integer and Code_Numeral.natural Potential INCOMPATIBILITY. - Lifting.invariant -> eq_onp INCOMPATIBILITY. * New internal SAT solver "cdclite" that produces models and proof traces. This solver replaces the internal SAT solvers "enumerate" and "dpll". Applications that explicitly used one of these two SAT solvers should use "cdclite" instead. In addition, "cdclite" is now the default SAT solver for the "sat" and "satx" proof methods and corresponding tactics; the old default can be restored using "declare [[sat_solver = zchaff_with_proofs]]". Minor INCOMPATIBILITY. * SMT module: A new version of the SMT module, temporarily called "SMT2", uses SMT-LIB 2 and supports recent versions of Z3 (e.g., 4.3). The new proof method is called "smt2". CVC3 and CVC4 are also supported as oracles. Yices is no longer supported, because no version of the solver can handle both SMT-LIB 2 and quantifiers. * Activation of Z3 now works via "z3_non_commercial" system option (without requiring restart), instead of former settings variable "Z3_NON_COMMERCIAL". The option can be edited in Isabelle/jEdit menu Plugin Options / Isabelle / General. * Sledgehammer: - Z3 can now produce Isar proofs. - MaSh overhaul: . New SML-based learning algorithms eliminate the dependency on Python and increase performance and reliability. . MaSh and MeSh are now used by default together with the traditional MePo (Meng-Paulson) relevance filter. To disable MaSh, set the "MaSh" system option in Isabelle/jEdit Plugin Options / Isabelle / General to "none". - New option: smt_proofs - Renamed options: isar_compress ~> compress isar_try0 ~> try0 INCOMPATIBILITY. * Removed solvers remote_cvc3 and remote_z3. Use cvc3 and z3 instead. * Nitpick: - Fixed soundness bug whereby mutually recursive datatypes could take infinite values. - Fixed soundness bug with low-level number functions such as "Abs_Integ" and "Rep_Integ". - Removed "std" option. - Renamed "show_datatypes" to "show_types" and "hide_datatypes" to "hide_types". * Metis: Removed legacy proof method 'metisFT'. Use 'metis (full_types)' instead. INCOMPATIBILITY. * Try0: Added 'algebra' and 'meson' to the set of proof methods. * Adjustion of INF and SUP operations: - Elongated constants INFI and SUPR to INFIMUM and SUPREMUM. - Consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly. - More aggressive normalization of expressions involving INF and Inf or SUP and Sup. - INF_image and SUP_image do not unfold composition. - Dropped facts INF_comp, SUP_comp. - Default congruence rules strong_INF_cong and strong_SUP_cong, with simplifier implication in premises. Generalize and replace former INT_cong, SUP_cong INCOMPATIBILITY. * SUP and INF generalized to conditionally_complete_lattice. * Swapped orientation of facts image_comp and vimage_comp: image_compose ~> image_comp [symmetric] image_comp ~> image_comp [symmetric] vimage_compose ~> vimage_comp [symmetric] vimage_comp ~> vimage_comp [symmetric] INCOMPATIBILITY. * Theory reorganization: split of Big_Operators.thy into Groups_Big.thy and Lattices_Big.thy. * Consolidated some facts about big group operators: setsum_0' ~> setsum.neutral setsum_0 ~> setsum.neutral_const setsum_addf ~> setsum.distrib setsum_cartesian_product ~> setsum.cartesian_product setsum_cases ~> setsum.If_cases setsum_commute ~> setsum.commute setsum_cong ~> setsum.cong setsum_delta ~> setsum.delta setsum_delta' ~> setsum.delta' setsum_diff1' ~> setsum.remove setsum_empty ~> setsum.empty setsum_infinite ~> setsum.infinite setsum_insert ~> setsum.insert setsum_inter_restrict'' ~> setsum.inter_filter setsum_mono_zero_cong_left ~> setsum.mono_neutral_cong_left setsum_mono_zero_cong_right ~> setsum.mono_neutral_cong_right setsum_mono_zero_left ~> setsum.mono_neutral_left setsum_mono_zero_right ~> setsum.mono_neutral_right setsum_reindex ~> setsum.reindex setsum_reindex_cong ~> setsum.reindex_cong setsum_reindex_nonzero ~> setsum.reindex_nontrivial setsum_restrict_set ~> setsum.inter_restrict setsum_Plus ~> setsum.Plus setsum_setsum_restrict ~> setsum.commute_restrict setsum_Sigma ~> setsum.Sigma setsum_subset_diff ~> setsum.subset_diff setsum_Un_disjoint ~> setsum.union_disjoint setsum_UN_disjoint ~> setsum.UNION_disjoint setsum_Un_Int ~> setsum.union_inter setsum_Union_disjoint ~> setsum.Union_disjoint setsum_UNION_zero ~> setsum.Union_comp setsum_Un_zero ~> setsum.union_inter_neutral strong_setprod_cong ~> setprod.strong_cong strong_setsum_cong ~> setsum.strong_cong setprod_1' ~> setprod.neutral setprod_1 ~> setprod.neutral_const setprod_cartesian_product ~> setprod.cartesian_product setprod_cong ~> setprod.cong setprod_delta ~> setprod.delta setprod_delta' ~> setprod.delta' setprod_empty ~> setprod.empty setprod_infinite ~> setprod.infinite setprod_insert ~> setprod.insert setprod_mono_one_cong_left ~> setprod.mono_neutral_cong_left setprod_mono_one_cong_right ~> setprod.mono_neutral_cong_right setprod_mono_one_left ~> setprod.mono_neutral_left setprod_mono_one_right ~> setprod.mono_neutral_right setprod_reindex ~> setprod.reindex setprod_reindex_cong ~> setprod.reindex_cong setprod_reindex_nonzero ~> setprod.reindex_nontrivial setprod_Sigma ~> setprod.Sigma setprod_subset_diff ~> setprod.subset_diff setprod_timesf ~> setprod.distrib setprod_Un2 ~> setprod.union_diff2 setprod_Un_disjoint ~> setprod.union_disjoint setprod_UN_disjoint ~> setprod.UNION_disjoint setprod_Un_Int ~> setprod.union_inter setprod_Union_disjoint ~> setprod.Union_disjoint setprod_Un_one ~> setprod.union_inter_neutral Dropped setsum_cong2 (simple variant of setsum.cong). Dropped setsum_inter_restrict' (simple variant of setsum.inter_restrict) Dropped setsum_reindex_id, setprod_reindex_id (simple variants of setsum.reindex [symmetric], setprod.reindex [symmetric]). INCOMPATIBILITY. * Abolished slightly odd global lattice interpretation for min/max. Fact consolidations: min_max.inf_assoc ~> min.assoc min_max.inf_commute ~> min.commute min_max.inf_left_commute ~> min.left_commute min_max.inf_idem ~> min.idem min_max.inf_left_idem ~> min.left_idem min_max.inf_right_idem ~> min.right_idem min_max.sup_assoc ~> max.assoc min_max.sup_commute ~> max.commute min_max.sup_left_commute ~> max.left_commute min_max.sup_idem ~> max.idem min_max.sup_left_idem ~> max.left_idem min_max.sup_inf_distrib1 ~> max_min_distrib2 min_max.sup_inf_distrib2 ~> max_min_distrib1 min_max.inf_sup_distrib1 ~> min_max_distrib2 min_max.inf_sup_distrib2 ~> min_max_distrib1 min_max.distrib ~> min_max_distribs min_max.inf_absorb1 ~> min.absorb1 min_max.inf_absorb2 ~> min.absorb2 min_max.sup_absorb1 ~> max.absorb1 min_max.sup_absorb2 ~> max.absorb2 min_max.le_iff_inf ~> min.absorb_iff1 min_max.le_iff_sup ~> max.absorb_iff2 min_max.inf_le1 ~> min.cobounded1 min_max.inf_le2 ~> min.cobounded2 le_maxI1, min_max.sup_ge1 ~> max.cobounded1 le_maxI2, min_max.sup_ge2 ~> max.cobounded2 min_max.le_infI1 ~> min.coboundedI1 min_max.le_infI2 ~> min.coboundedI2 min_max.le_supI1 ~> max.coboundedI1 min_max.le_supI2 ~> max.coboundedI2 min_max.less_infI1 ~> min.strict_coboundedI1 min_max.less_infI2 ~> min.strict_coboundedI2 min_max.less_supI1 ~> max.strict_coboundedI1 min_max.less_supI2 ~> max.strict_coboundedI2 min_max.inf_mono ~> min.mono min_max.sup_mono ~> max.mono min_max.le_infI, min_max.inf_greatest ~> min.boundedI min_max.le_supI, min_max.sup_least ~> max.boundedI min_max.le_inf_iff ~> min.bounded_iff min_max.le_sup_iff ~> max.bounded_iff For min_max.inf_sup_aci, prefer (one of) min.commute, min.assoc, min.left_commute, min.left_idem, max.commute, max.assoc, max.left_commute, max.left_idem directly. For min_max.inf_sup_ord, prefer (one of) min.cobounded1, min.cobounded2, max.cobounded1m max.cobounded2 directly. For min_ac or max_ac, prefer more general collection ac_simps. INCOMPATIBILITY. * Theorem disambiguation Inf_le_Sup (on finite sets) ~> Inf_fin_le_Sup_fin. INCOMPATIBILITY. * Qualified constant names Wellfounded.acc, Wellfounded.accp. INCOMPATIBILITY. * Fact generalization and consolidation: neq_one_mod_two, mod_2_not_eq_zero_eq_one_int ~> not_mod_2_eq_0_eq_1 INCOMPATIBILITY. * Purely algebraic definition of even. Fact generalization and consolidation: nat_even_iff_2_dvd, int_even_iff_2_dvd ~> even_iff_2_dvd even_zero_(nat|int) ~> even_zero INCOMPATIBILITY. * Abolished neg_numeral. - Canonical representation for minus one is "- 1". - Canonical representation for other negative numbers is "- (numeral _)". - When devising rule sets for number calculation, consider the following canonical cases: 0, 1, numeral _, - 1, - numeral _. - HOLogic.dest_number also recognizes numerals in non-canonical forms like "numeral One", "- numeral One", "- 0" and even "- ... - _". - Syntax for negative numerals is mere input syntax. INCOMPATIBILITY. * Reduced name variants for rules on associativity and commutativity: add_assoc ~> add.assoc add_commute ~> add.commute add_left_commute ~> add.left_commute mult_assoc ~> mult.assoc mult_commute ~> mult.commute mult_left_commute ~> mult.left_commute nat_add_assoc ~> add.assoc nat_add_commute ~> add.commute nat_add_left_commute ~> add.left_commute nat_mult_assoc ~> mult.assoc nat_mult_commute ~> mult.commute eq_assoc ~> iff_assoc eq_left_commute ~> iff_left_commute INCOMPATIBILITY. * Fact collections add_ac and mult_ac are considered old-fashioned. Prefer ac_simps instead, or specify rules (add|mult).(assoc|commute|left_commute) individually. * Elimination of fact duplicates: equals_zero_I ~> minus_unique diff_eq_0_iff_eq ~> right_minus_eq nat_infinite ~> infinite_UNIV_nat int_infinite ~> infinite_UNIV_int INCOMPATIBILITY. * Fact name consolidation: diff_def, diff_minus, ab_diff_minus ~> diff_conv_add_uminus minus_le_self_iff ~> neg_less_eq_nonneg le_minus_self_iff ~> less_eq_neg_nonpos neg_less_nonneg ~> neg_less_pos less_minus_self_iff ~> less_neg_neg [simp] INCOMPATIBILITY. * More simplification rules on unary and binary minus: add_diff_cancel, add_diff_cancel_left, add_le_same_cancel1, add_le_same_cancel2, add_less_same_cancel1, add_less_same_cancel2, add_minus_cancel, diff_add_cancel, le_add_same_cancel1, le_add_same_cancel2, less_add_same_cancel1, less_add_same_cancel2, minus_add_cancel, uminus_add_conv_diff. These correspondingly have been taken away from fact collections algebra_simps and field_simps. INCOMPATIBILITY. To restore proofs, the following patterns are helpful: a) Arbitrary failing proof not involving "diff_def": Consider simplification with algebra_simps or field_simps. b) Lifting rules from addition to subtraction: Try with "using of [... "- _" ...]" by simp". c) Simplification with "diff_def": just drop "diff_def". Consider simplification with algebra_simps or field_simps; or the brute way with "simp add: diff_conv_add_uminus del: add_uminus_conv_diff". * Introduce bdd_above and bdd_below in theory Conditionally_Complete_Lattices, use them instead of explicitly stating boundedness of sets. * ccpo.admissible quantifies only over non-empty chains to allow more syntax-directed proof rules; the case of the empty chain shows up as additional case in fixpoint induction proofs. INCOMPATIBILITY. * Removed and renamed theorems in Series: summable_le ~> suminf_le suminf_le ~> suminf_le_const series_pos_le ~> setsum_le_suminf series_pos_less ~> setsum_less_suminf suminf_ge_zero ~> suminf_nonneg suminf_gt_zero ~> suminf_pos suminf_gt_zero_iff ~> suminf_pos_iff summable_sumr_LIMSEQ_suminf ~> summable_LIMSEQ suminf_0_le ~> suminf_nonneg [rotate] pos_summable ~> summableI_nonneg_bounded ratio_test ~> summable_ratio_test removed series_zero, replaced by sums_finite removed auxiliary lemmas: sumr_offset, sumr_offset2, sumr_offset3, sumr_offset4, sumr_group, half, le_Suc_ex_iff, lemma_realpow_diff_sumr, real_setsum_nat_ivl_bounded, summable_le2, ratio_test_lemma2, sumr_minus_one_realpow_zerom, sumr_one_lb_realpow_zero, summable_convergent_sumr_iff, sumr_diff_mult_const INCOMPATIBILITY. * Replace (F)DERIV syntax by has_derivative: - "(f has_derivative f') (at x within s)" replaces "FDERIV f x : s : f'" - "(f has_field_derivative f') (at x within s)" replaces "DERIV f x : s : f'" - "f differentiable at x within s" replaces "_ differentiable _ in _" syntax - removed constant isDiff - "DERIV f x : f'" and "FDERIV f x : f'" syntax is only available as input syntax. - "DERIV f x : s : f'" and "FDERIV f x : s : f'" syntax removed. - Renamed FDERIV_... lemmas to has_derivative_... - renamed deriv (the syntax constant used for "DERIV _ _ :> _") to DERIV - removed DERIV_intros, has_derivative_eq_intros - introduced derivative_intros and deriative_eq_intros which includes now rules for DERIV, has_derivative and has_vector_derivative. - Other renamings: differentiable_def ~> real_differentiable_def differentiableE ~> real_differentiableE fderiv_def ~> has_derivative_at field_fderiv_def ~> field_has_derivative_at isDiff_der ~> differentiable_def deriv_fderiv ~> has_field_derivative_def deriv_def ~> DERIV_def INCOMPATIBILITY. * Include more theorems in continuous_intros. Remove the continuous_on_intros, isCont_intros collections, these facts are now in continuous_intros. * Theorems about complex numbers are now stated only using Re and Im, the Complex constructor is not used anymore. It is possible to use primcorec to defined the behaviour of a complex-valued function. Removed theorems about the Complex constructor from the simpset, they are available as the lemma collection legacy_Complex_simps. This especially removes i_complex_of_real: "ii * complex_of_real r = Complex 0 r". Instead the reverse direction is supported with Complex_eq: "Complex a b = a + \ * b" Moved csqrt from Fundamental_Algebra_Theorem to Complex. Renamings: Re/Im ~> complex.sel complex_Re/Im_zero ~> zero_complex.sel complex_Re/Im_add ~> plus_complex.sel complex_Re/Im_minus ~> uminus_complex.sel complex_Re/Im_diff ~> minus_complex.sel complex_Re/Im_one ~> one_complex.sel complex_Re/Im_mult ~> times_complex.sel complex_Re/Im_inverse ~> inverse_complex.sel complex_Re/Im_scaleR ~> scaleR_complex.sel complex_Re/Im_i ~> ii.sel complex_Re/Im_cnj ~> cnj.sel Re/Im_cis ~> cis.sel complex_divide_def ~> divide_complex_def complex_norm_def ~> norm_complex_def cmod_def ~> norm_complex_de Removed theorems: complex_zero_def complex_add_def complex_minus_def complex_diff_def complex_one_def complex_mult_def complex_inverse_def complex_scaleR_def INCOMPATIBILITY. * Theory Lubs moved HOL image to HOL-Library. It is replaced by Conditionally_Complete_Lattices. INCOMPATIBILITY. * HOL-Library: new theory src/HOL/Library/Tree.thy. * HOL-Library: removed theory src/HOL/Library/Kleene_Algebra.thy; it is subsumed by session Kleene_Algebra in AFP. * HOL-Library / theory RBT: various constants and facts are hidden; lifting setup is unregistered. INCOMPATIBILITY. * HOL-Cardinals: new theory src/HOL/Cardinals/Ordinal_Arithmetic.thy. * HOL-Word: bit representations prefer type bool over type bit. INCOMPATIBILITY. * HOL-Word: - Abandoned fact collection "word_arith_alts", which is a duplicate of "word_arith_wis". - Dropped first (duplicated) element in fact collections "sint_word_ariths", "word_arith_alts", "uint_word_ariths", "uint_word_arith_bintrs". * HOL-Number_Theory: - consolidated the proofs of the binomial theorem - the function fib is again of type nat => nat and not overloaded - no more references to Old_Number_Theory in the HOL libraries (except the AFP) INCOMPATIBILITY. * HOL-Multivariate_Analysis: - Type class ordered_real_vector for ordered vector spaces. - New theory Complex_Basic_Analysis defining complex derivatives, holomorphic functions, etc., ported from HOL Light's canal.ml. - Changed order of ordered_euclidean_space to be compatible with pointwise ordering on products. Therefore instance of conditionally_complete_lattice and ordered_real_vector. INCOMPATIBILITY: use box instead of greaterThanLessThan or explicit set-comprehensions with eucl_less for other (half-)open intervals. - removed dependencies on type class ordered_euclidean_space with introduction of "cbox" on euclidean_space - renamed theorems: interval ~> box mem_interval ~> mem_box interval_eq_empty ~> box_eq_empty interval_ne_empty ~> box_ne_empty interval_sing(1) ~> cbox_sing interval_sing(2) ~> box_sing subset_interval_imp ~> subset_box_imp subset_interval ~> subset_box open_interval ~> open_box closed_interval ~> closed_cbox interior_closed_interval ~> interior_cbox bounded_closed_interval ~> bounded_cbox compact_interval ~> compact_cbox bounded_subset_closed_interval_symmetric ~> bounded_subset_cbox_symmetric bounded_subset_closed_interval ~> bounded_subset_cbox mem_interval_componentwiseI ~> mem_box_componentwiseI convex_box ~> convex_prod rel_interior_real_interval ~> rel_interior_real_box convex_interval ~> convex_box convex_hull_eq_real_interval ~> convex_hull_eq_real_cbox frechet_derivative_within_closed_interval ~> frechet_derivative_within_cbox content_closed_interval' ~> content_cbox' elementary_subset_interval ~> elementary_subset_box diameter_closed_interval ~> diameter_cbox frontier_closed_interval ~> frontier_cbox frontier_open_interval ~> frontier_box bounded_subset_open_interval_symmetric ~> bounded_subset_box_symmetric closure_open_interval ~> closure_box open_closed_interval_convex ~> open_cbox_convex open_interval_midpoint ~> box_midpoint content_image_affinity_interval ~> content_image_affinity_cbox is_interval_interval ~> is_interval_cbox + is_interval_box + is_interval_closed_interval bounded_interval ~> bounded_closed_interval + bounded_boxes - respective theorems for intervals over the reals: content_closed_interval + content_cbox has_integral + has_integral_real fine_division_exists + fine_division_exists_real has_integral_null + has_integral_null_real tagged_division_union_interval + tagged_division_union_interval_real has_integral_const + has_integral_const_real integral_const + integral_const_real has_integral_bound + has_integral_bound_real integrable_continuous + integrable_continuous_real integrable_subinterval + integrable_subinterval_real has_integral_reflect_lemma + has_integral_reflect_lemma_real integrable_reflect + integrable_reflect_real integral_reflect + integral_reflect_real image_affinity_interval + image_affinity_cbox image_smult_interval + image_smult_cbox integrable_const + integrable_const_ivl integrable_on_subinterval + integrable_on_subcbox - renamed theorems: derivative_linear ~> has_derivative_bounded_linear derivative_is_linear ~> has_derivative_linear bounded_linear_imp_linear ~> bounded_linear.linear * HOL-Probability: - Renamed positive_integral to nn_integral: . Renamed all lemmas "*positive_integral*" to *nn_integral*" positive_integral_positive ~> nn_integral_nonneg . Renamed abbreviation integral\<^sup>P to integral\<^sup>N. - replaced the Lebesgue integral on real numbers by the more general Bochner integral for functions into a real-normed vector space. integral_zero ~> integral_zero / integrable_zero integral_minus ~> integral_minus / integrable_minus integral_add ~> integral_add / integrable_add integral_diff ~> integral_diff / integrable_diff integral_setsum ~> integral_setsum / integrable_setsum integral_multc ~> integral_mult_left / integrable_mult_left integral_cmult ~> integral_mult_right / integrable_mult_right integral_triangle_inequality~> integral_norm_bound integrable_nonneg ~> integrableI_nonneg integral_positive ~> integral_nonneg_AE integrable_abs_iff ~> integrable_abs_cancel positive_integral_lim_INF ~> nn_integral_liminf lebesgue_real_affine ~> lborel_real_affine borel_integral_has_integral ~> has_integral_lebesgue_integral integral_indicator ~> integral_real_indicator / integrable_real_indicator positive_integral_fst ~> nn_integral_fst' positive_integral_fst_measurable ~> nn_integral_fst positive_integral_snd_measurable ~> nn_integral_snd integrable_fst_measurable ~> integral_fst / integrable_fst / AE_integrable_fst integrable_snd_measurable ~> integral_snd / integrable_snd / AE_integrable_snd integral_monotone_convergence ~> integral_monotone_convergence / integrable_monotone_convergence integral_monotone_convergence_at_top ~> integral_monotone_convergence_at_top / integrable_monotone_convergence_at_top has_integral_iff_positive_integral_lebesgue ~> has_integral_iff_has_bochner_integral_lebesgue_nonneg lebesgue_integral_has_integral ~> has_integral_integrable_lebesgue_nonneg positive_integral_lebesgue_has_integral ~> integral_has_integral_lebesgue_nonneg / integrable_has_integral_lebesgue_nonneg lebesgue_integral_real_affine ~> nn_integral_real_affine has_integral_iff_positive_integral_lborel ~> integral_has_integral_nonneg / integrable_has_integral_nonneg The following theorems where removed: lebesgue_integral_nonneg lebesgue_integral_uminus lebesgue_integral_cmult lebesgue_integral_multc lebesgue_integral_cmult_nonneg integral_cmul_indicator integral_real - Formalized properties about exponentially, Erlang, and normal distributed random variables. * HOL-Decision_Procs: Separate command 'approximate' for approximative computation in src/HOL/Decision_Procs/Approximation. Minor INCOMPATIBILITY. *** Scala *** * The signature and semantics of Document.Snapshot.cumulate_markup / select_markup have been clarified. Markup is now traversed in the order of reports given by the prover: later markup is usually more specific and may override results accumulated so far. The elements guard is mandatory and checked precisely. Subtle INCOMPATIBILITY. * Substantial reworking of internal PIDE protocol communication channels. INCOMPATIBILITY. *** ML *** * Subtle change of semantics of Thm.eq_thm: theory stamps are not compared (according to Thm.thm_ord), but assumed to be covered by the current background theory. Thus equivalent data produced in different branches of the theory graph usually coincides (e.g. relevant for theory merge). Note that the softer Thm.eq_thm_prop is often more appropriate than Thm.eq_thm. * Proper context for basic Simplifier operations: rewrite_rule, rewrite_goals_rule, rewrite_goals_tac etc. INCOMPATIBILITY, need to pass runtime Proof.context (and ensure that the simplified entity actually belongs to it). * Proper context discipline for read_instantiate and instantiate_tac: variables that are meant to become schematic need to be given as fixed, and are generalized by the explicit context of local variables. This corresponds to Isar attributes "where" and "of" with 'for' declaration. INCOMPATIBILITY, also due to potential change of indices of schematic variables. * Moved ML_Compiler.exn_trace and other operations on exceptions to structure Runtime. Minor INCOMPATIBILITY. * Discontinued old Toplevel.debug in favour of system option "ML_exception_trace", which may be also declared within the context via "declare [[ML_exception_trace = true]]". Minor INCOMPATIBILITY. * Renamed configuration option "ML_trace" to "ML_source_trace". Minor INCOMPATIBILITY. * Configuration option "ML_print_depth" controls the pretty-printing depth of the ML compiler within the context. The old print_depth in ML is still available as default_print_depth, but rarely used. Minor INCOMPATIBILITY. * Toplevel function "use" refers to raw ML bootstrap environment, without Isar context nor antiquotations. Potential INCOMPATIBILITY. Note that 'ML_file' is the canonical command to load ML files into the formal context. * Simplified programming interface to define ML antiquotations, see structure ML_Antiquotation. Minor INCOMPATIBILITY. * ML antiquotation @{here} refers to its source position, which is occasionally useful for experimentation and diagnostic purposes. * ML antiquotation @{path} produces a Path.T value, similarly to Path.explode, but with compile-time check against the file-system and some PIDE markup. Note that unlike theory source, ML does not have a well-defined master directory, so an absolute symbolic path specification is usually required, e.g. "~~/src/HOL". * ML antiquotation @{print} inlines a function to print an arbitrary ML value, which is occasionally useful for diagnostic or demonstration purposes. *** System *** * Proof General with its traditional helper scripts is now an optional Isabelle component, e.g. see ProofGeneral-4.2-2 from the Isabelle component repository http://isabelle.in.tum.de/components/. Note that the "system" manual provides general explanations about add-on components, especially those that are not bundled with the release. * The raw Isabelle process executable has been renamed from "isabelle-process" to "isabelle_process", which conforms to common shell naming conventions, and allows to define a shell function within the Isabelle environment to avoid dynamic path lookup. Rare incompatibility for old tools that do not use the ISABELLE_PROCESS settings variable. * Former "isabelle tty" has been superseded by "isabelle console", with implicit build like "isabelle jedit", and without the mostly obsolete Isar TTY loop. * Simplified "isabelle display" tool. Settings variables DVI_VIEWER and PDF_VIEWER now refer to the actual programs, not shell command-lines. Discontinued option -c: invocation may be asynchronous via desktop environment, without any special precautions. Potential INCOMPATIBILITY with ambitious private settings. * Removed obsolete "isabelle unsymbolize". Note that the usual format for email communication is the Unicode rendering of Isabelle symbols, as produced by Isabelle/jEdit, for example. * Removed obsolete tool "wwwfind". Similar functionality may be integrated into Isabelle/jEdit eventually. * Improved 'display_drafts' concerning desktop integration and repeated invocation in PIDE front-end: re-use single file $ISABELLE_HOME_USER/tmp/drafts.pdf and corresponding views. * Session ROOT specifications require explicit 'document_files' for robust dependencies on LaTeX sources. Only these explicitly given files are copied to the document output directory, before document processing is started. * Windows: support for regular TeX installation (e.g. MiKTeX) instead of TeX Live from Cygwin. New in Isabelle2013-2 (December 2013) ------------------------------------- *** Prover IDE -- Isabelle/Scala/jEdit *** * More robust editing of running commands with internal forks, e.g. non-terminating 'by' steps. * More relaxed Sledgehammer panel: avoid repeated application of query after edits surrounding the command location. * More status information about commands that are interrupted accidentally (via physical event or Poly/ML runtime system signal, e.g. out-of-memory). *** System *** * More robust termination of external processes managed by Isabelle/ML: support cancellation of tasks within the range of milliseconds, as required for PIDE document editing with automatically tried tools (e.g. Sledgehammer). * Reactivated Isabelle/Scala kill command for external processes on Mac OS X, which was accidentally broken in Isabelle2013-1 due to a workaround for some Debian/Ubuntu Linux versions from 2013. New in Isabelle2013-1 (November 2013) ------------------------------------- *** General *** * Discontinued obsolete 'uses' within theory header. Note that commands like 'ML_file' work without separate declaration of file dependencies. Minor INCOMPATIBILITY. * Discontinued redundant 'use' command, which was superseded by 'ML_file' in Isabelle2013. Minor INCOMPATIBILITY. * Simplified subscripts within identifiers, using plain \<^sub> instead of the second copy \<^isub> and \<^isup>. Superscripts are only for literal tokens within notation; explicit mixfix annotations for consts or fixed variables may be used as fall-back for unusual names. Obsolete \ has been expanded to \<^sup>2 in Isabelle/HOL. INCOMPATIBILITY, use "isabelle update_sub_sup" to standardize symbols as a starting point for further manual cleanup. The ML reference variable "legacy_isub_isup" may be set as temporary workaround, to make the prover accept a subset of the old identifier syntax. * Document antiquotations: term style "isub" has been renamed to "sub". Minor INCOMPATIBILITY. * Uniform management of "quick_and_dirty" as system option (see also "isabelle options"), configuration option within the context (see also Config.get in Isabelle/ML), and attribute in Isabelle/Isar. Minor INCOMPATIBILITY, need to use more official Isabelle means to access quick_and_dirty, instead of historical poking into mutable reference. * Renamed command 'print_configs' to 'print_options'. Minor INCOMPATIBILITY. * Proper diagnostic command 'print_state'. Old 'pr' (with its implicit change of some global references) is retained for now as control command, e.g. for ProofGeneral 3.7.x. * Discontinued 'print_drafts' command with its old-fashioned PS output and Unix command-line print spooling. Minor INCOMPATIBILITY: use 'display_drafts' instead and print via the regular document viewer. * Updated and extended "isar-ref" and "implementation" manual, eliminated old "ref" manual. *** Prover IDE -- Isabelle/Scala/jEdit *** * New manual "jedit" for Isabelle/jEdit, see isabelle doc or Documentation panel. * Dockable window "Documentation" provides access to Isabelle documentation. * Dockable window "Find" provides query operations for formal entities (GUI front-end to 'find_theorems' command). * Dockable window "Sledgehammer" manages asynchronous / parallel sledgehammer runs over existing document sources, independently of normal editing and checking process. * Dockable window "Timing" provides an overview of relevant command timing information, depending on option jedit_timing_threshold. The same timing information is shown in the extended tooltip of the command keyword, when hovering the mouse over it while the CONTROL or COMMAND modifier is pressed. * Improved dockable window "Theories": Continuous checking of proof document (visible and required parts) may be controlled explicitly, using check box or shortcut "C+e ENTER". Individual theory nodes may be marked explicitly as required and checked in full, using check box or shortcut "C+e SPACE". * Improved completion mechanism, which is now managed by the Isabelle/jEdit plugin instead of SideKick. Refined table of Isabelle symbol abbreviations (see $ISABELLE_HOME/etc/symbols). * Standard jEdit keyboard shortcut C+b complete-word is remapped to isabelle.complete for explicit completion in Isabelle sources. INCOMPATIBILITY wrt. jEdit defaults, may have to invent new shortcuts to resolve conflict. * Improved support of various "minor modes" for Isabelle NEWS, options, session ROOT etc., with completion and SideKick tree view. * Strictly monotonic document update, without premature cancellation of running transactions that are still needed: avoid reset/restart of such command executions while editing. * Support for asynchronous print functions, as overlay to existing document content. * Support for automatic tools in HOL, which try to prove or disprove toplevel theorem statements. * Action isabelle.reset-font-size resets main text area font size according to Isabelle/Scala plugin option "jedit_font_reset_size" (see also "Plugin Options / Isabelle / General"). It can be bound to some keyboard shortcut by the user (e.g. C+0 and/or C+NUMPAD0). * File specifications in jEdit (e.g. file browser) may refer to $ISABELLE_HOME and $ISABELLE_HOME_USER on all platforms. Discontinued obsolete $ISABELLE_HOME_WINDOWS variable. * Improved support for Linux look-and-feel "GTK+", see also "Utilities / Global Options / Appearance". * Improved support of native Mac OS X functionality via "MacOSX" plugin, which is now enabled by default. *** Pure *** * Commands 'interpretation' and 'sublocale' are now target-sensitive. In particular, 'interpretation' allows for non-persistent interpretation within "context ... begin ... end" blocks offering a light-weight alternative to 'sublocale'. See "isar-ref" manual for details. * Improved locales diagnostic command 'print_dependencies'. * Discontinued obsolete 'axioms' command, which has been marked as legacy since Isabelle2009-2. INCOMPATIBILITY, use 'axiomatization' instead, while observing its uniform scope for polymorphism. * Discontinued empty name bindings in 'axiomatization'. INCOMPATIBILITY. * System option "proofs" has been discontinued. Instead the global state of Proofterm.proofs is persistently compiled into logic images as required, notably HOL-Proofs. Users no longer need to change Proofterm.proofs dynamically. Minor INCOMPATIBILITY. * Syntax translation functions (print_translation etc.) always depend on Proof.context. Discontinued former "(advanced)" option -- this is now the default. Minor INCOMPATIBILITY. * Former global reference trace_unify_fail is now available as configuration option "unify_trace_failure" (global context only). * SELECT_GOAL now retains the syntactic context of the overall goal state (schematic variables etc.). Potential INCOMPATIBILITY in rare situations. *** HOL *** * Stronger precedence of syntax for big intersection and union on sets, in accordance with corresponding lattice operations. INCOMPATIBILITY. * Notation "{p:A. P}" now allows tuple patterns as well. * Nested case expressions are now translated in a separate check phase rather than during parsing. The data for case combinators is separated from the datatype package. The declaration attribute "case_translation" can be used to register new case combinators: declare [[case_translation case_combinator constructor1 ... constructorN]] * Code generator: - 'code_printing' unifies 'code_const' / 'code_type' / 'code_class' / 'code_instance'. - 'code_identifier' declares name hints for arbitrary identifiers in generated code, subsuming 'code_modulename'. See the isar-ref manual for syntax diagrams, and the HOL theories for examples. * Attibute 'code': 'code' now declares concrete and abstract code equations uniformly. Use explicit 'code equation' and 'code abstract' to distinguish both when desired. * Discontinued theories Code_Integer and Efficient_Nat by a more fine-grain stack of theories Code_Target_Int, Code_Binary_Nat, Code_Target_Nat and Code_Target_Numeral. See the tutorial on code generation for details. INCOMPATIBILITY. * Numeric types are mapped by default to target language numerals: natural (replaces former code_numeral) and integer (replaces former code_int). Conversions are available as integer_of_natural / natural_of_integer / integer_of_nat / nat_of_integer (in HOL) and Code_Numeral.integer_of_natural / Code_Numeral.natural_of_integer (in ML). INCOMPATIBILITY. * Function package: For mutually recursive functions f and g, separate cases rules f.cases and g.cases are generated instead of unusable f_g.cases which exposed internal sum types. Potential INCOMPATIBILITY, in the case that the unusable rule was used nevertheless. * Function package: For each function f, new rules f.elims are generated, which eliminate equalities of the form "f x = t". * New command 'fun_cases' derives ad-hoc elimination rules for function equations as simplified instances of f.elims, analogous to inductive_cases. See ~~/src/HOL/ex/Fundefs.thy for some examples. * Lifting: - parametrized correspondence relations are now supported: + parametricity theorems for the raw term can be specified in the command lift_definition, which allow us to generate stronger transfer rules + setup_lifting generates stronger transfer rules if parametric correspondence relation can be generated + various new properties of the relator must be specified to support parametricity + parametricity theorem for the Quotient relation can be specified - setup_lifting generates domain rules for the Transfer package - stronger reflexivity prover of respectfulness theorems for type copies - ===> and --> are now local. The symbols can be introduced by interpreting the locale lifting_syntax (typically in an anonymous context) - Lifting/Transfer relevant parts of Library/Quotient_* are now in Main. Potential INCOMPATIBILITY - new commands for restoring and deleting Lifting/Transfer context: lifting_forget, lifting_update - the command print_quotmaps was renamed to print_quot_maps. INCOMPATIBILITY * Transfer: - better support for domains in Transfer: replace Domainp T by the actual invariant in a transferred goal - transfer rules can have as assumptions other transfer rules - Experimental support for transferring from the raw level to the abstract level: Transfer.transferred attribute - Attribute version of the transfer method: untransferred attribute * Reification and reflection: - Reification is now directly available in HOL-Main in structure "Reification". - Reflection now handles multiple lists with variables also. - The whole reflection stack has been decomposed into conversions. INCOMPATIBILITY. * Revised devices for recursive definitions over finite sets: - Only one fundamental fold combinator on finite set remains: Finite_Set.fold :: ('a => 'b => 'b) => 'b => 'a set => 'b This is now identity on infinite sets. - Locales ("mini packages") for fundamental definitions with Finite_Set.fold: folding, folding_idem. - Locales comm_monoid_set, semilattice_order_set and semilattice_neutr_order_set for big operators on sets. See theory Big_Operators for canonical examples. Note that foundational constants comm_monoid_set.F and semilattice_set.F correspond to former combinators fold_image and fold1 respectively. These are now gone. You may use those foundational constants as substitutes, but it is preferable to interpret the above locales accordingly. - Dropped class ab_semigroup_idem_mult (special case of lattice, no longer needed in connection with Finite_Set.fold etc.) - Fact renames: card.union_inter ~> card_Un_Int [symmetric] card.union_disjoint ~> card_Un_disjoint INCOMPATIBILITY. * Locale hierarchy for abstract orderings and (semi)lattices. * Complete_Partial_Order.admissible is defined outside the type class ccpo, but with mandatory prefix ccpo. Admissibility theorems lose the class predicate assumption or sort constraint when possible. INCOMPATIBILITY. * Introduce type class "conditionally_complete_lattice": Like a complete lattice but does not assume the existence of the top and bottom elements. Allows to generalize some lemmas about reals and extended reals. Removed SupInf and replaced it by the instantiation of conditionally_complete_lattice for real. Renamed lemmas about conditionally-complete lattice from Sup_... to cSup_... and from Inf_... to cInf_... to avoid hidding of similar complete lattice lemmas. * Introduce type class linear_continuum as combination of conditionally-complete lattices and inner dense linorders which have more than one element. INCOMPATIBILITY. * Introduced type classes order_top and order_bot. The old classes top and bot only contain the syntax without assumptions. INCOMPATIBILITY: Rename bot -> order_bot, top -> order_top * Introduce type classes "no_top" and "no_bot" for orderings without top and bottom elements. * Split dense_linorder into inner_dense_order and no_top, no_bot. * Complex_Main: Unify and move various concepts from HOL-Multivariate_Analysis to HOL-Complex_Main. - Introduce type class (lin)order_topology and linear_continuum_topology. Allows to generalize theorems about limits and order. Instances are reals and extended reals. - continuous and continuos_on from Multivariate_Analysis: "continuous" is the continuity of a function at a filter. "isCont" is now an abbrevitation: "isCont x f == continuous (at _) f". Generalized continuity lemmas from isCont to continuous on an arbitrary filter. - compact from Multivariate_Analysis. Use Bolzano's lemma to prove compactness of closed intervals on reals. Continuous functions attain infimum and supremum on compact sets. The inverse of a continuous function is continuous, when the function is continuous on a compact set. - connected from Multivariate_Analysis. Use it to prove the intermediate value theorem. Show connectedness of intervals on linear_continuum_topology). - first_countable_topology from Multivariate_Analysis. Is used to show equivalence of properties on the neighbourhood filter of x and on all sequences converging to x. - FDERIV: Definition of has_derivative moved to Deriv.thy. Moved theorems from Library/FDERIV.thy to Deriv.thy and base the definition of DERIV on FDERIV. Add variants of DERIV and FDERIV which are restricted to sets, i.e. to represent derivatives from left or right. - Removed the within-filter. It is replaced by the principal filter: F within X = inf F (principal X) - Introduce "at x within U" as a single constant, "at x" is now an abbreviation for "at x within UNIV" - Introduce named theorem collections tendsto_intros, continuous_intros, continuous_on_intros and FDERIV_intros. Theorems in tendsto_intros (or FDERIV_intros) are also available as tendsto_eq_intros (or FDERIV_eq_intros) where the right-hand side is replaced by a congruence rule. This allows to apply them as intro rules and then proving equivalence by the simplifier. - Restructured theories in HOL-Complex_Main: + Moved RealDef and RComplete into Real + Introduced Topological_Spaces and moved theorems about topological spaces, filters, limits and continuity to it + Renamed RealVector to Real_Vector_Spaces + Split Lim, SEQ, Series into Topological_Spaces, Real_Vector_Spaces, and Limits + Moved Ln and Log to Transcendental + Moved theorems about continuity from Deriv to Topological_Spaces - Remove various auxiliary lemmas. INCOMPATIBILITY. * Nitpick: - Added option "spy". - Reduce incidence of "too high arity" errors. * Sledgehammer: - Renamed option: isar_shrink ~> isar_compress INCOMPATIBILITY. - Added options "isar_try0", "spy". - Better support for "isar_proofs". - MaSh has been fined-tuned and now runs as a local server. * Improved support for ad hoc overloading of constants (see also isar-ref manual and ~~/src/HOL/ex/Adhoc_Overloading_Examples.thy). * Library/Polynomial.thy: - Use lifting for primitive definitions. - Explicit conversions from and to lists of coefficients, used for generated code. - Replaced recursion operator poly_rec by fold_coeffs. - Prefer pre-existing gcd operation for gcd. - Fact renames: poly_eq_iff ~> poly_eq_poly_eq_iff poly_ext ~> poly_eqI expand_poly_eq ~> poly_eq_iff IMCOMPATIBILITY. * New Library/Simps_Case_Conv.thy: Provides commands simps_of_case and case_of_simps to convert function definitions between a list of equations with patterns on the lhs and a single equation with case expressions on the rhs. See also Ex/Simps_Case_Conv_Examples.thy. * New Library/FSet.thy: type of finite sets defined as a subtype of sets defined by Lifting/Transfer. * Discontinued theory src/HOL/Library/Eval_Witness. INCOMPATIBILITY. * Consolidation of library theories on product orders: Product_Lattice ~> Product_Order -- pointwise order on products Product_ord ~> Product_Lexorder -- lexicographic order on products INCOMPATIBILITY. * Imperative-HOL: The MREC combinator is considered legacy and no longer included by default. INCOMPATIBILITY, use partial_function instead, or import theory Legacy_Mrec as a fallback. * HOL-Algebra: Discontinued theories ~~/src/HOL/Algebra/abstract and ~~/src/HOL/Algebra/poly. Existing theories should be based on ~~/src/HOL/Library/Polynomial instead. The latter provides integration with HOL's type classes for rings. INCOMPATIBILITY. * HOL-BNF: - Various improvements to BNF-based (co)datatype package, including new commands "primrec_new", "primcorec", and "datatype_new_compat", as well as documentation. See "datatypes.pdf" for details. - New "coinduction" method to avoid some boilerplate (compared to coinduct). - Renamed keywords: data ~> datatype_new codata ~> codatatype bnf_def ~> bnf - Renamed many generated theorems, including discs ~> disc map_comp' ~> map_comp map_id' ~> map_id sels ~> sel set_map' ~> set_map sets ~> set IMCOMPATIBILITY. *** ML *** * Spec_Check is a Quickcheck tool for Isabelle/ML. The ML function "check_property" allows to check specifications of the form "ALL x y z. prop x y z". See also ~~/src/Tools/Spec_Check/ with its Examples.thy in particular. * Improved printing of exception trace in Poly/ML 5.5.1, with regular tracing output in the command transaction context instead of physical stdout. See also Toplevel.debug, Toplevel.debugging and ML_Compiler.exn_trace. * ML type "theory" is now immutable, without any special treatment of drafts or linear updates (which could lead to "stale theory" errors in the past). Discontinued obsolete operations like Theory.copy, Theory.checkpoint, and the auxiliary type theory_ref. Minor INCOMPATIBILITY. * More uniform naming of goal functions for skipped proofs: Skip_Proof.prove ~> Goal.prove_sorry Skip_Proof.prove_global ~> Goal.prove_sorry_global Minor INCOMPATIBILITY. * Simplifier tactics and tools use proper Proof.context instead of historic type simpset. Old-style declarations like addsimps, addsimprocs etc. operate directly on Proof.context. Raw type simpset retains its use as snapshot of the main Simplifier context, using simpset_of and put_simpset on Proof.context. INCOMPATIBILITY -- port old tools by making them depend on (ctxt : Proof.context) instead of (ss : simpset), then turn (simpset_of ctxt) into ctxt. * Modifiers for classical wrappers (e.g. addWrapper, delWrapper) operate on Proof.context instead of claset, for uniformity with addIs, addEs, addDs etc. Note that claset_of and put_claset allow to manage clasets separately from the context. * Discontinued obsolete ML antiquotations @{claset} and @{simpset}. INCOMPATIBILITY, use @{context} instead. * Antiquotation @{theory_context A} is similar to @{theory A}, but presents the result as initial Proof.context. *** System *** * Discontinued obsolete isabelle usedir, mkdir, make -- superseded by "isabelle build" in Isabelle2013. INCOMPATIBILITY. * Discontinued obsolete isabelle-process options -f and -u (former administrative aliases of option -e). Minor INCOMPATIBILITY. * Discontinued obsolete isabelle print tool, and PRINT_COMMAND settings variable. * Discontinued ISABELLE_DOC_FORMAT settings variable and historic document formats: dvi.gz, ps, ps.gz -- the default document format is always pdf. * Isabelle settings variable ISABELLE_BUILD_JAVA_OPTIONS allows to specify global resources of the JVM process run by isabelle build. * Toplevel executable $ISABELLE_HOME/bin/isabelle_scala_script allows to run Isabelle/Scala source files as standalone programs. * Improved "isabelle keywords" tool (for old-style ProofGeneral keyword tables): use Isabelle/Scala operations, which inspect outer syntax without requiring to build sessions first. * Sessions may be organized via 'chapter' specifications in the ROOT file, which determines a two-level hierarchy of browser info. The old tree-like organization via implicit sub-session relation (with its tendency towards erratic fluctuation of URLs) has been discontinued. The default chapter is called "Unsorted". Potential INCOMPATIBILITY for HTML presentation of theories. New in Isabelle2013 (February 2013) ----------------------------------- *** General *** * Theorem status about oracles and unfinished/failed future proofs is no longer printed by default, since it is incompatible with incremental / parallel checking of the persistent document model. ML function Thm.peek_status may be used to inspect a snapshot of the ongoing evaluation process. Note that in batch mode --- notably isabelle build --- the system ensures that future proofs of all accessible theorems in the theory context are finished (as before). * Configuration option show_markup controls direct inlining of markup into the printed representation of formal entities --- notably type and sort constraints. This enables Prover IDE users to retrieve that information via tooltips in the output window, for example. * Command 'ML_file' evaluates ML text from a file directly within the theory, without any predeclaration via 'uses' in the theory header. * Old command 'use' command and corresponding keyword 'uses' in the theory header are legacy features and will be discontinued soon. Tools that load their additional source files may imitate the 'ML_file' implementation, such that the system can take care of dependencies properly. * Discontinued obsolete method fastsimp / tactic fast_simp_tac, which is called fastforce / fast_force_tac already since Isabelle2011-1. * Updated and extended "isar-ref" and "implementation" manual, reduced remaining material in old "ref" manual. * Improved support for auxiliary contexts that indicate block structure for specifications. Nesting of "context fixes ... context assumes ..." and "class ... context ...". * Attribute "consumes" allows a negative value as well, which is interpreted relatively to the total number of premises of the rule in the target context. This form of declaration is stable when exported from a nested 'context' with additional assumptions. It is the preferred form for definitional packages, notably cases/rules produced in HOL/inductive and HOL/function. * More informative error messages for Isar proof commands involving lazy enumerations (method applications etc.). * Refined 'help' command to retrieve outer syntax commands according to name patterns (with clickable results). *** Prover IDE -- Isabelle/Scala/jEdit *** * Parallel terminal proofs ('by') are enabled by default, likewise proofs that are built into packages like 'datatype', 'function'. This allows to "run ahead" checking the theory specifications on the surface, while the prover is still crunching on internal justifications. Unfinished / cancelled proofs are restarted as required to complete full proof checking eventually. * Improved output panel with tooltips, hyperlinks etc. based on the same Rich_Text_Area as regular Isabelle/jEdit buffers. Activation of tooltips leads to some window that supports the same recursively, which can lead to stacks of tooltips as the semantic document content is explored. ESCAPE closes the whole stack, individual windows may be closed separately, or detached to become independent jEdit dockables. * Improved support for commands that produce graph output: the text message contains a clickable area to open a new instance of the graph browser on demand. * More robust incremental parsing of outer syntax (partial comments, malformed symbols). Changing the balance of open/close quotes and comment delimiters works more conveniently with unfinished situations that frequently occur in user interaction. * More efficient painting and improved reactivity when editing large files. More scalable management of formal document content. * Smarter handling of tracing messages: prover process pauses after certain number of messages per command transaction, with some user dialog to stop or continue. This avoids swamping the front-end with potentially infinite message streams. * More plugin options and preferences, based on Isabelle/Scala. The jEdit plugin option panel provides access to some Isabelle/Scala options, including tuning parameters for editor reactivity and color schemes. * Dockable window "Symbols" provides some editing support for Isabelle symbols. * Dockable window "Monitor" shows ML runtime statistics. Note that continuous display of the chart slows down the system. * Improved editing support for control styles: subscript, superscript, bold, reset of style -- operating on single symbols or text selections. Cf. keyboard shortcuts C+e DOWN/UP/RIGHT/LEFT. * Actions isabelle.increase-font-size and isabelle.decrease-font-size adjust the main text area font size, and its derivatives for output, tooltips etc. Cf. keyboard shortcuts C-PLUS and C-MINUS, which often need to be adapted to local keyboard layouts. * More reactive completion popup by default: use \t (TAB) instead of \n (NEWLINE) to minimize intrusion into regular flow of editing. See also "Plugin Options / SideKick / General / Code Completion Options". * Implicit check and build dialog of the specified logic session image. For example, HOL, HOLCF, HOL-Nominal can be produced on demand, without bundling big platform-dependent heap images in the Isabelle distribution. * Uniform Java 7 platform on Linux, Mac OS X, Windows: recent updates from Oracle provide better multi-platform experience. This version is now bundled exclusively with Isabelle. *** Pure *** * Code generation for Haskell: restrict unqualified imports from Haskell Prelude to a small set of fundamental operations. * Command 'export_code': relative file names are interpreted relatively to master directory of current theory rather than the rather arbitrary current working directory. INCOMPATIBILITY. * Discontinued obsolete attribute "COMP". Potential INCOMPATIBILITY, use regular rule composition via "OF" / "THEN", or explicit proof structure instead. Note that Isabelle/ML provides a variety of operators like COMP, INCR_COMP, COMP_INCR, which need to be applied with some care where this is really required. * Command 'typ' supports an additional variant with explicit sort constraint, to infer and check the most general type conforming to a given sort. Example (in HOL): typ "_ * _ * bool * unit" :: finite * Command 'locale_deps' visualizes all locales and their relations as a Hasse diagram. *** HOL *** * Sledgehammer: - Added MaSh relevance filter based on machine-learning; see the Sledgehammer manual for details. - Polished Isar proofs generated with "isar_proofs" option. - Rationalized type encodings ("type_enc" option). - Renamed "kill_provers" subcommand to "kill_all". - Renamed options: isar_proof ~> isar_proofs isar_shrink_factor ~> isar_shrink max_relevant ~> max_facts relevance_thresholds ~> fact_thresholds * Quickcheck: added an optimisation for equality premises. It is switched on by default, and can be switched off by setting the configuration quickcheck_optimise_equality to false. * Quotient: only one quotient can be defined by quotient_type INCOMPATIBILITY. * Lifting: - generation of an abstraction function equation in lift_definition - quot_del attribute - renamed no_abs_code -> no_code (INCOMPATIBILITY.) * Simproc "finite_Collect" rewrites set comprehensions into pointfree expressions. * Preprocessing of the code generator rewrites set comprehensions into pointfree expressions. * The SMT solver Z3 has now by default a restricted set of directly supported features. For the full set of features (div/mod, nonlinear arithmetic, datatypes/records) with potential proof reconstruction failures, enable the configuration option "z3_with_extensions". Minor INCOMPATIBILITY. * Simplified 'typedef' specifications: historical options for implicit set definition and alternative name have been discontinued. The former behavior of "typedef (open) t = A" is now the default, but written just "typedef t = A". INCOMPATIBILITY, need to adapt theories accordingly. * Removed constant "chars"; prefer "Enum.enum" on type "char" directly. INCOMPATIBILITY. * Moved operation product, sublists and n_lists from theory Enum to List. INCOMPATIBILITY. * Theorem UN_o generalized to SUP_comp. INCOMPATIBILITY. * Class "comm_monoid_diff" formalises properties of bounded subtraction, with natural numbers and multisets as typical instances. * Added combinator "Option.these" with type "'a option set => 'a set". * Theory "Transitive_Closure": renamed lemmas reflcl_tranclp -> reflclp_tranclp rtranclp_reflcl -> rtranclp_reflclp INCOMPATIBILITY. * Theory "Rings": renamed lemmas (in class semiring) left_distrib ~> distrib_right right_distrib ~> distrib_left INCOMPATIBILITY. * Generalized the definition of limits: - Introduced the predicate filterlim (LIM x F. f x :> G) which expresses that when the input values x converge to F then the output f x converges to G. - Added filters for convergence to positive (at_top) and negative infinity (at_bot). - Moved infinity in the norm (at_infinity) from Multivariate_Analysis to Complex_Main. - Removed real_tendsto_inf, it is superseded by "LIM x F. f x :> at_top". INCOMPATIBILITY. * Theory "Library/Option_ord" provides instantiation of option type to lattice type classes. * Theory "Library/Multiset": renamed constant fold_mset ~> Multiset.fold fact fold_mset_commute ~> fold_mset_comm INCOMPATIBILITY. * Renamed theory Library/List_Prefix to Library/Sublist, with related changes as follows. - Renamed constants (and related lemmas) prefix ~> prefixeq strict_prefix ~> prefix - Replaced constant "postfix" by "suffixeq" with swapped argument order (i.e., "postfix xs ys" is now "suffixeq ys xs") and dropped old infix syntax "xs >>= ys"; use "suffixeq ys xs" instead. Renamed lemmas accordingly. - Added constant "list_hembeq" for homeomorphic embedding on lists. Added abbreviation "sublisteq" for special case "list_hembeq (op =)". - Theory Library/Sublist no longer provides "order" and "bot" type class instances for the prefix order (merely corresponding locale interpretations). The type class instances are now in theory Library/Prefix_Order. - The sublist relation of theory Library/Sublist_Order is now based on "Sublist.sublisteq". Renamed lemmas accordingly: le_list_append_le_same_iff ~> Sublist.sublisteq_append_le_same_iff le_list_append_mono ~> Sublist.list_hembeq_append_mono le_list_below_empty ~> Sublist.list_hembeq_Nil, Sublist.list_hembeq_Nil2 le_list_Cons_EX ~> Sublist.list_hembeq_ConsD le_list_drop_Cons2 ~> Sublist.sublisteq_Cons2' le_list_drop_Cons_neq ~> Sublist.sublisteq_Cons2_neq le_list_drop_Cons ~> Sublist.sublisteq_Cons' le_list_drop_many ~> Sublist.sublisteq_drop_many le_list_filter_left ~> Sublist.sublisteq_filter_left le_list_rev_drop_many ~> Sublist.sublisteq_rev_drop_many le_list_rev_take_iff ~> Sublist.sublisteq_append le_list_same_length ~> Sublist.sublisteq_same_length le_list_take_many_iff ~> Sublist.sublisteq_append' less_eq_list.drop ~> less_eq_list_drop less_eq_list.induct ~> less_eq_list_induct not_le_list_length ~> Sublist.not_sublisteq_length INCOMPATIBILITY. * New theory Library/Countable_Set. * Theory Library/Debug and Library/Parallel provide debugging and parallel execution for code generated towards Isabelle/ML. * Theory Library/FuncSet: Extended support for Pi and extensional and introduce the extensional dependent function space "PiE". Replaced extensional_funcset by an abbreviation, and renamed lemmas from extensional_funcset to PiE as follows: extensional_empty ~> PiE_empty extensional_funcset_empty_domain ~> PiE_empty_domain extensional_funcset_empty_range ~> PiE_empty_range extensional_funcset_arb ~> PiE_arb extensional_funcset_mem ~> PiE_mem extensional_funcset_extend_domainI ~> PiE_fun_upd extensional_funcset_restrict_domain ~> fun_upd_in_PiE extensional_funcset_extend_domain_eq ~> PiE_insert_eq card_extensional_funcset ~> card_PiE finite_extensional_funcset ~> finite_PiE INCOMPATIBILITY. * Theory Library/FinFun: theory of almost everywhere constant functions (supersedes the AFP entry "Code Generation for Functions as Data"). * Theory Library/Phantom: generic phantom type to make a type parameter appear in a constant's type. This alternative to adding TYPE('a) as another parameter avoids unnecessary closures in generated code. * Theory Library/RBT_Impl: efficient construction of red-black trees from sorted associative lists. Merging two trees with rbt_union may return a structurally different tree than before. Potential INCOMPATIBILITY. * Theory Library/IArray: immutable arrays with code generation. * Theory Library/Finite_Lattice: theory of finite lattices. * HOL/Multivariate_Analysis: replaced "basis :: 'a::euclidean_space => nat => real" "\\ :: (nat => real) => 'a::euclidean_space" on euclidean spaces by using the inner product "_ \ _" with vectors from the Basis set: "\\ i. f i" is superseded by "SUM i : Basis. f i * r i". With this change the following constants are also changed or removed: DIM('a) :: nat ~> card (Basis :: 'a set) (is an abbreviation) a $$ i ~> inner a i (where i : Basis) cart_base i removed \, \' removed Theorems about these constants where removed. Renamed lemmas: component_le_norm ~> Basis_le_norm euclidean_eq ~> euclidean_eq_iff differential_zero_maxmin_component ~> differential_zero_maxmin_cart euclidean_simps ~> inner_simps independent_basis ~> independent_Basis span_basis ~> span_Basis in_span_basis ~> in_span_Basis norm_bound_component_le ~> norm_boound_Basis_le norm_bound_component_lt ~> norm_boound_Basis_lt component_le_infnorm ~> Basis_le_infnorm INCOMPATIBILITY. * HOL/Probability: - Added simproc "measurable" to automatically prove measurability. - Added induction rules for sigma sets with disjoint union (sigma_sets_induct_disjoint) and for Borel-measurable functions (borel_measurable_induct). - Added the Daniell-Kolmogorov theorem (the existence the limit of a projective family). * HOL/Cardinals: Theories of ordinals and cardinals (supersedes the AFP entry "Ordinals_and_Cardinals"). * HOL/BNF: New (co)datatype package based on bounded natural functors with support for mixed, nested recursion and interesting non-free datatypes. * HOL/Finite_Set and Relation: added new set and relation operations expressed by Finite_Set.fold. * New theory HOL/Library/RBT_Set: implementation of sets by red-black trees for the code generator. * HOL/Library/RBT and HOL/Library/Mapping have been converted to Lifting/Transfer. possible INCOMPATIBILITY. * HOL/Set: renamed Set.project -> Set.filter INCOMPATIBILITY. *** Document preparation *** * Dropped legacy antiquotations "term_style" and "thm_style", since styles may be given as arguments to "term" and "thm" already. Discontinued legacy styles "prem1" .. "prem19". * Default LaTeX rendering for \ is now based on eurosym package, instead of slightly exotic babel/greek. * Document variant NAME may use different LaTeX entry point document/root_NAME.tex if that file exists, instead of the common document/root.tex. * Simplified custom document/build script, instead of old-style document/IsaMakefile. Minor INCOMPATIBILITY. *** ML *** * The default limit for maximum number of worker threads is now 8, instead of 4, in correspondence to capabilities of contemporary hardware and Poly/ML runtime system. * Type Seq.results and related operations support embedded error messages within lazy enumerations, and thus allow to provide informative errors in the absence of any usable results. * Renamed Position.str_of to Position.here to emphasize that this is a formal device to inline positions into message text, but not necessarily printing visible text. *** System *** * Advanced support for Isabelle sessions and build management, see "system" manual for the chapter of that name, especially the "isabelle build" tool and its examples. The "isabelle mkroot" tool prepares session root directories for use with "isabelle build", similar to former "isabelle mkdir" for "isabelle usedir". Note that this affects document preparation as well. INCOMPATIBILITY, isabelle usedir / mkdir / make are rendered obsolete. * Discontinued obsolete Isabelle/build script, it is superseded by the regular isabelle build tool. For example: isabelle build -s -b HOL * Discontinued obsolete "isabelle makeall". * Discontinued obsolete IsaMakefile and ROOT.ML files from the Isabelle distribution, except for rudimentary src/HOL/IsaMakefile that provides some traditional targets that invoke "isabelle build". Note that this is inefficient! Applications of Isabelle/HOL involving "isabelle make" should be upgraded to use "isabelle build" directly. * The "isabelle options" tool prints Isabelle system options, as required for "isabelle build", for example. * The "isabelle logo" tool produces EPS and PDF format simultaneously. Minor INCOMPATIBILITY in command-line options. * The "isabelle install" tool has now a simpler command-line. Minor INCOMPATIBILITY. * The "isabelle components" tool helps to resolve add-on components that are not bundled, or referenced from a bare-bones repository version of Isabelle. * Settings variable ISABELLE_PLATFORM_FAMILY refers to the general platform family: "linux", "macos", "windows". * The ML system is configured as regular component, and no longer picked up from some surrounding directory. Potential INCOMPATIBILITY for home-made settings. * Improved ML runtime statistics (heap, threads, future tasks etc.). * Discontinued support for Poly/ML 5.2.1, which was the last version without exception positions and advanced ML compiler/toplevel configuration. * Discontinued special treatment of Proof General -- no longer guess PROOFGENERAL_HOME based on accidental file-system layout. Minor INCOMPATIBILITY: provide PROOFGENERAL_HOME and PROOFGENERAL_OPTIONS settings manually, or use a Proof General version that has been bundled as Isabelle component. New in Isabelle2012 (May 2012) ------------------------------ *** General *** * Prover IDE (PIDE) improvements: - more robust Sledgehammer integration (as before the sledgehammer command-line needs to be typed into the source buffer) - markup for bound variables - markup for types of term variables (displayed as tooltips) - support for user-defined Isar commands within the running session - improved support for Unicode outside original 16bit range e.g. glyph for \ (thanks to jEdit 4.5.1) * Forward declaration of outer syntax keywords within the theory header -- minor INCOMPATIBILITY for user-defined commands. Allow new commands to be used in the same theory where defined. * Auxiliary contexts indicate block structure for specifications with additional parameters and assumptions. Such unnamed contexts may be nested within other targets, like 'theory', 'locale', 'class', 'instantiation' etc. Results from the local context are generalized accordingly and applied to the enclosing target context. Example: context fixes x y z :: 'a assumes xy: "x = y" and yz: "y = z" begin lemma my_trans: "x = z" using xy yz by simp end thm my_trans The most basic application is to factor-out context elements of several fixes/assumes/shows theorem statements, e.g. see ~~/src/HOL/Isar_Examples/Group_Context.thy Any other local theory specification element works within the "context ... begin ... end" block as well. * Bundled declarations associate attributed fact expressions with a given name in the context. These may be later included in other contexts. This allows to manage context extensions casually, without the logical dependencies of locales and locale interpretation. See commands 'bundle', 'include', 'including' etc. in the isar-ref manual. * Commands 'lemmas' and 'theorems' allow local variables using 'for' declaration, and results are standardized before being stored. Thus old-style "standard" after instantiation or composition of facts becomes obsolete. Minor INCOMPATIBILITY, due to potential change of indices of schematic variables. * Rule attributes in local theory declarations (e.g. locale or class) are now statically evaluated: the resulting theorem is stored instead of the original expression. INCOMPATIBILITY in rare situations, where the historic accident of dynamic re-evaluation in interpretations etc. was exploited. * New tutorial "Programming and Proving in Isabelle/HOL" ("prog-prove"). It completely supersedes "A Tutorial Introduction to Structured Isar Proofs" ("isar-overview"), which has been removed. It also supersedes "Isabelle/HOL, A Proof Assistant for Higher-Order Logic" as the recommended beginners tutorial, but does not cover all of the material of that old tutorial. * Updated and extended reference manuals: "isar-ref", "implementation", "system"; reduced remaining material in old "ref" manual. *** Pure *** * Command 'definition' no longer exports the foundational "raw_def" into the user context. Minor INCOMPATIBILITY, may use the regular "def" result with attribute "abs_def" to imitate the old version. * Attribute "abs_def" turns an equation of the form "f x y == t" into "f == %x y. t", which ensures that "simp" or "unfold" steps always expand it. This also works for object-logic equality. (Formerly undocumented feature.) * Sort constraints are now propagated in simultaneous statements, just like type constraints. INCOMPATIBILITY in rare situations, where distinct sorts used to be assigned accidentally. For example: lemma "P (x::'a::foo)" and "Q (y::'a::bar)" -- "now illegal" lemma "P (x::'a)" and "Q (y::'a::bar)" -- "now uniform 'a::bar instead of default sort for first occurrence (!)" * Rule composition via attribute "OF" (or ML functions OF/MRS) is more tolerant against multiple unifiers, as long as the final result is unique. (As before, rules are composed in canonical right-to-left order to accommodate newly introduced premises.) * Renamed some inner syntax categories: num ~> num_token xnum ~> xnum_token xstr ~> str_token Minor INCOMPATIBILITY. Note that in practice "num_const" or "num_position" etc. are mainly used instead (which also include position information via constraints). * Simplified configuration options for syntax ambiguity: see "syntax_ambiguity_warning" and "syntax_ambiguity_limit" in isar-ref manual. Minor INCOMPATIBILITY. * Discontinued configuration option "syntax_positions": atomic terms in parse trees are always annotated by position constraints. * Old code generator for SML and its commands 'code_module', 'code_library', 'consts_code', 'types_code' have been discontinued. Use commands of the generic code generator instead. INCOMPATIBILITY. * Redundant attribute "code_inline" has been discontinued. Use "code_unfold" instead. INCOMPATIBILITY. * Dropped attribute "code_unfold_post" in favor of the its dual "code_abbrev", which yields a common pattern in definitions like definition [code_abbrev]: "f = t" INCOMPATIBILITY. * Obsolete 'types' command has been discontinued. Use 'type_synonym' instead. INCOMPATIBILITY. * Discontinued old "prems" fact, which used to refer to the accidental collection of foundational premises in the context (already marked as legacy since Isabelle2011). *** HOL *** * Type 'a set is now a proper type constructor (just as before Isabelle2008). Definitions mem_def and Collect_def have disappeared. Non-trivial INCOMPATIBILITY. For developments keeping predicates and sets separate, it is often sufficient to rephrase some set S that has been accidentally used as predicates by "%x. x : S", and some predicate P that has been accidentally used as set by "{x. P x}". Corresponding proofs in a first step should be pruned from any tinkering with former theorems mem_def and Collect_def as far as possible. For developments which deliberately mix predicates and sets, a planning step is necessary to determine what should become a predicate and what a set. It can be helpful to carry out that step in Isabelle2011-1 before jumping right into the current release. * Code generation by default implements sets as container type rather than predicates. INCOMPATIBILITY. * New type synonym 'a rel = ('a * 'a) set * The representation of numerals has changed. Datatype "num" represents strictly positive binary numerals, along with functions "numeral :: num => 'a" and "neg_numeral :: num => 'a" to represent positive and negated numeric literals, respectively. See also definitions in ~~/src/HOL/Num.thy. Potential INCOMPATIBILITY, some user theories may require adaptations as follows: - Theorems with number_ring or number_semiring constraints: These classes are gone; use comm_ring_1 or comm_semiring_1 instead. - Theories defining numeric types: Remove number, number_semiring, and number_ring instances. Defer all theorems about numerals until after classes one and semigroup_add have been instantiated. - Numeral-only simp rules: Replace each rule having a "number_of v" pattern with two copies, one for numeral and one for neg_numeral. - Theorems about subclasses of semiring_1 or ring_1: These classes automatically support numerals now, so more simp rules and simprocs may now apply within the proof. - Definitions and theorems using old constructors Pls/Min/Bit0/Bit1: Redefine using other integer operations. * Transfer: New package intended to generalize the existing "descending" method and related theorem attributes from the Quotient package. (Not all functionality is implemented yet, but future development will focus on Transfer as an eventual replacement for the corresponding parts of the Quotient package.) - transfer_rule attribute: Maintains a collection of transfer rules, which relate constants at two different types. Transfer rules may relate different type instances of the same polymorphic constant, or they may relate an operation on a raw type to a corresponding operation on an abstract type (quotient or subtype). For example: ((A ===> B) ===> list_all2 A ===> list_all2 B) map map (cr_int ===> cr_int ===> cr_int) (%(x,y) (u,v). (x+u, y+v)) plus_int - transfer method: Replaces a subgoal on abstract types with an equivalent subgoal on the corresponding raw types. Constants are replaced with corresponding ones according to the transfer rules. Goals are generalized over all free variables by default; this is necessary for variables whose types change, but can be overridden for specific variables with e.g. "transfer fixing: x y z". The variant transfer' method allows replacing a subgoal with one that is logically stronger (rather than equivalent). - relator_eq attribute: Collects identity laws for relators of various type constructors, e.g. "list_all2 (op =) = (op =)". The transfer method uses these lemmas to infer transfer rules for non-polymorphic constants on the fly. - transfer_prover method: Assists with proving a transfer rule for a new constant, provided the constant is defined in terms of other constants that already have transfer rules. It should be applied after unfolding the constant definitions. - HOL/ex/Transfer_Int_Nat.thy: Example theory demonstrating transfer from type nat to type int. * Lifting: New package intended to generalize the quotient_definition facility of the Quotient package; designed to work with Transfer. - lift_definition command: Defines operations on an abstract type in terms of a corresponding operation on a representation type. Example syntax: lift_definition dlist_insert :: "'a => 'a dlist => 'a dlist" is List.insert Users must discharge a respectfulness proof obligation when each constant is defined. (For a type copy, i.e. a typedef with UNIV, the proof is discharged automatically.) The obligation is presented in a user-friendly, readable form; a respectfulness theorem in the standard format and a transfer rule are generated by the package. - Integration with code_abstype: For typedefs (e.g. subtypes corresponding to a datatype invariant, such as dlist), lift_definition generates a code certificate theorem and sets up code generation for each constant. - setup_lifting command: Sets up the Lifting package to work with a user-defined type. The user must provide either a quotient theorem or a type_definition theorem. The package configures transfer rules for equality and quantifiers on the type, and sets up the lift_definition command to work with the type. - Usage examples: See Quotient_Examples/Lift_DList.thy, Quotient_Examples/Lift_RBT.thy, Quotient_Examples/Lift_FSet.thy, Word/Word.thy and Library/Float.thy. * Quotient package: - The 'quotient_type' command now supports a 'morphisms' option with rep and abs functions, similar to typedef. - 'quotient_type' sets up new types to work with the Lifting and Transfer packages, as with 'setup_lifting'. - The 'quotient_definition' command now requires the user to prove a respectfulness property at the point where the constant is defined, similar to lift_definition; INCOMPATIBILITY. - Renamed predicate 'Quotient' to 'Quotient3', and renamed theorems accordingly, INCOMPATIBILITY. * New diagnostic command 'find_unused_assms' to find potentially superfluous assumptions in theorems using Quickcheck. * Quickcheck: - Quickcheck returns variable assignments as counterexamples, which allows to reveal the underspecification of functions under test. For example, refuting "hd xs = x", it presents the variable assignment xs = [] and x = a1 as a counterexample, assuming that any property is false whenever "hd []" occurs in it. These counterexample are marked as potentially spurious, as Quickcheck also returns "xs = []" as a counterexample to the obvious theorem "hd xs = hd xs". After finding a potentially spurious counterexample, Quickcheck continues searching for genuine ones. By default, Quickcheck shows potentially spurious and genuine counterexamples. The option "genuine_only" sets quickcheck to only show genuine counterexamples. - The command 'quickcheck_generator' creates random and exhaustive value generators for a given type and operations. It generates values by using the operations as if they were constructors of that type. - Support for multisets. - Added "use_subtype" options. - Added "quickcheck_locale" configuration to specify how to process conjectures in a locale context. * Nitpick: Fixed infinite loop caused by the 'peephole_optim' option and affecting 'rat' and 'real'. * Sledgehammer: - Integrated more tightly with SPASS, as described in the ITP 2012 paper "More SPASS with Isabelle". - Made it try "smt" as a fallback if "metis" fails or times out. - Added support for the following provers: Alt-Ergo (via Why3 and TFF1), iProver, iProver-Eq. - Sped up the minimizer. - Added "lam_trans", "uncurry_aliases", and "minimize" options. - Renamed "slicing" ("no_slicing") option to "slice" ("dont_slice"). - Renamed "sound" option to "strict". * Metis: Added possibility to specify lambda translations scheme as a parenthesized argument (e.g., "by (metis (lifting) ...)"). * SMT: Renamed "smt_fixed" option to "smt_read_only_certificates". * Command 'try0': Renamed from 'try_methods'. INCOMPATIBILITY. * New "case_product" attribute to generate a case rule doing multiple case distinctions at the same time. E.g. list.exhaust [case_product nat.exhaust] produces a rule which can be used to perform case distinction on both a list and a nat. * New "eventually_elim" method as a generalized variant of the eventually_elim* rules. Supports structured proofs. * Typedef with implicit set definition is considered legacy. Use "typedef (open)" form instead, which will eventually become the default. * Record: code generation can be switched off manually with declare [[record_coden = false]] -- "default true" * Datatype: type parameters allow explicit sort constraints. * Concrete syntax for case expressions includes constraints for source positions, and thus produces Prover IDE markup for its bindings. INCOMPATIBILITY for old-style syntax translations that augment the pattern notation; e.g. see src/HOL/HOLCF/One.thy for translations of one_case. * Clarified attribute "mono_set": pure declaration without modifying the result of the fact expression. * More default pred/set conversions on a couple of relation operations and predicates. Added powers of predicate relations. Consolidation of some relation theorems: converse_def ~> converse_unfold rel_comp_def ~> relcomp_unfold symp_def ~> (modified, use symp_def and sym_def instead) transp_def ~> transp_trans Domain_def ~> Domain_unfold Range_def ~> Domain_converse [symmetric] Generalized theorems INF_INT_eq, INF_INT_eq2, SUP_UN_eq, SUP_UN_eq2. See theory "Relation" for examples for making use of pred/set conversions by means of attributes "to_set" and "to_pred". INCOMPATIBILITY. * Renamed facts about the power operation on relations, i.e., relpow to match the constant's name: rel_pow_1 ~> relpow_1 rel_pow_0_I ~> relpow_0_I rel_pow_Suc_I ~> relpow_Suc_I rel_pow_Suc_I2 ~> relpow_Suc_I2 rel_pow_0_E ~> relpow_0_E rel_pow_Suc_E ~> relpow_Suc_E rel_pow_E ~> relpow_E rel_pow_Suc_D2 ~> relpow_Suc_D2 rel_pow_Suc_E2 ~> relpow_Suc_E2 rel_pow_Suc_D2' ~> relpow_Suc_D2' rel_pow_E2 ~> relpow_E2 rel_pow_add ~> relpow_add rel_pow_commute ~> relpow rel_pow_empty ~> relpow_empty: rtrancl_imp_UN_rel_pow ~> rtrancl_imp_UN_relpow rel_pow_imp_rtrancl ~> relpow_imp_rtrancl rtrancl_is_UN_rel_pow ~> rtrancl_is_UN_relpow rtrancl_imp_rel_pow ~> rtrancl_imp_relpow rel_pow_fun_conv ~> relpow_fun_conv rel_pow_finite_bounded1 ~> relpow_finite_bounded1 rel_pow_finite_bounded ~> relpow_finite_bounded rtrancl_finite_eq_rel_pow ~> rtrancl_finite_eq_relpow trancl_finite_eq_rel_pow ~> trancl_finite_eq_relpow single_valued_rel_pow ~> single_valued_relpow INCOMPATIBILITY. * Theory Relation: Consolidated constant name for relation composition and corresponding theorem names: - Renamed constant rel_comp to relcomp. - Dropped abbreviation pred_comp. Use relcompp instead. - Renamed theorems: rel_compI ~> relcompI rel_compEpair ~> relcompEpair rel_compE ~> relcompE pred_comp_rel_comp_eq ~> relcompp_relcomp_eq rel_comp_empty1 ~> relcomp_empty1 rel_comp_mono ~> relcomp_mono rel_comp_subset_Sigma ~> relcomp_subset_Sigma rel_comp_distrib ~> relcomp_distrib rel_comp_distrib2 ~> relcomp_distrib2 rel_comp_UNION_distrib ~> relcomp_UNION_distrib rel_comp_UNION_distrib2 ~> relcomp_UNION_distrib2 single_valued_rel_comp ~> single_valued_relcomp rel_comp_def ~> relcomp_unfold converse_rel_comp ~> converse_relcomp pred_compI ~> relcomppI pred_compE ~> relcomppE pred_comp_bot1 ~> relcompp_bot1 pred_comp_bot2 ~> relcompp_bot2 transp_pred_comp_less_eq ~> transp_relcompp_less_eq pred_comp_mono ~> relcompp_mono pred_comp_distrib ~> relcompp_distrib pred_comp_distrib2 ~> relcompp_distrib2 converse_pred_comp ~> converse_relcompp finite_rel_comp ~> finite_relcomp set_rel_comp ~> set_relcomp INCOMPATIBILITY. * Theory Divides: Discontinued redundant theorems about div and mod. INCOMPATIBILITY, use the corresponding generic theorems instead. DIVISION_BY_ZERO ~> div_by_0, mod_by_0 zdiv_self ~> div_self zmod_self ~> mod_self zdiv_zero ~> div_0 zmod_zero ~> mod_0 zdiv_zmod_equality ~> div_mod_equality2 zdiv_zmod_equality2 ~> div_mod_equality zmod_zdiv_trivial ~> mod_div_trivial zdiv_zminus_zminus ~> div_minus_minus zmod_zminus_zminus ~> mod_minus_minus zdiv_zminus2 ~> div_minus_right zmod_zminus2 ~> mod_minus_right zdiv_minus1_right ~> div_minus1_right zmod_minus1_right ~> mod_minus1_right zdvd_mult_div_cancel ~> dvd_mult_div_cancel zmod_zmult1_eq ~> mod_mult_right_eq zpower_zmod ~> power_mod zdvd_zmod ~> dvd_mod zdvd_zmod_imp_zdvd ~> dvd_mod_imp_dvd mod_mult_distrib ~> mult_mod_left mod_mult_distrib2 ~> mult_mod_right * Removed redundant theorems nat_mult_2 and nat_mult_2_right; use generic mult_2 and mult_2_right instead. INCOMPATIBILITY. * Finite_Set.fold now qualified. INCOMPATIBILITY. * Consolidated theorem names concerning fold combinators: inf_INFI_fold_inf ~> inf_INF_fold_inf sup_SUPR_fold_sup ~> sup_SUP_fold_sup INFI_fold_inf ~> INF_fold_inf SUPR_fold_sup ~> SUP_fold_sup union_set ~> union_set_fold minus_set ~> minus_set_fold INFI_set_fold ~> INF_set_fold SUPR_set_fold ~> SUP_set_fold INF_code ~> INF_set_foldr SUP_code ~> SUP_set_foldr foldr.simps ~> foldr.simps (in point-free formulation) foldr_fold_rev ~> foldr_conv_fold foldl_fold ~> foldl_conv_fold foldr_foldr ~> foldr_conv_foldl foldl_foldr ~> foldl_conv_foldr fold_set_remdups ~> fold_set_fold_remdups fold_set ~> fold_set_fold fold1_set ~> fold1_set_fold INCOMPATIBILITY. * Dropped rarely useful theorems concerning fold combinators: foldl_apply, foldl_fun_comm, foldl_rev, fold_weak_invariant, rev_foldl_cons, fold_set_remdups, fold_set, fold_set1, concat_conv_foldl, foldl_weak_invariant, foldl_invariant, foldr_invariant, foldl_absorb0, foldl_foldr1_lemma, foldl_foldr1, listsum_conv_fold, listsum_foldl, sort_foldl_insort, foldl_assoc, foldr_conv_foldl, start_le_sum, elem_le_sum, sum_eq_0_conv. INCOMPATIBILITY. For the common phrases "%xs. List.foldr plus xs 0" and "List.foldl plus 0", prefer "List.listsum". Otherwise it can be useful to boil down "List.foldr" and "List.foldl" to "List.fold" by unfolding "foldr_conv_fold" and "foldl_conv_fold". * Dropped lemmas minus_set_foldr, union_set_foldr, union_coset_foldr, inter_coset_foldr, Inf_fin_set_foldr, Sup_fin_set_foldr, Min_fin_set_foldr, Max_fin_set_foldr, Inf_set_foldr, Sup_set_foldr, INF_set_foldr, SUP_set_foldr. INCOMPATIBILITY. Prefer corresponding lemmas over fold rather than foldr, or make use of lemmas fold_conv_foldr and fold_rev. * Congruence rules Option.map_cong and Option.bind_cong for recursion through option types. * "Transitive_Closure.ntrancl": bounded transitive closure on relations. * Constant "Set.not_member" now qualified. INCOMPATIBILITY. * Theory Int: Discontinued many legacy theorems specific to type int. INCOMPATIBILITY, use the corresponding generic theorems instead. zminus_zminus ~> minus_minus zminus_0 ~> minus_zero zminus_zadd_distrib ~> minus_add_distrib zadd_commute ~> add_commute zadd_assoc ~> add_assoc zadd_left_commute ~> add_left_commute zadd_ac ~> add_ac zmult_ac ~> mult_ac zadd_0 ~> add_0_left zadd_0_right ~> add_0_right zadd_zminus_inverse2 ~> left_minus zmult_zminus ~> mult_minus_left zmult_commute ~> mult_commute zmult_assoc ~> mult_assoc zadd_zmult_distrib ~> left_distrib zadd_zmult_distrib2 ~> right_distrib zdiff_zmult_distrib ~> left_diff_distrib zdiff_zmult_distrib2 ~> right_diff_distrib zmult_1 ~> mult_1_left zmult_1_right ~> mult_1_right zle_refl ~> order_refl zle_trans ~> order_trans zle_antisym ~> order_antisym zle_linear ~> linorder_linear zless_linear ~> linorder_less_linear zadd_left_mono ~> add_left_mono zadd_strict_right_mono ~> add_strict_right_mono zadd_zless_mono ~> add_less_le_mono int_0_less_1 ~> zero_less_one int_0_neq_1 ~> zero_neq_one zless_le ~> less_le zpower_zadd_distrib ~> power_add zero_less_zpower_abs_iff ~> zero_less_power_abs_iff zero_le_zpower_abs ~> zero_le_power_abs * Theory Deriv: Renamed DERIV_nonneg_imp_nonincreasing ~> DERIV_nonneg_imp_nondecreasing * Theory Library/Multiset: Improved code generation of multisets. * Theory HOL/Library/Set_Algebras: Addition and multiplication on sets are expressed via type classes again. The special syntax \/\ has been replaced by plain +/*. Removed constant setsum_set, which is now subsumed by Big_Operators.setsum. INCOMPATIBILITY. * Theory HOL/Library/Diagonalize has been removed. INCOMPATIBILITY, use theory HOL/Library/Nat_Bijection instead. * Theory HOL/Library/RBT_Impl: Backing implementation of red-black trees is now inside a type class context. Names of affected operations and lemmas have been prefixed by rbt_. INCOMPATIBILITY for theories working directly with raw red-black trees, adapt the names as follows: Operations: bulkload -> rbt_bulkload del_from_left -> rbt_del_from_left del_from_right -> rbt_del_from_right del -> rbt_del delete -> rbt_delete ins -> rbt_ins insert -> rbt_insert insertw -> rbt_insert_with insert_with_key -> rbt_insert_with_key map_entry -> rbt_map_entry lookup -> rbt_lookup sorted -> rbt_sorted tree_greater -> rbt_greater tree_less -> rbt_less tree_less_symbol -> rbt_less_symbol union -> rbt_union union_with -> rbt_union_with union_with_key -> rbt_union_with_key Lemmas: balance_left_sorted -> balance_left_rbt_sorted balance_left_tree_greater -> balance_left_rbt_greater balance_left_tree_less -> balance_left_rbt_less balance_right_sorted -> balance_right_rbt_sorted balance_right_tree_greater -> balance_right_rbt_greater balance_right_tree_less -> balance_right_rbt_less balance_sorted -> balance_rbt_sorted balance_tree_greater -> balance_rbt_greater balance_tree_less -> balance_rbt_less bulkload_is_rbt -> rbt_bulkload_is_rbt combine_sorted -> combine_rbt_sorted combine_tree_greater -> combine_rbt_greater combine_tree_less -> combine_rbt_less delete_in_tree -> rbt_delete_in_tree delete_is_rbt -> rbt_delete_is_rbt del_from_left_tree_greater -> rbt_del_from_left_rbt_greater del_from_left_tree_less -> rbt_del_from_left_rbt_less del_from_right_tree_greater -> rbt_del_from_right_rbt_greater del_from_right_tree_less -> rbt_del_from_right_rbt_less del_in_tree -> rbt_del_in_tree del_inv1_inv2 -> rbt_del_inv1_inv2 del_sorted -> rbt_del_rbt_sorted del_tree_greater -> rbt_del_rbt_greater del_tree_less -> rbt_del_rbt_less dom_lookup_Branch -> dom_rbt_lookup_Branch entries_lookup -> entries_rbt_lookup finite_dom_lookup -> finite_dom_rbt_lookup insert_sorted -> rbt_insert_rbt_sorted insertw_is_rbt -> rbt_insertw_is_rbt insertwk_is_rbt -> rbt_insertwk_is_rbt insertwk_sorted -> rbt_insertwk_rbt_sorted insertw_sorted -> rbt_insertw_rbt_sorted ins_sorted -> ins_rbt_sorted ins_tree_greater -> ins_rbt_greater ins_tree_less -> ins_rbt_less is_rbt_sorted -> is_rbt_rbt_sorted lookup_balance -> rbt_lookup_balance lookup_bulkload -> rbt_lookup_rbt_bulkload lookup_delete -> rbt_lookup_rbt_delete lookup_Empty -> rbt_lookup_Empty lookup_from_in_tree -> rbt_lookup_from_in_tree lookup_in_tree -> rbt_lookup_in_tree lookup_ins -> rbt_lookup_ins lookup_insert -> rbt_lookup_rbt_insert lookup_insertw -> rbt_lookup_rbt_insertw lookup_insertwk -> rbt_lookup_rbt_insertwk lookup_keys -> rbt_lookup_keys lookup_map -> rbt_lookup_map lookup_map_entry -> rbt_lookup_rbt_map_entry lookup_tree_greater -> rbt_lookup_rbt_greater lookup_tree_less -> rbt_lookup_rbt_less lookup_union -> rbt_lookup_rbt_union map_entry_color_of -> rbt_map_entry_color_of map_entry_inv1 -> rbt_map_entry_inv1 map_entry_inv2 -> rbt_map_entry_inv2 map_entry_is_rbt -> rbt_map_entry_is_rbt map_entry_sorted -> rbt_map_entry_rbt_sorted map_entry_tree_greater -> rbt_map_entry_rbt_greater map_entry_tree_less -> rbt_map_entry_rbt_less map_tree_greater -> map_rbt_greater map_tree_less -> map_rbt_less map_sorted -> map_rbt_sorted paint_sorted -> paint_rbt_sorted paint_lookup -> paint_rbt_lookup paint_tree_greater -> paint_rbt_greater paint_tree_less -> paint_rbt_less sorted_entries -> rbt_sorted_entries tree_greater_eq_trans -> rbt_greater_eq_trans tree_greater_nit -> rbt_greater_nit tree_greater_prop -> rbt_greater_prop tree_greater_simps -> rbt_greater_simps tree_greater_trans -> rbt_greater_trans tree_less_eq_trans -> rbt_less_eq_trans tree_less_nit -> rbt_less_nit tree_less_prop -> rbt_less_prop tree_less_simps -> rbt_less_simps tree_less_trans -> rbt_less_trans tree_ord_props -> rbt_ord_props union_Branch -> rbt_union_Branch union_is_rbt -> rbt_union_is_rbt unionw_is_rbt -> rbt_unionw_is_rbt unionwk_is_rbt -> rbt_unionwk_is_rbt unionwk_sorted -> rbt_unionwk_rbt_sorted * Theory HOL/Library/Float: Floating point numbers are now defined as a subset of the real numbers. All operations are defined using the lifing-framework and proofs use the transfer method. INCOMPATIBILITY. Changed Operations: float_abs -> abs float_nprt -> nprt float_pprt -> pprt pow2 -> use powr round_down -> float_round_down round_up -> float_round_up scale -> exponent Removed Operations: ceiling_fl, lb_mult, lb_mod, ub_mult, ub_mod Renamed Lemmas: abs_float_def -> Float.compute_float_abs bitlen_ge0 -> bitlen_nonneg bitlen.simps -> Float.compute_bitlen float_components -> Float_mantissa_exponent float_divl.simps -> Float.compute_float_divl float_divr.simps -> Float.compute_float_divr float_eq_odd -> mult_powr_eq_mult_powr_iff float_power -> real_of_float_power lapprox_posrat_def -> Float.compute_lapprox_posrat lapprox_rat.simps -> Float.compute_lapprox_rat le_float_def' -> Float.compute_float_le le_float_def -> less_eq_float.rep_eq less_float_def' -> Float.compute_float_less less_float_def -> less_float.rep_eq normfloat_def -> Float.compute_normfloat normfloat_imp_odd_or_zero -> mantissa_not_dvd and mantissa_noteq_0 normfloat -> normfloat_def normfloat_unique -> use normfloat_def number_of_float_Float -> Float.compute_float_numeral, Float.compute_float_neg_numeral one_float_def -> Float.compute_float_one plus_float_def -> Float.compute_float_plus rapprox_posrat_def -> Float.compute_rapprox_posrat rapprox_rat.simps -> Float.compute_rapprox_rat real_of_float_0 -> zero_float.rep_eq real_of_float_1 -> one_float.rep_eq real_of_float_abs -> abs_float.rep_eq real_of_float_add -> plus_float.rep_eq real_of_float_minus -> uminus_float.rep_eq real_of_float_mult -> times_float.rep_eq real_of_float_simp -> Float.rep_eq real_of_float_sub -> minus_float.rep_eq round_down.simps -> Float.compute_float_round_down round_up.simps -> Float.compute_float_round_up times_float_def -> Float.compute_float_times uminus_float_def -> Float.compute_float_uminus zero_float_def -> Float.compute_float_zero Lemmas not necessary anymore, use the transfer method: bitlen_B0, bitlen_B1, bitlen_ge1, bitlen_Min, bitlen_Pls, float_divl, float_divr, float_le_simp, float_less1_mantissa_bound, float_less_simp, float_less_zero, float_le_zero, float_pos_less1_e_neg, float_pos_m_pos, float_split, float_split2, floor_pos_exp, lapprox_posrat, lapprox_posrat_bottom, lapprox_rat, lapprox_rat_bottom, normalized_float, rapprox_posrat, rapprox_posrat_le1, rapprox_rat, real_of_float_ge0_exp, real_of_float_neg_exp, real_of_float_nge0_exp, round_down floor_fl, round_up, zero_le_float, zero_less_float * New theory HOL/Library/DAList provides an abstract type for association lists with distinct keys. * Session HOL/IMP: Added new theory of abstract interpretation of annotated commands. * Session HOL-Import: Re-implementation from scratch is faster, simpler, and more scalable. Requires a proof bundle, which is available as an external component. Discontinued old (and mostly dead) Importer for HOL4 and HOL Light. INCOMPATIBILITY. * Session HOL-Word: Discontinued many redundant theorems specific to type 'a word. INCOMPATIBILITY, use the corresponding generic theorems instead. word_sub_alt ~> word_sub_wi word_add_alt ~> word_add_def word_mult_alt ~> word_mult_def word_minus_alt ~> word_minus_def word_0_alt ~> word_0_wi word_1_alt ~> word_1_wi word_add_0 ~> add_0_left word_add_0_right ~> add_0_right word_mult_1 ~> mult_1_left word_mult_1_right ~> mult_1_right word_add_commute ~> add_commute word_add_assoc ~> add_assoc word_add_left_commute ~> add_left_commute word_mult_commute ~> mult_commute word_mult_assoc ~> mult_assoc word_mult_left_commute ~> mult_left_commute word_left_distrib ~> left_distrib word_right_distrib ~> right_distrib word_left_minus ~> left_minus word_diff_0_right ~> diff_0_right word_diff_self ~> diff_self word_sub_def ~> diff_minus word_diff_minus ~> diff_minus word_add_ac ~> add_ac word_mult_ac ~> mult_ac word_plus_ac0 ~> add_0_left add_0_right add_ac word_times_ac1 ~> mult_1_left mult_1_right mult_ac word_order_trans ~> order_trans word_order_refl ~> order_refl word_order_antisym ~> order_antisym word_order_linear ~> linorder_linear lenw1_zero_neq_one ~> zero_neq_one word_number_of_eq ~> number_of_eq word_of_int_add_hom ~> wi_hom_add word_of_int_sub_hom ~> wi_hom_sub word_of_int_mult_hom ~> wi_hom_mult word_of_int_minus_hom ~> wi_hom_neg word_of_int_succ_hom ~> wi_hom_succ word_of_int_pred_hom ~> wi_hom_pred word_of_int_0_hom ~> word_0_wi word_of_int_1_hom ~> word_1_wi * Session HOL-Word: New proof method "word_bitwise" for splitting machine word equalities and inequalities into logical circuits, defined in HOL/Word/WordBitwise.thy. Supports addition, subtraction, multiplication, shifting by constants, bitwise operators and numeric constants. Requires fixed-length word types, not 'a word. Solves many standard word identities outright and converts more into first order problems amenable to blast or similar. See also examples in HOL/Word/Examples/WordExamples.thy. * Session HOL-Probability: Introduced the type "'a measure" to represent measures, this replaces the records 'a algebra and 'a measure_space. The locales based on subset_class now have two locale-parameters the space \ and the set of measurable sets M. The product of probability spaces uses now the same constant as the finite product of sigma-finite measure spaces "PiM :: ('i => 'a) measure". Most constants are defined now outside of locales and gain an additional parameter, like null_sets, almost_eventually or \'. Measure space constructions for distributions and densities now got their own constants distr and density. Instead of using locales to describe measure spaces with a finite space, the measure count_space and point_measure is introduced. INCOMPATIBILITY. Renamed constants: measure -> emeasure finite_measure.\' -> measure product_algebra_generator -> prod_algebra product_prob_space.emb -> prod_emb product_prob_space.infprod_algebra -> PiM Removed locales: completeable_measure_space finite_measure_space finite_prob_space finite_product_finite_prob_space finite_product_sigma_algebra finite_sigma_algebra measure_space pair_finite_prob_space pair_finite_sigma_algebra pair_finite_space pair_sigma_algebra product_sigma_algebra Removed constants: conditional_space distribution -> use distr measure, or distributed predicate image_space joint_distribution -> use distr measure, or distributed predicate pair_measure_generator product_prob_space.infprod_algebra -> use PiM subvimage Replacement theorems: finite_additivity_sufficient -> ring_of_sets.countably_additiveI_finite finite_measure.empty_measure -> measure_empty finite_measure.finite_continuity_from_above -> finite_measure.finite_Lim_measure_decseq finite_measure.finite_continuity_from_below -> finite_measure.finite_Lim_measure_incseq finite_measure.finite_measure_countably_subadditive -> finite_measure.finite_measure_subadditive_countably finite_measure.finite_measure_eq -> finite_measure.emeasure_eq_measure finite_measure.finite_measure -> finite_measure.emeasure_finite finite_measure.finite_measure_finite_singleton -> finite_measure.finite_measure_eq_setsum_singleton finite_measure.positive_measure' -> measure_nonneg finite_measure.real_measure -> finite_measure.emeasure_real finite_product_prob_space.finite_measure_times -> finite_product_prob_space.finite_measure_PiM_emb finite_product_sigma_algebra.in_P -> sets_PiM_I_finite finite_product_sigma_algebra.P_empty -> space_PiM_empty, sets_PiM_empty information_space.conditional_entropy_eq -> information_space.conditional_entropy_simple_distributed information_space.conditional_entropy_positive -> information_space.conditional_entropy_nonneg_simple information_space.conditional_mutual_information_eq_mutual_information -> information_space.conditional_mutual_information_eq_mutual_information_simple information_space.conditional_mutual_information_generic_positive -> information_space.conditional_mutual_information_nonneg_simple information_space.conditional_mutual_information_positive -> information_space.conditional_mutual_information_nonneg_simple information_space.entropy_commute -> information_space.entropy_commute_simple information_space.entropy_eq -> information_space.entropy_simple_distributed information_space.entropy_generic_eq -> information_space.entropy_simple_distributed information_space.entropy_positive -> information_space.entropy_nonneg_simple information_space.entropy_uniform_max -> information_space.entropy_uniform information_space.KL_eq_0_imp -> information_space.KL_eq_0_iff_eq information_space.KL_eq_0 -> information_space.KL_same_eq_0 information_space.KL_ge_0 -> information_space.KL_nonneg information_space.mutual_information_eq -> information_space.mutual_information_simple_distributed information_space.mutual_information_positive -> information_space.mutual_information_nonneg_simple Int_stable_cuboids -> Int_stable_atLeastAtMost Int_stable_product_algebra_generator -> positive_integral measure_preserving -> equality "distr M N f = N" "f : measurable M N" measure_space.additive -> emeasure_additive measure_space.AE_iff_null_set -> AE_iff_null measure_space.almost_everywhere_def -> eventually_ae_filter measure_space.almost_everywhere_vimage -> AE_distrD measure_space.continuity_from_above -> INF_emeasure_decseq measure_space.continuity_from_above_Lim -> Lim_emeasure_decseq measure_space.continuity_from_below_Lim -> Lim_emeasure_incseq measure_space.continuity_from_below -> SUP_emeasure_incseq measure_space_density -> emeasure_density measure_space.density_is_absolutely_continuous -> absolutely_continuousI_density measure_space.integrable_vimage -> integrable_distr measure_space.integral_translated_density -> integral_density measure_space.integral_vimage -> integral_distr measure_space.measure_additive -> plus_emeasure measure_space.measure_compl -> emeasure_compl measure_space.measure_countable_increasing -> emeasure_countable_increasing measure_space.measure_countably_subadditive -> emeasure_subadditive_countably measure_space.measure_decseq -> decseq_emeasure measure_space.measure_Diff -> emeasure_Diff measure_space.measure_Diff_null_set -> emeasure_Diff_null_set measure_space.measure_eq_0 -> emeasure_eq_0 measure_space.measure_finitely_subadditive -> emeasure_subadditive_finite measure_space.measure_finite_singleton -> emeasure_eq_setsum_singleton measure_space.measure_incseq -> incseq_emeasure measure_space.measure_insert -> emeasure_insert measure_space.measure_mono -> emeasure_mono measure_space.measure_not_negative -> emeasure_not_MInf measure_space.measure_preserving_Int_stable -> measure_eqI_generator_eq measure_space.measure_setsum -> setsum_emeasure measure_space.measure_setsum_split -> setsum_emeasure_cover measure_space.measure_space_vimage -> emeasure_distr measure_space.measure_subadditive_finite -> emeasure_subadditive_finite measure_space.measure_subadditive -> subadditive measure_space.measure_top -> emeasure_space measure_space.measure_UN_eq_0 -> emeasure_UN_eq_0 measure_space.measure_Un_null_set -> emeasure_Un_null_set measure_space.positive_integral_translated_density -> positive_integral_density measure_space.positive_integral_vimage -> positive_integral_distr measure_space.real_continuity_from_above -> Lim_measure_decseq measure_space.real_continuity_from_below -> Lim_measure_incseq measure_space.real_measure_countably_subadditive -> measure_subadditive_countably measure_space.real_measure_Diff -> measure_Diff measure_space.real_measure_finite_Union -> measure_finite_Union measure_space.real_measure_setsum_singleton -> measure_eq_setsum_singleton measure_space.real_measure_subadditive -> measure_subadditive measure_space.real_measure_Union -> measure_Union measure_space.real_measure_UNION -> measure_UNION measure_space.simple_function_vimage -> simple_function_comp measure_space.simple_integral_vimage -> simple_integral_distr measure_space.simple_integral_vimage -> simple_integral_distr measure_unique_Int_stable -> measure_eqI_generator_eq measure_unique_Int_stable_vimage -> measure_eqI_generator_eq pair_sigma_algebra.measurable_cut_fst -> sets_Pair1 pair_sigma_algebra.measurable_cut_snd -> sets_Pair2 pair_sigma_algebra.measurable_pair_image_fst -> measurable_Pair1 pair_sigma_algebra.measurable_pair_image_snd -> measurable_Pair2 pair_sigma_algebra.measurable_product_swap -> measurable_pair_swap_iff pair_sigma_algebra.pair_sigma_algebra_measurable -> measurable_pair_swap pair_sigma_algebra.pair_sigma_algebra_swap_measurable -> measurable_pair_swap' pair_sigma_algebra.sets_swap -> sets_pair_swap pair_sigma_finite.measure_cut_measurable_fst -> pair_sigma_finite.measurable_emeasure_Pair1 pair_sigma_finite.measure_cut_measurable_snd -> pair_sigma_finite.measurable_emeasure_Pair2 pair_sigma_finite.measure_preserving_swap -> pair_sigma_finite.distr_pair_swap pair_sigma_finite.pair_measure_alt2 -> pair_sigma_finite.emeasure_pair_measure_alt2 pair_sigma_finite.pair_measure_alt -> pair_sigma_finite.emeasure_pair_measure_alt pair_sigma_finite.pair_measure_times -> pair_sigma_finite.emeasure_pair_measure_Times prob_space.indep_distribution_eq_measure -> prob_space.indep_vars_iff_distr_eq_PiM prob_space.indep_var_distributionD -> prob_space.indep_var_distribution_eq prob_space.measure_space_1 -> prob_space.emeasure_space_1 prob_space.prob_space_vimage -> prob_space_distr prob_space.random_variable_restrict -> measurable_restrict prob_space_unique_Int_stable -> measure_eqI_prob_space product_algebraE -> prod_algebraE_all product_algebra_generator_der -> prod_algebra_eq_finite product_algebra_generator_into_space -> prod_algebra_sets_into_space product_algebraI -> sets_PiM_I_finite product_measure_exists -> product_sigma_finite.sigma_finite product_prob_space.finite_index_eq_finite_product -> product_prob_space.sets_PiM_generator product_prob_space.finite_measure_infprod_emb_Pi -> product_prob_space.measure_PiM_emb product_prob_space.infprod_spec -> product_prob_space.emeasure_PiM_emb_not_empty product_prob_space.measurable_component -> measurable_component_singleton product_prob_space.measurable_emb -> measurable_prod_emb product_prob_space.measurable_into_infprod_algebra -> measurable_PiM_single product_prob_space.measurable_singleton_infprod -> measurable_component_singleton product_prob_space.measure_emb -> emeasure_prod_emb product_prob_space.measure_preserving_restrict -> product_prob_space.distr_restrict product_sigma_algebra.product_algebra_into_space -> space_closed product_sigma_finite.measure_fold -> product_sigma_finite.distr_merge product_sigma_finite.measure_preserving_component_singelton -> product_sigma_finite.distr_singleton product_sigma_finite.measure_preserving_merge -> product_sigma_finite.distr_merge sequence_space.measure_infprod -> sequence_space.measure_PiM_countable sets_product_algebra -> sets_PiM sigma_algebra.measurable_sigma -> measurable_measure_of sigma_finite_measure.disjoint_sigma_finite -> sigma_finite_disjoint sigma_finite_measure.RN_deriv_vimage -> sigma_finite_measure.RN_deriv_distr sigma_product_algebra_sigma_eq -> sigma_prod_algebra_sigma_eq space_product_algebra -> space_PiM * Session HOL-TPTP: support to parse and import TPTP problems (all languages) into Isabelle/HOL. *** FOL *** * New "case_product" attribute (see HOL). *** ZF *** * Greater support for structured proofs involving induction or case analysis. * Much greater use of mathematical symbols. * Removal of many ML theorem bindings. INCOMPATIBILITY. *** ML *** * Antiquotation @{keyword "name"} produces a parser for outer syntax from a minor keyword introduced via theory header declaration. * Antiquotation @{command_spec "name"} produces the Outer_Syntax.command_spec from a major keyword introduced via theory header declaration; it can be passed to Outer_Syntax.command etc. * Local_Theory.define no longer hard-wires default theorem name "foo_def", but retains the binding as given. If that is Binding.empty / Attrib.empty_binding, the result is not registered as user-level fact. The Local_Theory.define_internal variant allows to specify a non-empty name (used for the foundation in the background theory), while omitting the fact binding in the user-context. Potential INCOMPATIBILITY for derived definitional packages: need to specify naming policy for primitive definitions more explicitly. * Renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic. * Antiquotation @{attributes [...]} embeds attribute source representation into the ML text, which is particularly useful with declarations like Local_Theory.note. * Structure Proof_Context follows standard naming scheme. Old ProofContext has been discontinued. INCOMPATIBILITY. * Refined Local_Theory.declaration {syntax, pervasive}, with subtle change of semantics: update is applied to auxiliary local theory context as well. * Modernized some old-style infix operations: addeqcongs ~> Simplifier.add_eqcong deleqcongs ~> Simplifier.del_eqcong addcongs ~> Simplifier.add_cong delcongs ~> Simplifier.del_cong setmksimps ~> Simplifier.set_mksimps setmkcong ~> Simplifier.set_mkcong setmksym ~> Simplifier.set_mksym setmkeqTrue ~> Simplifier.set_mkeqTrue settermless ~> Simplifier.set_termless setsubgoaler ~> Simplifier.set_subgoaler addsplits ~> Splitter.add_split delsplits ~> Splitter.del_split *** System *** * USER_HOME settings variable points to cross-platform user home directory, which coincides with HOME on POSIX systems only. Likewise, the Isabelle path specification "~" now expands to $USER_HOME, instead of former $HOME. A different default for USER_HOME may be set explicitly in shell environment, before Isabelle settings are evaluated. Minor INCOMPATIBILITY: need to adapt Isabelle path where the generic user home was intended. * ISABELLE_HOME_WINDOWS refers to ISABELLE_HOME in windows file name notation, which is useful for the jEdit file browser, for example. * ISABELLE_JDK_HOME settings variable points to JDK with javac and jar (not just JRE). New in Isabelle2011-1 (October 2011) ------------------------------------ *** General *** * Improved Isabelle/jEdit Prover IDE (PIDE), which can be invoked as "isabelle jedit" or "ISABELLE_HOME/Isabelle" on the command line. - Management of multiple theory files directly from the editor buffer store -- bypassing the file-system (no requirement to save files for checking). - Markup of formal entities within the text buffer, with semantic highlighting, tooltips and hyperlinks to jump to defining source positions. - Improved text rendering, with sub/superscripts in the source buffer (including support for copy/paste wrt. output panel, HTML theory output and other non-Isabelle text boxes). - Refined scheduling of proof checking and printing of results, based on interactive editor view. (Note: jEdit folding and narrowing allows to restrict buffer perspectives explicitly.) - Reduced CPU performance requirements, usable on machines with few cores. - Reduced memory requirements due to pruning of unused document versions (garbage collection). See also ~~/src/Tools/jEdit/README.html for further information, including some remaining limitations. * Theory loader: source files are exclusively located via the master directory of each theory node (where the .thy file itself resides). The global load path (such as src/HOL/Library) has been discontinued. Note that the path element ~~ may be used to reference theories in the Isabelle home folder -- for instance, "~~/src/HOL/Library/FuncSet". INCOMPATIBILITY. * Theory loader: source files are identified by content via SHA1 digests. Discontinued former path/modtime identification and optional ISABELLE_FILE_IDENT plugin scripts. * Parallelization of nested Isar proofs is subject to Goal.parallel_proofs_threshold (default 100). See also isabelle usedir option -Q. * Name space: former unsynchronized references are now proper configuration options, with more conventional names: long_names ~> names_long short_names ~> names_short unique_names ~> names_unique Minor INCOMPATIBILITY, need to declare options in context like this: declare [[names_unique = false]] * Literal facts `prop` may contain dummy patterns, e.g. `_ = _`. Note that the result needs to be unique, which means fact specifications may have to be refined after enriching a proof context. * Attribute "case_names" has been refined: the assumptions in each case can be named now by following the case name with [name1 name2 ...]. * Isabelle/Isar reference manual has been updated and extended: - "Synopsis" provides a catalog of main Isar language concepts. - Formal references in syntax diagrams, via @{rail} antiquotation. - Updated material from classic "ref" manual, notably about "Classical Reasoner". *** HOL *** * Class bot and top require underlying partial order rather than preorder: uniqueness of bot and top is guaranteed. INCOMPATIBILITY. * Class complete_lattice: generalized a couple of lemmas from sets; generalized theorems INF_cong and SUP_cong. New type classes for complete boolean algebras and complete linear orders. Lemmas Inf_less_iff, less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder. Changed proposition of lemmas Inf_bool_def, Sup_bool_def, Inf_fun_def, Sup_fun_def, Inf_apply, Sup_apply. Removed redundant lemmas (the right hand side gives hints how to replace them for (metis ...), or (simp only: ...) proofs): Inf_singleton ~> Inf_insert [where A="{}", unfolded Inf_empty inf_top_right] Sup_singleton ~> Sup_insert [where A="{}", unfolded Sup_empty sup_bot_right] Inf_binary ~> Inf_insert, Inf_empty, and inf_top_right Sup_binary ~> Sup_insert, Sup_empty, and sup_bot_right Int_eq_Inter ~> Inf_insert, Inf_empty, and inf_top_right Un_eq_Union ~> Sup_insert, Sup_empty, and sup_bot_right Inter_def ~> INF_def, image_def Union_def ~> SUP_def, image_def INT_eq ~> INF_def, and image_def UN_eq ~> SUP_def, and image_def INF_subset ~> INF_superset_mono [OF _ order_refl] More consistent and comprehensive names: INTER_eq_Inter_image ~> INF_def UNION_eq_Union_image ~> SUP_def INFI_def ~> INF_def SUPR_def ~> SUP_def INF_leI ~> INF_lower INF_leI2 ~> INF_lower2 le_INFI ~> INF_greatest le_SUPI ~> SUP_upper le_SUPI2 ~> SUP_upper2 SUP_leI ~> SUP_least INFI_bool_eq ~> INF_bool_eq SUPR_bool_eq ~> SUP_bool_eq INFI_apply ~> INF_apply SUPR_apply ~> SUP_apply INTER_def ~> INTER_eq UNION_def ~> UNION_eq INCOMPATIBILITY. * Renamed theory Complete_Lattice to Complete_Lattices. INCOMPATIBILITY. * Theory Complete_Lattices: lemmas Inf_eq_top_iff, INF_eq_top_iff, INF_image, Inf_insert, INF_top, Inf_top_conv, INF_top_conv, SUP_bot, Sup_bot_conv, SUP_bot_conv, Sup_eq_top_iff, SUP_eq_top_iff, SUP_image, Sup_insert are now declared as [simp]. INCOMPATIBILITY. * Theory Lattice: lemmas compl_inf_bot, compl_le_comp_iff, compl_sup_top, inf_idem, inf_left_idem, inf_sup_absorb, sup_idem, sup_inf_absob, sup_left_idem are now declared as [simp]. Minor INCOMPATIBILITY. * Added syntactic classes "inf" and "sup" for the respective constants. INCOMPATIBILITY: Changes in the argument order of the (mostly internal) locale predicates for some derived classes. * Theorem collections ball_simps and bex_simps do not contain theorems referring to UNION any longer; these have been moved to collection UN_ball_bex_simps. INCOMPATIBILITY. * Theory Archimedean_Field: floor now is defined as parameter of a separate type class floor_ceiling. * Theory Finite_Set: more coherent development of fold_set locales: locale fun_left_comm ~> locale comp_fun_commute locale fun_left_comm_idem ~> locale comp_fun_idem Both use point-free characterization; interpretation proofs may need adjustment. INCOMPATIBILITY. * Theory Limits: Type "'a net" has been renamed to "'a filter", in accordance with standard mathematical terminology. INCOMPATIBILITY. * Theory Complex_Main: The locale interpretations for the bounded_linear and bounded_bilinear locales have been removed, in order to reduce the number of duplicate lemmas. Users must use the original names for distributivity theorems, potential INCOMPATIBILITY. divide.add ~> add_divide_distrib divide.diff ~> diff_divide_distrib divide.setsum ~> setsum_divide_distrib mult.add_right ~> right_distrib mult.diff_right ~> right_diff_distrib mult_right.setsum ~> setsum_right_distrib mult_left.diff ~> left_diff_distrib * Theory Complex_Main: Several redundant theorems have been removed or replaced by more general versions. INCOMPATIBILITY. real_diff_def ~> minus_real_def real_divide_def ~> divide_real_def real_less_def ~> less_le real_abs_def ~> abs_real_def real_sgn_def ~> sgn_real_def real_mult_commute ~> mult_commute real_mult_assoc ~> mult_assoc real_mult_1 ~> mult_1_left real_add_mult_distrib ~> left_distrib real_zero_not_eq_one ~> zero_neq_one real_mult_inverse_left ~> left_inverse INVERSE_ZERO ~> inverse_zero real_le_refl ~> order_refl real_le_antisym ~> order_antisym real_le_trans ~> order_trans real_le_linear ~> linear real_le_eq_diff ~> le_iff_diff_le_0 real_add_left_mono ~> add_left_mono real_mult_order ~> mult_pos_pos real_mult_less_mono2 ~> mult_strict_left_mono real_of_int_real_of_nat ~> real_of_int_of_nat_eq real_0_le_divide_iff ~> zero_le_divide_iff realpow_two_disj ~> power2_eq_iff real_squared_diff_one_factored ~> square_diff_one_factored realpow_two_diff ~> square_diff_square_factored reals_complete2 ~> complete_real real_sum_squared_expand ~> power2_sum exp_ln_eq ~> ln_unique expi_add ~> exp_add expi_zero ~> exp_zero lemma_DERIV_subst ~> DERIV_cong LIMSEQ_Zfun_iff ~> tendsto_Zfun_iff LIMSEQ_const ~> tendsto_const LIMSEQ_norm ~> tendsto_norm LIMSEQ_add ~> tendsto_add LIMSEQ_minus ~> tendsto_minus LIMSEQ_minus_cancel ~> tendsto_minus_cancel LIMSEQ_diff ~> tendsto_diff bounded_linear.LIMSEQ ~> bounded_linear.tendsto bounded_bilinear.LIMSEQ ~> bounded_bilinear.tendsto LIMSEQ_mult ~> tendsto_mult LIMSEQ_inverse ~> tendsto_inverse LIMSEQ_divide ~> tendsto_divide LIMSEQ_pow ~> tendsto_power LIMSEQ_setsum ~> tendsto_setsum LIMSEQ_setprod ~> tendsto_setprod LIMSEQ_norm_zero ~> tendsto_norm_zero_iff LIMSEQ_rabs_zero ~> tendsto_rabs_zero_iff LIMSEQ_imp_rabs ~> tendsto_rabs LIMSEQ_add_minus ~> tendsto_add [OF _ tendsto_minus] LIMSEQ_add_const ~> tendsto_add [OF _ tendsto_const] LIMSEQ_diff_const ~> tendsto_diff [OF _ tendsto_const] LIMSEQ_Complex ~> tendsto_Complex LIM_ident ~> tendsto_ident_at LIM_const ~> tendsto_const LIM_add ~> tendsto_add LIM_add_zero ~> tendsto_add_zero LIM_minus ~> tendsto_minus LIM_diff ~> tendsto_diff LIM_norm ~> tendsto_norm LIM_norm_zero ~> tendsto_norm_zero LIM_norm_zero_cancel ~> tendsto_norm_zero_cancel LIM_norm_zero_iff ~> tendsto_norm_zero_iff LIM_rabs ~> tendsto_rabs LIM_rabs_zero ~> tendsto_rabs_zero LIM_rabs_zero_cancel ~> tendsto_rabs_zero_cancel LIM_rabs_zero_iff ~> tendsto_rabs_zero_iff LIM_compose ~> tendsto_compose LIM_mult ~> tendsto_mult LIM_scaleR ~> tendsto_scaleR LIM_of_real ~> tendsto_of_real LIM_power ~> tendsto_power LIM_inverse ~> tendsto_inverse LIM_sgn ~> tendsto_sgn isCont_LIM_compose ~> isCont_tendsto_compose bounded_linear.LIM ~> bounded_linear.tendsto bounded_linear.LIM_zero ~> bounded_linear.tendsto_zero bounded_bilinear.LIM ~> bounded_bilinear.tendsto bounded_bilinear.LIM_prod_zero ~> bounded_bilinear.tendsto_zero bounded_bilinear.LIM_left_zero ~> bounded_bilinear.tendsto_left_zero bounded_bilinear.LIM_right_zero ~> bounded_bilinear.tendsto_right_zero LIM_inverse_fun ~> tendsto_inverse [OF tendsto_ident_at] * Theory Complex_Main: The definition of infinite series was generalized. Now it is defined on the type class {topological_space, comm_monoid_add}. Hence it is useable also for extended real numbers. * Theory Complex_Main: The complex exponential function "expi" is now a type-constrained abbreviation for "exp :: complex => complex"; thus several polymorphic lemmas about "exp" are now applicable to "expi". * Code generation: - Theory Library/Code_Char_ord provides native ordering of characters in the target language. - Commands code_module and code_library are legacy, use export_code instead. - Method "evaluation" is legacy, use method "eval" instead. - Legacy evaluator "SML" is deactivated by default. May be reactivated by the following theory command: setup {* Value.add_evaluator ("SML", Codegen.eval_term) *} * Declare ext [intro] by default. Rare INCOMPATIBILITY. * New proof method "induction" that gives induction hypotheses the name "IH", thus distinguishing them from further hypotheses that come from rule induction. The latter are still called "hyps". Method "induction" is a thin wrapper around "induct" and follows the same syntax. * Method "fastsimp" has been renamed to "fastforce", but "fastsimp" is still available as a legacy feature for some time. * Nitpick: - Added "need" and "total_consts" options. - Reintroduced "show_skolems" option by popular demand. - Renamed attribute: nitpick_def ~> nitpick_unfold. INCOMPATIBILITY. * Sledgehammer: - Use quasi-sound (and efficient) translations by default. - Added support for the following provers: E-ToFoF, LEO-II, Satallax, SNARK, Waldmeister, and Z3 with TPTP syntax. - Automatically preplay and minimize proofs before showing them if this can be done within reasonable time. - sledgehammer available_provers ~> sledgehammer supported_provers. INCOMPATIBILITY. - Added "preplay_timeout", "slicing", "type_enc", "sound", "max_mono_iters", and "max_new_mono_instances" options. - Removed "explicit_apply" and "full_types" options as well as "Full Types" Proof General menu item. INCOMPATIBILITY. * Metis: - Removed "metisF" -- use "metis" instead. INCOMPATIBILITY. - Obsoleted "metisFT" -- use "metis (full_types)" instead. INCOMPATIBILITY. * Command 'try': - Renamed 'try_methods' and added "simp:", "intro:", "dest:", and "elim:" options. INCOMPATIBILITY. - Introduced 'try' that not only runs 'try_methods' but also 'solve_direct', 'sledgehammer', 'quickcheck', and 'nitpick'. * Quickcheck: - Added "eval" option to evaluate terms for the found counterexample (currently only supported by the default (exhaustive) tester). - Added post-processing of terms to obtain readable counterexamples (currently only supported by the default (exhaustive) tester). - New counterexample generator quickcheck[narrowing] enables narrowing-based testing. Requires the Glasgow Haskell compiler with its installation location defined in the Isabelle settings environment as ISABELLE_GHC. - Removed quickcheck tester "SML" based on the SML code generator (formly in HOL/Library). * Function package: discontinued option "tailrec". INCOMPATIBILITY, use 'partial_function' instead. * Theory Library/Extended_Reals replaces now the positive extended reals found in probability theory. This file is extended by Multivariate_Analysis/Extended_Real_Limits. * Theory Library/Old_Recdef: old 'recdef' package has been moved here, from where it must be imported explicitly if it is really required. INCOMPATIBILITY. * Theory Library/Wfrec: well-founded recursion combinator "wfrec" has been moved here. INCOMPATIBILITY. * Theory Library/Saturated provides type of numbers with saturated arithmetic. * Theory Library/Product_Lattice defines a pointwise ordering for the product type 'a * 'b, and provides instance proofs for various order and lattice type classes. * Theory Library/Countable now provides the "countable_datatype" proof method for proving "countable" class instances for datatypes. * Theory Library/Cset_Monad allows do notation for computable sets (cset) via the generic monad ad-hoc overloading facility. * Library: Theories of common data structures are split into theories for implementation, an invariant-ensuring type, and connection to an abstract type. INCOMPATIBILITY. - RBT is split into RBT and RBT_Mapping. - AssocList is split and renamed into AList and AList_Mapping. - DList is split into DList_Impl, DList, and DList_Cset. - Cset is split into Cset and List_Cset. * Theory Library/Nat_Infinity has been renamed to Library/Extended_Nat, with name changes of the following types and constants: type inat ~> type enat Fin ~> enat Infty ~> infinity (overloaded) iSuc ~> eSuc the_Fin ~> the_enat Every theorem name containing "inat", "Fin", "Infty", or "iSuc" has been renamed accordingly. INCOMPATIBILITY. * Session Multivariate_Analysis: The euclidean_space type class now fixes a constant "Basis :: 'a set" consisting of the standard orthonormal basis for the type. Users now have the option of quantifying over this set instead of using the "basis" function, e.g. "ALL x:Basis. P x" vs "ALL i vec_eq_iff dist_nth_le_cart ~> dist_vec_nth_le tendsto_vector ~> vec_tendstoI Cauchy_vector ~> vec_CauchyI * Session Multivariate_Analysis: Several duplicate theorems have been removed, and other theorems have been renamed or replaced with more general versions. INCOMPATIBILITY. finite_choice ~> finite_set_choice eventually_conjI ~> eventually_conj eventually_and ~> eventually_conj_iff eventually_false ~> eventually_False setsum_norm ~> norm_setsum Lim_sequentially ~> LIMSEQ_def Lim_ident_at ~> LIM_ident Lim_const ~> tendsto_const Lim_cmul ~> tendsto_scaleR [OF tendsto_const] Lim_neg ~> tendsto_minus Lim_add ~> tendsto_add Lim_sub ~> tendsto_diff Lim_mul ~> tendsto_scaleR Lim_vmul ~> tendsto_scaleR [OF _ tendsto_const] Lim_null_norm ~> tendsto_norm_zero_iff [symmetric] Lim_linear ~> bounded_linear.tendsto Lim_component ~> tendsto_euclidean_component Lim_component_cart ~> tendsto_vec_nth Lim_inner ~> tendsto_inner [OF tendsto_const] dot_lsum ~> inner_setsum_left dot_rsum ~> inner_setsum_right continuous_cmul ~> continuous_scaleR [OF continuous_const] continuous_neg ~> continuous_minus continuous_sub ~> continuous_diff continuous_vmul ~> continuous_scaleR [OF _ continuous_const] continuous_mul ~> continuous_scaleR continuous_inv ~> continuous_inverse continuous_at_within_inv ~> continuous_at_within_inverse continuous_at_inv ~> continuous_at_inverse continuous_at_norm ~> continuous_norm [OF continuous_at_id] continuous_at_infnorm ~> continuous_infnorm [OF continuous_at_id] continuous_at_component ~> continuous_component [OF continuous_at_id] continuous_on_neg ~> continuous_on_minus continuous_on_sub ~> continuous_on_diff continuous_on_cmul ~> continuous_on_scaleR [OF continuous_on_const] continuous_on_vmul ~> continuous_on_scaleR [OF _ continuous_on_const] continuous_on_mul ~> continuous_on_scaleR continuous_on_mul_real ~> continuous_on_mult continuous_on_inner ~> continuous_on_inner [OF continuous_on_const] continuous_on_norm ~> continuous_on_norm [OF continuous_on_id] continuous_on_inverse ~> continuous_on_inv uniformly_continuous_on_neg ~> uniformly_continuous_on_minus uniformly_continuous_on_sub ~> uniformly_continuous_on_diff subset_interior ~> interior_mono subset_closure ~> closure_mono closure_univ ~> closure_UNIV real_arch_lt ~> reals_Archimedean2 real_arch ~> reals_Archimedean3 real_abs_norm ~> abs_norm_cancel real_abs_sub_norm ~> norm_triangle_ineq3 norm_cauchy_schwarz_abs ~> Cauchy_Schwarz_ineq2 * Session HOL-Probability: - Caratheodory's extension lemma is now proved for ring_of_sets. - Infinite products of probability measures are now available. - Sigma closure is independent, if the generator is independent - Use extended reals instead of positive extended reals. INCOMPATIBILITY. * Session HOLCF: Discontinued legacy theorem names, INCOMPATIBILITY. expand_fun_below ~> fun_below_iff below_fun_ext ~> fun_belowI expand_cfun_eq ~> cfun_eq_iff ext_cfun ~> cfun_eqI expand_cfun_below ~> cfun_below_iff below_cfun_ext ~> cfun_belowI monofun_fun_fun ~> fun_belowD monofun_fun_arg ~> monofunE monofun_lub_fun ~> adm_monofun [THEN admD] cont_lub_fun ~> adm_cont [THEN admD] cont2cont_Rep_CFun ~> cont2cont_APP cont_Rep_CFun_app ~> cont_APP_app cont_Rep_CFun_app_app ~> cont_APP_app_app cont_cfun_fun ~> cont_Rep_cfun1 [THEN contE] cont_cfun_arg ~> cont_Rep_cfun2 [THEN contE] contlub_cfun ~> lub_APP [symmetric] contlub_LAM ~> lub_LAM [symmetric] thelubI ~> lub_eqI UU_I ~> bottomI lift_distinct1 ~> lift.distinct(1) lift_distinct2 ~> lift.distinct(2) Def_not_UU ~> lift.distinct(2) Def_inject ~> lift.inject below_UU_iff ~> below_bottom_iff eq_UU_iff ~> eq_bottom_iff *** Document preparation *** * Antiquotation @{rail} layouts railroad syntax diagrams, see also isar-ref manual, both for description and actual application of the same. * Antiquotation @{value} evaluates the given term and presents its result. * Antiquotations: term style "isub" provides ad-hoc conversion of variables x1, y23 into subscripted form x\<^isub>1, y\<^isub>2\<^isub>3. * Predefined LaTeX macros for Isabelle symbols \ and \ (e.g. see ~~/src/HOL/Library/Monad_Syntax.thy). * Localized \isabellestyle switch can be used within blocks or groups like this: \isabellestyle{it} %preferred default {\isabellestylett @{text "typewriter stuff"}} * Discontinued special treatment of hard tabulators. Implicit tab-width is now defined as 1. Potential INCOMPATIBILITY for visual layouts. *** ML *** * The inner syntax of sort/type/term/prop supports inlined YXML representations within quoted string tokens. By encoding logical entities via Term_XML (in ML or Scala) concrete syntax can be bypassed, which is particularly useful for producing bits of text under external program control. * Antiquotations for ML and document preparation are managed as theory data, which requires explicit setup. * Isabelle_Process.is_active allows tools to check if the official process wrapper is running (Isabelle/Scala/jEdit) or the old TTY loop (better known as Proof General). * Structure Proof_Context follows standard naming scheme. Old ProofContext is still available for some time as legacy alias. * Structure Timing provides various operations for timing; supersedes former start_timing/end_timing etc. * Path.print is the official way to show file-system paths to users (including quotes etc.). * Inner syntax: identifiers in parse trees of generic categories "logic", "aprop", "idt" etc. carry position information (disguised as type constraints). Occasional INCOMPATIBILITY with non-compliant translations that choke on unexpected type constraints. Positions can be stripped in ML translations via Syntax.strip_positions / Syntax.strip_positions_ast, or via the syntax constant "_strip_positions" within parse trees. As last resort, positions can be disabled via the configuration option Syntax.positions, which is called "syntax_positions" in Isar attribute syntax. * Discontinued special status of various ML structures that contribute to structure Syntax (Ast, Lexicon, Mixfix, Parser, Printer etc.): less pervasive content, no inclusion in structure Syntax. INCOMPATIBILITY, refer directly to Ast.Constant, Lexicon.is_identifier, Syntax_Trans.mk_binder_tr etc. * Typed print translation: discontinued show_sorts argument, which is already available via context of "advanced" translation. * Refined PARALLEL_GOALS tactical: degrades gracefully for schematic goal states; body tactic needs to address all subgoals uniformly. * Slightly more special eq_list/eq_set, with shortcut involving pointer equality (assumes that eq relation is reflexive). * Classical tactics use proper Proof.context instead of historic types claset/clasimpset. Old-style declarations like addIs, addEs, addDs operate directly on Proof.context. Raw type claset retains its use as snapshot of the classical context, which can be recovered via (put_claset HOL_cs) etc. Type clasimpset has been discontinued. INCOMPATIBILITY, classical tactics and derived proof methods require proper Proof.context. *** System *** * Discontinued support for Poly/ML 5.2, which was the last version without proper multithreading and TimeLimit implementation. * Discontinued old lib/scripts/polyml-platform, which has been obsolete since Isabelle2009-2. * Various optional external tools are referenced more robustly and uniformly by explicit Isabelle settings as follows: ISABELLE_CSDP (formerly CSDP_EXE) ISABELLE_GHC (formerly EXEC_GHC or GHC_PATH) ISABELLE_OCAML (formerly EXEC_OCAML) ISABELLE_SWIPL (formerly EXEC_SWIPL) ISABELLE_YAP (formerly EXEC_YAP) Note that automated detection from the file-system or search path has been discontinued. INCOMPATIBILITY. * Scala layer provides JVM method invocation service for static methods of type (String)String, see Invoke_Scala.method in ML. For example: Invoke_Scala.method "java.lang.System.getProperty" "java.home" Together with YXML.string_of_body/parse_body and XML.Encode/Decode this allows to pass structured values between ML and Scala. * The IsabelleText fonts includes some further glyphs to support the Prover IDE. Potential INCOMPATIBILITY: users who happen to have installed a local copy (which is normally *not* required) need to delete or update it from ~~/lib/fonts/. New in Isabelle2011 (January 2011) ---------------------------------- *** General *** * Experimental Prover IDE based on Isabelle/Scala and jEdit (see src/Tools/jEdit). This also serves as IDE for Isabelle/ML, with useful tooltips and hyperlinks produced from its static analysis. The bundled component provides an executable Isabelle tool that can be run like this: Isabelle2011/bin/isabelle jedit * Significantly improved Isabelle/Isar implementation manual. * System settings: ISABELLE_HOME_USER now includes ISABELLE_IDENTIFIER (and thus refers to something like $HOME/.isabelle/Isabelle2011), while the default heap location within that directory lacks that extra suffix. This isolates multiple Isabelle installations from each other, avoiding problems with old settings in new versions. INCOMPATIBILITY, need to copy/upgrade old user settings manually. * Source files are always encoded as UTF-8, instead of old-fashioned ISO-Latin-1. INCOMPATIBILITY. Isabelle LaTeX documents might require the following package declarations: \usepackage[utf8]{inputenc} \usepackage{textcomp} * Explicit treatment of UTF-8 sequences as Isabelle symbols, such that a Unicode character is treated as a single symbol, not a sequence of non-ASCII bytes as before. Since Isabelle/ML string literals may contain symbols without further backslash escapes, Unicode can now be used here as well. Recall that Symbol.explode in ML provides a consistent view on symbols, while raw explode (or String.explode) merely give a byte-oriented representation. * Theory loader: source files are primarily located via the master directory of each theory node (where the .thy file itself resides). The global load path is still partially available as legacy feature. Minor INCOMPATIBILITY due to subtle change in file lookup: use explicit paths, relatively to the theory. * Special treatment of ML file names has been discontinued. Historically, optional extensions .ML or .sml were added on demand -- at the cost of clarity of file dependencies. Recall that Isabelle/ML files exclusively use the .ML extension. Minor INCOMPATIBILITY. * Various options that affect pretty printing etc. are now properly handled within the context via configuration options, instead of unsynchronized references or print modes. There are both ML Config.T entities and Isar declaration attributes to access these. ML (Config.T) Isar (attribute) eta_contract eta_contract show_brackets show_brackets show_sorts show_sorts show_types show_types show_question_marks show_question_marks show_consts show_consts show_abbrevs show_abbrevs Syntax.ast_trace syntax_ast_trace Syntax.ast_stat syntax_ast_stat Syntax.ambiguity_level syntax_ambiguity_level Goal_Display.goals_limit goals_limit Goal_Display.show_main_goal show_main_goal Method.rule_trace rule_trace Thy_Output.display thy_output_display Thy_Output.quotes thy_output_quotes Thy_Output.indent thy_output_indent Thy_Output.source thy_output_source Thy_Output.break thy_output_break Note that corresponding "..._default" references in ML may only be changed globally at the ROOT session setup, but *not* within a theory. The option "show_abbrevs" supersedes the former print mode "no_abbrevs" with inverted meaning. * More systematic naming of some configuration options. INCOMPATIBILITY. trace_simp ~> simp_trace debug_simp ~> simp_debug * Support for real valued configuration options, using simplistic floating-point notation that coincides with the inner syntax for float_token. * Support for real valued preferences (with approximative PGIP type): front-ends need to accept "pgint" values in float notation. INCOMPATIBILITY. * The IsabelleText font now includes Cyrillic, Hebrew, Arabic from DejaVu Sans. * Discontinued support for Poly/ML 5.0 and 5.1 versions. *** Pure *** * Command 'type_synonym' (with single argument) replaces somewhat outdated 'types', which is still available as legacy feature for some time. * Command 'nonterminal' (with 'and' separated list of arguments) replaces somewhat outdated 'nonterminals'. INCOMPATIBILITY. * Command 'notepad' replaces former 'example_proof' for experimentation in Isar without any result. INCOMPATIBILITY. * Locale interpretation commands 'interpret' and 'sublocale' accept lists of equations to map definitions in a locale to appropriate entities in the context of the interpretation. The 'interpretation' command already provided this functionality. * Diagnostic command 'print_dependencies' prints the locale instances that would be activated if the specified expression was interpreted in the current context. Variant "print_dependencies!" assumes a context without interpretations. * Diagnostic command 'print_interps' prints interpretations in proofs in addition to interpretations in theories. * Discontinued obsolete 'global' and 'local' commands to manipulate the theory name space. Rare INCOMPATIBILITY. The ML functions Sign.root_path and Sign.local_path may be applied directly where this feature is still required for historical reasons. * Discontinued obsolete 'constdefs' command. INCOMPATIBILITY, use 'definition' instead. * The "prems" fact, which refers to the accidental collection of foundational premises in the context, is now explicitly marked as legacy feature and will be discontinued soon. Consider using "assms" of the head statement or reference facts by explicit names. * Document antiquotations @{class} and @{type} print classes and type constructors. * Document antiquotation @{file} checks file/directory entries within the local file system. *** HOL *** * Coercive subtyping: functions can be declared as coercions and type inference will add them as necessary upon input of a term. Theory Complex_Main declares real :: nat => real and real :: int => real as coercions. A coercion function f is declared like this: declare [[coercion f]] To lift coercions through type constructors (e.g. from nat => real to nat list => real list), map functions can be declared, e.g. declare [[coercion_map map]] Currently coercion inference is activated only in theories including real numbers, i.e. descendants of Complex_Main. This is controlled by the configuration option "coercion_enabled", e.g. it can be enabled in other theories like this: declare [[coercion_enabled]] * Command 'partial_function' provides basic support for recursive function definitions over complete partial orders. Concrete instances are provided for i) the option type, ii) tail recursion on arbitrary types, and iii) the heap monad of Imperative_HOL. See src/HOL/ex/Fundefs.thy and src/HOL/Imperative_HOL/ex/Linked_Lists.thy for examples. * Function package: f.psimps rules are no longer implicitly declared as [simp]. INCOMPATIBILITY. * Datatype package: theorems generated for executable equality (class "eq") carry proper names and are treated as default code equations. * Inductive package: now offers command 'inductive_simps' to automatically derive instantiated and simplified equations for inductive predicates, similar to 'inductive_cases'. * Command 'enriched_type' allows to register properties of the functorial structure of types. * Improved infrastructure for term evaluation using code generator techniques, in particular static evaluation conversions. * Code generator: Scala (2.8 or higher) has been added to the target languages. * Code generator: globbing constant expressions "*" and "Theory.*" have been replaced by the more idiomatic "_" and "Theory._". INCOMPATIBILITY. * Code generator: export_code without explicit file declaration prints to standard output. INCOMPATIBILITY. * Code generator: do not print function definitions for case combinators any longer. * Code generator: simplification with rules determined with src/Tools/Code/code_simp.ML and method "code_simp". * Code generator for records: more idiomatic representation of record types. Warning: records are not covered by ancient SML code generation any longer. INCOMPATIBILITY. In cases of need, a suitable rep_datatype declaration helps to succeed then: record 'a foo = ... ... rep_datatype foo_ext ... * Records: logical foundation type for records does not carry a '_type' suffix any longer (obsolete due to authentic syntax). INCOMPATIBILITY. * Quickcheck now by default uses exhaustive testing instead of random testing. Random testing can be invoked by "quickcheck [random]", exhaustive testing by "quickcheck [exhaustive]". * Quickcheck instantiates polymorphic types with small finite datatypes by default. This enables a simple execution mechanism to handle quantifiers and function equality over the finite datatypes. * Quickcheck random generator has been renamed from "code" to "random". INCOMPATIBILITY. * Quickcheck now has a configurable time limit which is set to 30 seconds by default. This can be changed by adding [timeout = n] to the quickcheck command. The time limit for Auto Quickcheck is still set independently. * Quickcheck in locales considers interpretations of that locale for counter example search. * Sledgehammer: - Added "smt" and "remote_smt" provers based on the "smt" proof method. See the Sledgehammer manual for details ("isabelle doc sledgehammer"). - Renamed commands: sledgehammer atp_info ~> sledgehammer running_provers sledgehammer atp_kill ~> sledgehammer kill_provers sledgehammer available_atps ~> sledgehammer available_provers INCOMPATIBILITY. - Renamed options: sledgehammer [atps = ...] ~> sledgehammer [provers = ...] sledgehammer [atp = ...] ~> sledgehammer [prover = ...] sledgehammer [timeout = 77 s] ~> sledgehammer [timeout = 77] (and "ms" and "min" are no longer supported) INCOMPATIBILITY. * Nitpick: - Renamed options: nitpick [timeout = 77 s] ~> nitpick [timeout = 77] nitpick [tac_timeout = 777 ms] ~> nitpick [tac_timeout = 0.777] INCOMPATIBILITY. - Added support for partial quotient types. - Added local versions of the "Nitpick.register_xxx" functions. - Added "whack" option. - Allow registration of quotient types as codatatypes. - Improved "merge_type_vars" option to merge more types. - Removed unsound "fast_descrs" option. - Added custom symmetry breaking for datatypes, making it possible to reach higher cardinalities. - Prevent the expansion of too large definitions. * Proof methods "metis" and "meson" now have configuration options "meson_trace", "metis_trace", and "metis_verbose" that can be enabled to diagnose these tools. E.g. using [[metis_trace = true]] * Auto Solve: Renamed "Auto Solve Direct". The tool is now available manually as command 'solve_direct'. * The default SMT solver Z3 must be enabled explicitly (due to licensing issues) by setting the environment variable Z3_NON_COMMERCIAL in etc/settings of the component, for example. For commercial applications, the SMT solver CVC3 is provided as fall-back; changing the SMT solver is done via the configuration option "smt_solver". * Remote SMT solvers need to be referred to by the "remote_" prefix, i.e. "remote_cvc3" and "remote_z3". * Added basic SMT support for datatypes, records, and typedefs using the oracle mode (no proofs). Direct support of pairs has been dropped in exchange (pass theorems fst_conv snd_conv pair_collapse to the SMT support for a similar behavior). Minor INCOMPATIBILITY. * Changed SMT configuration options: - Renamed: z3_proofs ~> smt_oracle (with inverted meaning) z3_trace_assms ~> smt_trace_used_facts INCOMPATIBILITY. - Added: smt_verbose smt_random_seed smt_datatypes smt_infer_triggers smt_monomorph_limit cvc3_options remote_cvc3_options remote_z3_options yices_options * Boogie output files (.b2i files) need to be declared in the theory header. * Simplification procedure "list_to_set_comprehension" rewrites list comprehensions applied to List.set to set comprehensions. Occasional INCOMPATIBILITY, may be deactivated like this: declare [[simproc del: list_to_set_comprehension]] * Removed old version of primrec package. INCOMPATIBILITY. * Removed simplifier congruence rule of "prod_case", as has for long been the case with "split". INCOMPATIBILITY. * String.literal is a type, but not a datatype. INCOMPATIBILITY. * Removed [split_format ... and ... and ...] version of [split_format]. Potential INCOMPATIBILITY. * Predicate "sorted" now defined inductively, with nice induction rules. INCOMPATIBILITY: former sorted.simps now named sorted_simps. * Constant "contents" renamed to "the_elem", to free the generic name contents for other uses. INCOMPATIBILITY. * Renamed class eq and constant eq (for code generation) to class equal and constant equal, plus renaming of related facts and various tuning. INCOMPATIBILITY. * Dropped type classes mult_mono and mult_mono1. INCOMPATIBILITY. * Removed output syntax "'a ~=> 'b" for "'a => 'b option". INCOMPATIBILITY. * Renamed theory Fset to Cset, type Fset.fset to Cset.set, in order to avoid confusion with finite sets. INCOMPATIBILITY. * Abandoned locales equiv, congruent and congruent2 for equivalence relations. INCOMPATIBILITY: use equivI rather than equiv_intro (same for congruent(2)). * Some previously unqualified names have been qualified: types bool ~> HOL.bool nat ~> Nat.nat constants Trueprop ~> HOL.Trueprop True ~> HOL.True False ~> HOL.False op & ~> HOL.conj op | ~> HOL.disj op --> ~> HOL.implies op = ~> HOL.eq Not ~> HOL.Not The ~> HOL.The All ~> HOL.All Ex ~> HOL.Ex Ex1 ~> HOL.Ex1 Let ~> HOL.Let If ~> HOL.If Ball ~> Set.Ball Bex ~> Set.Bex Suc ~> Nat.Suc Pair ~> Product_Type.Pair fst ~> Product_Type.fst snd ~> Product_Type.snd curry ~> Product_Type.curry op : ~> Set.member Collect ~> Set.Collect INCOMPATIBILITY. * More canonical naming convention for some fundamental definitions: bot_bool_eq ~> bot_bool_def top_bool_eq ~> top_bool_def inf_bool_eq ~> inf_bool_def sup_bool_eq ~> sup_bool_def bot_fun_eq ~> bot_fun_def top_fun_eq ~> top_fun_def inf_fun_eq ~> inf_fun_def sup_fun_eq ~> sup_fun_def INCOMPATIBILITY. * More stylized fact names: expand_fun_eq ~> fun_eq_iff expand_set_eq ~> set_eq_iff set_ext ~> set_eqI nat_number ~> eval_nat_numeral INCOMPATIBILITY. * Refactoring of code-generation specific operations in theory List: constants null ~> List.null facts mem_iff ~> member_def null_empty ~> null_def INCOMPATIBILITY. Note that these were not supposed to be used regularly unless for striking reasons; their main purpose was code generation. Various operations from the Haskell prelude are used for generating Haskell code. * Term "bij f" is now an abbreviation of "bij_betw f UNIV UNIV". Term "surj f" is now an abbreviation of "range f = UNIV". The theorems bij_def and surj_def are unchanged. INCOMPATIBILITY. * Abolished some non-alphabetic type names: "prod" and "sum" replace "*" and "+" respectively. INCOMPATIBILITY. * Name "Plus" of disjoint sum operator "<+>" is now hidden. Write "Sum_Type.Plus" instead. * Constant "split" has been merged with constant "prod_case"; names of ML functions, facts etc. involving split have been retained so far, though. INCOMPATIBILITY. * Dropped old infix syntax "_ mem _" for List.member; use "_ : set _" instead. INCOMPATIBILITY. * Removed lemma "Option.is_none_none" which duplicates "is_none_def". INCOMPATIBILITY. * Former theory Library/Enum is now part of the HOL-Main image. INCOMPATIBILITY: all constants of the Enum theory now have to be referred to by its qualified name. enum ~> Enum.enum nlists ~> Enum.nlists product ~> Enum.product * Theory Library/Monad_Syntax provides do-syntax for monad types. Syntax in Library/State_Monad has been changed to avoid ambiguities. INCOMPATIBILITY. * Theory Library/SetsAndFunctions has been split into Library/Function_Algebras and Library/Set_Algebras; canonical names for instance definitions for functions; various improvements. INCOMPATIBILITY. * Theory Library/Multiset provides stable quicksort implementation of sort_key. * Theory Library/Multiset: renamed empty_idemp ~> empty_neutral. INCOMPATIBILITY. * Session Multivariate_Analysis: introduced a type class for euclidean space. Most theorems are now stated in terms of euclidean spaces instead of finite cartesian products. types real ^ 'n ~> 'a::real_vector ~> 'a::euclidean_space ~> 'a::ordered_euclidean_space (depends on your needs) constants _ $ _ ~> _ $$ _ \ x. _ ~> \\ x. _ CARD('n) ~> DIM('a) Also note that the indices are now natural numbers and not from some finite type. Finite cartesian products of euclidean spaces, products of euclidean spaces the real and complex numbers are instantiated to be euclidean_spaces. INCOMPATIBILITY. * Session Probability: introduced pextreal as positive extended real numbers. Use pextreal as value for measures. Introduce the Radon-Nikodym derivative, product spaces and Fubini's theorem for arbitrary sigma finite measures. Introduces Lebesgue measure based on the integral in Multivariate Analysis. INCOMPATIBILITY. * Session Imperative_HOL: revamped, corrected dozens of inadequacies. INCOMPATIBILITY. * Session SPARK (with image HOL-SPARK) provides commands to load and prove verification conditions generated by the SPARK Ada program verifier. See also src/HOL/SPARK and src/HOL/SPARK/Examples. *** HOL-Algebra *** * Theorems for additive ring operations (locale abelian_monoid and descendants) are generated by interpretation from their multiplicative counterparts. Names (in particular theorem names) have the mandatory qualifier 'add'. Previous theorem names are redeclared for compatibility. * Structure "int_ring" is now an abbreviation (previously a definition). This fits more natural with advanced interpretations. *** HOLCF *** * The domain package now runs in definitional mode by default: The former command 'new_domain' is now called 'domain'. To use the domain package in its original axiomatic mode, use 'domain (unsafe)'. INCOMPATIBILITY. * The new class "domain" is now the default sort. Class "predomain" is an unpointed version of "domain". Theories can be updated by replacing sort annotations as shown below. INCOMPATIBILITY. 'a::type ~> 'a::countable 'a::cpo ~> 'a::predomain 'a::pcpo ~> 'a::domain * The old type class "rep" has been superseded by class "domain". Accordingly, users of the definitional package must remove any "default_sort rep" declarations. INCOMPATIBILITY. * The domain package (definitional mode) now supports unpointed predomain argument types, as long as they are marked 'lazy'. (Strict arguments must be in class "domain".) For example, the following domain definition now works: domain natlist = nil | cons (lazy "nat discr") (lazy "natlist") * Theory HOLCF/Library/HOL_Cpo provides cpo and predomain class instances for types from main HOL: bool, nat, int, char, 'a + 'b, 'a option, and 'a list. Additionally, it configures fixrec and the domain package to work with these types. For example: fixrec isInl :: "('a + 'b) u -> tr" where "isInl$(up$(Inl x)) = TT" | "isInl$(up$(Inr y)) = FF" domain V = VFun (lazy "V -> V") | VCon (lazy "nat") (lazy "V list") * The "(permissive)" option of fixrec has been replaced with a per-equation "(unchecked)" option. See src/HOL/HOLCF/Tutorial/Fixrec_ex.thy for examples. INCOMPATIBILITY. * The "bifinite" class no longer fixes a constant "approx"; the class now just asserts that such a function exists. INCOMPATIBILITY. * Former type "alg_defl" has been renamed to "defl". HOLCF no longer defines an embedding of type 'a defl into udom by default; instances of "bifinite" and "domain" classes are available in src/HOL/HOLCF/Library/Defl_Bifinite.thy. * The syntax "REP('a)" has been replaced with "DEFL('a)". * The predicate "directed" has been removed. INCOMPATIBILITY. * The type class "finite_po" has been removed. INCOMPATIBILITY. * The function "cprod_map" has been renamed to "prod_map". INCOMPATIBILITY. * The monadic bind operator on each powerdomain has new binder syntax similar to sets, e.g. "\\x\xs. t" represents "upper_bind\xs\(\ x. t)". * The infix syntax for binary union on each powerdomain has changed from e.g. "+\" to "\\", for consistency with set syntax. INCOMPATIBILITY. * The constant "UU" has been renamed to "bottom". The syntax "UU" is still supported as an input translation. * Renamed some theorems (the original names are also still available). expand_fun_below ~> fun_below_iff below_fun_ext ~> fun_belowI expand_cfun_eq ~> cfun_eq_iff ext_cfun ~> cfun_eqI expand_cfun_below ~> cfun_below_iff below_cfun_ext ~> cfun_belowI cont2cont_Rep_CFun ~> cont2cont_APP * The Abs and Rep functions for various types have changed names. Related theorem names have also changed to match. INCOMPATIBILITY. Rep_CFun ~> Rep_cfun Abs_CFun ~> Abs_cfun Rep_Sprod ~> Rep_sprod Abs_Sprod ~> Abs_sprod Rep_Ssum ~> Rep_ssum Abs_Ssum ~> Abs_ssum * Lemmas with names of the form *_defined_iff or *_strict_iff have been renamed to *_bottom_iff. INCOMPATIBILITY. * Various changes to bisimulation/coinduction with domain package: - Definitions of "bisim" constants no longer mention definedness. - With mutual recursion, "bisim" predicate is now curried. - With mutual recursion, each type gets a separate coind theorem. - Variable names in bisim_def and coinduct rules have changed. INCOMPATIBILITY. * Case combinators generated by the domain package for type "foo" are now named "foo_case" instead of "foo_when". INCOMPATIBILITY. * Several theorems have been renamed to more accurately reflect the names of constants and types involved. INCOMPATIBILITY. thelub_const ~> lub_const lub_const ~> is_lub_const thelubI ~> lub_eqI is_lub_lub ~> is_lubD2 lubI ~> is_lub_lub unique_lub ~> is_lub_unique is_ub_lub ~> is_lub_rangeD1 lub_bin_chain ~> is_lub_bin_chain lub_fun ~> is_lub_fun thelub_fun ~> lub_fun thelub_cfun ~> lub_cfun thelub_Pair ~> lub_Pair lub_cprod ~> is_lub_prod thelub_cprod ~> lub_prod minimal_cprod ~> minimal_prod inst_cprod_pcpo ~> inst_prod_pcpo UU_I ~> bottomI compact_UU ~> compact_bottom deflation_UU ~> deflation_bottom finite_deflation_UU ~> finite_deflation_bottom * Many legacy theorem names have been discontinued. INCOMPATIBILITY. sq_ord_less_eq_trans ~> below_eq_trans sq_ord_eq_less_trans ~> eq_below_trans refl_less ~> below_refl trans_less ~> below_trans antisym_less ~> below_antisym antisym_less_inverse ~> po_eq_conv [THEN iffD1] box_less ~> box_below rev_trans_less ~> rev_below_trans not_less2not_eq ~> not_below2not_eq less_UU_iff ~> below_UU_iff flat_less_iff ~> flat_below_iff adm_less ~> adm_below adm_not_less ~> adm_not_below adm_compact_not_less ~> adm_compact_not_below less_fun_def ~> below_fun_def expand_fun_less ~> fun_below_iff less_fun_ext ~> fun_belowI less_discr_def ~> below_discr_def discr_less_eq ~> discr_below_eq less_unit_def ~> below_unit_def less_cprod_def ~> below_prod_def prod_lessI ~> prod_belowI Pair_less_iff ~> Pair_below_iff fst_less_iff ~> fst_below_iff snd_less_iff ~> snd_below_iff expand_cfun_less ~> cfun_below_iff less_cfun_ext ~> cfun_belowI injection_less ~> injection_below less_up_def ~> below_up_def not_Iup_less ~> not_Iup_below Iup_less ~> Iup_below up_less ~> up_below Def_inject_less_eq ~> Def_below_Def Def_less_is_eq ~> Def_below_iff spair_less_iff ~> spair_below_iff less_sprod ~> below_sprod spair_less ~> spair_below sfst_less_iff ~> sfst_below_iff ssnd_less_iff ~> ssnd_below_iff fix_least_less ~> fix_least_below dist_less_one ~> dist_below_one less_ONE ~> below_ONE ONE_less_iff ~> ONE_below_iff less_sinlD ~> below_sinlD less_sinrD ~> below_sinrD *** FOL and ZF *** * All constant names are now qualified internally and use proper identifiers, e.g. "IFOL.eq" instead of "op =". INCOMPATIBILITY. *** ML *** * Antiquotation @{assert} inlines a function bool -> unit that raises Fail if the argument is false. Due to inlining the source position of failed assertions is included in the error output. * Discontinued antiquotation @{theory_ref}, which is obsolete since ML text is in practice always evaluated with a stable theory checkpoint. Minor INCOMPATIBILITY, use (Theory.check_thy @{theory}) instead. * Antiquotation @{theory A} refers to theory A from the ancestry of the current context, not any accidental theory loader state as before. Potential INCOMPATIBILITY, subtle change in semantics. * Syntax.pretty_priority (default 0) configures the required priority of pretty-printed output and thus affects insertion of parentheses. * Syntax.default_root (default "any") configures the inner syntax category (nonterminal symbol) for parsing of terms. * Former exception Library.UnequalLengths now coincides with ListPair.UnequalLengths. * Renamed structure MetaSimplifier to Raw_Simplifier. Note that the main functionality is provided by structure Simplifier. * Renamed raw "explode" function to "raw_explode" to emphasize its meaning. Note that internally to Isabelle, Symbol.explode is used in almost all situations. * Discontinued obsolete function sys_error and exception SYS_ERROR. See implementation manual for further details on exceptions in Isabelle/ML. * Renamed setmp_noncritical to Unsynchronized.setmp to emphasize its meaning. * Renamed structure PureThy to Pure_Thy and moved most of its operations to structure Global_Theory, to emphasize that this is rarely-used global-only stuff. * Discontinued Output.debug. Minor INCOMPATIBILITY, use plain writeln instead (or tracing for high-volume output). * Configuration option show_question_marks only affects regular pretty printing of types and terms, not raw Term.string_of_vname. * ML_Context.thm and ML_Context.thms are no longer pervasive. Rare INCOMPATIBILITY, superseded by static antiquotations @{thm} and @{thms} for most purposes. * ML structure Unsynchronized is never opened, not even in Isar interaction mode as before. Old Unsynchronized.set etc. have been discontinued -- use plain := instead. This should be *rare* anyway, since modern tools always work via official context data, notably configuration options. * Parallel and asynchronous execution requires special care concerning interrupts. Structure Exn provides some convenience functions that avoid working directly with raw Interrupt. User code must not absorb interrupts -- intermediate handling (for cleanup etc.) needs to be followed by re-raising of the original exception. Another common source of mistakes are "handle _" patterns, which make the meaning of the program subject to physical effects of the environment. New in Isabelle2009-2 (June 2010) --------------------------------- *** General *** * Authentic syntax for *all* logical entities (type classes, type constructors, term constants): provides simple and robust correspondence between formal entities and concrete syntax. Within the parse tree / AST representations, "constants" are decorated by their category (class, type, const) and spelled out explicitly with their full internal name. Substantial INCOMPATIBILITY concerning low-level syntax declarations and translations (translation rules and translation functions in ML). Some hints on upgrading: - Many existing uses of 'syntax' and 'translations' can be replaced by more modern 'type_notation', 'notation' and 'abbreviation', which are independent of this issue. - 'translations' require markup within the AST; the term syntax provides the following special forms: CONST c -- produces syntax version of constant c from context XCONST c -- literally c, checked as constant from context c -- literally c, if declared by 'syntax' Plain identifiers are treated as AST variables -- occasionally the system indicates accidental variables via the error "rhs contains extra variables". Type classes and type constructors are marked according to their concrete syntax. Some old translations rules need to be written for the "type" category, using type constructor application instead of pseudo-term application of the default category "logic". - 'parse_translation' etc. in ML may use the following antiquotations: @{class_syntax c} -- type class c within parse tree / AST @{term_syntax c} -- type constructor c within parse tree / AST @{const_syntax c} -- ML version of "CONST c" above @{syntax_const c} -- literally c (checked wrt. 'syntax' declarations) - Literal types within 'typed_print_translations', i.e. those *not* represented as pseudo-terms are represented verbatim. Use @{class c} or @{type_name c} here instead of the above syntax antiquotations. Note that old non-authentic syntax was based on unqualified base names, so all of the above "constant" names would coincide. Recall that 'print_syntax' and ML_command "set Syntax.trace_ast" help to diagnose syntax problems. * Type constructors admit general mixfix syntax, not just infix. * Concrete syntax may be attached to local entities without a proof body, too. This works via regular mixfix annotations for 'fix', 'def', 'obtain' etc. or via the explicit 'write' command, which is similar to the 'notation' command in theory specifications. * Discontinued unnamed infix syntax (legacy feature for many years) -- need to specify constant name and syntax separately. Internal ML datatype constructors have been renamed from InfixName to Infix etc. Minor INCOMPATIBILITY. * Schematic theorem statements need to be explicitly markup as such, via commands 'schematic_lemma', 'schematic_theorem', 'schematic_corollary'. Thus the relevance of the proof is made syntactically clear, which impacts performance in a parallel or asynchronous interactive environment. Minor INCOMPATIBILITY. * Use of cumulative prems via "!" in some proof methods has been discontinued (old legacy feature). * References 'trace_simp' and 'debug_simp' have been replaced by configuration options stored in the context. Enabling tracing (the case of debugging is similar) in proofs works via using [[trace_simp = true]] Tracing is then active for all invocations of the simplifier in subsequent goal refinement steps. Tracing may also still be enabled or disabled via the ProofGeneral settings menu. * Separate commands 'hide_class', 'hide_type', 'hide_const', 'hide_fact' replace the former 'hide' KIND command. Minor INCOMPATIBILITY. * Improved parallelism of proof term normalization: usedir -p2 -q0 is more efficient than combinations with -q1 or -q2. *** Pure *** * Proofterms record type-class reasoning explicitly, using the "unconstrain" operation internally. This eliminates all sort constraints from a theorem and proof, introducing explicit OFCLASS-premises. On the proof term level, this operation is automatically applied at theorem boundaries, such that closed proofs are always free of sort constraints. INCOMPATIBILITY for tools that inspect proof terms. * Local theory specifications may depend on extra type variables that are not present in the result type -- arguments TYPE('a) :: 'a itself are added internally. For example: definition unitary :: bool where "unitary = (ALL (x::'a) y. x = y)" * Predicates of locales introduced by classes carry a mandatory "class" prefix. INCOMPATIBILITY. * Vacuous class specifications observe default sort. INCOMPATIBILITY. * Old 'axclass' command has been discontinued. INCOMPATIBILITY, use 'class' instead. * Command 'code_reflect' allows to incorporate generated ML code into runtime environment; replaces immature code_datatype antiquotation. INCOMPATIBILITY. * Code generator: simple concept for abstract datatypes obeying invariants. * Code generator: details of internal data cache have no impact on the user space functionality any longer. * Methods "unfold_locales" and "intro_locales" ignore non-locale subgoals. This is more appropriate for interpretations with 'where'. INCOMPATIBILITY. * Command 'example_proof' opens an empty proof body. This allows to experiment with Isar, without producing any persistent result. * Commands 'type_notation' and 'no_type_notation' declare type syntax within a local theory context, with explicit checking of the constructors involved (in contrast to the raw 'syntax' versions). * Commands 'types' and 'typedecl' now work within a local theory context -- without introducing dependencies on parameters or assumptions, which is not possible in Isabelle/Pure. * Command 'defaultsort' has been renamed to 'default_sort', it works within a local theory context. Minor INCOMPATIBILITY. *** HOL *** * Command 'typedef' now works within a local theory context -- without introducing dependencies on parameters or assumptions, which is not possible in Isabelle/Pure/HOL. Note that the logical environment may contain multiple interpretations of local typedefs (with different non-emptiness proofs), even in a global theory context. * New package for quotient types. Commands 'quotient_type' and 'quotient_definition' may be used for defining types and constants by quotient constructions. An example is the type of integers created by quotienting pairs of natural numbers: fun intrel :: "(nat * nat) => (nat * nat) => bool" where "intrel (x, y) (u, v) = (x + v = u + y)" quotient_type int = "nat * nat" / intrel by (auto simp add: equivp_def expand_fun_eq) quotient_definition "0::int" is "(0::nat, 0::nat)" The method "lifting" can be used to lift of theorems from the underlying "raw" type to the quotient type. The example src/HOL/Quotient_Examples/FSet.thy includes such a quotient construction and provides a reasoning infrastructure for finite sets. * Renamed Library/Quotient.thy to Library/Quotient_Type.thy to avoid clash with new theory Quotient in Main HOL. * Moved the SMT binding into the main HOL session, eliminating separate HOL-SMT session. * List membership infix mem operation is only an input abbreviation. INCOMPATIBILITY. * Theory Library/Word.thy has been removed. Use library Word/Word.thy for future developements; former Library/Word.thy is still present in the AFP entry RSAPPS. * Theorem Int.int_induct renamed to Int.int_of_nat_induct and is no longer shadowed. INCOMPATIBILITY. * Dropped theorem duplicate comp_arith; use semiring_norm instead. INCOMPATIBILITY. * Dropped theorem RealPow.real_sq_order; use power2_le_imp_le instead. INCOMPATIBILITY. * Dropped normalizing_semiring etc; use the facts in semiring classes instead. INCOMPATIBILITY. * Dropped several real-specific versions of lemmas about floor and ceiling; use the generic lemmas from theory "Archimedean_Field" instead. INCOMPATIBILITY. floor_number_of_eq ~> floor_number_of le_floor_eq_number_of ~> number_of_le_floor le_floor_eq_zero ~> zero_le_floor le_floor_eq_one ~> one_le_floor floor_less_eq_number_of ~> floor_less_number_of floor_less_eq_zero ~> floor_less_zero floor_less_eq_one ~> floor_less_one less_floor_eq_number_of ~> number_of_less_floor less_floor_eq_zero ~> zero_less_floor less_floor_eq_one ~> one_less_floor floor_le_eq_number_of ~> floor_le_number_of floor_le_eq_zero ~> floor_le_zero floor_le_eq_one ~> floor_le_one floor_subtract_number_of ~> floor_diff_number_of floor_subtract_one ~> floor_diff_one ceiling_number_of_eq ~> ceiling_number_of ceiling_le_eq_number_of ~> ceiling_le_number_of ceiling_le_zero_eq ~> ceiling_le_zero ceiling_le_eq_one ~> ceiling_le_one less_ceiling_eq_number_of ~> number_of_less_ceiling less_ceiling_eq_zero ~> zero_less_ceiling less_ceiling_eq_one ~> one_less_ceiling ceiling_less_eq_number_of ~> ceiling_less_number_of ceiling_less_eq_zero ~> ceiling_less_zero ceiling_less_eq_one ~> ceiling_less_one le_ceiling_eq_number_of ~> number_of_le_ceiling le_ceiling_eq_zero ~> zero_le_ceiling le_ceiling_eq_one ~> one_le_ceiling ceiling_subtract_number_of ~> ceiling_diff_number_of ceiling_subtract_one ~> ceiling_diff_one * Theory "Finite_Set": various folding_XXX locales facilitate the application of the various fold combinators on finite sets. * Library theory "RBT" renamed to "RBT_Impl"; new library theory "RBT" provides abstract red-black tree type which is backed by "RBT_Impl" as implementation. INCOMPATIBILITY. * Theory Library/Coinductive_List has been removed -- superseded by AFP/thys/Coinductive. * Theory PReal, including the type "preal" and related operations, has been removed. INCOMPATIBILITY. * Real: new development using Cauchy Sequences. * Split off theory "Big_Operators" containing setsum, setprod, Inf_fin, Sup_fin, Min, Max from theory Finite_Set. INCOMPATIBILITY. * Theory "Rational" renamed to "Rat", for consistency with "Nat", "Int" etc. INCOMPATIBILITY. * Constant Rat.normalize needs to be qualified. INCOMPATIBILITY. * New set of rules "ac_simps" provides combined assoc / commute rewrites for all interpretations of the appropriate generic locales. * Renamed theory "OrderedGroup" to "Groups" and split theory "Ring_and_Field" into theories "Rings" and "Fields"; for more appropriate and more consistent names suitable for name prefixes within the HOL theories. INCOMPATIBILITY. * Some generic constants have been put to appropriate theories: - less_eq, less: Orderings - zero, one, plus, minus, uminus, times, abs, sgn: Groups - inverse, divide: Rings INCOMPATIBILITY. * More consistent naming of type classes involving orderings (and lattices): lower_semilattice ~> semilattice_inf upper_semilattice ~> semilattice_sup dense_linear_order ~> dense_linorder pordered_ab_group_add ~> ordered_ab_group_add pordered_ab_group_add_abs ~> ordered_ab_group_add_abs pordered_ab_semigroup_add ~> ordered_ab_semigroup_add pordered_ab_semigroup_add_imp_le ~> ordered_ab_semigroup_add_imp_le pordered_cancel_ab_semigroup_add ~> ordered_cancel_ab_semigroup_add pordered_cancel_comm_semiring ~> ordered_cancel_comm_semiring pordered_cancel_semiring ~> ordered_cancel_semiring pordered_comm_monoid_add ~> ordered_comm_monoid_add pordered_comm_ring ~> ordered_comm_ring pordered_comm_semiring ~> ordered_comm_semiring pordered_ring ~> ordered_ring pordered_ring_abs ~> ordered_ring_abs pordered_semiring ~> ordered_semiring ordered_ab_group_add ~> linordered_ab_group_add ordered_ab_semigroup_add ~> linordered_ab_semigroup_add ordered_cancel_ab_semigroup_add ~> linordered_cancel_ab_semigroup_add ordered_comm_semiring_strict ~> linordered_comm_semiring_strict ordered_field ~> linordered_field ordered_field_no_lb ~> linordered_field_no_lb ordered_field_no_ub ~> linordered_field_no_ub ordered_field_dense_linear_order ~> dense_linordered_field ordered_idom ~> linordered_idom ordered_ring ~> linordered_ring ordered_ring_le_cancel_factor ~> linordered_ring_le_cancel_factor ordered_ring_less_cancel_factor ~> linordered_ring_less_cancel_factor ordered_ring_strict ~> linordered_ring_strict ordered_semidom ~> linordered_semidom ordered_semiring ~> linordered_semiring ordered_semiring_1 ~> linordered_semiring_1 ordered_semiring_1_strict ~> linordered_semiring_1_strict ordered_semiring_strict ~> linordered_semiring_strict The following slightly odd type classes have been moved to a separate theory Library/Lattice_Algebras: lordered_ab_group_add ~> lattice_ab_group_add lordered_ab_group_add_abs ~> lattice_ab_group_add_abs lordered_ab_group_add_meet ~> semilattice_inf_ab_group_add lordered_ab_group_add_join ~> semilattice_sup_ab_group_add lordered_ring ~> lattice_ring INCOMPATIBILITY. * Refined field classes: - classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero include rule inverse 0 = 0 -- subsumes former division_by_zero class; - numerous lemmas have been ported from field to division_ring. INCOMPATIBILITY. * Refined algebra theorem collections: - dropped theorem group group_simps, use algebra_simps instead; - dropped theorem group ring_simps, use field_simps instead; - proper theorem collection field_simps subsumes former theorem groups field_eq_simps and field_simps; - dropped lemma eq_minus_self_iff which is a duplicate for equal_neg_zero. INCOMPATIBILITY. * Theory Finite_Set and List: some lemmas have been generalized from sets to lattices: fun_left_comm_idem_inter ~> fun_left_comm_idem_inf fun_left_comm_idem_union ~> fun_left_comm_idem_sup inter_Inter_fold_inter ~> inf_Inf_fold_inf union_Union_fold_union ~> sup_Sup_fold_sup Inter_fold_inter ~> Inf_fold_inf Union_fold_union ~> Sup_fold_sup inter_INTER_fold_inter ~> inf_INFI_fold_inf union_UNION_fold_union ~> sup_SUPR_fold_sup INTER_fold_inter ~> INFI_fold_inf UNION_fold_union ~> SUPR_fold_sup * Theory "Complete_Lattice": lemmas top_def and bot_def have been replaced by the more convenient lemmas Inf_empty and Sup_empty. Dropped lemmas Inf_insert_simp and Sup_insert_simp, which are subsumed by Inf_insert and Sup_insert. Lemmas Inf_UNIV and Sup_UNIV replace former Inf_Univ and Sup_Univ. Lemmas inf_top_right and sup_bot_right subsume inf_top and sup_bot respectively. INCOMPATIBILITY. * Reorganized theory Multiset: swapped notation of pointwise and multiset order: - pointwise ordering is instance of class order with standard syntax <= and <; - multiset ordering has syntax <=# and <#; partial order properties are provided by means of interpretation with prefix multiset_order; - less duplication, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation; - use insert_DiffM2 [symmetric] instead of elem_imp_eq_diff_union, if needed. Renamed: multiset_eq_conv_count_eq ~> multiset_ext_iff multi_count_ext ~> multiset_ext diff_union_inverse2 ~> diff_union_cancelR INCOMPATIBILITY. * Theory Permutation: replaced local "remove" by List.remove1. * Code generation: ML and OCaml code is decorated with signatures. * Theory List: added transpose. * Library/Nat_Bijection.thy is a collection of bijective functions between nat and other types, which supersedes the older libraries Library/Nat_Int_Bij.thy and HOLCF/NatIso.thy. INCOMPATIBILITY. Constants: Nat_Int_Bij.nat2_to_nat ~> prod_encode Nat_Int_Bij.nat_to_nat2 ~> prod_decode Nat_Int_Bij.int_to_nat_bij ~> int_encode Nat_Int_Bij.nat_to_int_bij ~> int_decode Countable.pair_encode ~> prod_encode NatIso.prod2nat ~> prod_encode NatIso.nat2prod ~> prod_decode NatIso.sum2nat ~> sum_encode NatIso.nat2sum ~> sum_decode NatIso.list2nat ~> list_encode NatIso.nat2list ~> list_decode NatIso.set2nat ~> set_encode NatIso.nat2set ~> set_decode Lemmas: Nat_Int_Bij.bij_nat_to_int_bij ~> bij_int_decode Nat_Int_Bij.nat2_to_nat_inj ~> inj_prod_encode Nat_Int_Bij.nat2_to_nat_surj ~> surj_prod_encode Nat_Int_Bij.nat_to_nat2_inj ~> inj_prod_decode Nat_Int_Bij.nat_to_nat2_surj ~> surj_prod_decode Nat_Int_Bij.i2n_n2i_id ~> int_encode_inverse Nat_Int_Bij.n2i_i2n_id ~> int_decode_inverse Nat_Int_Bij.surj_nat_to_int_bij ~> surj_int_encode Nat_Int_Bij.surj_int_to_nat_bij ~> surj_int_decode Nat_Int_Bij.inj_nat_to_int_bij ~> inj_int_encode Nat_Int_Bij.inj_int_to_nat_bij ~> inj_int_decode Nat_Int_Bij.bij_nat_to_int_bij ~> bij_int_encode Nat_Int_Bij.bij_int_to_nat_bij ~> bij_int_decode * Sledgehammer: - Renamed ATP commands: atp_info ~> sledgehammer running_atps atp_kill ~> sledgehammer kill_atps atp_messages ~> sledgehammer messages atp_minimize ~> sledgehammer minimize print_atps ~> sledgehammer available_atps INCOMPATIBILITY. - Added user's manual ("isabelle doc sledgehammer"). - Added option syntax and "sledgehammer_params" to customize Sledgehammer's behavior. See the manual for details. - Modified the Isar proof reconstruction code so that it produces direct proofs rather than proofs by contradiction. (This feature is still experimental.) - Made Isar proof reconstruction work for SPASS, remote ATPs, and in full-typed mode. - Added support for TPTP syntax for SPASS via the "spass_tptp" ATP. * Nitpick: - Added and implemented "binary_ints" and "bits" options. - Added "std" option and implemented support for nonstandard models. - Added and implemented "finitize" option to improve the precision of infinite datatypes based on a monotonicity analysis. - Added support for quotient types. - Added support for "specification" and "ax_specification" constructs. - Added support for local definitions (for "function" and "termination" proofs). - Added support for term postprocessors. - Optimized "Multiset.multiset" and "FinFun.finfun". - Improved efficiency of "destroy_constrs" optimization. - Fixed soundness bugs related to "destroy_constrs" optimization and record getters. - Fixed soundness bug related to higher-order constructors. - Fixed soundness bug when "full_descrs" is enabled. - Improved precision of set constructs. - Added "atoms" option. - Added cache to speed up repeated Kodkod invocations on the same problems. - Renamed "MiniSatJNI", "zChaffJNI", "BerkMinAlloy", and "SAT4JLight" to "MiniSat_JNI", "zChaff_JNI", "BerkMin_Alloy", and "SAT4J_Light". INCOMPATIBILITY. - Removed "skolemize", "uncurry", "sym_break", "flatten_prop", "sharing_depth", and "show_skolems" options. INCOMPATIBILITY. - Removed "nitpick_intro" attribute. INCOMPATIBILITY. * Method "induct" now takes instantiations of the form t, where t is not a variable, as a shorthand for "x == t", where x is a fresh variable. If this is not intended, t has to be enclosed in parentheses. By default, the equalities generated by definitional instantiations are pre-simplified, which may cause parameters of inductive cases to disappear, or may even delete some of the inductive cases. Use "induct (no_simp)" instead of "induct" to restore the old behaviour. The (no_simp) option is also understood by the "cases" and "nominal_induct" methods, which now perform pre-simplification, too. INCOMPATIBILITY. *** HOLCF *** * Variable names in lemmas generated by the domain package have changed; the naming scheme is now consistent with the HOL datatype package. Some proof scripts may be affected, INCOMPATIBILITY. * The domain package no longer defines the function "foo_copy" for recursive domain "foo". The reach lemma is now stated directly in terms of "foo_take". Lemmas and proofs that mention "foo_copy" must be reformulated in terms of "foo_take", INCOMPATIBILITY. * Most definedness lemmas generated by the domain package (previously of the form "x ~= UU ==> foo$x ~= UU") now have an if-and-only-if form like "foo$x = UU <-> x = UU", which works better as a simp rule. Proofs that used definedness lemmas as intro rules may break, potential INCOMPATIBILITY. * Induction and casedist rules generated by the domain package now declare proper case_names (one called "bottom", and one named for each constructor). INCOMPATIBILITY. * For mutually-recursive domains, separate "reach" and "take_lemma" rules are generated for each domain, INCOMPATIBILITY. foo_bar.reach ~> foo.reach bar.reach foo_bar.take_lemmas ~> foo.take_lemma bar.take_lemma * Some lemmas generated by the domain package have been renamed for consistency with the datatype package, INCOMPATIBILITY. foo.ind ~> foo.induct foo.finite_ind ~> foo.finite_induct foo.coind ~> foo.coinduct foo.casedist ~> foo.exhaust foo.exhaust ~> foo.nchotomy * For consistency with other definition packages, the fixrec package now generates qualified theorem names, INCOMPATIBILITY. foo_simps ~> foo.simps foo_unfold ~> foo.unfold foo_induct ~> foo.induct * The "fixrec_simp" attribute has been removed. The "fixrec_simp" method and internal fixrec proofs now use the default simpset instead. INCOMPATIBILITY. * The "contlub" predicate has been removed. Proof scripts should use lemma contI2 in place of monocontlub2cont, INCOMPATIBILITY. * The "admw" predicate has been removed, INCOMPATIBILITY. * The constants cpair, cfst, and csnd have been removed in favor of Pair, fst, and snd from Isabelle/HOL, INCOMPATIBILITY. *** ML *** * Antiquotations for basic formal entities: @{class NAME} -- type class @{class_syntax NAME} -- syntax representation of the above @{type_name NAME} -- logical type @{type_abbrev NAME} -- type abbreviation @{nonterminal NAME} -- type of concrete syntactic category @{type_syntax NAME} -- syntax representation of any of the above @{const_name NAME} -- logical constant (INCOMPATIBILITY) @{const_abbrev NAME} -- abbreviated constant @{const_syntax NAME} -- syntax representation of any of the above * Antiquotation @{syntax_const NAME} ensures that NAME refers to a raw syntax constant (cf. 'syntax' command). * Antiquotation @{make_string} inlines a function to print arbitrary values similar to the ML toplevel. The result is compiler dependent and may fall back on "?" in certain situations. * Diagnostic commands 'ML_val' and 'ML_command' may refer to antiquotations @{Isar.state} and @{Isar.goal}. This replaces impure Isar.state() and Isar.goal(), which belong to the old TTY loop and do not work with the asynchronous Isar document model. * Configuration options now admit dynamic default values, depending on the context or even global references. * SHA1.digest digests strings according to SHA-1 (see RFC 3174). It uses an efficient external library if available (for Poly/ML). * Renamed some important ML structures, while keeping the old names for some time as aliases within the structure Legacy: OuterKeyword ~> Keyword OuterLex ~> Token OuterParse ~> Parse OuterSyntax ~> Outer_Syntax PrintMode ~> Print_Mode SpecParse ~> Parse_Spec ThyInfo ~> Thy_Info ThyLoad ~> Thy_Load ThyOutput ~> Thy_Output TypeInfer ~> Type_Infer Note that "open Legacy" simplifies porting of sources, but forgetting to remove it again will complicate porting again in the future. * Most operations that refer to a global context are named accordingly, e.g. Simplifier.global_context or ProofContext.init_global. There are some situations where a global context actually works, but under normal circumstances one needs to pass the proper local context through the code! * Discontinued old TheoryDataFun with its copy/init operation -- data needs to be pure. Functor Theory_Data_PP retains the traditional Pretty.pp argument to merge, which is absent in the standard Theory_Data version. * Sorts.certify_sort and derived "cert" operations for types and terms no longer minimize sorts. Thus certification at the boundary of the inference kernel becomes invariant under addition of class relations, which is an important monotonicity principle. Sorts are now minimized in the syntax layer only, at the boundary between the end-user and the system. Subtle INCOMPATIBILITY, may have to use Sign.minimize_sort explicitly in rare situations. * Renamed old-style Drule.standard to Drule.export_without_context, to emphasize that this is in no way a standard operation. INCOMPATIBILITY. * Subgoal.FOCUS (and variants): resulting goal state is normalized as usual for resolution. Rare INCOMPATIBILITY. * Renamed varify/unvarify operations to varify_global/unvarify_global to emphasize that these only work in a global situation (which is quite rare). * Curried take and drop in library.ML; negative length is interpreted as infinity (as in chop). Subtle INCOMPATIBILITY. * Proof terms: type substitutions on proof constants now use canonical order of type variables. INCOMPATIBILITY for tools working with proof terms. * Raw axioms/defs may no longer carry sort constraints, and raw defs may no longer carry premises. User-level specifications are transformed accordingly by Thm.add_axiom/add_def. *** System *** * Discontinued special HOL_USEDIR_OPTIONS for the main HOL image; ISABELLE_USEDIR_OPTIONS applies uniformly to all sessions. Note that proof terms are enabled unconditionally in the new HOL-Proofs image. * Discontinued old ISABELLE and ISATOOL environment settings (legacy feature since Isabelle2009). Use ISABELLE_PROCESS and ISABELLE_TOOL, respectively. * Old lib/scripts/polyml-platform is superseded by the ISABELLE_PLATFORM setting variable, which defaults to the 32 bit variant, even on a 64 bit machine. The following example setting prefers 64 bit if available: ML_PLATFORM="${ISABELLE_PLATFORM64:-$ISABELLE_PLATFORM}" * The preliminary Isabelle/jEdit application demonstrates the emerging Isabelle/Scala layer for advanced prover interaction and integration. See src/Tools/jEdit or "isabelle jedit" provided by the properly built component. * "IsabelleText" is a Unicode font derived from Bitstream Vera Mono and Bluesky TeX fonts. It provides the usual Isabelle symbols, similar to the default assignment of the document preparation system (cf. isabellesym.sty). The Isabelle/Scala class Isabelle_System provides some operations for direct access to the font without asking the user for manual installation. New in Isabelle2009-1 (December 2009) ------------------------------------- *** General *** * Discontinued old form of "escaped symbols" such as \\. Only one backslash should be used, even in ML sources. *** Pure *** * Locale interpretation propagates mixins along the locale hierarchy. The currently only available mixins are the equations used to map local definitions to terms of the target domain of an interpretation. * Reactivated diagnostic command 'print_interps'. Use "print_interps loc" to print all interpretations of locale "loc" in the theory. Interpretations in proofs are not shown. * Thoroughly revised locales tutorial. New section on conditional interpretation. * On instantiation of classes, remaining undefined class parameters are formally declared. INCOMPATIBILITY. *** Document preparation *** * New generalized style concept for printing terms: @{foo (style) ...} instead of @{foo_style style ...} (old form is still retained for backward compatibility). Styles can be also applied for antiquotations prop, term_type and typeof. *** HOL *** * New proof method "smt" for a combination of first-order logic with equality, linear and nonlinear (natural/integer/real) arithmetic, and fixed-size bitvectors; there is also basic support for higher-order features (esp. lambda abstractions). It is an incomplete decision procedure based on external SMT solvers using the oracle mechanism; for the SMT solver Z3, this method is proof-producing. Certificates are provided to avoid calling the external solvers solely for re-checking proofs. Due to a remote SMT service there is no need for installing SMT solvers locally. See src/HOL/SMT. * New commands to load and prove verification conditions generated by the Boogie program verifier or derived systems (e.g. the Verifying C Compiler (VCC) or Spec#). See src/HOL/Boogie. * New counterexample generator tool 'nitpick' based on the Kodkod relational model finder. See src/HOL/Tools/Nitpick and src/HOL/Nitpick_Examples. * New commands 'code_pred' and 'values' to invoke the predicate compiler and to enumerate values of inductive predicates. * A tabled implementation of the reflexive transitive closure. * New implementation of quickcheck uses generic code generator; default generators are provided for all suitable HOL types, records and datatypes. Old quickcheck can be re-activated importing theory Library/SML_Quickcheck. * New testing tool Mirabelle for automated proof tools. Applies several tools and tactics like sledgehammer, metis, or quickcheck, to every proof step in a theory. To be used in batch mode via the "mirabelle" utility. * New proof method "sos" (sum of squares) for nonlinear real arithmetic (originally due to John Harison). It requires theory Library/Sum_Of_Squares. It is not a complete decision procedure but works well in practice on quantifier-free real arithmetic with +, -, *, ^, =, <= and <, i.e. boolean combinations of equalities and inequalities between polynomials. It makes use of external semidefinite programming solvers. Method "sos" generates a certificate that can be pasted into the proof thus avoiding the need to call an external tool every time the proof is checked. See src/HOL/Library/Sum_Of_Squares. * New method "linarith" invokes existing linear arithmetic decision procedure only. * New command 'atp_minimal' reduces result produced by Sledgehammer. * New Sledgehammer option "Full Types" in Proof General settings menu. Causes full type information to be output to the ATPs. This slows ATPs down considerably but eliminates a source of unsound "proofs" that fail later. * New method "metisFT": A version of metis that uses full type information in order to avoid failures of proof reconstruction. * New evaluator "approximate" approximates an real valued term using the same method as the approximation method. * Method "approximate" now supports arithmetic expressions as boundaries of intervals and implements interval splitting and Taylor series expansion. * ML antiquotation @{code_datatype} inserts definition of a datatype generated by the code generator; e.g. see src/HOL/Predicate.thy. * New theory SupInf of the supremum and infimum operators for sets of reals. * New theory Probability, which contains a development of measure theory, eventually leading to Lebesgue integration and probability. * Extended Multivariate Analysis to include derivation and Brouwer's fixpoint theorem. * Reorganization of number theory, INCOMPATIBILITY: - new number theory development for nat and int, in theories Divides and GCD as well as in new session Number_Theory - some constants and facts now suffixed with _nat and _int accordingly - former session NumberTheory now named Old_Number_Theory, including theories Legacy_GCD and Primes (prefer Number_Theory if possible) - moved theory Pocklington from src/HOL/Library to src/HOL/Old_Number_Theory * Theory GCD includes functions Gcd/GCD and Lcm/LCM for the gcd and lcm of finite and infinite sets. It is shown that they form a complete lattice. * Class semiring_div requires superclass no_zero_divisors and proof of div_mult_mult1; theorems div_mult_mult1, div_mult_mult2, div_mult_mult1_if, div_mult_mult1 and div_mult_mult2 have been generalized to class semiring_div, subsuming former theorems zdiv_zmult_zmult1, zdiv_zmult_zmult1_if, zdiv_zmult_zmult1 and zdiv_zmult_zmult2. div_mult_mult1 is now [simp] by default. INCOMPATIBILITY. * Refinements to lattice classes and sets: - less default intro/elim rules in locale variant, more default intro/elim rules in class variant: more uniformity - lemma ge_sup_conv renamed to le_sup_iff, in accordance with le_inf_iff - dropped lemma alias inf_ACI for inf_aci (same for sup_ACI and sup_aci) - renamed ACI to inf_sup_aci - new class "boolean_algebra" - class "complete_lattice" moved to separate theory "Complete_Lattice"; corresponding constants (and abbreviations) renamed and with authentic syntax: Set.Inf ~> Complete_Lattice.Inf Set.Sup ~> Complete_Lattice.Sup Set.INFI ~> Complete_Lattice.INFI Set.SUPR ~> Complete_Lattice.SUPR Set.Inter ~> Complete_Lattice.Inter Set.Union ~> Complete_Lattice.Union Set.INTER ~> Complete_Lattice.INTER Set.UNION ~> Complete_Lattice.UNION - authentic syntax for Set.Pow Set.image - mere abbreviations: Set.empty (for bot) Set.UNIV (for top) Set.inter (for inf, formerly Set.Int) Set.union (for sup, formerly Set.Un) Complete_Lattice.Inter (for Inf) Complete_Lattice.Union (for Sup) Complete_Lattice.INTER (for INFI) Complete_Lattice.UNION (for SUPR) - object-logic definitions as far as appropriate INCOMPATIBILITY. Care is required when theorems Int_subset_iff or Un_subset_iff are explicitly deleted as default simp rules; then also their lattice counterparts le_inf_iff and le_sup_iff have to be deleted to achieve the desired effect. * Rules inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer; the same applies to min_max.inf_absorb1 etc. INCOMPATIBILITY. * Rules sup_Int_eq and sup_Un_eq are no longer declared as pred_set_conv by default. INCOMPATIBILITY. * Power operations on relations and functions are now one dedicated constant "compow" with infix syntax "^^". Power operation on multiplicative monoids retains syntax "^" and is now defined generic in class power. INCOMPATIBILITY. * Relation composition "R O S" now has a more standard argument order: "R O S = {(x, z). EX y. (x, y) : R & (y, z) : S}". INCOMPATIBILITY, rewrite propositions with "S O R" --> "R O S". Proofs may occasionally break, since the O_assoc rule was not rewritten like this. Fix using O_assoc[symmetric]. The same applies to the curried version "R OO S". * Function "Inv" is renamed to "inv_into" and function "inv" is now an abbreviation for "inv_into UNIV". Lemmas are renamed accordingly. INCOMPATIBILITY. * Most rules produced by inductive and datatype package have mandatory prefixes. INCOMPATIBILITY. * Changed "DERIV_intros" to a dynamic fact, which can be augmented by the attribute of the same name. Each of the theorems in the list DERIV_intros assumes composition with an additional function and matches a variable to the derivative, which has to be solved by the Simplifier. Hence (auto intro!: DERIV_intros) computes the derivative of most elementary terms. Former Maclauren.DERIV_tac and Maclauren.deriv_tac should be replaced by (auto intro!: DERIV_intros). INCOMPATIBILITY. * Code generator attributes follow the usual underscore convention: code_unfold replaces code unfold code_post replaces code post etc. INCOMPATIBILITY. * Renamed methods: sizechange -> size_change induct_scheme -> induction_schema INCOMPATIBILITY. * Discontinued abbreviation "arbitrary" of constant "undefined". INCOMPATIBILITY, use "undefined" directly. * Renamed theorems: Suc_eq_add_numeral_1 -> Suc_eq_plus1 Suc_eq_add_numeral_1_left -> Suc_eq_plus1_left Suc_plus1 -> Suc_eq_plus1 *anti_sym -> *antisym* vector_less_eq_def -> vector_le_def INCOMPATIBILITY. * Added theorem List.map_map as [simp]. Removed List.map_compose. INCOMPATIBILITY. * Removed predicate "M hassize n" (<--> card M = n & finite M). INCOMPATIBILITY. *** HOLCF *** * Theory Representable defines a class "rep" of domains that are representable (via an ep-pair) in the universal domain type "udom". Instances are provided for all type constructors defined in HOLCF. * The 'new_domain' command is a purely definitional version of the domain package, for representable domains. Syntax is identical to the old domain package. The 'new_domain' package also supports indirect recursion using previously-defined type constructors. See src/HOLCF/ex/New_Domain.thy for examples. * Method "fixrec_simp" unfolds one step of a fixrec-defined constant on the left-hand side of an equation, and then performs simplification. Rewriting is done using rules declared with the "fixrec_simp" attribute. The "fixrec_simp" method is intended as a replacement for "fixpat"; see src/HOLCF/ex/Fixrec_ex.thy for examples. * The pattern-match compiler in 'fixrec' can now handle constructors with HOL function types. Pattern-match combinators for the Pair constructor are pre-configured. * The 'fixrec' package now produces better fixed-point induction rules for mutually-recursive definitions: Induction rules have conclusions of the form "P foo bar" instead of "P ". * The constant "sq_le" (with infix syntax "<<" or "\") has been renamed to "below". The name "below" now replaces "less" in many theorem names. (Legacy theorem names using "less" are still supported as well.) * The 'fixrec' package now supports "bottom patterns". Bottom patterns can be used to generate strictness rules, or to make functions more strict (much like the bang-patterns supported by the Glasgow Haskell Compiler). See src/HOLCF/ex/Fixrec_ex.thy for examples. *** ML *** * Support for Poly/ML 5.3.0, with improved reporting of compiler errors and run-time exceptions, including detailed source positions. * Structure Name_Space (formerly NameSpace) now manages uniquely identified entries, with some additional information such as source position, logical grouping etc. * Theory and context data is now introduced by the simplified and modernized functors Theory_Data, Proof_Data, Generic_Data. Data needs to be pure, but the old TheoryDataFun for mutable data (with explicit copy operation) is still available for some time. * Structure Synchronized (cf. src/Pure/Concurrent/synchronized.ML) provides a high-level programming interface to synchronized state variables with atomic update. This works via pure function application within a critical section -- its runtime should be as short as possible; beware of deadlocks if critical code is nested, either directly or indirectly via other synchronized variables! * Structure Unsynchronized (cf. src/Pure/ML-Systems/unsynchronized.ML) wraps raw ML references, explicitly indicating their non-thread-safe behaviour. The Isar toplevel keeps this structure open, to accommodate Proof General as well as quick and dirty interactive experiments with references. * PARALLEL_CHOICE and PARALLEL_GOALS provide basic support for parallel tactical reasoning. * Tacticals Subgoal.FOCUS, Subgoal.FOCUS_PREMS, Subgoal.FOCUS_PARAMS are similar to SUBPROOF, but are slightly more flexible: only the specified parts of the subgoal are imported into the context, and the body tactic may introduce new subgoals and schematic variables. * Old tactical METAHYPS, which does not observe the proof context, has been renamed to Old_Goals.METAHYPS and awaits deletion. Use SUBPROOF or Subgoal.FOCUS etc. * Renamed functor TableFun to Table, and GraphFun to Graph. (Since functors have their own ML name space there is no point to mark them separately.) Minor INCOMPATIBILITY. * Renamed NamedThmsFun to Named_Thms. INCOMPATIBILITY. * Renamed several structures FooBar to Foo_Bar. Occasional, INCOMPATIBILITY. * Operations of structure Skip_Proof no longer require quick_and_dirty mode, which avoids critical setmp. * Eliminated old Attrib.add_attributes, Method.add_methods and related combinators for "args". INCOMPATIBILITY, need to use simplified Attrib/Method.setup introduced in Isabelle2009. * Proper context for simpset_of, claset_of, clasimpset_of. May fall back on global_simpset_of, global_claset_of, global_clasimpset_of as last resort. INCOMPATIBILITY. * Display.pretty_thm now requires a proper context (cf. former ProofContext.pretty_thm). May fall back on Display.pretty_thm_global or even Display.pretty_thm_without_context as last resort. INCOMPATIBILITY. * Discontinued Display.pretty_ctyp/cterm etc. INCOMPATIBILITY, use Syntax.pretty_typ/term directly, preferably with proper context instead of global theory. *** System *** * Further fine tuning of parallel proof checking, scales up to 8 cores (max. speedup factor 5.0). See also Goal.parallel_proofs in ML and usedir option -q. * Support for additional "Isabelle components" via etc/components, see also the system manual. * The isabelle makeall tool now operates on all components with IsaMakefile, not just hardwired "logics". * Removed "compress" option from isabelle-process and isabelle usedir; this is always enabled. * Discontinued support for Poly/ML 4.x versions. * Isabelle tool "wwwfind" provides web interface for 'find_theorems' on a given logic image. This requires the lighttpd webserver and is currently supported on Linux only. New in Isabelle2009 (April 2009) -------------------------------- *** General *** * Simplified main Isabelle executables, with less surprises on case-insensitive file-systems (such as Mac OS). - The main Isabelle tool wrapper is now called "isabelle" instead of "isatool." - The former "isabelle" alias for "isabelle-process" has been removed (should rarely occur to regular users). - The former "isabelle-interface" and its alias "Isabelle" have been removed (interfaces are now regular Isabelle tools). Within scripts and make files, the Isabelle environment variables ISABELLE_TOOL and ISABELLE_PROCESS replace old ISATOOL and ISABELLE, respectively. (The latter are still available as legacy feature.) The old isabelle-interface wrapper could react in confusing ways if the interface was uninstalled or changed otherwise. Individual interface tool configuration is now more explicit, see also the Isabelle system manual. In particular, Proof General is now available via "isabelle emacs". INCOMPATIBILITY, need to adapt derivative scripts. Users may need to purge installed copies of Isabelle executables and re-run "isabelle install -p ...", or use symlinks. * The default for ISABELLE_HOME_USER is now ~/.isabelle instead of the old ~/isabelle, which was slightly non-standard and apt to cause surprises on case-insensitive file-systems (such as Mac OS). INCOMPATIBILITY, need to move existing ~/isabelle/etc, ~/isabelle/heaps, ~/isabelle/browser_info to the new place. Special care is required when using older releases of Isabelle. Note that ISABELLE_HOME_USER can be changed in Isabelle/etc/settings of any Isabelle distribution, in order to use the new ~/.isabelle uniformly. * Proofs of fully specified statements are run in parallel on multi-core systems. A speedup factor of 2.5 to 3.2 can be expected on a regular 4-core machine, if the initial heap space is made reasonably large (cf. Poly/ML option -H). (Requires Poly/ML 5.2.1 or later.) * The main reference manuals ("isar-ref", "implementation", and "system") have been updated and extended. Formally checked references as hyperlinks are now available uniformly. *** Pure *** * Complete re-implementation of locales. INCOMPATIBILITY in several respects. The most important changes are listed below. See the Tutorial on Locales ("locales" manual) for details. - In locale expressions, instantiation replaces renaming. Parameters must be declared in a for clause. To aid compatibility with previous parameter inheritance, in locale declarations, parameters that are not 'touched' (instantiation position "_" or omitted) are implicitly added with their syntax at the beginning of the for clause. - Syntax from abbreviations and definitions in locales is available in locale expressions and context elements. The latter is particularly useful in locale declarations. - More flexible mechanisms to qualify names generated by locale expressions. Qualifiers (prefixes) may be specified in locale expressions, and can be marked as mandatory (syntax: "name!:") or optional (syntax "name?:"). The default depends for plain "name:" depends on the situation where a locale expression is used: in commands 'locale' and 'sublocale' prefixes are optional, in 'interpretation' and 'interpret' prefixes are mandatory. The old implicit qualifiers derived from the parameter names of a locale are no longer generated. - Command "sublocale l < e" replaces "interpretation l < e". The instantiation clause in "interpretation" and "interpret" (square brackets) is no longer available. Use locale expressions. - When converting proof scripts, mandatory qualifiers in 'interpretation' and 'interpret' should be retained by default, even if this is an INCOMPATIBILITY compared to former behavior. In the worst case, use the "name?:" form for non-mandatory ones. Qualifiers in locale expressions range over a single locale instance only. - Dropped locale element "includes". This is a major INCOMPATIBILITY. In existing theorem specifications replace the includes element by the respective context elements of the included locale, omitting those that are already present in the theorem specification. Multiple assume elements of a locale should be replaced by a single one involving the locale predicate. In the proof body, declarations (most notably theorems) may be regained by interpreting the respective locales in the proof context as required (command "interpret"). If using "includes" in replacement of a target solely because the parameter types in the theorem are not as general as in the target, consider declaring a new locale with additional type constraints on the parameters (context element "constrains"). - Discontinued "locale (open)". INCOMPATIBILITY. - Locale interpretation commands no longer attempt to simplify goal. INCOMPATIBILITY: in rare situations the generated goal differs. Use methods intro_locales and unfold_locales to clarify. - Locale interpretation commands no longer accept interpretation attributes. INCOMPATIBILITY. * Class declaration: so-called "base sort" must not be given in import list any longer, but is inferred from the specification. Particularly in HOL, write class foo = ... instead of class foo = type + ... * Class target: global versions of theorems stemming do not carry a parameter prefix any longer. INCOMPATIBILITY. * Class 'instance' command no longer accepts attached definitions. INCOMPATIBILITY, use proper 'instantiation' target instead. * Recovered hiding of consts, which was accidentally broken in Isabelle2007. Potential INCOMPATIBILITY, ``hide const c'' really makes c inaccessible; consider using ``hide (open) const c'' instead. * Slightly more coherent Pure syntax, with updated documentation in isar-ref manual. Removed locales meta_term_syntax and meta_conjunction_syntax: TERM and &&& (formerly &&) are now permanent, INCOMPATIBILITY in rare situations. Note that &&& should not be used directly in regular applications. * There is a new syntactic category "float_const" for signed decimal fractions (e.g. 123.45 or -123.45). * Removed exotic 'token_translation' command. INCOMPATIBILITY, use ML interface with 'setup' command instead. * Command 'local_setup' is similar to 'setup', but operates on a local theory context. * The 'axiomatization' command now only works within a global theory context. INCOMPATIBILITY. * Goal-directed proof now enforces strict proof irrelevance wrt. sort hypotheses. Sorts required in the course of reasoning need to be covered by the constraints in the initial statement, completed by the type instance information of the background theory. Non-trivial sort hypotheses, which rarely occur in practice, may be specified via vacuous propositions of the form SORT_CONSTRAINT('a::c). For example: lemma assumes "SORT_CONSTRAINT('a::empty)" shows False ... The result contains an implicit sort hypotheses as before -- SORT_CONSTRAINT premises are eliminated as part of the canonical rule normalization. * Generalized Isar history, with support for linear undo, direct state addressing etc. * Changed defaults for unify configuration options: unify_trace_bound = 50 (formerly 25) unify_search_bound = 60 (formerly 30) * Different bookkeeping for code equations (INCOMPATIBILITY): a) On theory merge, the last set of code equations for a particular constant is taken (in accordance with the policy applied by other parts of the code generator framework). b) Code equations stemming from explicit declarations (e.g. code attribute) gain priority over default code equations stemming from definition, primrec, fun etc. * Keyword 'code_exception' now named 'code_abort'. INCOMPATIBILITY. * Unified theorem tables for both code generators. Thus [code func] has disappeared and only [code] remains. INCOMPATIBILITY. * Command 'find_consts' searches for constants based on type and name patterns, e.g. find_consts "_ => bool" By default, matching is against subtypes, but it may be restricted to the whole type. Searching by name is possible. Multiple queries are conjunctive and queries may be negated by prefixing them with a hyphen: find_consts strict: "_ => bool" name: "Int" -"int => int" * New 'find_theorems' criterion "solves" matches theorems that directly solve the current goal (modulo higher-order unification). * Auto solve feature for main theorem statements: whenever a new goal is stated, "find_theorems solves" is called; any theorems that could solve the lemma directly are listed as part of the goal state. Cf. associated options in Proof General Isabelle settings menu, enabled by default, with reasonable timeout for pathological cases of higher-order unification. *** Document preparation *** * Antiquotation @{lemma} now imitates a regular terminal proof, demanding keyword 'by' and supporting the full method expression syntax just like the Isar command 'by'. *** HOL *** * Integrated main parts of former image HOL-Complex with HOL. Entry points Main and Complex_Main remain as before. * Logic image HOL-Plain provides a minimal HOL with the most important tools available (inductive, datatype, primrec, ...). This facilitates experimentation and tool development. Note that user applications (and library theories) should never refer to anything below theory Main, as before. * Logic image HOL-Main stops at theory Main, and thus facilitates experimentation due to shorter build times. * Logic image HOL-NSA contains theories of nonstandard analysis which were previously part of former HOL-Complex. Entry point Hyperreal remains valid, but theories formerly using Complex_Main should now use new entry point Hypercomplex. * Generic ATP manager for Sledgehammer, based on ML threads instead of Posix processes. Avoids potentially expensive forking of the ML process. New thread-based implementation also works on non-Unix platforms (Cygwin). Provers are no longer hardwired, but defined within the theory via plain ML wrapper functions. Basic Sledgehammer commands are covered in the isar-ref manual. * Wrapper scripts for remote SystemOnTPTP service allows to use sledgehammer without local ATP installation (Vampire etc.). Other provers may be included via suitable ML wrappers, see also src/HOL/ATP_Linkup.thy. * ATP selection (E/Vampire/Spass) is now via Proof General's settings menu. * The metis method no longer fails because the theorem is too trivial (contains the empty clause). * The metis method now fails in the usual manner, rather than raising an exception, if it determines that it cannot prove the theorem. * Method "coherent" implements a prover for coherent logic (see also src/Tools/coherent.ML). * Constants "undefined" and "default" replace "arbitrary". Usually "undefined" is the right choice to replace "arbitrary", though logically there is no difference. INCOMPATIBILITY. * Command "value" now integrates different evaluation mechanisms. The result of the first successful evaluation mechanism is printed. In square brackets a particular named evaluation mechanisms may be specified (currently, [SML], [code] or [nbe]). See further src/HOL/ex/Eval_Examples.thy. * Normalization by evaluation now allows non-leftlinear equations. Declare with attribute [code nbe]. * Methods "case_tac" and "induct_tac" now refer to the very same rules as the structured Isar versions "cases" and "induct", cf. the corresponding "cases" and "induct" attributes. Mutual induction rules are now presented as a list of individual projections (e.g. foo_bar.inducts for types foo and bar); the old format with explicit HOL conjunction is no longer supported. INCOMPATIBILITY, in rare situations a different rule is selected --- notably nested tuple elimination instead of former prod.exhaust: use explicit (case_tac t rule: prod.exhaust) here. * Attributes "cases", "induct", "coinduct" support "del" option. * Removed fact "case_split_thm", which duplicates "case_split". * The option datatype has been moved to a new theory Option. Renamed option_map to Option.map, and o2s to Option.set, INCOMPATIBILITY. * New predicate "strict_mono" classifies strict functions on partial orders. With strict functions on linear orders, reasoning about (in)equalities is facilitated by theorems "strict_mono_eq", "strict_mono_less_eq" and "strict_mono_less". * Some set operations are now proper qualified constants with authentic syntax. INCOMPATIBILITY: op Int ~> Set.Int op Un ~> Set.Un INTER ~> Set.INTER UNION ~> Set.UNION Inter ~> Set.Inter Union ~> Set.Union {} ~> Set.empty UNIV ~> Set.UNIV * Class complete_lattice with operations Inf, Sup, INFI, SUPR now in theory Set. * Auxiliary class "itself" has disappeared -- classes without any parameter are treated as expected by the 'class' command. * Leibnitz's Series for Pi and the arcus tangens and logarithm series. * Common decision procedures (Cooper, MIR, Ferrack, Approximation, Dense_Linear_Order) are now in directory HOL/Decision_Procs. * Theory src/HOL/Decision_Procs/Approximation provides the new proof method "approximation". It proves formulas on real values by using interval arithmetic. In the formulas are also the transcendental functions sin, cos, tan, atan, ln, exp and the constant pi are allowed. For examples see src/HOL/Descision_Procs/ex/Approximation_Ex.thy. * Theory "Reflection" now resides in HOL/Library. * Entry point to Word library now simply named "Word". INCOMPATIBILITY. * Made source layout more coherent with logical distribution structure: src/HOL/Library/RType.thy ~> src/HOL/Typerep.thy src/HOL/Library/Code_Message.thy ~> src/HOL/ src/HOL/Library/GCD.thy ~> src/HOL/ src/HOL/Library/Order_Relation.thy ~> src/HOL/ src/HOL/Library/Parity.thy ~> src/HOL/ src/HOL/Library/Univ_Poly.thy ~> src/HOL/ src/HOL/Real/ContNotDenum.thy ~> src/HOL/Library/ src/HOL/Real/Lubs.thy ~> src/HOL/ src/HOL/Real/PReal.thy ~> src/HOL/ src/HOL/Real/Rational.thy ~> src/HOL/ src/HOL/Real/RComplete.thy ~> src/HOL/ src/HOL/Real/RealDef.thy ~> src/HOL/ src/HOL/Real/RealPow.thy ~> src/HOL/ src/HOL/Real/Real.thy ~> src/HOL/ src/HOL/Complex/Complex_Main.thy ~> src/HOL/ src/HOL/Complex/Complex.thy ~> src/HOL/ src/HOL/Complex/FrechetDeriv.thy ~> src/HOL/Library/ src/HOL/Complex/Fundamental_Theorem_Algebra.thy ~> src/HOL/Library/ src/HOL/Hyperreal/Deriv.thy ~> src/HOL/ src/HOL/Hyperreal/Fact.thy ~> src/HOL/ src/HOL/Hyperreal/Integration.thy ~> src/HOL/ src/HOL/Hyperreal/Lim.thy ~> src/HOL/ src/HOL/Hyperreal/Ln.thy ~> src/HOL/ src/HOL/Hyperreal/Log.thy ~> src/HOL/ src/HOL/Hyperreal/MacLaurin.thy ~> src/HOL/ src/HOL/Hyperreal/NthRoot.thy ~> src/HOL/ src/HOL/Hyperreal/Series.thy ~> src/HOL/ src/HOL/Hyperreal/SEQ.thy ~> src/HOL/ src/HOL/Hyperreal/Taylor.thy ~> src/HOL/ src/HOL/Hyperreal/Transcendental.thy ~> src/HOL/ src/HOL/Real/Float ~> src/HOL/Library/ src/HOL/Real/HahnBanach ~> src/HOL/HahnBanach src/HOL/Real/RealVector.thy ~> src/HOL/ src/HOL/arith_data.ML ~> src/HOL/Tools src/HOL/hologic.ML ~> src/HOL/Tools src/HOL/simpdata.ML ~> src/HOL/Tools src/HOL/int_arith1.ML ~> src/HOL/Tools/int_arith.ML src/HOL/int_factor_simprocs.ML ~> src/HOL/Tools src/HOL/nat_simprocs.ML ~> src/HOL/Tools src/HOL/Real/float_arith.ML ~> src/HOL/Tools src/HOL/Real/float_syntax.ML ~> src/HOL/Tools src/HOL/Real/rat_arith.ML ~> src/HOL/Tools src/HOL/Real/real_arith.ML ~> src/HOL/Tools src/HOL/Library/Array.thy ~> src/HOL/Imperative_HOL src/HOL/Library/Heap_Monad.thy ~> src/HOL/Imperative_HOL src/HOL/Library/Heap.thy ~> src/HOL/Imperative_HOL src/HOL/Library/Imperative_HOL.thy ~> src/HOL/Imperative_HOL src/HOL/Library/Ref.thy ~> src/HOL/Imperative_HOL src/HOL/Library/Relational.thy ~> src/HOL/Imperative_HOL * If methods "eval" and "evaluation" encounter a structured proof state with !!/==>, only the conclusion is evaluated to True (if possible), avoiding strange error messages. * Method "sizechange" automates termination proofs using (a modification of) the size-change principle. Requires SAT solver. See src/HOL/ex/Termination.thy for examples. * Simplifier: simproc for let expressions now unfolds if bound variable occurs at most once in let expression body. INCOMPATIBILITY. * Method "arith": Linear arithmetic now ignores all inequalities when fast_arith_neq_limit is exceeded, instead of giving up entirely. * New attribute "arith" for facts that should always be used automatically by arithmetic. It is intended to be used locally in proofs, e.g. assumes [arith]: "x > 0" Global usage is discouraged because of possible performance impact. * New classes "top" and "bot" with corresponding operations "top" and "bot" in theory Orderings; instantiation of class "complete_lattice" requires instantiation of classes "top" and "bot". INCOMPATIBILITY. * Changed definition lemma "less_fun_def" in order to provide an instance for preorders on functions; use lemma "less_le" instead. INCOMPATIBILITY. * Theory Orderings: class "wellorder" moved here, with explicit induction rule "less_induct" as assumption. For instantiation of "wellorder" by means of predicate "wf", use rule wf_wellorderI. INCOMPATIBILITY. * Theory Orderings: added class "preorder" as superclass of "order". INCOMPATIBILITY: Instantiation proofs for order, linorder etc. slightly changed. Some theorems named order_class.* now named preorder_class.*. * Theory Relation: renamed "refl" to "refl_on", "reflexive" to "refl, "diag" to "Id_on". * Theory Finite_Set: added a new fold combinator of type ('a => 'b => 'b) => 'b => 'a set => 'b Occasionally this is more convenient than the old fold combinator which is now defined in terms of the new one and renamed to fold_image. * Theories Ring_and_Field and OrderedGroup: The lemmas "group_simps" and "ring_simps" have been replaced by "algebra_simps" (which can be extended with further lemmas!). At the moment both still exist but the former will disappear at some point. * Theory Power: Lemma power_Suc is now declared as a simp rule in class recpower. Type-specific simp rules for various recpower types have been removed. INCOMPATIBILITY, rename old lemmas as follows: rat_power_0 -> power_0 rat_power_Suc -> power_Suc realpow_0 -> power_0 realpow_Suc -> power_Suc complexpow_0 -> power_0 complexpow_Suc -> power_Suc power_poly_0 -> power_0 power_poly_Suc -> power_Suc * Theories Ring_and_Field and Divides: Definition of "op dvd" has been moved to separate class dvd in Ring_and_Field; a couple of lemmas on dvd has been generalized to class comm_semiring_1. Likewise a bunch of lemmas from Divides has been generalized from nat to class semiring_div. INCOMPATIBILITY. This involves the following theorem renames resulting from duplicate elimination: dvd_def_mod ~> dvd_eq_mod_eq_0 zero_dvd_iff ~> dvd_0_left_iff dvd_0 ~> dvd_0_right DIVISION_BY_ZERO_DIV ~> div_by_0 DIVISION_BY_ZERO_MOD ~> mod_by_0 mult_div ~> div_mult_self2_is_id mult_mod ~> mod_mult_self2_is_0 * Theory IntDiv: removed many lemmas that are instances of class-based generalizations (from Divides and Ring_and_Field). INCOMPATIBILITY, rename old lemmas as follows: dvd_diff -> nat_dvd_diff dvd_zminus_iff -> dvd_minus_iff mod_add1_eq -> mod_add_eq mod_mult1_eq -> mod_mult_right_eq mod_mult1_eq' -> mod_mult_left_eq mod_mult_distrib_mod -> mod_mult_eq nat_mod_add_left_eq -> mod_add_left_eq nat_mod_add_right_eq -> mod_add_right_eq nat_mod_div_trivial -> mod_div_trivial nat_mod_mod_trivial -> mod_mod_trivial zdiv_zadd_self1 -> div_add_self1 zdiv_zadd_self2 -> div_add_self2 zdiv_zmult_self1 -> div_mult_self2_is_id zdiv_zmult_self2 -> div_mult_self1_is_id zdvd_triv_left -> dvd_triv_left zdvd_triv_right -> dvd_triv_right zdvd_zmult_cancel_disj -> dvd_mult_cancel_left zmod_eq0_zdvd_iff -> dvd_eq_mod_eq_0[symmetric] zmod_zadd_left_eq -> mod_add_left_eq zmod_zadd_right_eq -> mod_add_right_eq zmod_zadd_self1 -> mod_add_self1 zmod_zadd_self2 -> mod_add_self2 zmod_zadd1_eq -> mod_add_eq zmod_zdiff1_eq -> mod_diff_eq zmod_zdvd_zmod -> mod_mod_cancel zmod_zmod_cancel -> mod_mod_cancel zmod_zmult_self1 -> mod_mult_self2_is_0 zmod_zmult_self2 -> mod_mult_self1_is_0 zmod_1 -> mod_by_1 zdiv_1 -> div_by_1 zdvd_abs1 -> abs_dvd_iff zdvd_abs2 -> dvd_abs_iff zdvd_refl -> dvd_refl zdvd_trans -> dvd_trans zdvd_zadd -> dvd_add zdvd_zdiff -> dvd_diff zdvd_zminus_iff -> dvd_minus_iff zdvd_zminus2_iff -> minus_dvd_iff zdvd_zmultD -> dvd_mult_right zdvd_zmultD2 -> dvd_mult_left zdvd_zmult_mono -> mult_dvd_mono zdvd_0_right -> dvd_0_right zdvd_0_left -> dvd_0_left_iff zdvd_1_left -> one_dvd zminus_dvd_iff -> minus_dvd_iff * Theory Rational: 'Fract k 0' now equals '0'. INCOMPATIBILITY. * The real numbers offer decimal input syntax: 12.34 is translated into 1234/10^2. This translation is not reversed upon output. * Theory Library/Polynomial defines an abstract type 'a poly of univariate polynomials with coefficients of type 'a. In addition to the standard ring operations, it also supports div and mod. Code generation is also supported, using list-style constructors. * Theory Library/Inner_Product defines a class of real_inner for real inner product spaces, with an overloaded operation inner :: 'a => 'a => real. Class real_inner is a subclass of real_normed_vector from theory RealVector. * Theory Library/Product_Vector provides instances for the product type 'a * 'b of several classes from RealVector and Inner_Product. Definitions of addition, subtraction, scalar multiplication, norms, and inner products are included. * Theory Library/Bit defines the field "bit" of integers modulo 2. In addition to the field operations, numerals and case syntax are also supported. * Theory Library/Diagonalize provides constructive version of Cantor's first diagonalization argument. * Theory Library/GCD: Curried operations gcd, lcm (for nat) and zgcd, zlcm (for int); carried together from various gcd/lcm developements in the HOL Distribution. Constants zgcd and zlcm replace former igcd and ilcm; corresponding theorems renamed accordingly. INCOMPATIBILITY, may recover tupled syntax as follows: hide (open) const gcd abbreviation gcd where "gcd == (%(a, b). GCD.gcd a b)" notation (output) GCD.gcd ("gcd '(_, _')") The same works for lcm, zgcd, zlcm. * Theory Library/Nat_Infinity: added addition, numeral syntax and more instantiations for algebraic structures. Removed some duplicate theorems. Changes in simp rules. INCOMPATIBILITY. * ML antiquotation @{code} takes a constant as argument and generates corresponding code in background and inserts name of the corresponding resulting ML value/function/datatype constructor binding in place. All occurrences of @{code} with a single ML block are generated simultaneously. Provides a generic and safe interface for instrumentalizing code generation. See src/HOL/Decision_Procs/Ferrack.thy for a more ambitious application. In future you ought to refrain from ad-hoc compiling generated SML code on the ML toplevel. Note that (for technical reasons) @{code} cannot refer to constants for which user-defined serializations are set. Refer to the corresponding ML counterpart directly in that cases. * Command 'rep_datatype': instead of theorem names the command now takes a list of terms denoting the constructors of the type to be represented as datatype. The characteristic theorems have to be proven. INCOMPATIBILITY. Also observe that the following theorems have disappeared in favour of existing ones: unit_induct ~> unit.induct prod_induct ~> prod.induct sum_induct ~> sum.induct Suc_Suc_eq ~> nat.inject Suc_not_Zero Zero_not_Suc ~> nat.distinct *** HOL-Algebra *** * New locales for orders and lattices where the equivalence relation is not restricted to equality. INCOMPATIBILITY: all order and lattice locales use a record structure with field eq for the equivalence. * New theory of factorial domains. * Units_l_inv and Units_r_inv are now simp rules by default. INCOMPATIBILITY. Simplifier proof that require deletion of l_inv and/or r_inv will now also require deletion of these lemmas. * Renamed the following theorems, INCOMPATIBILITY: UpperD ~> Upper_memD LowerD ~> Lower_memD least_carrier ~> least_closed greatest_carrier ~> greatest_closed greatest_Lower_above ~> greatest_Lower_below one_zero ~> carrier_one_zero one_not_zero ~> carrier_one_not_zero (collision with assumption) *** HOL-Nominal *** * Nominal datatypes can now contain type-variables. * Commands 'nominal_inductive' and 'equivariance' work with local theory targets. * Nominal primrec can now works with local theory targets and its specification syntax now conforms to the general format as seen in 'inductive' etc. * Method "perm_simp" honours the standard simplifier attributes (no_asm), (no_asm_use) etc. * The new predicate #* is defined like freshness, except that on the left hand side can be a set or list of atoms. * Experimental command 'nominal_inductive2' derives strong induction principles for inductive definitions. In contrast to 'nominal_inductive', which can only deal with a fixed number of binders, it can deal with arbitrary expressions standing for sets of atoms to be avoided. The only inductive definition we have at the moment that needs this generalisation is the typing rule for Lets in the algorithm W: Gamma |- t1 : T1 (x,close Gamma T1)::Gamma |- t2 : T2 x#Gamma ----------------------------------------------------------------- Gamma |- Let x be t1 in t2 : T2 In this rule one wants to avoid all the binders that are introduced by "close Gamma T1". We are looking for other examples where this feature might be useful. Please let us know. *** HOLCF *** * Reimplemented the simplification procedure for proving continuity subgoals. The new simproc is extensible; users can declare additional continuity introduction rules with the attribute [cont2cont]. * The continuity simproc now uses a different introduction rule for solving continuity subgoals on terms with lambda abstractions. In some rare cases the new simproc may fail to solve subgoals that the old one could solve, and "simp add: cont2cont_LAM" may be necessary. Potential INCOMPATIBILITY. * Command 'fixrec': specification syntax now conforms to the general format as seen in 'inductive' etc. See src/HOLCF/ex/Fixrec_ex.thy for examples. INCOMPATIBILITY. *** ZF *** * Proof of Zorn's Lemma for partial orders. *** ML *** * Multithreading for Poly/ML 5.1/5.2 is no longer supported, only for Poly/ML 5.2.1 or later. Important note: the TimeLimit facility depends on multithreading, so timouts will not work before Poly/ML 5.2.1! * High-level support for concurrent ML programming, see src/Pure/Cuncurrent. The data-oriented model of "future values" is particularly convenient to organize independent functional computations. The concept of "synchronized variables" provides a higher-order interface for components with shared state, avoiding the delicate details of mutexes and condition variables. (Requires Poly/ML 5.2.1 or later.) * ML bindings produced via Isar commands are stored within the Isar context (theory or proof). Consequently, commands like 'use' and 'ML' become thread-safe and work with undo as expected (concerning top-level bindings, not side-effects on global references). INCOMPATIBILITY, need to provide proper Isar context when invoking the compiler at runtime; really global bindings need to be given outside a theory. (Requires Poly/ML 5.2 or later.) * Command 'ML_prf' is analogous to 'ML' but works within a proof context. Top-level ML bindings are stored within the proof context in a purely sequential fashion, disregarding the nested proof structure. ML bindings introduced by 'ML_prf' are discarded at the end of the proof. (Requires Poly/ML 5.2 or later.) * Simplified ML attribute and method setup, cf. functions Attrib.setup and Method.setup, as well as Isar commands 'attribute_setup' and 'method_setup'. INCOMPATIBILITY for 'method_setup', need to simplify existing code accordingly, or use plain 'setup' together with old Method.add_method. * Simplified ML oracle interface Thm.add_oracle promotes 'a -> cterm to 'a -> thm, while results are always tagged with an authentic oracle name. The Isar command 'oracle' is now polymorphic, no argument type is specified. INCOMPATIBILITY, need to simplify existing oracle code accordingly. Note that extra performance may be gained by producing the cterm carefully, avoiding slow Thm.cterm_of. * Simplified interface for defining document antiquotations via ThyOutput.antiquotation, ThyOutput.output, and optionally ThyOutput.maybe_pretty_source. INCOMPATIBILITY, need to simplify user antiquotations accordingly, see src/Pure/Thy/thy_output.ML for common examples. * More systematic treatment of long names, abstract name bindings, and name space operations. Basic operations on qualified names have been move from structure NameSpace to Long_Name, e.g. Long_Name.base_name, Long_Name.append. Old type bstring has been mostly replaced by abstract type binding (see structure Binding), which supports precise qualification by packages and local theory targets, as well as proper tracking of source positions. INCOMPATIBILITY, need to wrap old bstring values into Binding.name, or better pass through abstract bindings everywhere. See further src/Pure/General/long_name.ML, src/Pure/General/binding.ML and src/Pure/General/name_space.ML * Result facts (from PureThy.note_thms, ProofContext.note_thms, LocalTheory.note etc.) now refer to the *full* internal name, not the bstring as before. INCOMPATIBILITY, not detected by ML type-checking! * Disposed old type and term read functions (Sign.read_def_typ, Sign.read_typ, Sign.read_def_terms, Sign.read_term, Thm.read_def_cterms, Thm.read_cterm etc.). INCOMPATIBILITY, should use regular Syntax.read_typ, Syntax.read_term, Syntax.read_typ_global, Syntax.read_term_global etc.; see also OldGoals.read_term as last resort for legacy applications. * Disposed old declarations, tactics, tactic combinators that refer to the simpset or claset of an implicit theory (such as Addsimps, Simp_tac, SIMPSET). INCOMPATIBILITY, should use @{simpset} etc. in embedded ML text, or local_simpset_of with a proper context passed as explicit runtime argument. * Rules and tactics that read instantiations (read_instantiate, res_inst_tac, thin_tac, subgoal_tac etc.) now demand a proper proof context, which is required for parsing and type-checking. Moreover, the variables are specified as plain indexnames, not string encodings thereof. INCOMPATIBILITY. * Generic Toplevel.add_hook interface allows to analyze the result of transactions. E.g. see src/Pure/ProofGeneral/proof_general_pgip.ML for theorem dependency output of transactions resulting in a new theory state. * ML antiquotations: block-structured compilation context indicated by \ ... \; additional antiquotation forms: @{binding name} - basic name binding @{let ?pat = term} - term abbreviation (HO matching) @{note name = fact} - fact abbreviation @{thm fact} - singleton fact (with attributes) @{thms fact} - general fact (with attributes) @{lemma prop by method} - singleton goal @{lemma prop by meth1 meth2} - singleton goal @{lemma prop1 ... propN by method} - general goal @{lemma prop1 ... propN by meth1 meth2} - general goal @{lemma (open) ...} - open derivation *** System *** * The Isabelle "emacs" tool provides a specific interface to invoke Proof General / Emacs, with more explicit failure if that is not installed (the old isabelle-interface script silently falls back on isabelle-process). The PROOFGENERAL_HOME setting determines the installation location of the Proof General distribution. * Isabelle/lib/classes/Pure.jar provides basic support to integrate the Isabelle process into a JVM/Scala application. See Isabelle/lib/jedit/plugin for a minimal example. (The obsolete Java process wrapper has been discontinued.) * Added homegrown Isabelle font with unicode layout, see lib/fonts. * Various status messages (with exact source position information) are emitted, if proper markup print mode is enabled. This allows user-interface components to provide detailed feedback on internal prover operations. New in Isabelle2008 (June 2008) ------------------------------- *** General *** * The Isabelle/Isar Reference Manual (isar-ref) has been reorganized and updated, with formally checked references as hyperlinks. * Theory loader: use_thy (and similar operations) no longer set the implicit ML context, which was occasionally hard to predict and in conflict with concurrency. INCOMPATIBILITY, use ML within Isar which provides a proper context already. * Theory loader: old-style ML proof scripts being *attached* to a thy file are no longer supported. INCOMPATIBILITY, regular 'uses' and 'use' within a theory file will do the job. * Name space merge now observes canonical order, i.e. the second space is inserted into the first one, while existing entries in the first space take precedence. INCOMPATIBILITY in rare situations, may try to swap theory imports. * Syntax: symbol \ is now considered a letter. Potential INCOMPATIBILITY in identifier syntax etc. * Outer syntax: string tokens no longer admit escaped white space, which was an accidental (undocumented) feature. INCOMPATIBILITY, use white space without escapes. * Outer syntax: string tokens may contain arbitrary character codes specified via 3 decimal digits (as in SML). E.g. "foo\095bar" for "foo_bar". *** Pure *** * Context-dependent token translations. Default setup reverts locally fixed variables, and adds hilite markup for undeclared frees. * Unused theorems can be found using the new command 'unused_thms'. There are three ways of invoking it: (1) unused_thms Only finds unused theorems in the current theory. (2) unused_thms thy_1 ... thy_n - Finds unused theorems in the current theory and all of its ancestors, excluding the theories thy_1 ... thy_n and all of their ancestors. (3) unused_thms thy_1 ... thy_n - thy'_1 ... thy'_m Finds unused theorems in the theories thy'_1 ... thy'_m and all of their ancestors, excluding the theories thy_1 ... thy_n and all of their ancestors. In order to increase the readability of the list produced by unused_thms, theorems that have been created by a particular instance of a theory command such as 'inductive' or 'function' are considered to belong to the same "group", meaning that if at least one theorem in this group is used, the other theorems in the same group are no longer reported as unused. Moreover, if all theorems in the group are unused, only one theorem in the group is displayed. Note that proof objects have to be switched on in order for unused_thms to work properly (i.e. !proofs must be >= 1, which is usually the case when using Proof General with the default settings). * Authentic naming of facts disallows ad-hoc overwriting of previous theorems within the same name space. INCOMPATIBILITY, need to remove duplicate fact bindings, or even accidental fact duplications. Note that tools may maintain dynamically scoped facts systematically, using PureThy.add_thms_dynamic. * Command 'hide' now allows to hide from "fact" name space as well. * Eliminated destructive theorem database, simpset, claset, and clasimpset. Potential INCOMPATIBILITY, really need to observe linear update of theories within ML code. * Eliminated theory ProtoPure and CPure, leaving just one Pure theory. INCOMPATIBILITY, object-logics depending on former Pure require additional setup PureThy.old_appl_syntax_setup; object-logics depending on former CPure need to refer to Pure. * Commands 'use' and 'ML' are now purely functional, operating on theory/local_theory. Removed former 'ML_setup' (on theory), use 'ML' instead. Added 'ML_val' as mere diagnostic replacement for 'ML'. INCOMPATIBILITY. * Command 'setup': discontinued implicit version with ML reference. * Instantiation target allows for simultaneous specification of class instance operations together with an instantiation proof. Type-checking phase allows to refer to class operations uniformly. See src/HOL/Complex/Complex.thy for an Isar example and src/HOL/Library/Eval.thy for an ML example. * Indexing of literal facts: be more serious about including only facts from the visible specification/proof context, but not the background context (locale etc.). Affects `prop` notation and method "fact". INCOMPATIBILITY: need to name facts explicitly in rare situations. * Method "cases", "induct", "coinduct": removed obsolete/undocumented "(open)" option, which used to expose internal bound variables to the proof text. * Isar statements: removed obsolete case "rule_context". INCOMPATIBILITY, better use explicit fixes/assumes. * Locale proofs: default proof step now includes 'unfold_locales'; hence 'proof' without argument may be used to unfold locale predicates. *** Document preparation *** * Simplified pdfsetup.sty: color/hyperref is used unconditionally for both pdf and dvi (hyperlinks usually work in xdvi as well); removed obsolete thumbpdf setup (contemporary PDF viewers do this on the spot); renamed link color from "darkblue" to "linkcolor" (default value unchanged, can be redefined via \definecolor); no longer sets "a4paper" option (unnecessary or even intrusive). * Antiquotation @{lemma A method} proves proposition A by the given method (either a method name or a method name plus (optional) method arguments in parentheses) and prints A just like @{prop A}. *** HOL *** * New primrec package. Specification syntax conforms in style to definition/function/.... No separate induction rule is provided. The "primrec" command distinguishes old-style and new-style specifications by syntax. The former primrec package is now named OldPrimrecPackage. When adjusting theories, beware: constants stemming from new-style primrec specifications have authentic syntax. * Metis prover is now an order of magnitude faster, and also works with multithreading. * Metis: the maximum number of clauses that can be produced from a theorem is now given by the attribute max_clauses. Theorems that exceed this number are ignored, with a warning printed. * Sledgehammer no longer produces structured proofs by default. To enable, declare [[sledgehammer_full = true]]. Attributes reconstruction_modulus, reconstruction_sorts renamed sledgehammer_modulus, sledgehammer_sorts. INCOMPATIBILITY. * Method "induct_scheme" derives user-specified induction rules from well-founded induction and completeness of patterns. This factors out some operations that are done internally by the function package and makes them available separately. See src/HOL/ex/Induction_Scheme.thy for examples. * More flexible generation of measure functions for termination proofs: Measure functions can be declared by proving a rule of the form "is_measure f" and giving it the [measure_function] attribute. The "is_measure" predicate is logically meaningless (always true), and just guides the heuristic. To find suitable measure functions, the termination prover sets up the goal "is_measure ?f" of the appropriate type and generates all solutions by Prolog-style backward proof using the declared rules. This setup also deals with rules like "is_measure f ==> is_measure (list_size f)" which accommodates nested datatypes that recurse through lists. Similar rules are predeclared for products and option types. * Turned the type of sets "'a set" into an abbreviation for "'a => bool" INCOMPATIBILITIES: - Definitions of overloaded constants on sets have to be replaced by definitions on => and bool. - Some definitions of overloaded operators on sets can now be proved using the definitions of the operators on => and bool. Therefore, the following theorems have been renamed: subset_def -> subset_eq psubset_def -> psubset_eq set_diff_def -> set_diff_eq Compl_def -> Compl_eq Sup_set_def -> Sup_set_eq Inf_set_def -> Inf_set_eq sup_set_def -> sup_set_eq inf_set_def -> inf_set_eq - Due to the incompleteness of the HO unification algorithm, some rules such as subst may require manual instantiation, if some of the unknowns in the rule is a set. - Higher order unification and forward proofs: The proof pattern have "P (S::'a set)" <...> then have "EX S. P S" .. no longer works (due to the incompleteness of the HO unification algorithm) and must be replaced by the pattern have "EX S. P S" proof show "P S" <...> qed - Calculational reasoning with subst (or similar rules): The proof pattern have "P (S::'a set)" <...> also have "S = T" <...> finally have "P T" . no longer works (for similar reasons as the previous example) and must be replaced by something like have "P (S::'a set)" <...> moreover have "S = T" <...> ultimately have "P T" by simp - Tactics or packages written in ML code: Code performing pattern matching on types via Type ("set", [T]) => ... must be rewritten. Moreover, functions like strip_type or binder_types no longer return the right value when applied to a type of the form T1 => ... => Tn => U => bool rather than T1 => ... => Tn => U set * Merged theories Wellfounded_Recursion, Accessible_Part and Wellfounded_Relations to theory Wellfounded. * Explicit class "eq" for executable equality. INCOMPATIBILITY. * Class finite no longer treats UNIV as class parameter. Use class enum from theory Library/Enum instead to achieve a similar effect. INCOMPATIBILITY. * Theory List: rule list_induct2 now has explicitly named cases "Nil" and "Cons". INCOMPATIBILITY. * HOL (and FOL): renamed variables in rules imp_elim and swap. Potential INCOMPATIBILITY. * Theory Product_Type: duplicated lemmas split_Pair_apply and injective_fst_snd removed, use split_eta and prod_eqI instead. Renamed upd_fst to apfst and upd_snd to apsnd. INCOMPATIBILITY. * Theory Nat: removed redundant lemmas that merely duplicate lemmas of the same name in theory Orderings: less_trans less_linear le_imp_less_or_eq le_less_trans less_le_trans less_not_sym less_asym Renamed less_imp_le to less_imp_le_nat, and less_irrefl to less_irrefl_nat. Potential INCOMPATIBILITY due to more general types and different variable names. * Library/Option_ord.thy: Canonical order on option type. * Library/RBT.thy: Red-black trees, an efficient implementation of finite maps. * Library/Countable.thy: Type class for countable types. * Theory Int: The representation of numerals has changed. The infix operator BIT and the bit datatype with constructors B0 and B1 have disappeared. INCOMPATIBILITY, use "Int.Bit0 x" and "Int.Bit1 y" in place of "x BIT bit.B0" and "y BIT bit.B1", respectively. Theorems involving BIT, B0, or B1 have been renamed with "Bit0" or "Bit1" accordingly. * Theory Nat: definition of <= and < on natural numbers no longer depend on well-founded relations. INCOMPATIBILITY. Definitions le_def and less_def have disappeared. Consider lemmas not_less [symmetric, where ?'a = nat] and less_eq [symmetric] instead. * Theory Finite_Set: locales ACf, ACe, ACIf, ACIfSL and ACIfSLlin (whose purpose mainly is for various fold_set functionals) have been abandoned in favor of the existing algebraic classes ab_semigroup_mult, comm_monoid_mult, ab_semigroup_idem_mult, lower_semilattice (resp. upper_semilattice) and linorder. INCOMPATIBILITY. * Theory Transitive_Closure: induct and cases rules now declare proper case_names ("base" and "step"). INCOMPATIBILITY. * Theorem Inductive.lfp_ordinal_induct generalized to complete lattices. The form set-specific version is available as Inductive.lfp_ordinal_induct_set. * Renamed theorems "power.simps" to "power_int.simps". INCOMPATIBILITY. * Class semiring_div provides basic abstract properties of semirings with division and modulo operations. Subsumes former class dvd_mod. * Merged theories IntDef, Numeral and IntArith into unified theory Int. INCOMPATIBILITY. * Theory Library/Code_Index: type "index" now represents natural numbers rather than integers. INCOMPATIBILITY. * New class "uminus" with operation "uminus" (split of from class "minus" which now only has operation "minus", binary). INCOMPATIBILITY. * Constants "card", "internal_split", "option_map" now with authentic syntax. INCOMPATIBILITY. * Definitions subset_def, psubset_def, set_diff_def, Compl_def, le_bool_def, less_bool_def, le_fun_def, less_fun_def, inf_bool_def, sup_bool_def, Inf_bool_def, Sup_bool_def, inf_fun_def, sup_fun_def, Inf_fun_def, Sup_fun_def, inf_set_def, sup_set_def, Inf_set_def, Sup_set_def, le_def, less_def, option_map_def now with object equality. INCOMPATIBILITY. * Records. Removed K_record, and replaced it by pure lambda term %x. c. The simplifier setup is now more robust against eta expansion. INCOMPATIBILITY: in cases explicitly referring to K_record. * Library/Multiset: {#a, b, c#} abbreviates {#a#} + {#b#} + {#c#}. * Library/ListVector: new theory of arithmetic vector operations. * Library/Order_Relation: new theory of various orderings as sets of pairs. Defines preorders, partial orders, linear orders and well-orders on sets and on types. *** ZF *** * Renamed some theories to allow to loading both ZF and HOL in the same session: Datatype -> Datatype_ZF Inductive -> Inductive_ZF Int -> Int_ZF IntDiv -> IntDiv_ZF Nat -> Nat_ZF List -> List_ZF Main -> Main_ZF INCOMPATIBILITY: ZF theories that import individual theories below Main might need to be adapted. Regular theory Main is still available, as trivial extension of Main_ZF. *** ML *** * ML within Isar: antiquotation @{const name} or @{const name(typargs)} produces statically-checked Const term. * Functor NamedThmsFun: data is available to the user as dynamic fact (of the same name). Removed obsolete print command. * Removed obsolete "use_legacy_bindings" function. * The ``print mode'' is now a thread-local value derived from a global template (the former print_mode reference), thus access becomes non-critical. The global print_mode reference is for session management only; user-code should use print_mode_value, print_mode_active, PrintMode.setmp etc. INCOMPATIBILITY. * Functions system/system_out provide a robust way to invoke external shell commands, with propagation of interrupts (requires Poly/ML 5.2.1). Do not use OS.Process.system etc. from the basis library! *** System *** * Default settings: PROOFGENERAL_OPTIONS no longer impose xemacs --- in accordance with Proof General 3.7, which prefers GNU emacs. * isatool tty runs Isabelle process with plain tty interaction; optional line editor may be specified via ISABELLE_LINE_EDITOR setting, the default settings attempt to locate "ledit" and "rlwrap". * isatool browser now works with Cygwin as well, using general "javapath" function defined in Isabelle process environment. * YXML notation provides a simple and efficient alternative to standard XML transfer syntax. See src/Pure/General/yxml.ML and isatool yxml as described in the Isabelle system manual. * JVM class isabelle.IsabelleProcess (located in Isabelle/lib/classes) provides general wrapper for managing an Isabelle process in a robust fashion, with ``cooked'' output from stdin/stderr. * Rudimentary Isabelle plugin for jEdit (see Isabelle/lib/jedit), based on Isabelle/JVM process wrapper (see Isabelle/lib/classes). * Removed obsolete THIS_IS_ISABELLE_BUILD feature. NB: the documented way of changing the user's settings is via ISABELLE_HOME_USER/etc/settings, which is a fully featured bash script. * Multithreading.max_threads := 0 refers to the number of actual CPU cores of the underlying machine, which is a good starting point for optimal performance tuning. The corresponding usedir option -M allows "max" as an alias for "0". WARNING: does not work on certain versions of Mac OS (with Poly/ML 5.1). * isabelle-process: non-ML sessions are run with "nice", to reduce the adverse effect of Isabelle flooding interactive front-ends (notably ProofGeneral / XEmacs). New in Isabelle2007 (November 2007) ----------------------------------- *** General *** * More uniform information about legacy features, notably a warning/error of "Legacy feature: ...", depending on the state of the tolerate_legacy_features flag (default true). FUTURE INCOMPATIBILITY: legacy features will disappear eventually. * Theory syntax: the header format ``theory A = B + C:'' has been discontinued in favour of ``theory A imports B C begin''. Use isatool fixheaders to convert existing theory files. INCOMPATIBILITY. * Theory syntax: the old non-Isar theory file format has been discontinued altogether. Note that ML proof scripts may still be used with Isar theories; migration is usually quite simple with the ML function use_legacy_bindings. INCOMPATIBILITY. * Theory syntax: some popular names (e.g. 'class', 'declaration', 'fun', 'help', 'if') are now keywords. INCOMPATIBILITY, use double quotes. * Theory loader: be more serious about observing the static theory header specifications (including optional directories), but not the accidental file locations of previously successful loads. The strict update policy of former update_thy is now already performed by use_thy, so the former has been removed; use_thys updates several theories simultaneously, just as 'imports' within a theory header specification, but without merging the results. Potential INCOMPATIBILITY: may need to refine theory headers and commands ROOT.ML which depend on load order. * Theory loader: optional support for content-based file identification, instead of the traditional scheme of full physical path plus date stamp; configured by the ISABELLE_FILE_IDENT setting (cf. the system manual). The new scheme allows to work with non-finished theories in persistent session images, such that source files may be moved later on without requiring reloads. * Theory loader: old-style ML proof scripts being *attached* to a thy file (with the same base name as the theory) are considered a legacy feature, which will disappear eventually. Even now, the theory loader no longer maintains dependencies on such files. * Syntax: the scope for resolving ambiguities via type-inference is now limited to individual terms, instead of whole simultaneous specifications as before. This greatly reduces the complexity of the syntax module and improves flexibility by separating parsing and type-checking. INCOMPATIBILITY: additional type-constraints (explicit 'fixes' etc.) are required in rare situations. * Syntax: constants introduced by new-style packages ('definition', 'abbreviation' etc.) are passed through the syntax module in ``authentic mode''. This means that associated mixfix annotations really stick to such constants, independently of potential name space ambiguities introduced later on. INCOMPATIBILITY: constants in parse trees are represented slightly differently, may need to adapt syntax translations accordingly. Use CONST marker in 'translations' and @{const_syntax} antiquotation in 'parse_translation' etc. * Legacy goal package: reduced interface to the bare minimum required to keep existing proof scripts running. Most other user-level functions are now part of the OldGoals structure, which is *not* open by default (consider isatool expandshort before open OldGoals). Removed top_sg, prin, printyp, pprint_term/typ altogether, because these tend to cause confusion about the actual goal (!) context being used here, which is not necessarily the same as the_context(). * Command 'find_theorems': supports "*" wild-card in "name:" criterion; "with_dups" option. Certain ProofGeneral versions might support a specific search form (see ProofGeneral/CHANGES). * The ``prems limit'' option (cf. ProofContext.prems_limit) is now -1 by default, which means that "prems" (and also "fixed variables") are suppressed from proof state output. Note that the ProofGeneral settings mechanism allows to change and save options persistently, but older versions of Isabelle will fail to start up if a negative prems limit is imposed. * Local theory targets may be specified by non-nested blocks of ``context/locale/class ... begin'' followed by ``end''. The body may contain definitions, theorems etc., including any derived mechanism that has been implemented on top of these primitives. This concept generalizes the existing ``theorem (in ...)'' towards more versatility and scalability. * Proof General interface: proper undo of final 'end' command; discontinued Isabelle/classic mode (ML proof scripts). *** Document preparation *** * Added antiquotation @{theory name} which prints the given name, after checking that it refers to a valid ancestor theory in the current context. * Added antiquotations @{ML_type text} and @{ML_struct text} which check the given source text as ML type/structure, printing verbatim. * Added antiquotation @{abbrev "c args"} which prints the abbreviation "c args == rhs" given in the current context. (Any number of arguments may be given on the LHS.) *** Pure *** * The 'class' package offers a combination of axclass and locale to achieve Haskell-like type classes in Isabelle. Definitions and theorems within a class context produce both relative results (with implicit parameters according to the locale context), and polymorphic constants with qualified polymorphism (according to the class context). Within the body context of a 'class' target, a separate syntax layer ("user space type system") takes care of converting between global polymorphic consts and internal locale representation. See src/HOL/ex/Classpackage.thy for examples (as well as main HOL). "isatool doc classes" provides a tutorial. * Generic code generator framework allows to generate executable code for ML and Haskell (including Isabelle classes). A short usage sketch: internal compilation: export_code in SML writing SML code to a file: export_code in SML writing OCaml code to a file: export_code in OCaml writing Haskell code to a bunch of files: export_code in Haskell evaluating closed propositions to True/False using code generation: method ``eval'' Reasonable default setup of framework in HOL. Theorem attributs for selecting and transforming function equations theorems: [code fun]: select a theorem as function equation for a specific constant [code fun del]: deselect a theorem as function equation for a specific constant [code inline]: select an equation theorem for unfolding (inlining) in place [code inline del]: deselect an equation theorem for unfolding (inlining) in place User-defined serializations (target in {SML, OCaml, Haskell}): code_const {(target) }+ code_type {(target) }+ code_instance {(target)}+ where instance ::= :: code_class {(target) }+ where class target syntax ::= {where { == }+}? code_instance and code_class only are effective to target Haskell. For example usage see src/HOL/ex/Codegenerator.thy and src/HOL/ex/Codegenerator_Pretty.thy. A separate tutorial on code generation from Isabelle/HOL theories is available via "isatool doc codegen". * Code generator: consts in 'consts_code' Isar commands are now referred to by usual term syntax (including optional type annotations). * Command 'no_translations' removes translation rules from theory syntax. * Overloaded definitions are now actually checked for acyclic dependencies. The overloading scheme is slightly more general than that of Haskell98, although Isabelle does not demand an exact correspondence to type class and instance declarations. INCOMPATIBILITY, use ``defs (unchecked overloaded)'' to admit more exotic versions of overloading -- at the discretion of the user! Polymorphic constants are represented via type arguments, i.e. the instantiation that matches an instance against the most general declaration given in the signature. For example, with the declaration c :: 'a => 'a => 'a, an instance c :: nat => nat => nat is represented as c(nat). Overloading is essentially simultaneous structural recursion over such type arguments. Incomplete specification patterns impose global constraints on all occurrences, e.g. c('a * 'a) on the LHS means that more general c('a * 'b) will be disallowed on any RHS. Command 'print_theory' outputs the normalized system of recursive equations, see section "definitions". * Configuration options are maintained within the theory or proof context (with name and type bool/int/string), providing a very simple interface to a poor-man's version of general context data. Tools may declare options in ML (e.g. using Attrib.config_int) and then refer to these values using Config.get etc. Users may change options via an associated attribute of the same name. This form of context declaration works particularly well with commands 'declare' or 'using', for example ``declare [[foo = 42]]''. Thus it has become very easy to avoid global references, which would not observe Isar toplevel undo/redo and fail to work with multithreading. Various global ML references of Pure and HOL have been turned into configuration options: Unify.search_bound unify_search_bound Unify.trace_bound unify_trace_bound Unify.trace_simp unify_trace_simp Unify.trace_types unify_trace_types Simplifier.simp_depth_limit simp_depth_limit Blast.depth_limit blast_depth_limit DatatypeProp.dtK datatype_distinctness_limit fast_arith_neq_limit fast_arith_neq_limit fast_arith_split_limit fast_arith_split_limit * Named collections of theorems may be easily installed as context data using the functor NamedThmsFun (see also src/Pure/Tools/named_thms.ML). The user may add or delete facts via attributes; there is also a toplevel print command. This facility is just a common case of general context data, which is the preferred way for anything more complex than just a list of facts in canonical order. * Isar: command 'declaration' augments a local theory by generic declaration functions written in ML. This enables arbitrary content being added to the context, depending on a morphism that tells the difference of the original declaration context wrt. the application context encountered later on. * Isar: proper interfaces for simplification procedures. Command 'simproc_setup' declares named simprocs (with match patterns, and body text in ML). Attribute "simproc" adds/deletes simprocs in the current context. ML antiquotation @{simproc name} retrieves named simprocs. * Isar: an extra pair of brackets around attribute declarations abbreviates a theorem reference involving an internal dummy fact, which will be ignored later --- only the effect of the attribute on the background context will persist. This form of in-place declarations is particularly useful with commands like 'declare' and 'using', for example ``have A using [[simproc a]] by simp''. * Isar: method "assumption" (and implicit closing of subproofs) now takes simple non-atomic goal assumptions into account: after applying an assumption as a rule the resulting subgoals are solved by atomic assumption steps. This is particularly useful to finish 'obtain' goals, such as "!!x. (!!x. P x ==> thesis) ==> P x ==> thesis", without referring to the original premise "!!x. P x ==> thesis" in the Isar proof context. POTENTIAL INCOMPATIBILITY: method "assumption" is more permissive. * Isar: implicit use of prems from the Isar proof context is considered a legacy feature. Common applications like ``have A .'' may be replaced by ``have A by fact'' or ``note `A`''. In general, referencing facts explicitly here improves readability and maintainability of proof texts. * Isar: improper proof element 'guess' is like 'obtain', but derives the obtained context from the course of reasoning! For example: assume "EX x y. A x & B y" -- "any previous fact" then guess x and y by clarify This technique is potentially adventurous, depending on the facts and proof tools being involved here. * Isar: known facts from the proof context may be specified as literal propositions, using ASCII back-quote syntax. This works wherever named facts used to be allowed so far, in proof commands, proof methods, attributes etc. Literal facts are retrieved from the context according to unification of type and term parameters. For example, provided that "A" and "A ==> B" and "!!x. P x ==> Q x" are known theorems in the current context, then these are valid literal facts: `A` and `A ==> B` and `!!x. P x ==> Q x" as well as `P a ==> Q a` etc. There is also a proof method "fact" which does the same composition for explicit goal states, e.g. the following proof texts coincide with certain special cases of literal facts: have "A" by fact == note `A` have "A ==> B" by fact == note `A ==> B` have "!!x. P x ==> Q x" by fact == note `!!x. P x ==> Q x` have "P a ==> Q a" by fact == note `P a ==> Q a` * Isar: ":" (colon) is no longer a symbolic identifier character in outer syntax. Thus symbolic identifiers may be used without additional white space in declarations like this: ``assume *: A''. * Isar: 'print_facts' prints all local facts of the current context, both named and unnamed ones. * Isar: 'def' now admits simultaneous definitions, e.g.: def x == "t" and y == "u" * Isar: added command 'unfolding', which is structurally similar to 'using', but affects both the goal state and facts by unfolding given rewrite rules. Thus many occurrences of the 'unfold' method or 'unfolded' attribute may be replaced by first-class proof text. * Isar: methods 'unfold' / 'fold', attributes 'unfolded' / 'folded', and command 'unfolding' now all support object-level equalities (potentially conditional). The underlying notion of rewrite rule is analogous to the 'rule_format' attribute, but *not* that of the Simplifier (which is usually more generous). * Isar: the new attribute [rotated n] (default n = 1) rotates the premises of a theorem by n. Useful in conjunction with drule. * Isar: the goal restriction operator [N] (default N = 1) evaluates a method expression within a sandbox consisting of the first N sub-goals, which need to exist. For example, ``simp_all [3]'' simplifies the first three sub-goals, while (rule foo, simp_all)[] simplifies all new goals that emerge from applying rule foo to the originally first one. * Isar: schematic goals are no longer restricted to higher-order patterns; e.g. ``lemma "?P(?x)" by (rule TrueI)'' now works as expected. * Isar: the conclusion of a long theorem statement is now either 'shows' (a simultaneous conjunction, as before), or 'obtains' (essentially a disjunction of cases with local parameters and assumptions). The latter allows to express general elimination rules adequately; in this notation common elimination rules look like this: lemma exE: -- "EX x. P x ==> (!!x. P x ==> thesis) ==> thesis" assumes "EX x. P x" obtains x where "P x" lemma conjE: -- "A & B ==> (A ==> B ==> thesis) ==> thesis" assumes "A & B" obtains A and B lemma disjE: -- "A | B ==> (A ==> thesis) ==> (B ==> thesis) ==> thesis" assumes "A | B" obtains A | B The subsequent classical rules even refer to the formal "thesis" explicitly: lemma classical: -- "(~ thesis ==> thesis) ==> thesis" obtains "~ thesis" lemma Peirce's_Law: -- "((thesis ==> something) ==> thesis) ==> thesis" obtains "thesis ==> something" The actual proof of an 'obtains' statement is analogous to that of the Isar proof element 'obtain', only that there may be several cases. Optional case names may be specified in parentheses; these will be available both in the present proof and as annotations in the resulting rule, for later use with the 'cases' method (cf. attribute case_names). * Isar: the assumptions of a long theorem statement are available as "assms" fact in the proof context. This is more appropriate than the (historical) "prems", which refers to all assumptions of the current context, including those from the target locale, proof body etc. * Isar: 'print_statement' prints theorems from the current theory or proof context in long statement form, according to the syntax of a top-level lemma. * Isar: 'obtain' takes an optional case name for the local context introduction rule (default "that"). * Isar: removed obsolete 'concl is' patterns. INCOMPATIBILITY, use explicit (is "_ ==> ?foo") in the rare cases where this still happens to occur. * Pure: syntax "CONST name" produces a fully internalized constant according to the current context. This is particularly useful for syntax translations that should refer to internal constant representations independently of name spaces. * Pure: syntax constant for foo (binder "FOO ") is called "foo_binder" instead of "FOO ". This allows multiple binder declarations to coexist in the same context. INCOMPATIBILITY. * Isar/locales: 'notation' provides a robust interface to the 'syntax' primitive that also works in a locale context (both for constants and fixed variables). Type declaration and internal syntactic representation of given constants retrieved from the context. Likewise, the 'no_notation' command allows to remove given syntax annotations from the current context. * Isar/locales: new derived specification elements 'axiomatization', 'definition', 'abbreviation', which support type-inference, admit object-level specifications (equality, equivalence). See also the isar-ref manual. Examples: axiomatization eq (infix "===" 50) where eq_refl: "x === x" and eq_subst: "x === y ==> P x ==> P y" definition "f x y = x + y + 1" definition g where "g x = f x x" abbreviation neq (infix "=!=" 50) where "x =!= y == ~ (x === y)" These specifications may be also used in a locale context. Then the constants being introduced depend on certain fixed parameters, and the constant name is qualified by the locale base name. An internal abbreviation takes care for convenient input and output, making the parameters implicit and using the original short name. See also src/HOL/ex/Abstract_NAT.thy for an example of deriving polymorphic entities from a monomorphic theory. Presently, abbreviations are only available 'in' a target locale, but not inherited by general import expressions. Also note that 'abbreviation' may be used as a type-safe replacement for 'syntax' + 'translations' in common applications. The "no_abbrevs" print mode prevents folding of abbreviations in term output. Concrete syntax is attached to specified constants in internal form, independently of name spaces. The parse tree representation is slightly different -- use 'notation' instead of raw 'syntax', and 'translations' with explicit "CONST" markup to accommodate this. * Pure/Isar: unified syntax for new-style specification mechanisms (e.g. 'definition', 'abbreviation', or 'inductive' in HOL) admits full type inference and dummy patterns ("_"). For example: definition "K x _ = x" inductive conj for A B where "A ==> B ==> conj A B" * Pure: command 'print_abbrevs' prints all constant abbreviations of the current context. Print mode "no_abbrevs" prevents inversion of abbreviations on output. * Isar/locales: improved parameter handling: use of locales "var" and "struct" no longer necessary; - parameter renamings are no longer required to be injective. For example, this allows to define endomorphisms as locale endom = homom mult mult h. * Isar/locales: changed the way locales with predicates are defined. Instead of accumulating the specification, the imported expression is now an interpretation. INCOMPATIBILITY: different normal form of locale expressions. In particular, in interpretations of locales with predicates, goals repesenting already interpreted fragments are not removed automatically. Use methods `intro_locales' and `unfold_locales'; see below. * Isar/locales: new methods `intro_locales' and `unfold_locales' provide backward reasoning on locales predicates. The methods are aware of interpretations and discharge corresponding goals. `intro_locales' is less aggressive then `unfold_locales' and does not unfold predicates to assumptions. * Isar/locales: the order in which locale fragments are accumulated has changed. This enables to override declarations from fragments due to interpretations -- for example, unwanted simp rules. * Isar/locales: interpretation in theories and proof contexts has been extended. One may now specify (and prove) equations, which are unfolded in interpreted theorems. This is useful for replacing defined concepts (constants depending on locale parameters) by concepts already existing in the target context. Example: interpretation partial_order ["op <= :: [int, int] => bool"] where "partial_order.less (op <=) (x::int) y = (x < y)" Typically, the constant `partial_order.less' is created by a definition specification element in the context of locale partial_order. * Method "induct": improved internal context management to support local fixes and defines on-the-fly. Thus explicit meta-level connectives !! and ==> are rarely required anymore in inductive goals (using object-logic connectives for this purpose has been long obsolete anyway). Common proof patterns are explained in src/HOL/Induct/Common_Patterns.thy, see also src/HOL/Isar_examples/Puzzle.thy and src/HOL/Lambda for realistic examples. * Method "induct": improved handling of simultaneous goals. Instead of introducing object-level conjunction, the statement is now split into several conclusions, while the corresponding symbolic cases are nested accordingly. INCOMPATIBILITY, proofs need to be structured explicitly, see src/HOL/Induct/Common_Patterns.thy, for example. * Method "induct": mutual induction rules are now specified as a list of rule sharing the same induction cases. HOL packages usually provide foo_bar.inducts for mutually defined items foo and bar (e.g. inductive predicates/sets or datatypes). INCOMPATIBILITY, users need to specify mutual induction rules differently, i.e. like this: (induct rule: foo_bar.inducts) (induct set: foo bar) (induct pred: foo bar) (induct type: foo bar) The ML function ProjectRule.projections turns old-style rules into the new format. * Method "coinduct": dual of induction, see src/HOL/Library/Coinductive_List.thy for various examples. * Method "cases", "induct", "coinduct": the ``(open)'' option is considered a legacy feature. * Attribute "symmetric" produces result with standardized schematic variables (index 0). Potential INCOMPATIBILITY. * Simplifier: by default the simplifier trace only shows top level rewrites now. That is, trace_simp_depth_limit is set to 1 by default. Thus there is less danger of being flooded by the trace. The trace indicates where parts have been suppressed. * Provers/classical: removed obsolete classical version of elim_format attribute; classical elim/dest rules are now treated uniformly when manipulating the claset. * Provers/classical: stricter checks to ensure that supplied intro, dest and elim rules are well-formed; dest and elim rules must have at least one premise. * Provers/classical: attributes dest/elim/intro take an optional weight argument for the rule (just as the Pure versions). Weights are ignored by automated tools, but determine the search order of single rule steps. * Syntax: input syntax now supports dummy variable binding "%_. b", where the body does not mention the bound variable. Note that dummy patterns implicitly depend on their context of bounds, which makes "{_. _}" match any set comprehension as expected. Potential INCOMPATIBILITY -- parse translations need to cope with syntactic constant "_idtdummy" in the binding position. * Syntax: removed obsolete syntactic constant "_K" and its associated parse translation. INCOMPATIBILITY -- use dummy abstraction instead, for example "A -> B" => "Pi A (%_. B)". * Pure: 'class_deps' command visualizes the subclass relation, using the graph browser tool. * Pure: 'print_theory' now suppresses certain internal declarations by default; use '!' option for full details. *** HOL *** * Method "metis" proves goals by applying the Metis general-purpose resolution prover (see also http://gilith.com/software/metis/). Examples are in the directory MetisExamples. WARNING: the Isabelle/HOL-Metis integration does not yet work properly with multi-threading. * Command 'sledgehammer' invokes external automatic theorem provers as background processes. It generates calls to the "metis" method if successful. These can be pasted into the proof. Users do not have to wait for the automatic provers to return. WARNING: does not really work with multi-threading. * New "auto_quickcheck" feature tests outermost goal statements for potential counter-examples. Controlled by ML references auto_quickcheck (default true) and auto_quickcheck_time_limit (default 5000 milliseconds). Fails silently if statements is outside of executable fragment, or any other codgenerator problem occurs. * New constant "undefined" with axiom "undefined x = undefined". * Added class "HOL.eq", allowing for code generation with polymorphic equality. * Some renaming of class constants due to canonical name prefixing in the new 'class' package: HOL.abs ~> HOL.abs_class.abs HOL.divide ~> HOL.divide_class.divide 0 ~> HOL.zero_class.zero 1 ~> HOL.one_class.one op + ~> HOL.plus_class.plus op - ~> HOL.minus_class.minus uminus ~> HOL.minus_class.uminus op * ~> HOL.times_class.times op < ~> HOL.ord_class.less op <= > HOL.ord_class.less_eq Nat.power ~> Power.power_class.power Nat.size ~> Nat.size_class.size Numeral.number_of ~> Numeral.number_class.number_of FixedPoint.Inf ~> Lattices.complete_lattice_class.Inf FixedPoint.Sup ~> Lattices.complete_lattice_class.Sup Orderings.min ~> Orderings.ord_class.min Orderings.max ~> Orderings.ord_class.max Divides.op div ~> Divides.div_class.div Divides.op mod ~> Divides.div_class.mod Divides.op dvd ~> Divides.div_class.dvd INCOMPATIBILITY. Adaptions may be required in the following cases: a) User-defined constants using any of the names "plus", "minus", "times", "less" or "less_eq". The standard syntax translations for "+", "-" and "*" may go wrong. INCOMPATIBILITY: use more specific names. b) Variables named "plus", "minus", "times", "less", "less_eq" INCOMPATIBILITY: use more specific names. c) Permutative equations (e.g. "a + b = b + a") Since the change of names also changes the order of terms, permutative rewrite rules may get applied in a different order. Experience shows that this is rarely the case (only two adaptions in the whole Isabelle distribution). INCOMPATIBILITY: rewrite proofs d) ML code directly refering to constant names This in general only affects hand-written proof tactics, simprocs and so on. INCOMPATIBILITY: grep your sourcecode and replace names. Consider using @{const_name} antiquotation. * New class "default" with associated constant "default". * Function "sgn" is now overloaded and available on int, real, complex (and other numeric types), using class "sgn". Two possible defs of sgn are given as equational assumptions in the classes sgn_if and sgn_div_norm; ordered_idom now also inherits from sgn_if. INCOMPATIBILITY. * Locale "partial_order" now unified with class "order" (cf. theory Orderings), added parameter "less". INCOMPATIBILITY. * Renamings in classes "order" and "linorder": facts "refl", "trans" and "cases" to "order_refl", "order_trans" and "linorder_cases", to avoid clashes with HOL "refl" and "trans". INCOMPATIBILITY. * Classes "order" and "linorder": potential INCOMPATIBILITY due to changed order of proof goals in instance proofs. * The transitivity reasoner for partial and linear orders is set up for classes "order" and "linorder". Instances of the reasoner are available in all contexts importing or interpreting the corresponding locales. Method "order" invokes the reasoner separately; the reasoner is also integrated with the Simplifier as a solver. Diagnostic command 'print_orders' shows the available instances of the reasoner in the current context. * Localized monotonicity predicate in theory "Orderings"; integrated lemmas max_of_mono and min_of_mono with this predicate. INCOMPATIBILITY. * Formulation of theorem "dense" changed slightly due to integration with new class dense_linear_order. * Uniform lattice theory development in HOL. constants "meet" and "join" now named "inf" and "sup" constant "Meet" now named "Inf" classes "meet_semilorder" and "join_semilorder" now named "lower_semilattice" and "upper_semilattice" class "lorder" now named "lattice" class "comp_lat" now named "complete_lattice" Instantiation of lattice classes allows explicit definitions for "inf" and "sup" operations (or "Inf" and "Sup" for complete lattices). INCOMPATIBILITY. Theorem renames: meet_left_le ~> inf_le1 meet_right_le ~> inf_le2 join_left_le ~> sup_ge1 join_right_le ~> sup_ge2 meet_join_le ~> inf_sup_ord le_meetI ~> le_infI join_leI ~> le_supI le_meet ~> le_inf_iff le_join ~> ge_sup_conv meet_idempotent ~> inf_idem join_idempotent ~> sup_idem meet_comm ~> inf_commute join_comm ~> sup_commute meet_leI1 ~> le_infI1 meet_leI2 ~> le_infI2 le_joinI1 ~> le_supI1 le_joinI2 ~> le_supI2 meet_assoc ~> inf_assoc join_assoc ~> sup_assoc meet_left_comm ~> inf_left_commute meet_left_idempotent ~> inf_left_idem join_left_comm ~> sup_left_commute join_left_idempotent ~> sup_left_idem meet_aci ~> inf_aci join_aci ~> sup_aci le_def_meet ~> le_iff_inf le_def_join ~> le_iff_sup join_absorp2 ~> sup_absorb2 join_absorp1 ~> sup_absorb1 meet_absorp1 ~> inf_absorb1 meet_absorp2 ~> inf_absorb2 meet_join_absorp ~> inf_sup_absorb join_meet_absorp ~> sup_inf_absorb distrib_join_le ~> distrib_sup_le distrib_meet_le ~> distrib_inf_le add_meet_distrib_left ~> add_inf_distrib_left add_join_distrib_left ~> add_sup_distrib_left is_join_neg_meet ~> is_join_neg_inf is_meet_neg_join ~> is_meet_neg_sup add_meet_distrib_right ~> add_inf_distrib_right add_join_distrib_right ~> add_sup_distrib_right add_meet_join_distribs ~> add_sup_inf_distribs join_eq_neg_meet ~> sup_eq_neg_inf meet_eq_neg_join ~> inf_eq_neg_sup add_eq_meet_join ~> add_eq_inf_sup meet_0_imp_0 ~> inf_0_imp_0 join_0_imp_0 ~> sup_0_imp_0 meet_0_eq_0 ~> inf_0_eq_0 join_0_eq_0 ~> sup_0_eq_0 neg_meet_eq_join ~> neg_inf_eq_sup neg_join_eq_meet ~> neg_sup_eq_inf join_eq_if ~> sup_eq_if mono_meet ~> mono_inf mono_join ~> mono_sup meet_bool_eq ~> inf_bool_eq join_bool_eq ~> sup_bool_eq meet_fun_eq ~> inf_fun_eq join_fun_eq ~> sup_fun_eq meet_set_eq ~> inf_set_eq join_set_eq ~> sup_set_eq meet1_iff ~> inf1_iff meet2_iff ~> inf2_iff meet1I ~> inf1I meet2I ~> inf2I meet1D1 ~> inf1D1 meet2D1 ~> inf2D1 meet1D2 ~> inf1D2 meet2D2 ~> inf2D2 meet1E ~> inf1E meet2E ~> inf2E join1_iff ~> sup1_iff join2_iff ~> sup2_iff join1I1 ~> sup1I1 join2I1 ~> sup2I1 join1I1 ~> sup1I1 join2I2 ~> sup1I2 join1CI ~> sup1CI join2CI ~> sup2CI join1E ~> sup1E join2E ~> sup2E is_meet_Meet ~> is_meet_Inf Meet_bool_def ~> Inf_bool_def Meet_fun_def ~> Inf_fun_def Meet_greatest ~> Inf_greatest Meet_lower ~> Inf_lower Meet_set_def ~> Inf_set_def Sup_def ~> Sup_Inf Sup_bool_eq ~> Sup_bool_def Sup_fun_eq ~> Sup_fun_def Sup_set_eq ~> Sup_set_def listsp_meetI ~> listsp_infI listsp_meet_eq ~> listsp_inf_eq meet_min ~> inf_min join_max ~> sup_max * Added syntactic class "size"; overloaded constant "size" now has type "'a::size ==> bool" * Internal reorganisation of `size' of datatypes: size theorems "foo.size" are no longer subsumed by "foo.simps" (but are still simplification rules by default!); theorems "prod.size" now named "*.size". * Class "div" now inherits from class "times" rather than "type". INCOMPATIBILITY. * HOL/Finite_Set: "name-space" locales Lattice, Distrib_lattice, Linorder etc. have disappeared; operations defined in terms of fold_set now are named Inf_fin, Sup_fin. INCOMPATIBILITY. * HOL/Nat: neq0_conv no longer declared as iff. INCOMPATIBILITY. * HOL-Word: New extensive library and type for generic, fixed size machine words, with arithmetic, bit-wise, shifting and rotating operations, reflection into int, nat, and bool lists, automation for linear arithmetic (by automatic reflection into nat or int), including lemmas on overflow and monotonicity. Instantiated to all appropriate arithmetic type classes, supporting automatic simplification of numerals on all operations. * Library/Boolean_Algebra: locales for abstract boolean algebras. * Library/Numeral_Type: numbers as types, e.g. TYPE(32). * Code generator library theories: - Code_Integer represents HOL integers by big integer literals in target languages. - Code_Char represents HOL characters by character literals in target languages. - Code_Char_chr like Code_Char, but also offers treatment of character codes; includes Code_Integer. - Executable_Set allows to generate code for finite sets using lists. - Executable_Rat implements rational numbers as triples (sign, enumerator, denominator). - Executable_Real implements a subset of real numbers, namly those representable by rational numbers. - Efficient_Nat implements natural numbers by integers, which in general will result in higher efficency; pattern matching with 0/Suc is eliminated; includes Code_Integer. - Code_Index provides an additional datatype index which is mapped to target-language built-in integers. - Code_Message provides an additional datatype message_string which is isomorphic to strings; messages are mapped to target-language strings. * New package for inductive predicates An n-ary predicate p with m parameters z_1, ..., z_m can now be defined via inductive p :: "U_1 => ... => U_m => T_1 => ... => T_n => bool" for z_1 :: U_1 and ... and z_n :: U_m where rule_1: "... ==> p z_1 ... z_m t_1_1 ... t_1_n" | ... with full support for type-inference, rather than consts s :: "U_1 => ... => U_m => (T_1 * ... * T_n) set" abbreviation p :: "U_1 => ... => U_m => T_1 => ... => T_n => bool" where "p z_1 ... z_m x_1 ... x_n == (x_1, ..., x_n) : s z_1 ... z_m" inductive "s z_1 ... z_m" intros rule_1: "... ==> (t_1_1, ..., t_1_n) : s z_1 ... z_m" ... For backward compatibility, there is a wrapper allowing inductive sets to be defined with the new package via inductive_set s :: "U_1 => ... => U_m => (T_1 * ... * T_n) set" for z_1 :: U_1 and ... and z_n :: U_m where rule_1: "... ==> (t_1_1, ..., t_1_n) : s z_1 ... z_m" | ... or inductive_set s :: "U_1 => ... => U_m => (T_1 * ... * T_n) set" and p :: "U_1 => ... => U_m => T_1 => ... => T_n => bool" for z_1 :: U_1 and ... and z_n :: U_m where "p z_1 ... z_m x_1 ... x_n == (x_1, ..., x_n) : s z_1 ... z_m" | rule_1: "... ==> p z_1 ... z_m t_1_1 ... t_1_n" | ... if the additional syntax "p ..." is required. Numerous examples can be found in the subdirectories src/HOL/Auth, src/HOL/Bali, src/HOL/Induct, and src/HOL/MicroJava. INCOMPATIBILITIES: - Since declaration and definition of inductive sets or predicates is no longer separated, abbreviations involving the newly introduced sets or predicates must be specified together with the introduction rules after the 'where' keyword (see above), rather than before the actual inductive definition. - The variables in induction and elimination rules are now quantified in the order of their occurrence in the introduction rules, rather than in alphabetical order. Since this may break some proofs, these proofs either have to be repaired, e.g. by reordering the variables a_i_1 ... a_i_{k_i} in Isar 'case' statements of the form case (rule_i a_i_1 ... a_i_{k_i}) or the old order of quantification has to be restored by explicitly adding meta-level quantifiers in the introduction rules, i.e. | rule_i: "!!a_i_1 ... a_i_{k_i}. ... ==> p z_1 ... z_m t_i_1 ... t_i_n" - The format of the elimination rules is now p z_1 ... z_m x_1 ... x_n ==> (!!a_1_1 ... a_1_{k_1}. x_1 = t_1_1 ==> ... ==> x_n = t_1_n ==> ... ==> P) ==> ... ==> P for predicates and (x_1, ..., x_n) : s z_1 ... z_m ==> (!!a_1_1 ... a_1_{k_1}. x_1 = t_1_1 ==> ... ==> x_n = t_1_n ==> ... ==> P) ==> ... ==> P for sets rather than x : s z_1 ... z_m ==> (!!a_1_1 ... a_1_{k_1}. x = (t_1_1, ..., t_1_n) ==> ... ==> P) ==> ... ==> P This may require terms in goals to be expanded to n-tuples (e.g. using case_tac or simplification with the split_paired_all rule) before the above elimination rule is applicable. - The elimination or case analysis rules for (mutually) inductive sets or predicates are now called "p_1.cases" ... "p_k.cases". The list of rules "p_1_..._p_k.elims" is no longer available. * New package "function"/"fun" for general recursive functions, supporting mutual and nested recursion, definitions in local contexts, more general pattern matching and partiality. See HOL/ex/Fundefs.thy for small examples, and the separate tutorial on the function package. The old recdef "package" is still available as before, but users are encouraged to use the new package. * Method "lexicographic_order" automatically synthesizes termination relations as lexicographic combinations of size measures. * Case-expressions allow arbitrary constructor-patterns (including "_") and take their order into account, like in functional programming. Internally, this is translated into nested case-expressions; missing cases are added and mapped to the predefined constant "undefined". In complicated cases printing may no longer show the original input but the internal form. Lambda-abstractions allow the same form of pattern matching: "% pat1 => e1 | ..." is an abbreviation for "%x. case x of pat1 => e1 | ..." where x is a new variable. * IntDef: The constant "int :: nat => int" has been removed; now "int" is an abbreviation for "of_nat :: nat => int". The simplification rules for "of_nat" have been changed to work like "int" did previously. Potential INCOMPATIBILITY: - "of_nat (Suc m)" simplifies to "1 + of_nat m" instead of "of_nat m + 1" - of_nat_diff and of_nat_mult are no longer default simp rules * Method "algebra" solves polynomial equations over (semi)rings using Groebner bases. The (semi)ring structure is defined by locales and the tool setup depends on that generic context. Installing the method for a specific type involves instantiating the locale and possibly adding declarations for computation on the coefficients. The method is already instantiated for natural numbers and for the axiomatic class of idoms with numerals. See also the paper by Chaieb and Wenzel at CALCULEMUS 2007 for the general principles underlying this architecture of context-aware proof-tools. * Method "ferrack" implements quantifier elimination over special-purpose dense linear orders using locales (analogous to "algebra"). The method is already installed for class {ordered_field,recpower,number_ring} which subsumes real, hyperreal, rat, etc. * Former constant "List.op @" now named "List.append". Use ML antiquotations @{const_name List.append} or @{term " ... @ ... "} to circumvent possible incompatibilities when working on ML level. * primrec: missing cases mapped to "undefined" instead of "arbitrary". * New function listsum :: 'a list => 'a for arbitrary monoids. Special syntax: "SUM x <- xs. f x" (and latex variants) * New syntax for Haskell-like list comprehension (input only), eg. [(x,y). x <- xs, y <- ys, x ~= y], see also src/HOL/List.thy. * The special syntax for function "filter" has changed from [x : xs. P] to [x <- xs. P] to avoid an ambiguity caused by list comprehension syntax, and for uniformity. INCOMPATIBILITY. * [a..b] is now defined for arbitrary linear orders. It used to be defined on nat only, as an abbreviation for [a.. B" for equality on bool (with priority 25 like -->); output depends on the "iff" print_mode, the default is "A = B" (with priority 50). * Relations less (<) and less_eq (<=) are also available on type bool. Modified syntax to disallow nesting without explicit parentheses, e.g. "(x < y) < z" or "x < (y < z)", but NOT "x < y < z". Potential INCOMPATIBILITY. * "LEAST x:A. P" expands to "LEAST x. x:A & P" (input only). * Relation composition operator "op O" now has precedence 75 and binds stronger than union and intersection. INCOMPATIBILITY. * The old set interval syntax "{m..n(}" (and relatives) has been removed. Use "{m.. ==> False", equivalences (i.e. "=" on type bool) are handled, variable names of the form "lit_" are no longer reserved, significant speedup. * Methods "sat" and "satx" can now replay MiniSat proof traces. zChaff is still supported as well. * 'inductive' and 'datatype': provide projections of mutual rules, bundled as foo_bar.inducts; * Library: moved theories Parity, GCD, Binomial, Infinite_Set to Library. * Library: moved theory Accessible_Part to main HOL. * Library: added theory Coinductive_List of potentially infinite lists as greatest fixed-point. * Library: added theory AssocList which implements (finite) maps as association lists. * Method "evaluation" solves goals (i.e. a boolean expression) efficiently by compiling it to ML. The goal is "proved" (via an oracle) if it evaluates to True. * Linear arithmetic now splits certain operators (e.g. min, max, abs) also when invoked by the simplifier. This results in the Simplifier being more powerful on arithmetic goals. INCOMPATIBILITY. Configuration option fast_arith_split_limit=0 recovers the old behavior. * Support for hex (0x20) and binary (0b1001) numerals. * New method: reify eqs (t), where eqs are equations for an interpretation I :: 'a list => 'b => 'c and t::'c is an optional parameter, computes a term s::'b and a list xs::'a list and proves the theorem I xs s = t. This is also known as reification or quoting. The resulting theorem is applied to the subgoal to substitute t with I xs s. If t is omitted, the subgoal itself is reified. * New method: reflection corr_thm eqs (t). The parameters eqs and (t) are as explained above. corr_thm is a theorem for I vs (f t) = I vs t, where f is supposed to be a computable function (in the sense of code generattion). The method uses reify to compute s and xs as above then applies corr_thm and uses normalization by evaluation to "prove" f s = r and finally gets the theorem t = r, which is again applied to the subgoal. An Example is available in src/HOL/ex/ReflectionEx.thy. * Reflection: Automatic reification now handels binding, an example is available in src/HOL/ex/ReflectionEx.thy * HOL-Statespace: ``State Spaces: The Locale Way'' introduces a command 'statespace' that is similar to 'record', but introduces an abstract specification based on the locale infrastructure instead of HOL types. This leads to extra flexibility in composing state spaces, in particular multiple inheritance and renaming of components. *** HOL-Complex *** * Hyperreal: Functions root and sqrt are now defined on negative real inputs so that root n (- x) = - root n x and sqrt (- x) = - sqrt x. Nonnegativity side conditions have been removed from many lemmas, so that more subgoals may now be solved by simplification; potential INCOMPATIBILITY. * Real: new type classes formalize real normed vector spaces and algebras, using new overloaded constants scaleR :: real => 'a => 'a and norm :: 'a => real. * Real: constant of_real :: real => 'a::real_algebra_1 injects from reals into other types. The overloaded constant Reals :: 'a set is now defined as range of_real; potential INCOMPATIBILITY. * Real: proper support for ML code generation, including 'quickcheck'. Reals are implemented as arbitrary precision rationals. * Hyperreal: Several constants that previously worked only for the reals have been generalized, so they now work over arbitrary vector spaces. Type annotations may need to be added in some cases; potential INCOMPATIBILITY. Infinitesimal :: ('a::real_normed_vector) star set HFinite :: ('a::real_normed_vector) star set HInfinite :: ('a::real_normed_vector) star set approx :: ('a::real_normed_vector) star => 'a star => bool monad :: ('a::real_normed_vector) star => 'a star set galaxy :: ('a::real_normed_vector) star => 'a star set (NS)LIMSEQ :: [nat => 'a::real_normed_vector, 'a] => bool (NS)convergent :: (nat => 'a::real_normed_vector) => bool (NS)Bseq :: (nat => 'a::real_normed_vector) => bool (NS)Cauchy :: (nat => 'a::real_normed_vector) => bool (NS)LIM :: ['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool is(NS)Cont :: ['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool deriv :: ['a::real_normed_field => 'a, 'a, 'a] => bool sgn :: 'a::real_normed_vector => 'a exp :: 'a::{recpower,real_normed_field,banach} => 'a * Complex: Some complex-specific constants are now abbreviations for overloaded ones: complex_of_real = of_real, cmod = norm, hcmod = hnorm. Other constants have been entirely removed in favor of the polymorphic versions (INCOMPATIBILITY): approx <-- capprox HFinite <-- CFinite HInfinite <-- CInfinite Infinitesimal <-- CInfinitesimal monad <-- cmonad galaxy <-- cgalaxy (NS)LIM <-- (NS)CLIM, (NS)CRLIM is(NS)Cont <-- is(NS)Contc, is(NS)contCR (ns)deriv <-- (ns)cderiv *** HOL-Algebra *** * Formalisation of ideals and the quotient construction over rings. * Order and lattice theory no longer based on records. INCOMPATIBILITY. * Renamed lemmas least_carrier -> least_closed and greatest_carrier -> greatest_closed. INCOMPATIBILITY. * Method algebra is now set up via an attribute. For examples see Ring.thy. INCOMPATIBILITY: the method is now weaker on combinations of algebraic structures. * Renamed theory CRing to Ring. *** HOL-Nominal *** * Substantial, yet incomplete support for nominal datatypes (binding structures) based on HOL-Nominal logic. See src/HOL/Nominal and src/HOL/Nominal/Examples. Prospective users should consult http://isabelle.in.tum.de/nominal/ *** ML *** * ML basics: just one true type int, which coincides with IntInf.int (even on SML/NJ). * ML within Isar: antiquotations allow to embed statically-checked formal entities in the source, referring to the context available at compile-time. For example: ML {* @{sort "{zero,one}"} *} ML {* @{typ "'a => 'b"} *} ML {* @{term "%x. x"} *} ML {* @{prop "x == y"} *} ML {* @{ctyp "'a => 'b"} *} ML {* @{cterm "%x. x"} *} ML {* @{cprop "x == y"} *} ML {* @{thm asm_rl} *} ML {* @{thms asm_rl} *} ML {* @{type_name c} *} ML {* @{type_syntax c} *} ML {* @{const_name c} *} ML {* @{const_syntax c} *} ML {* @{context} *} ML {* @{theory} *} ML {* @{theory Pure} *} ML {* @{theory_ref} *} ML {* @{theory_ref Pure} *} ML {* @{simpset} *} ML {* @{claset} *} ML {* @{clasimpset} *} The same works for sources being ``used'' within an Isar context. * ML in Isar: improved error reporting; extra verbosity with ML_Context.trace enabled. * Pure/General/table.ML: the join operations now works via exceptions DUP/SAME instead of type option. This is simpler in simple cases, and admits slightly more efficient complex applications. * Pure: 'advanced' translation functions (parse_translation etc.) now use Context.generic instead of just theory. * Pure: datatype Context.generic joins theory/Proof.context and provides some facilities for code that works in either kind of context, notably GenericDataFun for uniform theory and proof data. * Pure: simplified internal attribute type, which is now always Context.generic * thm -> Context.generic * thm. Global (theory) vs. local (Proof.context) attributes have been discontinued, while minimizing code duplication. Thm.rule_attribute and Thm.declaration_attribute build canonical attributes; see also structure Context for further operations on Context.generic, notably GenericDataFun. INCOMPATIBILITY, need to adapt attribute type declarations and definitions. * Context data interfaces (Theory/Proof/GenericDataFun): removed name/print, uninitialized data defaults to ad-hoc copy of empty value, init only required for impure data. INCOMPATIBILITY: empty really need to be empty (no dependencies on theory content!) * Pure/kernel: consts certification ignores sort constraints given in signature declarations. (This information is not relevant to the logic, but only for type inference.) SIGNIFICANT INTERNAL CHANGE, potential INCOMPATIBILITY. * Pure: axiomatic type classes are now purely definitional, with explicit proofs of class axioms and super class relations performed internally. See Pure/axclass.ML for the main internal interfaces -- notably AxClass.define_class supercedes AxClass.add_axclass, and AxClass.axiomatize_class/classrel/arity supersede Sign.add_classes/classrel/arities. * Pure/Isar: Args/Attrib parsers operate on Context.generic -- global/local versions on theory vs. Proof.context have been discontinued; Attrib.syntax and Method.syntax have been adapted accordingly. INCOMPATIBILITY, need to adapt parser expressions for attributes, methods, etc. * Pure: several functions of signature "... -> theory -> theory * ..." have been reoriented to "... -> theory -> ... * theory" in order to allow natural usage in combination with the ||>, ||>>, |-> and fold_map combinators. * Pure: official theorem names (closed derivations) and additional comments (tags) are now strictly separate. Name hints -- which are maintained as tags -- may be attached any time without affecting the derivation. * Pure: primitive rule lift_rule now takes goal cterm instead of an actual goal state (thm). Use Thm.lift_rule (Thm.cprem_of st i) to achieve the old behaviour. * Pure: the "Goal" constant is now called "prop", supporting a slightly more general idea of ``protecting'' meta-level rule statements. * Pure: Logic.(un)varify only works in a global context, which is now enforced instead of silently assumed. INCOMPATIBILITY, may use Logic.legacy_(un)varify as temporary workaround. * Pure: structure Name provides scalable operations for generating internal variable names, notably Name.variants etc. This replaces some popular functions from term.ML: Term.variant -> Name.variant Term.variantlist -> Name.variant_list Term.invent_names -> Name.invent_list Note that low-level renaming rarely occurs in new code -- operations from structure Variable are used instead (see below). * Pure: structure Variable provides fundamental operations for proper treatment of fixed/schematic variables in a context. For example, Variable.import introduces fixes for schematics of given facts and Variable.export reverses the effect (up to renaming) -- this replaces various freeze_thaw operations. * Pure: structure Goal provides simple interfaces for init/conclude/finish and tactical prove operations (replacing former Tactic.prove). Goal.prove is the canonical way to prove results within a given context; Goal.prove_global is a degraded version for theory level goals, including a global Drule.standard. Note that OldGoals.prove_goalw_cterm has long been obsolete, since it is ill-behaved in a local proof context (e.g. with local fixes/assumes or in a locale context). * Pure/Syntax: generic interfaces for parsing (Syntax.parse_term etc.) and type checking (Syntax.check_term etc.), with common combinations (Syntax.read_term etc.). These supersede former Sign.read_term etc. which are considered legacy and await removal. * Pure/Syntax: generic interfaces for type unchecking (Syntax.uncheck_terms etc.) and unparsing (Syntax.unparse_term etc.), with common combinations (Syntax.pretty_term, Syntax.string_of_term etc.). Former Sign.pretty_term, Sign.string_of_term etc. are still available for convenience, but refer to the very same operations using a mere theory instead of a full context. * Isar: simplified treatment of user-level errors, using exception ERROR of string uniformly. Function error now merely raises ERROR, without any side effect on output channels. The Isar toplevel takes care of proper display of ERROR exceptions. ML code may use plain handle/can/try; cat_error may be used to concatenate errors like this: ... handle ERROR msg => cat_error msg "..." Toplevel ML code (run directly or through the Isar toplevel) may be embedded into the Isar toplevel with exception display/debug like this: Isar.toplevel (fn () => ...) INCOMPATIBILITY, removed special transform_error facilities, removed obsolete variants of user-level exceptions (ERROR_MESSAGE, Context.PROOF, ProofContext.CONTEXT, Proof.STATE, ProofHistory.FAIL) -- use plain ERROR instead. * Isar: theory setup now has type (theory -> theory), instead of a list. INCOMPATIBILITY, may use #> to compose setup functions. * Isar: ML toplevel pretty printer for type Proof.context, subject to ProofContext.debug/verbose flags. * Isar: Toplevel.theory_to_proof admits transactions that modify the theory before entering a proof state. Transactions now always see a quasi-functional intermediate checkpoint, both in interactive and batch mode. * Isar: simplified interfaces for outer syntax. Renamed OuterSyntax.add_keywords to OuterSyntax.keywords. Removed OuterSyntax.add_parsers -- this functionality is now included in OuterSyntax.command etc. INCOMPATIBILITY. * Simplifier: the simpset of a running simplification process now contains a proof context (cf. Simplifier.the_context), which is the very context that the initial simpset has been retrieved from (by simpset_of/local_simpset_of). Consequently, all plug-in components (solver, looper etc.) may depend on arbitrary proof data. * Simplifier.inherit_context inherits the proof context (plus the local bounds) of the current simplification process; any simproc etc. that calls the Simplifier recursively should do this! Removed former Simplifier.inherit_bounds, which is already included here -- INCOMPATIBILITY. Tools based on low-level rewriting may even have to specify an explicit context using Simplifier.context/theory_context. * Simplifier/Classical Reasoner: more abstract interfaces change_simpset/claset for modifying the simpset/claset reference of a theory; raw versions simpset/claset_ref etc. have been discontinued -- INCOMPATIBILITY. * Provers: more generic wrt. syntax of object-logics, avoid hardwired "Trueprop" etc. *** System *** * settings: the default heap location within ISABELLE_HOME_USER now includes ISABELLE_IDENTIFIER. This simplifies use of multiple Isabelle installations. * isabelle-process: option -S (secure mode) disables some critical operations, notably runtime compilation and evaluation of ML source code. * Basic Isabelle mode for jEdit, see Isabelle/lib/jedit/. * Support for parallel execution, using native multicore support of Poly/ML 5.1. The theory loader exploits parallelism when processing independent theories, according to the given theory header specifications. The maximum number of worker threads is specified via usedir option -M or the "max-threads" setting in Proof General. A speedup factor of 1.5--3.5 can be expected on a 4-core machine, and up to 6 on a 8-core machine. User-code needs to observe certain guidelines for thread-safe programming, see appendix A in the Isar Implementation manual. New in Isabelle2005 (October 2005) ---------------------------------- *** General *** * Theory headers: the new header syntax for Isar theories is theory imports ... uses ... begin where the 'uses' part is optional. The previous syntax theory = + ... + : will disappear in the next release. Use isatool fixheaders to convert existing theory files. Note that there is no change in ancient non-Isar theories now, but these will disappear soon. * Theory loader: parent theories can now also be referred to via relative and absolute paths. * Command 'find_theorems' searches for a list of criteria instead of a list of constants. Known criteria are: intro, elim, dest, name:string, simp:term, and any term. Criteria can be preceded by '-' to select theorems that do not match. Intro, elim, dest select theorems that match the current goal, name:s selects theorems whose fully qualified name contain s, and simp:term selects all simplification rules whose lhs match term. Any other term is interpreted as pattern and selects all theorems matching the pattern. Available in ProofGeneral under 'ProofGeneral -> Find Theorems' or C-c C-f. Example: C-c C-f (100) "(_::nat) + _ + _" intro -name: "HOL." prints the last 100 theorems matching the pattern "(_::nat) + _ + _", matching the current goal as introduction rule and not having "HOL." in their name (i.e. not being defined in theory HOL). * Command 'thms_containing' has been discontinued in favour of 'find_theorems'; INCOMPATIBILITY. * Communication with Proof General is now 8bit clean, which means that Unicode text in UTF-8 encoding may be used within theory texts (both formal and informal parts). Cf. option -U of the Isabelle Proof General interface. Here are some simple examples (cf. src/HOL/ex): http://isabelle.in.tum.de/library/HOL/ex/Hebrew.html http://isabelle.in.tum.de/library/HOL/ex/Chinese.html * Improved efficiency of the Simplifier and, to a lesser degree, the Classical Reasoner. Typical big applications run around 2 times faster. *** Document preparation *** * Commands 'display_drafts' and 'print_drafts' perform simple output of raw sources. Only those symbols that do not require additional LaTeX packages (depending on comments in isabellesym.sty) are displayed properly, everything else is left verbatim. isatool display and isatool print are used as front ends (these are subject to the DVI/PDF_VIEWER and PRINT_COMMAND settings, respectively). * Command tags control specific markup of certain regions of text, notably folding and hiding. Predefined tags include "theory" (for theory begin and end), "proof" for proof commands, and "ML" for commands involving ML code; the additional tags "visible" and "invisible" are unused by default. Users may give explicit tag specifications in the text, e.g. ''by %invisible (auto)''. The interpretation of tags is determined by the LaTeX job during document preparation: see option -V of isatool usedir, or options -n and -t of isatool document, or even the LaTeX macros \isakeeptag, \isafoldtag, \isadroptag. Several document versions may be produced at the same time via isatool usedir (the generated index.html will link all of them). Typical specifications include ''-V document=theory,proof,ML'' to present theory/proof/ML parts faithfully, ''-V outline=/proof,/ML'' to fold proof and ML commands, and ''-V mutilated=-theory,-proof,-ML'' to omit these parts without any formal replacement text. The Isabelle site default settings produce ''document'' and ''outline'' versions as specified above. * Several new antiquotations: @{term_type term} prints a term with its type annotated; @{typeof term} prints the type of a term; @{const const} is the same as @{term const}, but checks that the argument is a known logical constant; @{term_style style term} and @{thm_style style thm} print a term or theorem applying a "style" to it @{ML text} Predefined styles are 'lhs' and 'rhs' printing the lhs/rhs of definitions, equations, inequations etc., 'concl' printing only the conclusion of a meta-logical statement theorem, and 'prem1' .. 'prem19' to print the specified premise. TermStyle.add_style provides an ML interface for introducing further styles. See also the "LaTeX Sugar" document practical applications. The ML antiquotation prints type-checked ML expressions verbatim. * Markup commands 'chapter', 'section', 'subsection', 'subsubsection', and 'text' support optional locale specification '(in loc)', which specifies the default context for interpreting antiquotations. For example: 'text (in lattice) {* @{thm inf_assoc}*}'. * Option 'locale=NAME' of antiquotations specifies an alternative context interpreting the subsequent argument. For example: @{thm [locale=lattice] inf_assoc}. * Proper output of proof terms (@{prf ...} and @{full_prf ...}) within a proof context. * Proper output of antiquotations for theory commands involving a proof context (such as 'locale' or 'theorem (in loc) ...'). * Delimiters of outer tokens (string etc.) now produce separate LaTeX macros (\isachardoublequoteopen, isachardoublequoteclose etc.). * isatool usedir: new option -C (default true) controls whether option -D should include a copy of the original document directory; -C false prevents unwanted effects such as copying of administrative CVS data. *** Pure *** * Considerably improved version of 'constdefs' command. Now performs automatic type-inference of declared constants; additional support for local structure declarations (cf. locales and HOL records), see also isar-ref manual. Potential INCOMPATIBILITY: need to observe strictly sequential dependencies of definitions within a single 'constdefs' section; moreover, the declared name needs to be an identifier. If all fails, consider to fall back on 'consts' and 'defs' separately. * Improved indexed syntax and implicit structures. First of all, indexed syntax provides a notational device for subscripted application, using the new syntax \<^bsub>term\<^esub> for arbitrary expressions. Secondly, in a local context with structure declarations, number indexes \<^sub>n or the empty index (default number 1) refer to a certain fixed variable implicitly; option show_structs controls printing of implicit structures. Typical applications of these concepts involve record types and locales. * New command 'no_syntax' removes grammar declarations (and translations) resulting from the given syntax specification, which is interpreted in the same manner as for the 'syntax' command. * 'Advanced' translation functions (parse_translation etc.) may depend on the signature of the theory context being presently used for parsing/printing, see also isar-ref manual. * Improved 'oracle' command provides a type-safe interface to turn an ML expression of type theory -> T -> term into a primitive rule of type theory -> T -> thm (i.e. the functionality of Thm.invoke_oracle is already included here); see also FOL/ex/IffExample.thy; INCOMPATIBILITY. * axclass: name space prefix for class "c" is now "c_class" (was "c" before); "cI" is no longer bound, use "c.intro" instead. INCOMPATIBILITY. This change avoids clashes of fact bindings for axclasses vs. locales. * Improved internal renaming of symbolic identifiers -- attach primes instead of base 26 numbers. * New flag show_question_marks controls printing of leading question marks in schematic variable names. * In schematic variable names, *any* symbol following \<^isub> or \<^isup> is now treated as part of the base name. For example, the following works without printing of awkward ".0" indexes: lemma "x\<^isub>1 = x\<^isub>2 ==> x\<^isub>2 = x\<^isub>1" by simp * Inner syntax includes (*(*nested*) comments*). * Pretty printer now supports unbreakable blocks, specified in mixfix annotations as "(00...)". * Clear separation of logical types and nonterminals, where the latter may only occur in 'syntax' specifications or type abbreviations. Before that distinction was only partially implemented via type class "logic" vs. "{}". Potential INCOMPATIBILITY in rare cases of improper use of 'types'/'consts' instead of 'nonterminals'/'syntax'. Some very exotic syntax specifications may require further adaption (e.g. Cube/Cube.thy). * Removed obsolete type class "logic", use the top sort {} instead. Note that non-logical types should be declared as 'nonterminals' rather than 'types'. INCOMPATIBILITY for new object-logic specifications. * Attributes 'induct' and 'cases': type or set names may now be locally fixed variables as well. * Simplifier: can now control the depth to which conditional rewriting is traced via the PG menu Isabelle -> Settings -> Trace Simp Depth Limit. * Simplifier: simplification procedures may now take the current simpset into account (cf. Simplifier.simproc(_i) / mk_simproc interface), which is very useful for calling the Simplifier recursively. Minor INCOMPATIBILITY: the 'prems' argument of simprocs is gone -- use prems_of_ss on the simpset instead. Moreover, the low-level mk_simproc no longer applies Logic.varify internally, to allow for use in a context of fixed variables. * thin_tac now works even if the assumption being deleted contains !! or ==>. More generally, erule now works even if the major premise of the elimination rule contains !! or ==>. * Method 'rules' has been renamed to 'iprover'. INCOMPATIBILITY. * Reorganized bootstrapping of the Pure theories; CPure is now derived from Pure, which contains all common declarations already. Both theories are defined via plain Isabelle/Isar .thy files. INCOMPATIBILITY: elements of CPure (such as the CPure.intro / CPure.elim / CPure.dest attributes) now appear in the Pure name space; use isatool fixcpure to adapt your theory and ML sources. * New syntax 'name(i-j, i-, i, ...)' for referring to specific selections of theorems in named facts via index ranges. * 'print_theorems': in theory mode, really print the difference wrt. the last state (works for interactive theory development only), in proof mode print all local facts (cf. 'print_facts'); * 'hide': option '(open)' hides only base names. * More efficient treatment of intermediate checkpoints in interactive theory development. * Code generator is now invoked via code_module (incremental code generation) and code_library (modular code generation, ML structures for each theory). INCOMPATIBILITY: new keywords 'file' and 'contains' must be quoted when used as identifiers. * New 'value' command for reading, evaluating and printing terms using the code generator. INCOMPATIBILITY: command keyword 'value' must be quoted when used as identifier. *** Locales *** * New commands for the interpretation of locale expressions in theories (1), locales (2) and proof contexts (3). These generate proof obligations from the expression specification. After the obligations have been discharged, theorems of the expression are added to the theory, target locale or proof context. The synopsis of the commands is a follows: (1) interpretation expr inst (2) interpretation target < expr (3) interpret expr inst Interpretation in theories and proof contexts require a parameter instantiation of terms from the current context. This is applied to specifications and theorems of the interpreted expression. Interpretation in locales only permits parameter renaming through the locale expression. Interpretation is smart in that interpretations that are active already do not occur in proof obligations, neither are instantiated theorems stored in duplicate. Use 'print_interps' to inspect active interpretations of a particular locale. For details, see the Isar Reference manual. Examples can be found in HOL/Finite_Set.thy and HOL/Algebra/UnivPoly.thy. INCOMPATIBILITY: former 'instantiate' has been withdrawn, use 'interpret' instead. * New context element 'constrains' for adding type constraints to parameters. * Context expressions: renaming of parameters with syntax redeclaration. * Locale declaration: 'includes' disallowed. * Proper static binding of attribute syntax -- i.e. types / terms / facts mentioned as arguments are always those of the locale definition context, independently of the context of later invocations. Moreover, locale operations (renaming and type / term instantiation) are applied to attribute arguments as expected. INCOMPATIBILITY of the ML interface: always pass Attrib.src instead of actual attributes; rare situations may require Attrib.attribute to embed those attributes into Attrib.src that lack concrete syntax. Attribute implementations need to cooperate properly with the static binding mechanism. Basic parsers Args.XXX_typ/term/prop and Attrib.XXX_thm etc. already do the right thing without further intervention. Only unusual applications -- such as "where" or "of" (cf. src/Pure/Isar/attrib.ML), which process arguments depending both on the context and the facts involved -- may have to assign parsed values to argument tokens explicitly. * Changed parameter management in theorem generation for long goal statements with 'includes'. INCOMPATIBILITY: produces a different theorem statement in rare situations. * Locale inspection command 'print_locale' omits notes elements. Use 'print_locale!' to have them included in the output. *** Provers *** * Provers/hypsubst.ML: improved version of the subst method, for single-step rewriting: it now works in bound variable contexts. New is 'subst (asm)', for rewriting an assumption. INCOMPATIBILITY: may rewrite a different subterm than the original subst method, which is still available as 'simplesubst'. * Provers/quasi.ML: new transitivity reasoners for transitivity only and quasi orders. * Provers/trancl.ML: new transitivity reasoner for transitive and reflexive-transitive closure of relations. * Provers/blast.ML: new reference depth_limit to make blast's depth limit (previously hard-coded with a value of 20) user-definable. * Provers/simplifier.ML has been moved to Pure, where Simplifier.setup is peformed already. Object-logics merely need to finish their initial simpset configuration as before. INCOMPATIBILITY. *** HOL *** * Symbolic syntax of Hilbert Choice Operator is now as follows: syntax (epsilon) "_Eps" :: "[pttrn, bool] => 'a" ("(3\_./ _)" [0, 10] 10) The symbol \ is displayed as the alternative epsilon of LaTeX and x-symbol; use option '-m epsilon' to get it actually printed. Moreover, the mathematically important symbolic identifier \ becomes available as variable, constant etc. INCOMPATIBILITY, * "x > y" abbreviates "y < x" and "x >= y" abbreviates "y <= x". Similarly for all quantifiers: "ALL x > y" etc. The x-symbol for >= is \. New transitivity rules have been added to HOL/Orderings.thy to support corresponding Isar calculations. * "{x:A. P}" abbreviates "{x. x:A & P}", and similarly for "\" instead of ":". * theory SetInterval: changed the syntax for open intervals: Old New {..n(} {.. {\1<\.\.} \.\.\([^(}]*\)(} -> \.\.<\1} * Theory Commutative_Ring (in Library): method comm_ring for proving equalities in commutative rings; method 'algebra' provides a generic interface. * Theory Finite_Set: changed the syntax for 'setsum', summation over finite sets: "setsum (%x. e) A", which used to be "\x:A. e", is now either "SUM x:A. e" or "\x \ A. e". The bound variable can be a tuple pattern. Some new syntax forms are available: "\x | P. e" for "setsum (%x. e) {x. P}" "\x = a..b. e" for "setsum (%x. e) {a..b}" "\x = a..x < k. e" for "setsum (%x. e) {..x < k. e" used to be based on a separate function "Summation", which has been discontinued. * theory Finite_Set: in structured induction proofs, the insert case is now 'case (insert x F)' instead of the old counterintuitive 'case (insert F x)'. * The 'refute' command has been extended to support a much larger fragment of HOL, including axiomatic type classes, constdefs and typedefs, inductive datatypes and recursion. * New tactics 'sat' and 'satx' to prove propositional tautologies. Requires zChaff with proof generation to be installed. See HOL/ex/SAT_Examples.thy for examples. * Datatype induction via method 'induct' now preserves the name of the induction variable. For example, when proving P(xs::'a list) by induction on xs, the induction step is now P(xs) ==> P(a#xs) rather than P(list) ==> P(a#list) as previously. Potential INCOMPATIBILITY in unstructured proof scripts. * Reworked implementation of records. Improved scalability for records with many fields, avoiding performance problems for type inference. Records are no longer composed of nested field types, but of nested extension types. Therefore the record type only grows linear in the number of extensions and not in the number of fields. The top-level (users) view on records is preserved. Potential INCOMPATIBILITY only in strange cases, where the theory depends on the old record representation. The type generated for a record is called _ext_type. Flag record_quick_and_dirty_sensitive can be enabled to skip the proofs triggered by a record definition or a simproc (if quick_and_dirty is enabled). Definitions of large records can take quite long. New simproc record_upd_simproc for simplification of multiple record updates enabled by default. Moreover, trivial updates are also removed: r(|x := x r|) = r. INCOMPATIBILITY: old proofs break occasionally, since simplification is more powerful by default. * typedef: proper support for polymorphic sets, which contain extra type-variables in the term. * Simplifier: automatically reasons about transitivity chains involving "trancl" (r^+) and "rtrancl" (r^*) by setting up tactics provided by Provers/trancl.ML as additional solvers. INCOMPATIBILITY: old proofs break occasionally as simplification may now solve more goals than previously. * Simplifier: converts x <= y into x = y if assumption y <= x is present. Works for all partial orders (class "order"), in particular numbers and sets. For linear orders (e.g. numbers) it treats ~ x < y just like y <= x. * Simplifier: new simproc for "let x = a in f x". If a is a free or bound variable or a constant then the let is unfolded. Otherwise first a is simplified to b, and then f b is simplified to g. If possible we abstract b from g arriving at "let x = b in h x", otherwise we unfold the let and arrive at g. The simproc can be enabled/disabled by the reference use_let_simproc. Potential INCOMPATIBILITY since simplification is more powerful by default. * Classical reasoning: the meson method now accepts theorems as arguments. * Prover support: pre-release of the Isabelle-ATP linkup, which runs background jobs to provide advice on the provability of subgoals. * Theory OrderedGroup and Ring_and_Field: various additions and improvements to faciliate calculations involving equalities and inequalities. The following theorems have been eliminated or modified (INCOMPATIBILITY): abs_eq now named abs_of_nonneg abs_of_ge_0 now named abs_of_nonneg abs_minus_eq now named abs_of_nonpos imp_abs_id now named abs_of_nonneg imp_abs_neg_id now named abs_of_nonpos mult_pos now named mult_pos_pos mult_pos_le now named mult_nonneg_nonneg mult_pos_neg_le now named mult_nonneg_nonpos mult_pos_neg2_le now named mult_nonneg_nonpos2 mult_neg now named mult_neg_neg mult_neg_le now named mult_nonpos_nonpos * The following lemmas in Ring_and_Field have been added to the simplifier: zero_le_square not_square_less_zero The following lemmas have been deleted from Real/RealPow: realpow_zero_zero realpow_two realpow_less zero_le_power realpow_two_le abs_realpow_two realpow_two_abs * Theory Parity: added rules for simplifying exponents. * Theory List: The following theorems have been eliminated or modified (INCOMPATIBILITY): list_all_Nil now named list_all.simps(1) list_all_Cons now named list_all.simps(2) list_all_conv now named list_all_iff set_mem_eq now named mem_iff * Theories SetsAndFunctions and BigO (see HOL/Library) support asymptotic "big O" calculations. See the notes in BigO.thy. *** HOL-Complex *** * Theory RealDef: better support for embedding natural numbers and integers in the reals. The following theorems have been eliminated or modified (INCOMPATIBILITY): exp_ge_add_one_self now requires no hypotheses real_of_int_add reversed direction of equality (use [symmetric]) real_of_int_minus reversed direction of equality (use [symmetric]) real_of_int_diff reversed direction of equality (use [symmetric]) real_of_int_mult reversed direction of equality (use [symmetric]) * Theory RComplete: expanded support for floor and ceiling functions. * Theory Ln is new, with properties of the natural logarithm * Hyperreal: There is a new type constructor "star" for making nonstandard types. The old type names are now type synonyms: hypreal = real star hypnat = nat star hcomplex = complex star * Hyperreal: Many groups of similarly-defined constants have been replaced by polymorphic versions (INCOMPATIBILITY): star_of <-- hypreal_of_real, hypnat_of_nat, hcomplex_of_complex starset <-- starsetNat, starsetC *s* <-- *sNat*, *sc* starset_n <-- starsetNat_n, starsetC_n *sn* <-- *sNatn*, *scn* InternalSets <-- InternalNatSets, InternalCSets starfun <-- starfun{Nat,Nat2,C,RC,CR} *f* <-- *fNat*, *fNat2*, *fc*, *fRc*, *fcR* starfun_n <-- starfun{Nat,Nat2,C,RC,CR}_n *fn* <-- *fNatn*, *fNat2n*, *fcn*, *fRcn*, *fcRn* InternalFuns <-- InternalNatFuns, InternalNatFuns2, Internal{C,RC,CR}Funs * Hyperreal: Many type-specific theorems have been removed in favor of theorems specific to various axiomatic type classes (INCOMPATIBILITY): add_commute <-- {hypreal,hypnat,hcomplex}_add_commute add_assoc <-- {hypreal,hypnat,hcomplex}_add_assocs OrderedGroup.add_0 <-- {hypreal,hypnat,hcomplex}_add_zero_left OrderedGroup.add_0_right <-- {hypreal,hcomplex}_add_zero_right right_minus <-- hypreal_add_minus left_minus <-- {hypreal,hcomplex}_add_minus_left mult_commute <-- {hypreal,hypnat,hcomplex}_mult_commute mult_assoc <-- {hypreal,hypnat,hcomplex}_mult_assoc mult_1_left <-- {hypreal,hypnat}_mult_1, hcomplex_mult_one_left mult_1_right <-- hcomplex_mult_one_right mult_zero_left <-- hcomplex_mult_zero_left left_distrib <-- {hypreal,hypnat,hcomplex}_add_mult_distrib right_distrib <-- hypnat_add_mult_distrib2 zero_neq_one <-- {hypreal,hypnat,hcomplex}_zero_not_eq_one right_inverse <-- hypreal_mult_inverse left_inverse <-- hypreal_mult_inverse_left, hcomplex_mult_inv_left order_refl <-- {hypreal,hypnat}_le_refl order_trans <-- {hypreal,hypnat}_le_trans order_antisym <-- {hypreal,hypnat}_le_anti_sym order_less_le <-- {hypreal,hypnat}_less_le linorder_linear <-- {hypreal,hypnat}_le_linear add_left_mono <-- {hypreal,hypnat}_add_left_mono mult_strict_left_mono <-- {hypreal,hypnat}_mult_less_mono2 add_nonneg_nonneg <-- hypreal_le_add_order * Hyperreal: Separate theorems having to do with type-specific versions of constants have been merged into theorems that apply to the new polymorphic constants (INCOMPATIBILITY): STAR_UNIV_set <-- {STAR_real,NatStar_real,STARC_complex}_set STAR_empty_set <-- {STAR,NatStar,STARC}_empty_set STAR_Un <-- {STAR,NatStar,STARC}_Un STAR_Int <-- {STAR,NatStar,STARC}_Int STAR_Compl <-- {STAR,NatStar,STARC}_Compl STAR_subset <-- {STAR,NatStar,STARC}_subset STAR_mem <-- {STAR,NatStar,STARC}_mem STAR_mem_Compl <-- {STAR,STARC}_mem_Compl STAR_diff <-- {STAR,STARC}_diff STAR_star_of_image_subset <-- {STAR_hypreal_of_real, NatStar_hypreal_of_real, STARC_hcomplex_of_complex}_image_subset starset_n_Un <-- starset{Nat,C}_n_Un starset_n_Int <-- starset{Nat,C}_n_Int starset_n_Compl <-- starset{Nat,C}_n_Compl starset_n_diff <-- starset{Nat,C}_n_diff InternalSets_Un <-- Internal{Nat,C}Sets_Un InternalSets_Int <-- Internal{Nat,C}Sets_Int InternalSets_Compl <-- Internal{Nat,C}Sets_Compl InternalSets_diff <-- Internal{Nat,C}Sets_diff InternalSets_UNIV_diff <-- Internal{Nat,C}Sets_UNIV_diff InternalSets_starset_n <-- Internal{Nat,C}Sets_starset{Nat,C}_n starset_starset_n_eq <-- starset{Nat,C}_starset{Nat,C}_n_eq starset_n_starset <-- starset{Nat,C}_n_starset{Nat,C} starfun_n_starfun <-- starfun{Nat,Nat2,C,RC,CR}_n_starfun{Nat,Nat2,C,RC,CR} starfun <-- starfun{Nat,Nat2,C,RC,CR} starfun_mult <-- starfun{Nat,Nat2,C,RC,CR}_mult starfun_add <-- starfun{Nat,Nat2,C,RC,CR}_add starfun_minus <-- starfun{Nat,Nat2,C,RC,CR}_minus starfun_diff <-- starfun{C,RC,CR}_diff starfun_o <-- starfun{NatNat2,Nat2,_stafunNat,C,C_starfunRC,_starfunCR}_o starfun_o2 <-- starfun{NatNat2,_stafunNat,C,C_starfunRC,_starfunCR}_o2 starfun_const_fun <-- starfun{Nat,Nat2,C,RC,CR}_const_fun starfun_inverse <-- starfun{Nat,C,RC,CR}_inverse starfun_eq <-- starfun{Nat,Nat2,C,RC,CR}_eq starfun_eq_iff <-- starfun{C,RC,CR}_eq_iff starfun_Id <-- starfunC_Id starfun_approx <-- starfun{Nat,CR}_approx starfun_capprox <-- starfun{C,RC}_capprox starfun_abs <-- starfunNat_rabs starfun_lambda_cancel <-- starfun{C,CR,RC}_lambda_cancel starfun_lambda_cancel2 <-- starfun{C,CR,RC}_lambda_cancel2 starfun_mult_HFinite_approx <-- starfunCR_mult_HFinite_capprox starfun_mult_CFinite_capprox <-- starfun{C,RC}_mult_CFinite_capprox starfun_add_capprox <-- starfun{C,RC}_add_capprox starfun_add_approx <-- starfunCR_add_approx starfun_inverse_inverse <-- starfunC_inverse_inverse starfun_divide <-- starfun{C,CR,RC}_divide starfun_n <-- starfun{Nat,C}_n starfun_n_mult <-- starfun{Nat,C}_n_mult starfun_n_add <-- starfun{Nat,C}_n_add starfun_n_add_minus <-- starfunNat_n_add_minus starfun_n_const_fun <-- starfun{Nat,C}_n_const_fun starfun_n_minus <-- starfun{Nat,C}_n_minus starfun_n_eq <-- starfun{Nat,C}_n_eq star_n_add <-- {hypreal,hypnat,hcomplex}_add star_n_minus <-- {hypreal,hcomplex}_minus star_n_diff <-- {hypreal,hcomplex}_diff star_n_mult <-- {hypreal,hcomplex}_mult star_n_inverse <-- {hypreal,hcomplex}_inverse star_n_le <-- {hypreal,hypnat}_le star_n_less <-- {hypreal,hypnat}_less star_n_zero_num <-- {hypreal,hypnat,hcomplex}_zero_num star_n_one_num <-- {hypreal,hypnat,hcomplex}_one_num star_n_abs <-- hypreal_hrabs star_n_divide <-- hcomplex_divide star_of_add <-- {hypreal_of_real,hypnat_of_nat,hcomplex_of_complex}_add star_of_minus <-- {hypreal_of_real,hcomplex_of_complex}_minus star_of_diff <-- hypreal_of_real_diff star_of_mult <-- {hypreal_of_real,hypnat_of_nat,hcomplex_of_complex}_mult star_of_one <-- {hypreal_of_real,hcomplex_of_complex}_one star_of_zero <-- {hypreal_of_real,hypnat_of_nat,hcomplex_of_complex}_zero star_of_le <-- {hypreal_of_real,hypnat_of_nat}_le_iff star_of_less <-- {hypreal_of_real,hypnat_of_nat}_less_iff star_of_eq <-- {hypreal_of_real,hypnat_of_nat,hcomplex_of_complex}_eq_iff star_of_inverse <-- {hypreal_of_real,hcomplex_of_complex}_inverse star_of_divide <-- {hypreal_of_real,hcomplex_of_complex}_divide star_of_of_nat <-- {hypreal_of_real,hcomplex_of_complex}_of_nat star_of_of_int <-- {hypreal_of_real,hcomplex_of_complex}_of_int star_of_number_of <-- {hypreal,hcomplex}_number_of star_of_number_less <-- number_of_less_hypreal_of_real_iff star_of_number_le <-- number_of_le_hypreal_of_real_iff star_of_eq_number <-- hypreal_of_real_eq_number_of_iff star_of_less_number <-- hypreal_of_real_less_number_of_iff star_of_le_number <-- hypreal_of_real_le_number_of_iff star_of_power <-- hypreal_of_real_power star_of_eq_0 <-- hcomplex_of_complex_zero_iff * Hyperreal: new method "transfer" that implements the transfer principle of nonstandard analysis. With a subgoal that mentions nonstandard types like "'a star", the command "apply transfer" replaces it with an equivalent one that mentions only standard types. To be successful, all free variables must have standard types; non- standard variables must have explicit universal quantifiers. * Hyperreal: A theory of Taylor series. *** HOLCF *** * Discontinued special version of 'constdefs' (which used to support continuous functions) in favor of the general Pure one with full type-inference. * New simplification procedure for solving continuity conditions; it is much faster on terms with many nested lambda abstractions (cubic instead of exponential time). * New syntax for domain package: selector names are now optional. Parentheses should be omitted unless argument is lazy, for example: domain 'a stream = cons "'a" (lazy "'a stream") * New command 'fixrec' for defining recursive functions with pattern matching; defining multiple functions with mutual recursion is also supported. Patterns may include the constants cpair, spair, up, sinl, sinr, or any data constructor defined by the domain package. The given equations are proven as rewrite rules. See HOLCF/ex/Fixrec_ex.thy for syntax and examples. * New commands 'cpodef' and 'pcpodef' for defining predicate subtypes of cpo and pcpo types. Syntax is exactly like the 'typedef' command, but the proof obligation additionally includes an admissibility requirement. The packages generate instances of class cpo or pcpo, with continuity and strictness theorems for Rep and Abs. * HOLCF: Many theorems have been renamed according to a more standard naming scheme (INCOMPATIBILITY): foo_inject: "foo$x = foo$y ==> x = y" foo_eq: "(foo$x = foo$y) = (x = y)" foo_less: "(foo$x << foo$y) = (x << y)" foo_strict: "foo$UU = UU" foo_defined: "... ==> foo$x ~= UU" foo_defined_iff: "(foo$x = UU) = (x = UU)" *** ZF *** * ZF/ex: theories Group and Ring provide examples in abstract algebra, including the First Isomorphism Theorem (on quotienting by the kernel of a homomorphism). * ZF/Simplifier: install second copy of type solver that actually makes use of TC rules declared to Isar proof contexts (or locales); the old version is still required for ML proof scripts. *** Cube *** * Converted to Isar theory format; use locales instead of axiomatic theories. *** ML *** * Pure/library.ML: added ##>, ##>>, #>> -- higher-order counterparts for ||>, ||>>, |>>, * Pure/library.ML no longer defines its own option datatype, but uses that of the SML basis, which has constructors NONE and SOME instead of None and Some, as well as exception Option.Option instead of OPTION. The functions the, if_none, is_some, is_none have been adapted accordingly, while Option.map replaces apsome. * Pure/library.ML: the exception LIST has been given up in favour of the standard exceptions Empty and Subscript, as well as Library.UnequalLengths. Function like Library.hd and Library.tl are superceded by the standard hd and tl functions etc. A number of basic list functions are no longer exported to the ML toplevel, as they are variants of predefined functions. The following suggests how one can translate existing code: rev_append xs ys = List.revAppend (xs, ys) nth_elem (i, xs) = List.nth (xs, i) last_elem xs = List.last xs flat xss = List.concat xss seq fs = List.app fs partition P xs = List.partition P xs mapfilter f xs = List.mapPartial f xs * Pure/library.ML: several combinators for linear functional transformations, notably reverse application and composition: x |> f f #> g (x, y) |-> f f #-> g * Pure/library.ML: introduced/changed precedence of infix operators: infix 1 |> |-> ||> ||>> |>> |>>> #> #->; infix 2 ?; infix 3 o oo ooo oooo; infix 4 ~~ upto downto; Maybe INCOMPATIBILITY when any of those is used in conjunction with other infix operators. * Pure/library.ML: natural list combinators fold, fold_rev, and fold_map support linear functional transformations and nesting. For example: fold f [x1, ..., xN] y = y |> f x1 |> ... |> f xN (fold o fold) f [xs1, ..., xsN] y = y |> fold f xs1 |> ... |> fold f xsN fold f [x1, ..., xN] = f x1 #> ... #> f xN (fold o fold) f [xs1, ..., xsN] = fold f xs1 #> ... #> fold f xsN * Pure/library.ML: the following selectors on type 'a option are available: the: 'a option -> 'a (*partial*) these: 'a option -> 'a where 'a = 'b list the_default: 'a -> 'a option -> 'a the_list: 'a option -> 'a list * Pure/General: structure AList (cf. Pure/General/alist.ML) provides basic operations for association lists, following natural argument order; moreover the explicit equality predicate passed here avoids potentially expensive polymorphic runtime equality checks. The old functions may be expressed as follows: assoc = uncurry (AList.lookup (op =)) assocs = these oo AList.lookup (op =) overwrite = uncurry (AList.update (op =)) o swap * Pure/General: structure AList (cf. Pure/General/alist.ML) provides val make: ('a -> 'b) -> 'a list -> ('a * 'b) list val find: ('a * 'b -> bool) -> ('c * 'b) list -> 'a -> 'c list replacing make_keylist and keyfilter (occassionally used) Naive rewrites: make_keylist = AList.make keyfilter = AList.find (op =) * eq_fst and eq_snd now take explicit equality parameter, thus avoiding eqtypes. Naive rewrites: eq_fst = eq_fst (op =) eq_snd = eq_snd (op =) * Removed deprecated apl and apr (rarely used). Naive rewrites: apl (n, op) =>>= curry op n apr (op, m) =>>= fn n => op (n, m) * Pure/General: structure OrdList (cf. Pure/General/ord_list.ML) provides a reasonably efficient light-weight implementation of sets as lists. * Pure/General: generic tables (cf. Pure/General/table.ML) provide a few new operations; existing lookup and update are now curried to follow natural argument order (for use with fold etc.); INCOMPATIBILITY, use (uncurry Symtab.lookup) etc. as last resort. * Pure/General: output via the Isabelle channels of writeln/warning/error etc. is now passed through Output.output, with a hook for arbitrary transformations depending on the print_mode (cf. Output.add_mode -- the first active mode that provides a output function wins). Already formatted output may be embedded into further text via Output.raw; the result of Pretty.string_of/str_of and derived functions (string_of_term/cterm/thm etc.) is already marked raw to accommodate easy composition of diagnostic messages etc. Programmers rarely need to care about Output.output or Output.raw at all, with some notable exceptions: Output.output is required when bypassing the standard channels (writeln etc.), or in token translations to produce properly formatted results; Output.raw is required when capturing already output material that will eventually be presented to the user a second time. For the default print mode, both Output.output and Output.raw have no effect. * Pure/General: Output.time_accumulator NAME creates an operator ('a -> 'b) -> 'a -> 'b to measure runtime and count invocations; the cumulative results are displayed at the end of a batch session. * Pure/General: File.sysify_path and File.quote_sysify path have been replaced by File.platform_path and File.shell_path (with appropriate hooks). This provides a clean interface for unusual systems where the internal and external process view of file names are different. * Pure: more efficient orders for basic syntactic entities: added fast_string_ord, fast_indexname_ord, fast_term_ord; changed sort_ord and typ_ord to use fast_string_ord and fast_indexname_ord (term_ord is NOT affected); structures Symtab, Vartab, Typtab, Termtab use the fast orders now -- potential INCOMPATIBILITY for code that depends on a particular order for Symtab.keys, Symtab.dest, etc. (consider using Library.sort_strings on result). * Pure/term.ML: combinators fold_atyps, fold_aterms, fold_term_types, fold_types traverse types/terms from left to right, observing natural argument order. Supercedes previous foldl_XXX versions, add_frees, add_vars etc. have been adapted as well: INCOMPATIBILITY. * Pure: name spaces have been refined, with significant changes of the internal interfaces -- INCOMPATIBILITY. Renamed cond_extern(_table) to extern(_table). The plain name entry path is superceded by a general 'naming' context, which also includes the 'policy' to produce a fully qualified name and external accesses of a fully qualified name; NameSpace.extend is superceded by context dependent Sign.declare_name. Several theory and proof context operations modify the naming context. Especially note Theory.restore_naming and ProofContext.restore_naming to get back to a sane state; note that Theory.add_path is no longer sufficient to recover from Theory.absolute_path in particular. * Pure: new flags short_names (default false) and unique_names (default true) for controlling output of qualified names. If short_names is set, names are printed unqualified. If unique_names is reset, the name prefix is reduced to the minimum required to achieve the original result when interning again, even if there is an overlap with earlier declarations. * Pure/TheoryDataFun: change of the argument structure; 'prep_ext' is now 'extend', and 'merge' gets an additional Pretty.pp argument (useful for printing error messages). INCOMPATIBILITY. * Pure: major reorganization of the theory context. Type Sign.sg and Theory.theory are now identified, referring to the universal Context.theory (see Pure/context.ML). Actual signature and theory content is managed as theory data. The old code and interfaces were spread over many files and structures; the new arrangement introduces considerable INCOMPATIBILITY to gain more clarity: Context -- theory management operations (name, identity, inclusion, parents, ancestors, merge, etc.), plus generic theory data; Sign -- logical signature and syntax operations (declaring consts, types, etc.), plus certify/read for common entities; Theory -- logical theory operations (stating axioms, definitions, oracles), plus a copy of logical signature operations (consts, types, etc.); also a few basic management operations (Theory.copy, Theory.merge, etc.) The most basic sign_of operations (Theory.sign_of, Thm.sign_of_thm etc.) as well as the sign field in Thm.rep_thm etc. have been retained for convenience -- they merely return the theory. * Pure: type Type.tsig is superceded by theory in most interfaces. * Pure: the Isar proof context type is already defined early in Pure as Context.proof (note that ProofContext.context and Proof.context are aliases, where the latter is the preferred name). This enables other Isabelle components to refer to that type even before Isar is present. * Pure/sign/theory: discontinued named name spaces (i.e. classK, typeK, constK, axiomK, oracleK), but provide explicit operations for any of these kinds. For example, Sign.intern typeK is now Sign.intern_type, Theory.hide_space Sign.typeK is now Theory.hide_types. Also note that former Theory.hide_classes/types/consts are now Theory.hide_classes_i/types_i/consts_i, while the non '_i' versions internalize their arguments! INCOMPATIBILITY. * Pure: get_thm interface (of PureThy and ProofContext) expects datatype thmref (with constructors Name and NameSelection) instead of plain string -- INCOMPATIBILITY; * Pure: cases produced by proof methods specify options, where NONE means to remove case bindings -- INCOMPATIBILITY in (RAW_)METHOD_CASES. * Pure: the following operations retrieve axioms or theorems from a theory node or theory hierarchy, respectively: Theory.axioms_of: theory -> (string * term) list Theory.all_axioms_of: theory -> (string * term) list PureThy.thms_of: theory -> (string * thm) list PureThy.all_thms_of: theory -> (string * thm) list * Pure: print_tac now outputs the goal through the trace channel. * Isar toplevel: improved diagnostics, mostly for Poly/ML only. Reference Toplevel.debug (default false) controls detailed printing and tracing of low-level exceptions; Toplevel.profiling (default 0) controls execution profiling -- set to 1 for time and 2 for space (both increase the runtime). * Isar session: The initial use of ROOT.ML is now always timed, i.e. the log will show the actual process times, in contrast to the elapsed wall-clock time that the outer shell wrapper produces. * Simplifier: improved handling of bound variables (nameless representation, avoid allocating new strings). Simprocs that invoke the Simplifier recursively should use Simplifier.inherit_bounds to avoid local name clashes. Failure to do so produces warnings "Simplifier: renamed bound variable ..."; set Simplifier.debug_bounds for further details. * ML functions legacy_bindings and use_legacy_bindings produce ML fact bindings for all theorems stored within a given theory; this may help in porting non-Isar theories to Isar ones, while keeping ML proof scripts for the time being. * ML operator HTML.with_charset specifies the charset begin used for generated HTML files. For example: HTML.with_charset "utf-8" use_thy "Hebrew"; HTML.with_charset "utf-8" use_thy "Chinese"; *** System *** * Allow symlinks to all proper Isabelle executables (Isabelle, isabelle, isatool etc.). * ISABELLE_DOC_FORMAT setting specifies preferred document format (for isatool doc, isatool mkdir, display_drafts etc.). * isatool usedir: option -f allows specification of the ML file to be used by Isabelle; default is ROOT.ML. * New isatool version outputs the version identifier of the Isabelle distribution being used. * HOL: new isatool dimacs2hol converts files in DIMACS CNF format (containing Boolean satisfiability problems) into Isabelle/HOL theories. New in Isabelle2004 (April 2004) -------------------------------- *** General *** * Provers/order.ML: new efficient reasoner for partial and linear orders. Replaces linorder.ML. * Pure: Greek letters (except small lambda, \), as well as Gothic (\...\\...\), calligraphic (\...\), and Euler (\...\), are now considered normal letters, and can therefore be used anywhere where an ASCII letter (a...zA...Z) has until now. COMPATIBILITY: This obviously changes the parsing of some terms, especially where a symbol has been used as a binder, say '\x. ...', which is now a type error since \x will be parsed as an identifier. Fix it by inserting a space around former symbols. Call 'isatool fixgreek' to try to fix parsing errors in existing theory and ML files. * Pure: Macintosh and Windows line-breaks are now allowed in theory files. * Pure: single letter sub/superscripts (\<^isub> and \<^isup>) are now allowed in identifiers. Similar to Greek letters \<^isub> is now considered a normal (but invisible) letter. For multiple letter subscripts repeat \<^isub> like this: x\<^isub>1\<^isub>2. * Pure: There are now sub-/superscripts that can span more than one character. Text between \<^bsub> and \<^esub> is set in subscript in ProofGeneral and LaTeX, text between \<^bsup> and \<^esup> in superscript. The new control characters are not identifier parts. * Pure: Control-symbols of the form \<^raw:...> will literally print the content of "..." to the latex file instead of \isacntrl... . The "..." may consist of any printable characters excluding the end bracket >. * Pure: Using new Isar command "finalconsts" (or the ML functions Theory.add_finals or Theory.add_finals_i) it is now possible to declare constants "final", which prevents their being given a definition later. It is useful for constants whose behaviour is fixed axiomatically rather than definitionally, such as the meta-logic connectives. * Pure: 'instance' now handles general arities with general sorts (i.e. intersections of classes), * Presentation: generated HTML now uses a CSS style sheet to make layout (somewhat) independent of content. It is copied from lib/html/isabelle.css. It can be changed to alter the colors/layout of generated pages. *** Isar *** * Tactic emulation methods rule_tac, erule_tac, drule_tac, frule_tac, cut_tac, subgoal_tac and thin_tac: - Now understand static (Isar) contexts. As a consequence, users of Isar locales are no longer forced to write Isar proof scripts. For details see Isar Reference Manual, paragraph 4.3.2: Further tactic emulations. - INCOMPATIBILITY: names of variables to be instantiated may no longer be enclosed in quotes. Instead, precede variable name with `?'. This is consistent with the instantiation attribute "where". * Attributes "where" and "of": - Now take type variables of instantiated theorem into account when reading the instantiation string. This fixes a bug that caused instantiated theorems to have too special types in some circumstances. - "where" permits explicit instantiations of type variables. * Calculation commands "moreover" and "also" no longer interfere with current facts ("this"), admitting arbitrary combinations with "then" and derived forms. * Locales: - Goal statements involving the context element "includes" no longer generate theorems with internal delta predicates (those ending on "_axioms") in the premise. Resolve particular premise with .intro to obtain old form. - Fixed bug in type inference ("unify_frozen") that prevented mix of target specification and "includes" elements in goal statement. - Rule sets .intro and .axioms no longer declared as [intro?] and [elim?] (respectively) by default. - Experimental command for instantiation of locales in proof contexts: instantiate
: "g2 u \ (\x. g2 (2 * x - 1)) ` ({0..1} \ {x. \ x * 2 \ 1})" if "0 < u" "u \ 1" for u using that assms by (rule_tac x="(u+1)/2" in image_eqI) (auto simp: field_simps pathfinish_def pathstart_def) have "g2 0 \ (\x. g1 (2 * x)) ` ({0..1} \ {x. x * 2 \ 1})" using assms by (rule_tac x="1/2" in image_eqI) (auto simp: pathfinish_def pathstart_def) then have "path_image g2 \ path_image (g1 +++ g2)" by (auto simp: path_image_def joinpaths_def intro!: \
) ultimately show ?thesis using path_image_join_subset by blast qed lemma not_in_path_image_join: assumes "x \ path_image g1" and "x \ path_image g2" shows "x \ path_image (g1 +++ g2)" using assms and path_image_join_subset[of g1 g2] by auto lemma pathstart_compose: "pathstart(f \ p) = f(pathstart p)" by (simp add: pathstart_def) lemma pathfinish_compose: "pathfinish(f \ p) = f(pathfinish p)" by (simp add: pathfinish_def) lemma path_image_compose: "path_image (f \ p) = f ` (path_image p)" by (simp add: image_comp path_image_def) lemma path_compose_join: "f \ (p +++ q) = (f \ p) +++ (f \ q)" by (rule ext) (simp add: joinpaths_def) lemma path_compose_reversepath: "f \ reversepath p = reversepath(f \ p)" by (rule ext) (simp add: reversepath_def) lemma joinpaths_eq: "(\t. t \ {0..1} \ p t = p' t) \ (\t. t \ {0..1} \ q t = q' t) \ t \ {0..1} \ (p +++ q) t = (p' +++ q') t" by (auto simp: joinpaths_def) lemma simple_path_inj_on: "simple_path g \ inj_on g {0<..<1}" by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def) subsection\<^marker>\tag unimportant\\Simple paths with the endpoints removed\ lemma simple_path_endless: assumes "simple_path c" shows "path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" (is "?lhs = ?rhs") proof show "?lhs \ ?rhs" using less_eq_real_def by (auto simp: path_image_def pathstart_def pathfinish_def) show "?rhs \ ?lhs" using assms apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def) using less_eq_real_def zero_le_one by blast+ qed lemma connected_simple_path_endless: assumes "simple_path c" shows "connected(path_image c - {pathstart c,pathfinish c})" proof - have "continuous_on {0<..<1} c" using assms by (simp add: simple_path_def continuous_on_path path_def subset_iff) then have "connected (c ` {0<..<1})" using connected_Ioo connected_continuous_image by blast then show ?thesis using assms by (simp add: simple_path_endless) qed lemma nonempty_simple_path_endless: "simple_path c \ path_image c - {pathstart c,pathfinish c} \ {}" by (simp add: simple_path_endless) subsection\<^marker>\tag unimportant\\The operations on paths\ lemma path_image_subset_reversepath: "path_image(reversepath g) \ path_image g" by simp lemma path_imp_reversepath: "path g \ path(reversepath g)" by simp lemma half_bounded_equal: "1 \ x * 2 \ x * 2 \ 1 \ x = (1/2::real)" by simp lemma continuous_on_joinpaths: assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2" shows "continuous_on {0..1} (g1 +++ g2)" proof - have "{0..1::real} = {0..1/2} \ {1/2..1}" by auto then show ?thesis using assms by (metis path_def path_join) qed lemma path_join_imp: "\path g1; path g2; pathfinish g1 = pathstart g2\ \ path(g1 +++ g2)" by simp lemma simple_path_join_loop: assumes "arc g1" "arc g2" "pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1" "path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}" shows "simple_path(g1 +++ g2)" proof - have injg1: "inj_on g1 {0..1}" using assms by (simp add: arc_def) have injg2: "inj_on g2 {0..1}" using assms by (simp add: arc_def) have g12: "g1 1 = g2 0" and g21: "g2 1 = g1 0" and sb: "g1 ` {0..1} \ g2 ` {0..1} \ {g1 0, g2 0}" using assms by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) { fix x and y::real assume g2_eq: "g2 (2 * x - 1) = g1 (2 * y)" and xyI: "x \ 1 \ y \ 0" and xy: "x \ 1" "0 \ y" " y * 2 \ 1" "\ x * 2 \ 1" then consider "g1 (2 * y) = g1 0" | "g1 (2 * y) = g2 0" using sb by force then have False proof cases case 1 then have "y = 0" using xy g2_eq by (auto dest!: inj_onD [OF injg1]) then show ?thesis using xy g2_eq xyI by (auto dest: inj_onD [OF injg2] simp flip: g21) next case 2 then have "2*x = 1" using g2_eq g12 inj_onD [OF injg2] atLeastAtMost_iff xy(1) xy(4) by fastforce with xy show False by auto qed } note * = this { fix x and y::real assume xy: "g1 (2 * x) = g2 (2 * y - 1)" "y \ 1" "0 \ x" "\ y * 2 \ 1" "x * 2 \ 1" then have "x = 0 \ y = 1" using * xy by force } note ** = this show ?thesis using assms apply (simp add: arc_def simple_path_def) apply (auto simp: joinpaths_def split: if_split_asm dest!: * ** dest: inj_onD [OF injg1] inj_onD [OF injg2]) done qed lemma arc_join: assumes "arc g1" "arc g2" "pathfinish g1 = pathstart g2" "path_image g1 \ path_image g2 \ {pathstart g2}" shows "arc(g1 +++ g2)" proof - have injg1: "inj_on g1 {0..1}" using assms by (simp add: arc_def) have injg2: "inj_on g2 {0..1}" using assms by (simp add: arc_def) have g11: "g1 1 = g2 0" and sb: "g1 ` {0..1} \ g2 ` {0..1} \ {g2 0}" using assms by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) { fix x and y::real assume xy: "g2 (2 * x - 1) = g1 (2 * y)" "x \ 1" "0 \ y" " y * 2 \ 1" "\ x * 2 \ 1" then have "g1 (2 * y) = g2 0" using sb by force then have False using xy inj_onD injg2 by fastforce } note * = this show ?thesis using assms apply (simp add: arc_def inj_on_def) apply (auto simp: joinpaths_def arc_imp_path split: if_split_asm dest: * *[OF sym] inj_onD [OF injg1] inj_onD [OF injg2]) done qed lemma reversepath_joinpaths: "pathfinish g1 = pathstart g2 \ reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1" unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def by (rule ext) (auto simp: mult.commute) subsection\<^marker>\tag unimportant\\Some reversed and "if and only if" versions of joining theorems\ lemma path_join_path_ends: fixes g1 :: "real \ 'a::metric_space" assumes "path(g1 +++ g2)" "path g2" shows "pathfinish g1 = pathstart g2" proof (rule ccontr) define e where "e = dist (g1 1) (g2 0)" assume Neg: "pathfinish g1 \ pathstart g2" then have "0 < dist (pathfinish g1) (pathstart g2)" by auto then have "e > 0" by (metis e_def pathfinish_def pathstart_def) then have "\e>0. \d>0. \x'\{0..1}. dist x' 0 < d \ dist (g2 x') (g2 0) < e" using \path g2\ atLeastAtMost_iff zero_le_one unfolding path_def continuous_on_iff by blast then obtain d1 where "d1 > 0" and d1: "\x'. \x'\{0..1}; norm x' < d1\ \ dist (g2 x') (g2 0) < e/2" by (metis \0 < e\ half_gt_zero_iff norm_conv_dist) obtain d2 where "d2 > 0" and d2: "\x'. \x'\{0..1}; dist x' (1/2) < d2\ \ dist ((g1 +++ g2) x') (g1 1) < e/2" using assms(1) \e > 0\ unfolding path_def continuous_on_iff apply (drule_tac x="1/2" in bspec, simp) apply (drule_tac x="e/2" in spec, force simp: joinpaths_def) done have int01_1: "min (1/2) (min d1 d2) / 2 \ {0..1}" using \d1 > 0\ \d2 > 0\ by (simp add: min_def) have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1" using \d1 > 0\ \d2 > 0\ by (simp add: min_def dist_norm) have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \ {0..1}" using \d1 > 0\ \d2 > 0\ by (simp add: min_def) have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2" using \d1 > 0\ \d2 > 0\ by (simp add: min_def dist_norm) have [simp]: "\ min (1 / 2) (min d1 d2) \ 0" using \d1 > 0\ \d2 > 0\ by (simp add: min_def) have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2" "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2" using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def) then have "dist (g1 1) (g2 0) < e/2 + e/2" using dist_triangle_half_r e_def by blast then show False by (simp add: e_def [symmetric]) qed lemma path_join_eq [simp]: fixes g1 :: "real \ 'a::metric_space" assumes "path g1" "path g2" shows "path(g1 +++ g2) \ pathfinish g1 = pathstart g2" using assms by (metis path_join_path_ends path_join_imp) lemma simple_path_joinE: assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2" obtains "arc g1" "arc g2" "path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}" proof - have *: "\x y. \0 \ x; x \ 1; 0 \ y; y \ 1; (g1 +++ g2) x = (g1 +++ g2) y\ \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0" using assms by (simp add: simple_path_def) have "path g1" using assms path_join simple_path_imp_path by blast moreover have "inj_on g1 {0..1}" proof (clarsimp simp: inj_on_def) fix x y assume "g1 x = g1 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1" then show "x = y" using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs) qed ultimately have "arc g1" using assms by (simp add: arc_def) have [simp]: "g2 0 = g1 1" using assms by (metis pathfinish_def pathstart_def) have "path g2" using assms path_join simple_path_imp_path by blast moreover have "inj_on g2 {0..1}" proof (clarsimp simp: inj_on_def) fix x y assume "g2 x = g2 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1" then show "x = y" using * [of "(x + 1) / 2" "(y + 1) / 2"] by (force simp: joinpaths_def split_ifs field_split_simps) qed ultimately have "arc g2" using assms by (simp add: arc_def) have "g2 y = g1 0 \ g2 y = g1 1" if "g1 x = g2 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1" for x y using * [of "x / 2" "(y + 1) / 2"] that by (auto simp: joinpaths_def split_ifs field_split_simps) then have "path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}" by (fastforce simp: pathstart_def pathfinish_def path_image_def) with \arc g1\ \arc g2\ show ?thesis using that by blast qed lemma simple_path_join_loop_eq: assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2" shows "simple_path(g1 +++ g2) \ arc g1 \ arc g2 \ path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}" by (metis assms simple_path_joinE simple_path_join_loop) lemma arc_join_eq: assumes "pathfinish g1 = pathstart g2" shows "arc(g1 +++ g2) \ arc g1 \ arc g2 \ path_image g1 \ path_image g2 \ {pathstart g2}" (is "?lhs = ?rhs") proof assume ?lhs then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path) then have *: "\x y. \0 \ x; x \ 1; 0 \ y; y \ 1; (g1 +++ g2) x = (g1 +++ g2) y\ \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0" using assms by (simp add: simple_path_def) have False if "g1 0 = g2 u" "0 \ u" "u \ 1" for u using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \?lhs\] by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs field_split_simps) then have n1: "pathstart g1 \ path_image g2" unfolding pathstart_def path_image_def using atLeastAtMost_iff by blast show ?rhs using \?lhs\ using \simple_path (g1 +++ g2)\ assms n1 simple_path_joinE by auto next assume ?rhs then show ?lhs using assms by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join) qed lemma arc_join_eq_alt: "pathfinish g1 = pathstart g2 \ (arc(g1 +++ g2) \ arc g1 \ arc g2 \ path_image g1 \ path_image g2 = {pathstart g2})" using pathfinish_in_path_image by (fastforce simp: arc_join_eq) subsection\<^marker>\tag unimportant\\The joining of paths is associative\ lemma path_assoc: "\pathfinish p = pathstart q; pathfinish q = pathstart r\ \ path(p +++ (q +++ r)) \ path((p +++ q) +++ r)" by simp lemma simple_path_assoc: assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r" shows "simple_path (p +++ (q +++ r)) \ simple_path ((p +++ q) +++ r)" proof (cases "pathstart p = pathfinish r") case True show ?thesis proof assume "simple_path (p +++ q +++ r)" with assms True show "simple_path ((p +++ q) +++ r)" by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join dest: arc_distinct_ends [of r]) next assume 0: "simple_path ((p +++ q) +++ r)" with assms True have q: "pathfinish r \ path_image q" using arc_distinct_ends by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join) have "pathstart r \ path_image p" using assms by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff pathfinish_in_path_image pathfinish_join simple_path_joinE) with assms 0 q True show "simple_path (p +++ q +++ r)" by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join dest!: subsetD [OF _ IntI]) qed next case False { fix x :: 'a assume a: "path_image p \ path_image q \ {pathstart q}" "(path_image p \ path_image q) \ path_image r \ {pathstart r}" "x \ path_image p" "x \ path_image r" have "pathstart r \ path_image q" by (metis assms(2) pathfinish_in_path_image) with a have "x = pathstart q" by blast } with False assms show ?thesis by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join) qed lemma arc_assoc: "\pathfinish p = pathstart q; pathfinish q = pathstart r\ \ arc(p +++ (q +++ r)) \ arc((p +++ q) +++ r)" by (simp add: arc_simple_path simple_path_assoc) subsubsection\<^marker>\tag unimportant\\Symmetry and loops\ lemma path_sym: "\pathfinish p = pathstart q; pathfinish q = pathstart p\ \ path(p +++ q) \ path(q +++ p)" by auto lemma simple_path_sym: "\pathfinish p = pathstart q; pathfinish q = pathstart p\ \ simple_path(p +++ q) \ simple_path(q +++ p)" by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop) lemma path_image_sym: "\pathfinish p = pathstart q; pathfinish q = pathstart p\ \ path_image(p +++ q) = path_image(q +++ p)" by (simp add: path_image_join sup_commute) subsection\Subpath\ definition\<^marker>\tag important\ subpath :: "real \ real \ (real \ 'a) \ real \ 'a::real_normed_vector" where "subpath a b g \ \x. g((b - a) * x + a)" lemma path_image_subpath_gen: fixes g :: "_ \ 'a::real_normed_vector" shows "path_image(subpath u v g) = g ` (closed_segment u v)" by (auto simp add: closed_segment_real_eq path_image_def subpath_def) lemma path_image_subpath: fixes g :: "real \ 'a::real_normed_vector" shows "path_image(subpath u v g) = (if u \ v then g ` {u..v} else g ` {v..u})" by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl) lemma path_image_subpath_commute: fixes g :: "real \ 'a::real_normed_vector" shows "path_image(subpath u v g) = path_image(subpath v u g)" by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl) lemma path_subpath [simp]: fixes g :: "real \ 'a::real_normed_vector" assumes "path g" "u \ {0..1}" "v \ {0..1}" shows "path(subpath u v g)" proof - have "continuous_on {0..1} (g \ (\x. ((v-u) * x+ u)))" using assms apply (intro continuous_intros; simp add: image_affinity_atLeastAtMost [where c=u]) apply (auto simp: path_def continuous_on_subset) done then show ?thesis by (simp add: path_def subpath_def) qed lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)" by (simp add: pathstart_def subpath_def) lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)" by (simp add: pathfinish_def subpath_def) lemma subpath_trivial [simp]: "subpath 0 1 g = g" by (simp add: subpath_def) lemma subpath_reversepath: "subpath 1 0 g = reversepath g" by (simp add: reversepath_def subpath_def) lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g" by (simp add: reversepath_def subpath_def algebra_simps) lemma subpath_translation: "subpath u v ((\x. a + x) \ g) = (\x. a + x) \ subpath u v g" by (rule ext) (simp add: subpath_def) lemma subpath_image: "subpath u v (f \ g) = f \ subpath u v g" by (rule ext) (simp add: subpath_def) lemma affine_ineq: fixes x :: "'a::linordered_idom" assumes "x \ 1" "v \ u" shows "v + x * u \ u + x * v" proof - have "(1-x)*(u-v) \ 0" using assms by auto then show ?thesis by (simp add: algebra_simps) qed lemma sum_le_prod1: fixes a::real shows "\a \ 1; b \ 1\ \ a + b \ 1 + a * b" by (metis add.commute affine_ineq mult.right_neutral) lemma simple_path_subpath_eq: "simple_path(subpath u v g) \ path(subpath u v g) \ u\v \ (\x y. x \ closed_segment u v \ y \ closed_segment u v \ g x = g y \ x = y \ x = u \ y = v \ x = v \ y = u)" (is "?lhs = ?rhs") proof assume ?lhs then have p: "path (\x. g ((v - u) * x + u))" and sim: "(\x y. \x\{0..1}; y\{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\ \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0)" by (auto simp: simple_path_def subpath_def) { fix x y assume "x \ closed_segment u v" "y \ closed_segment u v" "g x = g y" then have "x = y \ x = u \ y = v \ x = v \ y = u" using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost) (simp_all add: field_split_simps) } moreover have "path(subpath u v g) \ u\v" using sim [of "1/3" "2/3"] p by (auto simp: subpath_def) ultimately show ?rhs by metis next assume ?rhs then have d1: "\x y. \g x = g y; u \ x; x \ v; u \ y; y \ v\ \ x = y \ x = u \ y = v \ x = v \ y = u" and d2: "\x y. \g x = g y; v \ x; x \ u; v \ y; y \ u\ \ x = y \ x = u \ y = v \ x = v \ y = u" and ne: "u < v \ v < u" and psp: "path (subpath u v g)" by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost) have [simp]: "\x. u + x * v = v + x * u \ u=v \ x=1" by algebra show ?lhs using psp ne unfolding simple_path_def subpath_def by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) qed lemma arc_subpath_eq: "arc(subpath u v g) \ path(subpath u v g) \ u\v \ inj_on g (closed_segment u v)" (is "?lhs = ?rhs") proof assume ?lhs then have p: "path (\x. g ((v - u) * x + u))" and sim: "(\x y. \x\{0..1}; y\{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\ \ x = y)" by (auto simp: arc_def inj_on_def subpath_def) { fix x y assume "x \ closed_segment u v" "y \ closed_segment u v" "g x = g y" then have "x = y" using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p by (cases "v = u") (simp_all split: if_split_asm add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost, simp add: field_simps) } moreover have "path(subpath u v g) \ u\v" using sim [of "1/3" "2/3"] p by (auto simp: subpath_def) ultimately show ?rhs unfolding inj_on_def by metis next assume ?rhs then have d1: "\x y. \g x = g y; u \ x; x \ v; u \ y; y \ v\ \ x = y" and d2: "\x y. \g x = g y; v \ x; x \ u; v \ y; y \ u\ \ x = y" and ne: "u < v \ v < u" and psp: "path (subpath u v g)" by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost) show ?lhs using psp ne unfolding arc_def subpath_def inj_on_def by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) qed lemma simple_path_subpath: assumes "simple_path g" "u \ {0..1}" "v \ {0..1}" "u \ v" shows "simple_path(subpath u v g)" using assms apply (simp add: simple_path_subpath_eq simple_path_imp_path) apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce) done lemma arc_simple_path_subpath: "\simple_path g; u \ {0..1}; v \ {0..1}; g u \ g v\ \ arc(subpath u v g)" by (force intro: simple_path_subpath simple_path_imp_arc) lemma arc_subpath_arc: "\arc g; u \ {0..1}; v \ {0..1}; u \ v\ \ arc(subpath u v g)" by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD) lemma arc_simple_path_subpath_interior: "\simple_path g; u \ {0..1}; v \ {0..1}; u \ v; \u-v\ < 1\ \ arc(subpath u v g)" by (force simp: simple_path_def intro: arc_simple_path_subpath) lemma path_image_subpath_subset: "\u \ {0..1}; v \ {0..1}\ \ path_image(subpath u v g) \ path_image g" by (metis atLeastAtMost_iff atLeastatMost_subset_iff path_image_def path_image_subpath subset_image_iff) lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p" by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps) subsection\<^marker>\tag unimportant\\There is a subpath to the frontier\ lemma subpath_to_frontier_explicit: fixes S :: "'a::metric_space set" assumes g: "path g" and "pathfinish g \ S" obtains u where "0 \ u" "u \ 1" "\x. 0 \ x \ x < u \ g x \ interior S" "(g u \ interior S)" "(u = 0 \ g u \ closure S)" proof - have gcon: "continuous_on {0..1} g" using g by (simp add: path_def) moreover have "bounded ({u. g u \ closure (- S)} \ {0..1})" using compact_eq_bounded_closed by fastforce ultimately have com: "compact ({0..1} \ {u. g u \ closure (- S)})" using closed_vimage_Int by (metis (full_types) Int_commute closed_atLeastAtMost closed_closure compact_eq_bounded_closed vimage_def) have "1 \ {u. g u \ closure (- S)}" using assms by (simp add: pathfinish_def closure_def) then have dis: "{0..1} \ {u. g u \ closure (- S)} \ {}" using atLeastAtMost_iff zero_le_one by blast then obtain u where "0 \ u" "u \ 1" and gu: "g u \ closure (- S)" and umin: "\t. \0 \ t; t \ 1; g t \ closure (- S)\ \ u \ t" using compact_attains_inf [OF com dis] by fastforce then have umin': "\t. \0 \ t; t \ 1; t < u\ \ g t \ S" using closure_def by fastforce have \
: "g u \ closure S" if "u \ 0" proof - have "u > 0" using that \0 \ u\ by auto { fix e::real assume "e > 0" obtain d where "d>0" and d: "\x'. \x' \ {0..1}; dist x' u \ d\ \ dist (g x') (g u) < e" using continuous_onE [OF gcon _ \e > 0\] \0 \ _\ \_ \ 1\ atLeastAtMost_iff by auto have *: "dist (max 0 (u - d / 2)) u \ d" using \0 \ u\ \u \ 1\ \d > 0\ by (simp add: dist_real_def) have "\y\S. dist y (g u) < e" using \0 < u\ \u \ 1\ \d > 0\ by (force intro: d [OF _ *] umin') } then show ?thesis by (simp add: frontier_def closure_approachable) qed show ?thesis proof show "\x. 0 \ x \ x < u \ g x \ interior S" using \u \ 1\ interior_closure umin by fastforce show "g u \ interior S" by (simp add: gu interior_closure) qed (use \0 \ u\ \u \ 1\ \
in auto) qed lemma subpath_to_frontier_strong: assumes g: "path g" and "pathfinish g \ S" obtains u where "0 \ u" "u \ 1" "g u \ interior S" "u = 0 \ (\x. 0 \ x \ x < 1 \ subpath 0 u g x \ interior S) \ g u \ closure S" proof - obtain u where "0 \ u" "u \ 1" and gxin: "\x. 0 \ x \ x < u \ g x \ interior S" and gunot: "(g u \ interior S)" and u0: "(u = 0 \ g u \ closure S)" using subpath_to_frontier_explicit [OF assms] by blast show ?thesis proof show "g u \ interior S" using gunot by blast qed (use \0 \ u\ \u \ 1\ u0 in \(force simp: subpath_def gxin)+\) qed lemma subpath_to_frontier: assumes g: "path g" and g0: "pathstart g \ closure S" and g1: "pathfinish g \ S" obtains u where "0 \ u" "u \ 1" "g u \ frontier S" "path_image(subpath 0 u g) - {g u} \ interior S" proof - obtain u where "0 \ u" "u \ 1" and notin: "g u \ interior S" and disj: "u = 0 \ (\x. 0 \ x \ x < 1 \ subpath 0 u g x \ interior S) \ g u \ closure S" (is "_ \ ?P") using subpath_to_frontier_strong [OF g g1] by blast show ?thesis proof show "g u \ frontier S" by (metis DiffI disj frontier_def g0 notin pathstart_def) show "path_image (subpath 0 u g) - {g u} \ interior S" using disj proof assume "u = 0" then show ?thesis by (simp add: path_image_subpath) next assume P: ?P show ?thesis proof (clarsimp simp add: path_image_subpath_gen) fix y assume y: "y \ closed_segment 0 u" "g y \ interior S" with \0 \ u\ have "0 \ y" "y \ u" by (auto simp: closed_segment_eq_real_ivl split: if_split_asm) then have "y=u \ subpath 0 u g (y/u) \ interior S" using P less_eq_real_def by force then show "g y = g u" using y by (auto simp: subpath_def split: if_split_asm) qed qed qed (use \0 \ u\ \u \ 1\ in auto) qed lemma exists_path_subpath_to_frontier: fixes S :: "'a::real_normed_vector set" assumes "path g" "pathstart g \ closure S" "pathfinish g \ S" obtains h where "path h" "pathstart h = pathstart g" "path_image h \ path_image g" "path_image h - {pathfinish h} \ interior S" "pathfinish h \ frontier S" proof - obtain u where u: "0 \ u" "u \ 1" "g u \ frontier S" "(path_image(subpath 0 u g) - {g u}) \ interior S" using subpath_to_frontier [OF assms] by blast show ?thesis proof show "path_image (subpath 0 u g) \ path_image g" by (simp add: path_image_subpath_subset u) show "pathstart (subpath 0 u g) = pathstart g" by (metis pathstart_def pathstart_subpath) qed (use assms u in \auto simp: path_image_subpath\) qed lemma exists_path_subpath_to_frontier_closed: fixes S :: "'a::real_normed_vector set" assumes S: "closed S" and g: "path g" and g0: "pathstart g \ S" and g1: "pathfinish g \ S" obtains h where "path h" "pathstart h = pathstart g" "path_image h \ path_image g \ S" "pathfinish h \ frontier S" proof - obtain h where h: "path h" "pathstart h = pathstart g" "path_image h \ path_image g" "path_image h - {pathfinish h} \ interior S" "pathfinish h \ frontier S" using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto show ?thesis proof show "path_image h \ path_image g \ S" using assms h interior_subset [of S] by (auto simp: frontier_def) qed (use h in auto) qed subsection \Shift Path to Start at Some Given Point\ definition\<^marker>\tag important\ shiftpath :: "real \ (real \ 'a::topological_space) \ real \ 'a" where "shiftpath a f = (\x. if (a + x) \ 1 then f (a + x) else f (a + x - 1))" lemma shiftpath_alt_def: "shiftpath a f = (\x. if x \ 1-a then f (a + x) else f (a + x - 1))" by (auto simp: shiftpath_def) lemma pathstart_shiftpath: "a \ 1 \ pathstart (shiftpath a g) = g a" unfolding pathstart_def shiftpath_def by auto lemma pathfinish_shiftpath: assumes "0 \ a" and "pathfinish g = pathstart g" shows "pathfinish (shiftpath a g) = g a" using assms unfolding pathstart_def pathfinish_def shiftpath_def by auto lemma endpoints_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0 .. 1}" shows "pathfinish (shiftpath a g) = g a" and "pathstart (shiftpath a g) = g a" using assms by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) lemma closed_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0..1}" shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" using endpoints_shiftpath[OF assms] by auto lemma path_shiftpath: assumes "path g" and "pathfinish g = pathstart g" and "a \ {0..1}" shows "path (shiftpath a g)" proof - have *: "{0 .. 1} = {0 .. 1-a} \ {1-a .. 1}" using assms(3) by auto have **: "\x. x + a = 1 \ g (x + a - 1) = g (x + a)" using assms(2)[unfolded pathfinish_def pathstart_def] by auto show ?thesis unfolding path_def shiftpath_def * proof (rule continuous_on_closed_Un) have contg: "continuous_on {0..1} g" using \path g\ path_def by blast show "continuous_on {0..1-a} (\x. if a + x \ 1 then g (a + x) else g (a + x - 1))" proof (rule continuous_on_eq) show "continuous_on {0..1-a} (g \ (+) a)" by (intro continuous_intros continuous_on_subset [OF contg]) (use \a \ {0..1}\ in auto) qed auto show "continuous_on {1-a..1} (\x. if a + x \ 1 then g (a + x) else g (a + x - 1))" proof (rule continuous_on_eq) show "continuous_on {1-a..1} (g \ (+) (a - 1))" by (intro continuous_intros continuous_on_subset [OF contg]) (use \a \ {0..1}\ in auto) qed (auto simp: "**" add.commute add_diff_eq) qed auto qed lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0..1}" and "x \ {0..1}" shows "shiftpath (1 - a) (shiftpath a g) x = g x" using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto lemma path_image_shiftpath: assumes a: "a \ {0..1}" and "pathfinish g = pathstart g" shows "path_image (shiftpath a g) = path_image g" proof - { fix x assume g: "g 1 = g 0" "x \ {0..1::real}" and gne: "\y. y\{0..1} \ {x. \ a + x \ 1} \ g x \ g (a + y - 1)" then have "\y\{0..1} \ {x. a + x \ 1}. g x = g (a + y)" proof (cases "a \ x") case False then show ?thesis apply (rule_tac x="1 + x - a" in bexI) using g gne[of "1 + x - a"] a by (force simp: field_simps)+ next case True then show ?thesis using g a by (rule_tac x="x - a" in bexI) (auto simp: field_simps) qed } then show ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def by (auto simp: image_iff) qed lemma simple_path_shiftpath: assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \ a" "a \ 1" shows "simple_path (shiftpath a g)" unfolding simple_path_def proof (intro conjI impI ballI) show "path (shiftpath a g)" by (simp add: assms path_shiftpath simple_path_imp_path) have *: "\x y. \g x = g y; x \ {0..1}; y \ {0..1}\ \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0" using assms by (simp add: simple_path_def) show "x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0" if "x \ {0..1}" "y \ {0..1}" "shiftpath a g x = shiftpath a g y" for x y using that a unfolding shiftpath_def by (force split: if_split_asm dest!: *) qed subsection \Straight-Line Paths\ definition\<^marker>\tag important\ linepath :: "'a::real_normed_vector \ 'a \ real \ 'a" where "linepath a b = (\x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" unfolding pathstart_def linepath_def by auto lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" unfolding pathfinish_def linepath_def by auto lemma linepath_inner: "linepath a b x \ v = linepath (a \ v) (b \ v) x" by (simp add: linepath_def algebra_simps) lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x" by (simp add: linepath_def) lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x" by (simp add: linepath_def) lemma linepath_0': "linepath a b 0 = a" by (simp add: linepath_def) lemma linepath_1': "linepath a b 1 = b" by (simp add: linepath_def) lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" unfolding linepath_def by (intro continuous_intros) lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)" using continuous_linepath_at by (auto intro!: continuous_at_imp_continuous_on) lemma path_linepath[iff]: "path (linepath a b)" unfolding path_def by (rule continuous_on_linepath) lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" unfolding path_image_def segment linepath_def by auto lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" unfolding reversepath_def linepath_def by auto lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b" by (simp add: linepath_def) lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x" by (simp add: linepath_def) lemma arc_linepath: assumes "a \ b" shows [simp]: "arc (linepath a b)" proof - { fix x y :: "real" assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps) with assms have "x = y" by simp } then show ?thesis unfolding arc_def inj_on_def by (fastforce simp: algebra_simps linepath_def) qed lemma simple_path_linepath[intro]: "a \ b \ simple_path (linepath a b)" by (simp add: arc_imp_simple_path) lemma linepath_trivial [simp]: "linepath a a x = a" by (simp add: linepath_def real_vector.scale_left_diff_distrib) lemma linepath_refl: "linepath a a = (\x. a)" by auto lemma subpath_refl: "subpath a a g = linepath (g a) (g a)" by (simp add: subpath_def linepath_def algebra_simps) lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)" by (simp add: scaleR_conv_of_real linepath_def) lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x" by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def) lemma inj_on_linepath: assumes "a \ b" shows "inj_on (linepath a b) {0..1}" proof (clarsimp simp: inj_on_def linepath_def) fix x y assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b" "0 \ x" "x \ 1" "0 \ y" "y \ 1" then have "x *\<^sub>R (a - b) = y *\<^sub>R (a - b)" by (auto simp: algebra_simps) then show "x=y" using assms by auto qed lemma linepath_le_1: fixes a::"'a::linordered_idom" shows "\a \ 1; b \ 1; 0 \ u; u \ 1\ \ (1 - u) * a + u * b \ 1" using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto lemma linepath_in_path: shows "x \ {0..1} \ linepath a b x \ closed_segment a b" by (auto simp: segment linepath_def) lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b" by (auto simp: segment linepath_def) lemma linepath_in_convex_hull: fixes x::real assumes a: "a \ convex hull S" and b: "b \ convex hull S" and x: "0\x" "x\1" shows "linepath a b x \ convex hull S" proof - have "linepath a b x \ closed_segment a b" using x by (auto simp flip: linepath_image_01) then show ?thesis using a b convex_contains_segment by blast qed lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b" by (simp add: linepath_def) lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0" by (simp add: linepath_def) lemma bounded_linear_linepath: assumes "bounded_linear f" shows "f (linepath a b x) = linepath (f a) (f b) x" proof - interpret f: bounded_linear f by fact show ?thesis by (simp add: linepath_def f.add f.scale) qed lemma bounded_linear_linepath': assumes "bounded_linear f" shows "f \ linepath a b = linepath (f a) (f b)" using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff) lemma linepath_cnj': "cnj \ linepath a b = linepath (cnj a) (cnj b)" by (simp add: linepath_def fun_eq_iff) lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A" by (auto simp: linepath_def) lemma has_vector_derivative_linepath_within: "(linepath a b has_vector_derivative (b - a)) (at x within S)" by (force intro: derivative_eq_intros simp add: linepath_def has_vector_derivative_def algebra_simps) subsection\<^marker>\tag unimportant\\Segments via convex hulls\ lemma segments_subset_convex_hull: "closed_segment a b \ (convex hull {a,b,c})" "closed_segment a c \ (convex hull {a,b,c})" "closed_segment b c \ (convex hull {a,b,c})" "closed_segment b a \ (convex hull {a,b,c})" "closed_segment c a \ (convex hull {a,b,c})" "closed_segment c b \ (convex hull {a,b,c})" by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono]) lemma midpoints_in_convex_hull: assumes "x \ convex hull s" "y \ convex hull s" shows "midpoint x y \ convex hull s" proof - have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \ convex hull s" by (rule convexD_alt) (use assms in auto) then show ?thesis by (simp add: midpoint_def algebra_simps) qed lemma not_in_interior_convex_hull_3: fixes a :: "complex" shows "a \ interior(convex hull {a,b,c})" "b \ interior(convex hull {a,b,c})" "c \ interior(convex hull {a,b,c})" by (auto simp: card_insert_le_m1 not_in_interior_convex_hull) lemma midpoint_in_closed_segment [simp]: "midpoint a b \ closed_segment a b" using midpoints_in_convex_hull segment_convex_hull by blast lemma midpoint_in_open_segment [simp]: "midpoint a b \ open_segment a b \ a \ b" by (simp add: open_segment_def) lemma continuous_IVT_local_extremum: fixes f :: "'a::euclidean_space \ real" assumes contf: "continuous_on (closed_segment a b) f" and "a \ b" "f a = f b" obtains z where "z \ open_segment a b" "(\w \ closed_segment a b. (f w) \ (f z)) \ (\w \ closed_segment a b. (f z) \ (f w))" proof - obtain c where "c \ closed_segment a b" and c: "\y. y \ closed_segment a b \ f y \ f c" using continuous_attains_sup [of "closed_segment a b" f] contf by auto obtain d where "d \ closed_segment a b" and d: "\y. y \ closed_segment a b \ f d \ f y" using continuous_attains_inf [of "closed_segment a b" f] contf by auto show ?thesis proof (cases "c \ open_segment a b \ d \ open_segment a b") case True then show ?thesis using c d that by blast next case False then have "(c = a \ c = b) \ (d = a \ d = b)" by (simp add: \c \ closed_segment a b\ \d \ closed_segment a b\ open_segment_def) with \a \ b\ \f a = f b\ c d show ?thesis by (rule_tac z = "midpoint a b" in that) (fastforce+) qed qed text\An injective map into R is also an open map w.r.T. the universe, and conversely. \ proposition injective_eq_1d_open_map_UNIV: fixes f :: "real \ real" assumes contf: "continuous_on S f" and S: "is_interval S" shows "inj_on f S \ (\T. open T \ T \ S \ open(f ` T))" (is "?lhs = ?rhs") proof safe fix T assume injf: ?lhs and "open T" and "T \ S" have "\U. open U \ f x \ U \ U \ f ` T" if "x \ T" for x proof - obtain \ where "\ > 0" and \: "cball x \ \ T" using \open T\ \x \ T\ open_contains_cball_eq by blast show ?thesis proof (intro exI conjI) have "closed_segment (x-\) (x+\) = {x-\..x+\}" using \0 < \\ by (auto simp: closed_segment_eq_real_ivl) also have "\ \ S" using \ \T \ S\ by (auto simp: dist_norm subset_eq) finally have "f ` (open_segment (x-\) (x+\)) = open_segment (f (x-\)) (f (x+\))" using continuous_injective_image_open_segment_1 by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf]) then show "open (f ` {x-\<..})" using \0 < \\ by (simp add: open_segment_eq_real_ivl) show "f x \ f ` {x - \<..}" by (auto simp: \\ > 0\) show "f ` {x - \<..} \ f ` T" using \ by (auto simp: dist_norm subset_iff) qed qed with open_subopen show "open (f ` T)" by blast next assume R: ?rhs have False if xy: "x \ S" "y \ S" and "f x = f y" "x \ y" for x y proof - have "open (f ` open_segment x y)" using R by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy) moreover have "continuous_on (closed_segment x y) f" by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that) then obtain \ where "\ \ open_segment x y" and \: "(\w \ closed_segment x y. (f w) \ (f \)) \ (\w \ closed_segment x y. (f \) \ (f w))" using continuous_IVT_local_extremum [of x y f] \f x = f y\ \x \ y\ by blast ultimately obtain e where "e>0" and e: "\u. dist u (f \) < e \ u \ f ` open_segment x y" using open_dist by (metis image_eqI) have fin: "f \ + (e/2) \ f ` open_segment x y" "f \ - (e/2) \ f ` open_segment x y" using e [of "f \ + (e/2)"] e [of "f \ - (e/2)"] \e > 0\ by (auto simp: dist_norm) show ?thesis using \ \0 < e\ fin open_closed_segment by fastforce qed then show ?lhs by (force simp: inj_on_def) qed subsection\<^marker>\tag unimportant\ \Bounding a point away from a path\ lemma not_on_path_ball: fixes g :: "real \ 'a::heine_borel" assumes "path g" and z: "z \ path_image g" shows "\e > 0. ball z e \ path_image g = {}" proof - have "closed (path_image g)" by (simp add: \path g\ closed_path_image) then obtain a where "a \ path_image g" "\y \ path_image g. dist z a \ dist z y" by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z]) then show ?thesis by (rule_tac x="dist z a" in exI) (use dist_commute z in auto) qed lemma not_on_path_cball: fixes g :: "real \ 'a::heine_borel" assumes "path g" and "z \ path_image g" shows "\e>0. cball z e \ (path_image g) = {}" proof - obtain e where "ball z e \ path_image g = {}" "e > 0" using not_on_path_ball[OF assms] by auto moreover have "cball z (e/2) \ ball z e" using \e > 0\ by auto ultimately show ?thesis by (rule_tac x="e/2" in exI) auto qed subsection \Path component\ text \Original formalization by Tom Hales\ definition\<^marker>\tag important\ "path_component S x y \ (\g. path g \ path_image g \ S \ pathstart g = x \ pathfinish g = y)" abbreviation\<^marker>\tag important\ "path_component_set S x \ Collect (path_component S x)" lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def lemma path_component_mem: assumes "path_component S x y" shows "x \ S" and "y \ S" using assms unfolding path_defs by auto lemma path_component_refl: assumes "x \ S" shows "path_component S x x" using assms unfolding path_defs by (metis (full_types) assms continuous_on_const image_subset_iff path_image_def) lemma path_component_refl_eq: "path_component S x x \ x \ S" by (auto intro!: path_component_mem path_component_refl) lemma path_component_sym: "path_component S x y \ path_component S y x" unfolding path_component_def by (metis (no_types) path_image_reversepath path_reversepath pathfinish_reversepath pathstart_reversepath) lemma path_component_trans: assumes "path_component S x y" and "path_component S y z" shows "path_component S x z" using assms unfolding path_component_def by (metis path_join pathfinish_join pathstart_join subset_path_image_join) lemma path_component_of_subset: "S \ T \ path_component S x y \ path_component T x y" unfolding path_component_def by auto lemma path_component_linepath: fixes S :: "'a::real_normed_vector set" shows "closed_segment a b \ S \ path_component S a b" unfolding path_component_def by (rule_tac x="linepath a b" in exI, auto) subsubsection\<^marker>\tag unimportant\ \Path components as sets\ lemma path_component_set: "path_component_set S x = {y. (\g. path g \ path_image g \ S \ pathstart g = x \ pathfinish g = y)}" by (auto simp: path_component_def) lemma path_component_subset: "path_component_set S x \ S" by (auto simp: path_component_mem(2)) lemma path_component_eq_empty: "path_component_set S x = {} \ x \ S" using path_component_mem path_component_refl_eq by fastforce lemma path_component_mono: "S \ T \ (path_component_set S x) \ (path_component_set T x)" by (simp add: Collect_mono path_component_of_subset) lemma path_component_eq: "y \ path_component_set S x \ path_component_set S y = path_component_set S x" by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans) subsection \Path connectedness of a space\ definition\<^marker>\tag important\ "path_connected S \ (\x\S. \y\S. \g. path g \ path_image g \ S \ pathstart g = x \ pathfinish g = y)" lemma path_connectedin_iff_path_connected_real [simp]: "path_connectedin euclideanreal S \ path_connected S" by (simp add: path_connectedin path_connected_def path_defs) lemma path_connected_component: "path_connected S \ (\x\S. \y\S. path_component S x y)" unfolding path_connected_def path_component_def by auto lemma path_connected_component_set: "path_connected S \ (\x\S. path_component_set S x = S)" unfolding path_connected_component path_component_subset using path_component_mem by blast lemma path_component_maximal: "\x \ T; path_connected T; T \ S\ \ T \ (path_component_set S x)" by (metis path_component_mono path_connected_component_set) lemma convex_imp_path_connected: fixes S :: "'a::real_normed_vector set" assumes "convex S" shows "path_connected S" unfolding path_connected_def using assms convex_contains_segment by fastforce lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)" by (simp add: convex_imp_path_connected) lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)" using path_connected_component_set by auto lemma path_connected_imp_connected: assumes "path_connected S" shows "connected S" proof (rule connectedI) fix e1 e2 assume as: "open e1" "open e2" "S \ e1 \ e2" "e1 \ e2 \ S = {}" "e1 \ S \ {}" "e2 \ S \ {}" then obtain x1 x2 where obt:"x1 \ e1 \ S" "x2 \ e2 \ S" by auto then obtain g where g: "path g" "path_image g \ S" "pathstart g = x1" "pathfinish g = x2" using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto have *: "connected {0..1::real}" by (auto intro!: convex_connected) have "{0..1} \ {x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2}" using as(3) g(2)[unfolded path_defs] by blast moreover have "{x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto moreover have "{x \ {0..1}. g x \ e1} \ {} \ {x \ {0..1}. g x \ e2} \ {}" using g(3,4)[unfolded path_defs] using obt by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) ultimately show False using *[unfolded connected_local not_ex, rule_format, of "{0..1} \ g -` e1" "{0..1} \ g -` e2"] using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)] using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)] by auto qed lemma open_path_component: fixes S :: "'a::real_normed_vector set" assumes "open S" shows "open (path_component_set S x)" unfolding open_contains_ball proof fix y assume as: "y \ path_component_set S x" then have "y \ S" by (simp add: path_component_mem(2)) then obtain e where e: "e > 0" "ball y e \ S" using assms openE by blast have "\u. dist y u < e \ path_component S x u" by (metis (full_types) as centre_in_ball convex_ball convex_imp_path_connected e mem_Collect_eq mem_ball path_component_eq path_component_of_subset path_connected_component) then show "\e > 0. ball y e \ path_component_set S x" using \e>0\ by auto qed lemma open_non_path_component: fixes S :: "'a::real_normed_vector set" assumes "open S" shows "open (S - path_component_set S x)" unfolding open_contains_ball proof fix y assume y: "y \ S - path_component_set S x" then obtain e where e: "e > 0" "ball y e \ S" using assms openE by auto show "\e>0. ball y e \ S - path_component_set S x" proof (intro exI conjI subsetI DiffI notI) show "\x. x \ ball y e \ x \ S" using e by blast show False if "z \ ball y e" "z \ path_component_set S x" for z proof - have "y \ path_component_set S z" by (meson assms convex_ball convex_imp_path_connected e open_contains_ball_eq open_path_component path_component_maximal that(1)) then have "y \ path_component_set S x" using path_component_eq that(2) by blast then show False using y by blast qed qed (use e in auto) qed lemma connected_open_path_connected: fixes S :: "'a::real_normed_vector set" assumes "open S" and "connected S" shows "path_connected S" unfolding path_connected_component_set proof (rule, rule, rule path_component_subset, rule) fix x y assume "x \ S" and "y \ S" show "y \ path_component_set S x" proof (rule ccontr) assume "\ ?thesis" moreover have "path_component_set S x \ S \ {}" using \x \ S\ path_component_eq_empty path_component_subset[of S x] by auto ultimately show False using \y \ S\ open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] using assms(2)[unfolded connected_def not_ex, rule_format, of "path_component_set S x" "S - path_component_set S x"] by auto qed qed lemma path_connected_continuous_image: assumes contf: "continuous_on S f" and "path_connected S" shows "path_connected (f ` S)" unfolding path_connected_def proof (rule, rule) fix x' y' assume "x' \ f ` S" "y' \ f ` S" then obtain x y where x: "x \ S" and y: "y \ S" and x': "x' = f x" and y': "y' = f y" by auto from x y obtain g where "path g \ path_image g \ S \ pathstart g = x \ pathfinish g = y" using assms(2)[unfolded path_connected_def] by fast then show "\g. path g \ path_image g \ f ` S \ pathstart g = x' \ pathfinish g = y'" unfolding x' y' path_defs by (fastforce intro: continuous_on_compose continuous_on_subset[OF contf]) qed lemma path_connected_translationI: fixes a :: "'a :: topological_group_add" assumes "path_connected S" shows "path_connected ((\x. a + x) ` S)" by (intro path_connected_continuous_image assms continuous_intros) lemma path_connected_translation: fixes a :: "'a :: topological_group_add" shows "path_connected ((\x. a + x) ` S) = path_connected S" proof - have "\x y. (+) (x::'a) ` (+) (0 - x) ` y = y" by (simp add: image_image) then show ?thesis by (metis (no_types) path_connected_translationI) qed lemma path_connected_segment [simp]: fixes a :: "'a::real_normed_vector" shows "path_connected (closed_segment a b)" by (simp add: convex_imp_path_connected) lemma path_connected_open_segment [simp]: fixes a :: "'a::real_normed_vector" shows "path_connected (open_segment a b)" by (simp add: convex_imp_path_connected) lemma homeomorphic_path_connectedness: "S homeomorphic T \ path_connected S \ path_connected T" unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image) lemma path_connected_empty [simp]: "path_connected {}" unfolding path_connected_def by auto lemma path_connected_singleton [simp]: "path_connected {a}" unfolding path_connected_def pathstart_def pathfinish_def path_image_def using path_def by fastforce lemma path_connected_Un: assumes "path_connected S" and "path_connected T" and "S \ T \ {}" shows "path_connected (S \ T)" unfolding path_connected_component proof (intro ballI) fix x y assume x: "x \ S \ T" and y: "y \ S \ T" from assms obtain z where z: "z \ S" "z \ T" by auto show "path_component (S \ T) x y" using x y proof safe assume "x \ S" "y \ S" then show "path_component (S \ T) x y" by (meson Un_upper1 \path_connected S\ path_component_of_subset path_connected_component) next assume "x \ S" "y \ T" then show "path_component (S \ T) x y" by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component) next assume "x \ T" "y \ S" then show "path_component (S \ T) x y" by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component) next assume "x \ T" "y \ T" then show "path_component (S \ T) x y" by (metis Un_upper1 assms(2) path_component_of_subset path_connected_component sup_commute) qed qed lemma path_connected_UNION: assumes "\i. i \ A \ path_connected (S i)" and "\i. i \ A \ z \ S i" shows "path_connected (\i\A. S i)" unfolding path_connected_component proof clarify fix x i y j assume *: "i \ A" "x \ S i" "j \ A" "y \ S j" then have "path_component (S i) x z" and "path_component (S j) z y" using assms by (simp_all add: path_connected_component) then have "path_component (\i\A. S i) x z" and "path_component (\i\A. S i) z y" using *(1,3) by (auto elim!: path_component_of_subset [rotated]) then show "path_component (\i\A. S i) x y" by (rule path_component_trans) qed lemma path_component_path_image_pathstart: assumes p: "path p" and x: "x \ path_image p" shows "path_component (path_image p) (pathstart p) x" proof - obtain y where x: "x = p y" and y: "0 \ y" "y \ 1" using x by (auto simp: path_image_def) show ?thesis unfolding path_component_def proof (intro exI conjI) have "continuous_on ((*) y ` {0..1}) p" by (simp add: continuous_on_path image_mult_atLeastAtMost_if p y) then have "continuous_on {0..1} (p \ ((*) y))" using continuous_on_compose continuous_on_mult_const by blast then show "path (\u. p (y * u))" by (simp add: path_def) show "path_image (\u. p (y * u)) \ path_image p" using y mult_le_one by (fastforce simp: path_image_def image_iff) qed (auto simp: pathstart_def pathfinish_def x) qed lemma path_connected_path_image: "path p \ path_connected(path_image p)" unfolding path_connected_component by (meson path_component_path_image_pathstart path_component_sym path_component_trans) lemma path_connected_path_component [simp]: "path_connected (path_component_set s x)" proof - { fix y z assume pa: "path_component s x y" "path_component s x z" then have pae: "path_component_set s x = path_component_set s y" using path_component_eq by auto have yz: "path_component s y z" using pa path_component_sym path_component_trans by blast then have "\g. path g \ path_image g \ path_component_set s x \ pathstart g = y \ pathfinish g = z" apply (simp add: path_component_def) by (metis pae path_component_maximal path_connected_path_image pathstart_in_path_image) } then show ?thesis by (simp add: path_connected_def) qed lemma path_component: "path_component S x y \ (\t. path_connected t \ t \ S \ x \ t \ y \ t)" apply (intro iffI) apply (metis path_connected_path_image path_defs(5) pathfinish_in_path_image pathstart_in_path_image) using path_component_of_subset path_connected_component by blast lemma path_component_path_component [simp]: "path_component_set (path_component_set S x) x = path_component_set S x" proof (cases "x \ S") case True show ?thesis by (metis True mem_Collect_eq path_component_refl path_connected_component_set path_connected_path_component) next case False then show ?thesis by (metis False empty_iff path_component_eq_empty) qed lemma path_component_subset_connected_component: "(path_component_set S x) \ (connected_component_set S x)" proof (cases "x \ S") case True show ?thesis by (simp add: True connected_component_maximal path_component_refl path_component_subset path_connected_imp_connected) next case False then show ?thesis using path_component_eq_empty by auto qed subsection\<^marker>\tag unimportant\\Lemmas about path-connectedness\ lemma path_connected_linear_image: fixes f :: "'a::real_normed_vector \ 'b::real_normed_vector" assumes "path_connected S" "bounded_linear f" shows "path_connected(f ` S)" by (auto simp: linear_continuous_on assms path_connected_continuous_image) lemma is_interval_path_connected: "is_interval S \ path_connected S" by (simp add: convex_imp_path_connected is_interval_convex) lemma path_connected_Ioi[simp]: "path_connected {a<..}" for a :: real by (simp add: convex_imp_path_connected) lemma path_connected_Ici[simp]: "path_connected {a..}" for a :: real by (simp add: convex_imp_path_connected) lemma path_connected_Iio[simp]: "path_connected {.. (\x \ topspace X. \y \ topspace X. \S. path_connectedin X S \ x \ S \ y \ S)" by (metis path_connectedin path_connectedin_topspace path_connected_space_def) lemma connectedin_path_image: "pathin X g \ connectedin X (g ` ({0..1}))" by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image) lemma compactin_path_image: "pathin X g \ compactin X (g ` ({0..1}))" unfolding pathin_def by (rule image_compactin [of "top_of_set {0..1}"]) auto lemma linear_homeomorphism_image: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "linear f" "inj f" obtains g where "homeomorphism (f ` S) S g f" proof - obtain g where "linear g" "g \ f = id" using assms linear_injective_left_inverse by blast then have "homeomorphism (f ` S) S g f" using assms unfolding homeomorphism_def by (auto simp: eq_id_iff [symmetric] image_comp linear_conv_bounded_linear linear_continuous_on) then show thesis .. qed lemma linear_homeomorphic_image: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "linear f" "inj f" shows "S homeomorphic f ` S" by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms]) lemma path_connected_Times: assumes "path_connected s" "path_connected t" shows "path_connected (s \ t)" proof (simp add: path_connected_def Sigma_def, clarify) fix x1 y1 x2 y2 assume "x1 \ s" "y1 \ t" "x2 \ s" "y2 \ t" obtain g where "path g" and g: "path_image g \ s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2" using \x1 \ s\ \x2 \ s\ assms by (force simp: path_connected_def) obtain h where "path h" and h: "path_image h \ t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2" using \y1 \ t\ \y2 \ t\ assms by (force simp: path_connected_def) have "path (\z. (x1, h z))" using \path h\ unfolding path_def by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force) moreover have "path (\z. (g z, y2))" using \path g\ unfolding path_def by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force) ultimately have 1: "path ((\z. (x1, h z)) +++ (\z. (g z, y2)))" by (metis hf gs path_join_imp pathstart_def pathfinish_def) have "path_image ((\z. (x1, h z)) +++ (\z. (g z, y2))) \ path_image (\z. (x1, h z)) \ path_image (\z. (g z, y2))" by (rule Path_Connected.path_image_join_subset) also have "\ \ (\x\s. \x1\t. {(x, x1)})" using g h \x1 \ s\ \y2 \ t\ by (force simp: path_image_def) finally have 2: "path_image ((\z. (x1, h z)) +++ (\z. (g z, y2))) \ (\x\s. \x1\t. {(x, x1)})" . show "\g. path g \ path_image g \ (\x\s. \x1\t. {(x, x1)}) \ pathstart g = (x1, y1) \ pathfinish g = (x2, y2)" using 1 2 gf hs by (metis (no_types, lifting) pathfinish_def pathfinish_join pathstart_def pathstart_join) qed lemma is_interval_path_connected_1: fixes s :: "real set" shows "is_interval s \ path_connected s" using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast subsection\<^marker>\tag unimportant\\Path components\ lemma Union_path_component [simp]: "Union {path_component_set S x |x. x \ S} = S" apply (rule subset_antisym) using path_component_subset apply force using path_component_refl by auto lemma path_component_disjoint: "disjnt (path_component_set S a) (path_component_set S b) \ (a \ path_component_set S b)" unfolding disjnt_iff using path_component_sym path_component_trans by blast lemma path_component_eq_eq: "path_component S x = path_component S y \ (x \ S) \ (y \ S) \ x \ S \ y \ S \ path_component S x y" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs by (metis (no_types) path_component_mem(1) path_component_refl) next assume ?rhs then show ?lhs proof assume "x \ S \ y \ S" then show ?lhs by (metis Collect_empty_eq_bot path_component_eq_empty) next assume S: "x \ S \ y \ S \ path_component S x y" show ?lhs by (rule ext) (metis S path_component_trans path_component_sym) qed qed lemma path_component_unique: assumes "x \ c" "c \ S" "path_connected c" "\c'. \x \ c'; c' \ S; path_connected c'\ \ c' \ c" shows "path_component_set S x = c" (is "?lhs = ?rhs") proof show "?lhs \ ?rhs" using assms by (metis mem_Collect_eq path_component_refl path_component_subset path_connected_path_component subsetD) qed (simp add: assms path_component_maximal) lemma path_component_intermediate_subset: "path_component_set u a \ t \ t \ u \ path_component_set t a = path_component_set u a" by (metis (no_types) path_component_mono path_component_path_component subset_antisym) lemma complement_path_component_Union: fixes x :: "'a :: topological_space" shows "S - path_component_set S x = \({path_component_set S y| y. y \ S} - {path_component_set S x})" proof - have *: "(\x. x \ S - {a} \ disjnt a x) \ \S - a = \(S - {a})" for a::"'a set" and S by (auto simp: disjnt_def) have "\y. y \ {path_component_set S x |x. x \ S} - {path_component_set S x} \ disjnt (path_component_set S x) y" using path_component_disjoint path_component_eq by fastforce then have "\{path_component_set S x |x. x \ S} - path_component_set S x = \({path_component_set S y |y. y \ S} - {path_component_set S x})" by (meson *) then show ?thesis by simp qed subsection\Path components\ definition path_component_of where "path_component_of X x y \ \g. pathin X g \ g 0 = x \ g 1 = y" abbreviation path_component_of_set where "path_component_of_set X x \ Collect (path_component_of X x)" definition path_components_of :: "'a topology \ 'a set set" where "path_components_of X \ path_component_of_set X ` topspace X" lemma pathin_canon_iff: "pathin (top_of_set T) g \ path g \ g ` {0..1} \ T" by (simp add: path_def pathin_def) lemma path_component_of_canon_iff [simp]: "path_component_of (top_of_set T) a b \ path_component T a b" by (simp add: path_component_of_def pathin_canon_iff path_defs) lemma path_component_in_topspace: "path_component_of X x y \ x \ topspace X \ y \ topspace X" by (auto simp: path_component_of_def pathin_def continuous_map_def) lemma path_component_of_refl: "path_component_of X x x \ x \ topspace X" by (metis path_component_in_topspace path_component_of_def pathin_const) lemma path_component_of_sym: assumes "path_component_of X x y" shows "path_component_of X y x" using assms apply (clarsimp simp: path_component_of_def pathin_def) apply (rule_tac x="g \ (\t. 1 - t)" in exI) apply (auto intro!: continuous_map_compose simp: continuous_map_in_subtopology continuous_on_op_minus) done lemma path_component_of_sym_iff: "path_component_of X x y \ path_component_of X y x" by (metis path_component_of_sym) lemma continuous_map_cases_le: assumes contp: "continuous_map X euclideanreal p" and contq: "continuous_map X euclideanreal q" and contf: "continuous_map (subtopology X {x. x \ topspace X \ p x \ q x}) Y f" and contg: "continuous_map (subtopology X {x. x \ topspace X \ q x \ p x}) Y g" and fg: "\x. \x \ topspace X; p x = q x\ \ f x = g x" shows "continuous_map X Y (\x. if p x \ q x then f x else g x)" proof - have "continuous_map X Y (\x. if q x - p x \ {0..} then f x else g x)" proof (rule continuous_map_cases_function) show "continuous_map X euclideanreal (\x. q x - p x)" by (intro contp contq continuous_intros) show "continuous_map (subtopology X {x \ topspace X. q x - p x \ euclideanreal closure_of {0..}}) Y f" by (simp add: contf) show "continuous_map (subtopology X {x \ topspace X. q x - p x \ euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g" by (simp add: contg flip: Compl_eq_Diff_UNIV) qed (auto simp: fg) then show ?thesis by simp qed lemma continuous_map_cases_lt: assumes contp: "continuous_map X euclideanreal p" and contq: "continuous_map X euclideanreal q" and contf: "continuous_map (subtopology X {x. x \ topspace X \ p x \ q x}) Y f" and contg: "continuous_map (subtopology X {x. x \ topspace X \ q x \ p x}) Y g" and fg: "\x. \x \ topspace X; p x = q x\ \ f x = g x" shows "continuous_map X Y (\x. if p x < q x then f x else g x)" proof - have "continuous_map X Y (\x. if q x - p x \ {0<..} then f x else g x)" proof (rule continuous_map_cases_function) show "continuous_map X euclideanreal (\x. q x - p x)" by (intro contp contq continuous_intros) show "continuous_map (subtopology X {x \ topspace X. q x - p x \ euclideanreal closure_of {0<..}}) Y f" by (simp add: contf) show "continuous_map (subtopology X {x \ topspace X. q x - p x \ euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g" by (simp add: contg flip: Compl_eq_Diff_UNIV) qed (auto simp: fg) then show ?thesis by simp qed lemma path_component_of_trans: assumes "path_component_of X x y" and "path_component_of X y z" shows "path_component_of X x z" unfolding path_component_of_def pathin_def proof - let ?T01 = "top_of_set {0..1::real}" obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1" and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1" using assms unfolding path_component_of_def pathin_def by blast let ?g = "\x. if x \ 1/2 then (g1 \ (\t. 2 * t)) x else (g2 \ (\t. 2 * t -1)) x" show "\g. continuous_map ?T01 X g \ g 0 = x \ g 1 = z" proof (intro exI conjI) show "continuous_map (subtopology euclideanreal {0..1}) X ?g" proof (intro continuous_map_cases_le continuous_map_compose, force, force) show "continuous_map (subtopology ?T01 {x \ topspace ?T01. x \ 1/2}) ?T01 ((*) 2)" by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology) have "continuous_map (subtopology (top_of_set {0..1}) {x. 0 \ x \ x \ 1 \ 1 \ x * 2}) euclideanreal (\t. 2 * t - 1)" by (intro continuous_intros) (force intro: continuous_map_from_subtopology) then show "continuous_map (subtopology ?T01 {x \ topspace ?T01. 1/2 \ x}) ?T01 (\t. 2 * t - 1)" by (force simp: continuous_map_in_subtopology) show "(g1 \ (*) 2) x = (g2 \ (\t. 2 * t - 1)) x" if "x \ topspace ?T01" "x = 1/2" for x using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology) qed (auto simp: g1 g2) qed (auto simp: g1 g2) qed lemma path_component_of_mono: "\path_component_of (subtopology X S) x y; S \ T\ \ path_component_of (subtopology X T) x y" unfolding path_component_of_def by (metis subsetD pathin_subtopology) lemma path_component_of: "path_component_of X x y \ (\T. path_connectedin X T \ x \ T \ y \ T)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs by (metis atLeastAtMost_iff image_eqI order_refl path_component_of_def path_connectedin_path_image zero_le_one) next assume ?rhs then show ?lhs by (metis path_component_of_def path_connectedin) qed lemma path_component_of_set: "path_component_of X x y \ (\g. pathin X g \ g 0 = x \ g 1 = y)" by (auto simp: path_component_of_def) lemma path_component_of_subset_topspace: "Collect(path_component_of X x) \ topspace X" using path_component_in_topspace by fastforce lemma path_component_of_eq_empty: "Collect(path_component_of X x) = {} \ (x \ topspace X)" using path_component_in_topspace path_component_of_refl by fastforce lemma path_connected_space_iff_path_component: "path_connected_space X \ (\x \ topspace X. \y \ topspace X. path_component_of X x y)" by (simp add: path_component_of path_connected_space_subconnected) lemma path_connected_space_imp_path_component_of: "\path_connected_space X; a \ topspace X; b \ topspace X\ \ path_component_of X a b" by (simp add: path_connected_space_iff_path_component) lemma path_connected_space_path_component_set: "path_connected_space X \ (\x \ topspace X. Collect(path_component_of X x) = topspace X)" using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce lemma path_component_of_maximal: "\path_connectedin X s; x \ s\ \ s \ Collect(path_component_of X x)" using path_component_of by fastforce lemma path_component_of_equiv: "path_component_of X x y \ x \ topspace X \ y \ topspace X \ path_component_of X x = path_component_of X y" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (simp add: fun_eq_iff path_component_in_topspace) apply (meson path_component_of_sym path_component_of_trans) done qed (simp add: path_component_of_refl) lemma path_component_of_disjoint: "disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) \ ~(path_component_of X x y)" by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv) lemma path_component_of_eq: "path_component_of X x = path_component_of X y \ (x \ topspace X) \ (y \ topspace X) \ x \ topspace X \ y \ topspace X \ path_component_of X x y" by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv) lemma path_component_of_aux: "path_component_of X x y \ path_component_of (subtopology X (Collect (path_component_of X x))) x y" by (meson path_component_of path_component_of_maximal path_connectedin_subtopology) lemma path_connectedin_path_component_of: "path_connectedin X (Collect (path_component_of X x))" proof - have "topspace (subtopology X (path_component_of_set X x)) = path_component_of_set X x" by (meson path_component_of_subset_topspace topspace_subtopology_subset) then have "path_connected_space (subtopology X (path_component_of_set X x))" by (metis (full_types) path_component_of_aux mem_Collect_eq path_component_of_equiv path_connected_space_iff_path_component) then show ?thesis by (simp add: path_component_of_subset_topspace path_connectedin_def) qed lemma path_connectedin_euclidean [simp]: "path_connectedin euclidean S \ path_connected S" by (auto simp: path_connectedin_def path_connected_space_iff_path_component path_connected_component) lemma path_connected_space_euclidean_subtopology [simp]: "path_connected_space(subtopology euclidean S) \ path_connected S" using path_connectedin_topspace by force lemma Union_path_components_of: "\(path_components_of X) = topspace X" by (auto simp: path_components_of_def path_component_of_equiv) lemma path_components_of_maximal: "\C \ path_components_of X; path_connectedin X S; ~disjnt C S\ \ S \ C" apply (auto simp: path_components_of_def path_component_of_equiv) using path_component_of_maximal path_connectedin_def apply fastforce by (meson disjnt_subset2 path_component_of_disjoint path_component_of_equiv path_component_of_maximal) lemma pairwise_disjoint_path_components_of: "pairwise disjnt (path_components_of X)" by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv) lemma complement_path_components_of_Union: "C \ path_components_of X \ topspace X - C = \(path_components_of X - {C})" by (metis Diff_cancel Diff_subset Union_path_components_of cSup_singleton diff_Union_pairwise_disjoint insert_subset pairwise_disjoint_path_components_of) lemma nonempty_path_components_of: assumes "C \ path_components_of X" shows "C \ {}" proof - have "C \ path_component_of_set X ` topspace X" using assms path_components_of_def by blast then show ?thesis using path_component_of_refl by fastforce qed lemma path_components_of_subset: "C \ path_components_of X \ C \ topspace X" by (auto simp: path_components_of_def path_component_of_equiv) lemma path_connectedin_path_components_of: "C \ path_components_of X \ path_connectedin X C" by (auto simp: path_components_of_def path_connectedin_path_component_of) lemma path_component_in_path_components_of: "Collect (path_component_of X a) \ path_components_of X \ a \ topspace X" by (metis imageI nonempty_path_components_of path_component_of_eq_empty path_components_of_def) lemma path_connectedin_Union: assumes \: "\S. S \ \ \ path_connectedin X S" "\\ \ {}" shows "path_connectedin X (\\)" proof - obtain a where "\S. S \ \ \ a \ S" using assms by blast then have "\x. x \ topspace (subtopology X (\\)) \ path_component_of (subtopology X (\\)) a x" by simp (meson Union_upper \ path_component_of path_connectedin_subtopology) then show ?thesis using \ unfolding path_connectedin_def by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component) qed lemma path_connectedin_Un: "\path_connectedin X S; path_connectedin X T; S \ T \ {}\ \ path_connectedin X (S \ T)" by (blast intro: path_connectedin_Union [of "{S,T}", simplified]) lemma path_connected_space_iff_components_eq: "path_connected_space X \ (\C \ path_components_of X. \C' \ path_components_of X. C = C')" unfolding path_components_of_def proof (intro iffI ballI) assume "\C \ path_component_of_set X ` topspace X. \C' \ path_component_of_set X ` topspace X. C = C'" then show "path_connected_space X" using path_component_of_refl path_connected_space_iff_path_component by fastforce qed (auto simp: path_connected_space_path_component_set) lemma path_components_of_eq_empty: "path_components_of X = {} \ topspace X = {}" using Union_path_components_of nonempty_path_components_of by fastforce lemma path_components_of_empty_space: "topspace X = {} \ path_components_of X = {}" by (simp add: path_components_of_eq_empty) lemma path_components_of_subset_singleton: "path_components_of X \ {S} \ path_connected_space X \ (topspace X = {} \ topspace X = S)" proof (cases "topspace X = {}") case True then show ?thesis by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty) next case False have "(path_components_of X = {S}) \ (path_connected_space X \ topspace X = S)" proof (intro iffI conjI) assume L: "path_components_of X = {S}" then show "path_connected_space X" by (simp add: path_connected_space_iff_components_eq) show "topspace X = S" by (metis L ccpo_Sup_singleton [of S] Union_path_components_of) next assume R: "path_connected_space X \ topspace X = S" then show "path_components_of X = {S}" using ccpo_Sup_singleton [of S] by (metis False all_not_in_conv insert_iff mk_disjoint_insert path_component_in_path_components_of path_connected_space_iff_components_eq path_connected_space_path_component_set) qed with False show ?thesis by (simp add: path_components_of_eq_empty subset_singleton_iff) qed lemma path_connected_space_iff_components_subset_singleton: "path_connected_space X \ (\a. path_components_of X \ {a})" by (simp add: path_components_of_subset_singleton) lemma path_components_of_eq_singleton: "path_components_of X = {S} \ path_connected_space X \ topspace X \ {} \ S = topspace X" by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff) lemma path_components_of_path_connected_space: "path_connected_space X \ path_components_of X = (if topspace X = {} then {} else {topspace X})" by (simp add: path_components_of_eq_empty path_components_of_eq_singleton) lemma path_component_subset_connected_component_of: "path_component_of_set X x \ connected_component_of_set X x" proof (cases "x \ topspace X") case True then show ?thesis by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of) next case False then show ?thesis using path_component_of_eq_empty by fastforce qed lemma exists_path_component_of_superset: assumes S: "path_connectedin X S" and ne: "topspace X \ {}" obtains C where "C \ path_components_of X" "S \ C" proof (cases "S = {}") case True then show ?thesis using ne path_components_of_eq_empty that by fastforce next case False then obtain a where "a \ S" by blast show ?thesis proof show "Collect (path_component_of X a) \ path_components_of X" by (meson \a \ S\ S subsetD path_component_in_path_components_of path_connectedin_subset_topspace) show "S \ Collect (path_component_of X a)" by (simp add: S \a \ S\ path_component_of_maximal) qed qed lemma path_component_of_eq_overlap: "path_component_of X x = path_component_of X y \ (x \ topspace X) \ (y \ topspace X) \ Collect (path_component_of X x) \ Collect (path_component_of X y) \ {}" by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty) lemma path_component_of_nonoverlap: "Collect (path_component_of X x) \ Collect (path_component_of X y) = {} \ (x \ topspace X) \ (y \ topspace X) \ path_component_of X x \ path_component_of X y" by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap) lemma path_component_of_overlap: "Collect (path_component_of X x) \ Collect (path_component_of X y) \ {} \ x \ topspace X \ y \ topspace X \ path_component_of X x = path_component_of X y" by (meson path_component_of_nonoverlap) lemma path_components_of_disjoint: "\C \ path_components_of X; C' \ path_components_of X\ \ disjnt C C' \ C \ C'" by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv) lemma path_components_of_overlap: "\C \ path_components_of X; C' \ path_components_of X\ \ C \ C' \ {} \ C = C'" by (auto simp: path_components_of_def path_component_of_equiv) lemma path_component_of_unique: "\x \ C; path_connectedin X C; \C'. \x \ C'; path_connectedin X C'\ \ C' \ C\ \ Collect (path_component_of X x) = C" by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of) lemma path_component_of_discrete_topology [simp]: "Collect (path_component_of (discrete_topology U) x) = (if x \ U then {x} else {})" proof - have "\C'. \x \ C'; path_connectedin (discrete_topology U) C'\ \ C' \ {x}" by (metis path_connectedin_discrete_topology subsetD singletonD) then have "x \ U \ Collect (path_component_of (discrete_topology U) x) = {x}" by (simp add: path_component_of_unique) then show ?thesis using path_component_in_topspace by fastforce qed lemma path_component_of_discrete_topology_iff [simp]: "path_component_of (discrete_topology U) x y \ x \ U \ y=x" by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD) lemma path_components_of_discrete_topology [simp]: "path_components_of (discrete_topology U) = (\x. {x}) ` U" by (auto simp: path_components_of_def image_def fun_eq_iff) lemma homeomorphic_map_path_component_of: assumes f: "homeomorphic_map X Y f" and x: "x \ topspace X" shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)" proof - obtain g where g: "homeomorphic_maps X Y f g" using f homeomorphic_map_maps by blast show ?thesis proof have "Collect (path_component_of Y (f x)) \ topspace Y" by (simp add: path_component_of_subset_topspace) moreover have "g ` Collect(path_component_of Y (f x)) \ Collect (path_component_of X (g (f x)))" using g x unfolding homeomorphic_maps_def by (metis f homeomorphic_imp_surjective_map imageI mem_Collect_eq path_component_of_maximal path_component_of_refl path_connectedin_continuous_map_image path_connectedin_path_component_of) ultimately show "Collect (path_component_of Y (f x)) \ f ` Collect (path_component_of X x)" using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff by metis show "f ` Collect (path_component_of X x) \ Collect (path_component_of Y (f x))" proof (rule path_component_of_maximal) show "path_connectedin Y (f ` Collect (path_component_of X x))" by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of) qed (simp add: path_component_of_refl x) qed qed lemma homeomorphic_map_path_components_of: assumes "homeomorphic_map X Y f" shows "path_components_of Y = (image f) ` (path_components_of X)" (is "?lhs = ?rhs") unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric] using assms homeomorphic_map_path_component_of by fastforce subsection \Sphere is path-connected\ lemma path_connected_punctured_universe: assumes "2 \ DIM('a::euclidean_space)" shows "path_connected (- {a::'a})" proof - let ?A = "{x::'a. \i\Basis. x \ i < a \ i}" let ?B = "{x::'a. \i\Basis. a \ i < x \ i}" have A: "path_connected ?A" unfolding Collect_bex_eq proof (rule path_connected_UNION) fix i :: 'a assume "i \ Basis" then show "(\i\Basis. (a \ i - 1)*\<^sub>R i) \ {x::'a. x \ i < a \ i}" by simp show "path_connected {x. x \ i < a \ i}" using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \ i"] by (simp add: inner_commute) qed have B: "path_connected ?B" unfolding Collect_bex_eq proof (rule path_connected_UNION) fix i :: 'a assume "i \ Basis" then show "(\i\Basis. (a \ i + 1) *\<^sub>R i) \ {x::'a. a \ i < x \ i}" by simp show "path_connected {x. a \ i < x \ i}" using convex_imp_path_connected [OF convex_halfspace_gt, of "a \ i" i] by (simp add: inner_commute) qed obtain S :: "'a set" where "S \ Basis" and "card S = Suc (Suc 0)" using ex_card[OF assms] by auto then obtain b0 b1 :: 'a where "b0 \ Basis" and "b1 \ Basis" and "b0 \ b1" unfolding card_Suc_eq by auto then have "a + b0 - b1 \ ?A \ ?B" by (auto simp: inner_simps inner_Basis) then have "?A \ ?B \ {}" by fast with A B have "path_connected (?A \ ?B)" by (rule path_connected_Un) also have "?A \ ?B = {x. \i\Basis. x \ i \ a \ i}" unfolding neq_iff bex_disj_distrib Collect_disj_eq .. also have "\ = {x. x \ a}" unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def) also have "\ = - {a}" by auto finally show ?thesis . qed corollary connected_punctured_universe: "2 \ DIM('N::euclidean_space) \ connected(- {a::'N})" by (simp add: path_connected_punctured_universe path_connected_imp_connected) proposition path_connected_sphere: fixes a :: "'a :: euclidean_space" assumes "2 \ DIM('a)" shows "path_connected(sphere a r)" proof (cases r "0::real" rule: linorder_cases) case less then show ?thesis by (simp) next case equal then show ?thesis by (simp) next case greater then have eq: "(sphere (0::'a) r) = (\x. (r / norm x) *\<^sub>R x) ` (- {0::'a})" by (force simp: image_iff split: if_split_asm) have "continuous_on (- {0::'a}) (\x. (r / norm x) *\<^sub>R x)" by (intro continuous_intros) auto then have "path_connected ((\x. (r / norm x) *\<^sub>R x) ` (- {0::'a}))" by (intro path_connected_continuous_image path_connected_punctured_universe assms) with eq have "path_connected (sphere (0::'a) r)" by auto then have "path_connected((+) a ` (sphere (0::'a) r))" by (simp add: path_connected_translation) then show ?thesis by (metis add.right_neutral sphere_translation) qed lemma connected_sphere: fixes a :: "'a :: euclidean_space" assumes "2 \ DIM('a)" shows "connected(sphere a r)" using path_connected_sphere [OF assms] by (simp add: path_connected_imp_connected) corollary path_connected_complement_bounded_convex: fixes S :: "'a :: euclidean_space set" assumes "bounded S" "convex S" and 2: "2 \ DIM('a)" shows "path_connected (- S)" proof (cases "S = {}") case True then show ?thesis using convex_imp_path_connected by auto next case False then obtain a where "a \ S" by auto have \
[rule_format]: "\y\S. \u. 0 \ u \ u \ 1 \ (1 - u) *\<^sub>R a + u *\<^sub>R y \ S" using \convex S\ \a \ S\ by (simp add: convex_alt) { fix x y assume "x \ S" "y \ S" then have "x \ a" "y \ a" using \a \ S\ by auto then have bxy: "bounded(insert x (insert y S))" by (simp add: \bounded S\) then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B" and "S \ ball a B" using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm) define C where "C = B / norm(x - a)" let ?Cxa = "a + C *\<^sub>R (x - a)" { fix u assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R ?Cxa \ S" and "0 \ u" "u \ 1" have CC: "1 \ 1 + (C - 1) * u" using \x \ a\ \0 \ u\ Bx by (auto simp add: C_def norm_minus_commute) have *: "\v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)" by (simp add: algebra_simps) have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) = (1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x" by (simp add: algebra_simps) also have "\ = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x" using CC by (simp add: field_simps) also have "\ = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x" by (simp add: algebra_simps) also have "\ = x + ((1 / (1 + C * u - u)) *\<^sub>R a + ((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))" using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x" by (simp add: algebra_simps) have False using \
[of "a + (1 + (C - 1) * u) *\<^sub>R (x - a)" "1 / (1 + (C - 1) * u)"] using u \x \ a\ \x \ S\ \0 \ u\ CC by (auto simp: xeq *) } then have pcx: "path_component (- S) x ?Cxa" by (force simp: closed_segment_def intro!: path_component_linepath) define D where "D = B / norm(y - a)" \ \massive duplication with the proof above\ let ?Dya = "a + D *\<^sub>R (y - a)" { fix u assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R ?Dya \ S" and "0 \ u" "u \ 1" have DD: "1 \ 1 + (D - 1) * u" using \y \ a\ \0 \ u\ By by (auto simp add: D_def norm_minus_commute) have *: "\v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)" by (simp add: algebra_simps) have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) = (1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y" by (simp add: algebra_simps) also have "\ = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y" using DD by (simp add: field_simps) also have "\ = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y" by (simp add: algebra_simps) also have "\ = y + ((1 / (1 + D * u - u)) *\<^sub>R a + ((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))" using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y" by (simp add: algebra_simps) have False using \
[of "a + (1 + (D - 1) * u) *\<^sub>R (y - a)" "1 / (1 + (D - 1) * u)"] using u \y \ a\ \y \ S\ \0 \ u\ DD by (auto simp: xeq *) } then have pdy: "path_component (- S) y ?Dya" by (force simp: closed_segment_def intro!: path_component_linepath) have pyx: "path_component (- S) ?Dya ?Cxa" proof (rule path_component_of_subset) show "sphere a B \ - S" using \S \ ball a B\ by (force simp: ball_def dist_norm norm_minus_commute) have aB: "?Dya \ sphere a B" "?Cxa \ sphere a B" using \x \ a\ using \y \ a\ B by (auto simp: dist_norm C_def D_def) then show "path_component (sphere a B) ?Dya ?Cxa" using path_connected_sphere [OF 2] path_connected_component by blast qed have "path_component (- S) x y" by (metis path_component_trans path_component_sym pcx pdy pyx) } then show ?thesis by (auto simp: path_connected_component) qed lemma connected_complement_bounded_convex: fixes S :: "'a :: euclidean_space set" assumes "bounded S" "convex S" "2 \ DIM('a)" shows "connected (- S)" using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast lemma connected_diff_ball: fixes S :: "'a :: euclidean_space set" assumes "connected S" "cball a r \ S" "2 \ DIM('a)" shows "connected (S - ball a r)" proof (rule connected_diff_open_from_closed [OF ball_subset_cball]) show "connected (cball a r - ball a r)" using assms connected_sphere by (auto simp: cball_diff_eq_sphere) qed (auto simp: assms dist_norm) proposition connected_open_delete: assumes "open S" "connected S" and 2: "2 \ DIM('N::euclidean_space)" shows "connected(S - {a::'N})" proof (cases "a \ S") case True with \open S\ obtain \ where "\ > 0" and \: "cball a \ \ S" using open_contains_cball_eq by blast define b where "b \ a + \ *\<^sub>R (SOME i. i \ Basis)" have "dist a b = \" by (simp add: b_def dist_norm SOME_Basis \0 < \\ less_imp_le) with \ have "b \ \{S - ball a r |r. 0 < r \ r < \}" by auto then have nonemp: "(\{S - ball a r |r. 0 < r \ r < \}) = {} \ False" by auto have con: "\r. r < \ \ connected (S - ball a r)" using \ by (force intro: connected_diff_ball [OF \connected S\ _ 2]) have "x \ \{S - ball a r |r. 0 < r \ r < \}" if "x \ S - {a}" for x using that \0 < \\ by (intro UnionI [of "S - ball a (min \ (dist a x) / 2)"]) auto then have "S - {a} = \{S - ball a r | r. 0 < r \ r < \}" by auto then show ?thesis by (auto intro: connected_Union con dest!: nonemp) next case False then show ?thesis by (simp add: \connected S\) qed corollary path_connected_open_delete: assumes "open S" "connected S" and 2: "2 \ DIM('N::euclidean_space)" shows "path_connected(S - {a::'N})" by (simp add: assms connected_open_delete connected_open_path_connected open_delete) corollary path_connected_punctured_ball: "2 \ DIM('N::euclidean_space) \ path_connected(ball a r - {a::'N})" by (simp add: path_connected_open_delete) corollary connected_punctured_ball: "2 \ DIM('N::euclidean_space) \ connected(ball a r - {a::'N})" by (simp add: connected_open_delete) corollary connected_open_delete_finite: fixes S T::"'a::euclidean_space set" assumes S: "open S" "connected S" and 2: "2 \ DIM('a)" and "finite T" shows "connected(S - T)" using \finite T\ S proof (induct T) case empty show ?case using \connected S\ by simp next case (insert x F) then have "connected (S-F)" by auto moreover have "open (S - F)" using finite_imp_closed[OF \finite F\] \open S\ by auto ultimately have "connected (S - F - {x})" using connected_open_delete[OF _ _ 2] by auto thus ?case by (metis Diff_insert) qed lemma sphere_1D_doubleton_zero: assumes 1: "DIM('a) = 1" and "r > 0" obtains x y::"'a::euclidean_space" where "sphere 0 r = {x,y} \ dist x y = 2*r" proof - obtain b::'a where b: "Basis = {b}" using 1 card_1_singletonE by blast show ?thesis proof (intro that conjI) have "x = norm x *\<^sub>R b \ x = - norm x *\<^sub>R b" if "r = norm x" for x proof - have xb: "(x \ b) *\<^sub>R b = x" using euclidean_representation [of x, unfolded b] by force then have "norm ((x \ b) *\<^sub>R b) = norm x" by simp with b have "\x \ b\ = norm x" using norm_Basis by (simp add: b) with xb show ?thesis by (metis (mono_tags, opaque_lifting) abs_eq_iff abs_norm_cancel) qed with \r > 0\ b show "sphere 0 r = {r *\<^sub>R b, - r *\<^sub>R b}" by (force simp: sphere_def dist_norm) have "dist (r *\<^sub>R b) (- r *\<^sub>R b) = norm (r *\<^sub>R b + r *\<^sub>R b)" by (simp add: dist_norm) also have "\ = norm ((2*r) *\<^sub>R b)" by (metis mult_2 scaleR_add_left) also have "\ = 2*r" using \r > 0\ b norm_Basis by fastforce finally show "dist (r *\<^sub>R b) (- r *\<^sub>R b) = 2*r" . qed qed lemma sphere_1D_doubleton: fixes a :: "'a :: euclidean_space" assumes "DIM('a) = 1" and "r > 0" obtains x y where "sphere a r = {x,y} \ dist x y = 2*r" proof - have "sphere a r = (+) a ` sphere 0 r" by (metis add.right_neutral sphere_translation) then show ?thesis using sphere_1D_doubleton_zero [OF assms] by (metis (mono_tags, lifting) dist_add_cancel image_empty image_insert that) qed lemma psubset_sphere_Compl_connected: fixes S :: "'a::euclidean_space set" assumes S: "S \ sphere a r" and "0 < r" and 2: "2 \ DIM('a)" shows "connected(- S)" proof - have "S \ sphere a r" using S by blast obtain b where "dist a b = r" and "b \ S" using S mem_sphere by blast have CS: "- S = {x. dist a x \ r \ (x \ S)} \ {x. r \ dist a x \ (x \ S)}" by auto have "{x. dist a x \ r \ x \ S} \ {x. r \ dist a x \ x \ S} \ {}" using \b \ S\ \dist a b = r\ by blast moreover have "connected {x. dist a x \ r \ x \ S}" using assms by (force intro: connected_intermediate_closure [of "ball a r"]) moreover have "connected {x. r \ dist a x \ x \ S}" proof (rule connected_intermediate_closure [of "- cball a r"]) show "{x. r \ dist a x \ x \ S} \ closure (- cball a r)" using interior_closure by (force intro: connected_complement_bounded_convex) qed (use assms connected_complement_bounded_convex in auto) ultimately show ?thesis by (simp add: CS connected_Un) qed subsection\Every annulus is a connected set\ lemma path_connected_2DIM_I: fixes a :: "'N::euclidean_space" assumes 2: "2 \ DIM('N)" and pc: "path_connected {r. 0 \ r \ P r}" shows "path_connected {x. P(norm(x - a))}" proof - have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}" by force moreover have "path_connected {x::'N. P(norm x)}" proof - let ?D = "{x. 0 \ x \ P x} \ sphere (0::'N) 1" have "x \ (\z. fst z *\<^sub>R snd z) ` ?D" if "P (norm x)" for x::'N proof (cases "x=0") case True with that show ?thesis apply (simp add: image_iff) by (metis (no_types) mem_sphere_0 order_refl vector_choose_size zero_le_one) next case False with that show ?thesis by (rule_tac x="(norm x, x /\<^sub>R norm x)" in image_eqI) auto qed then have *: "{x::'N. P(norm x)} = (\z. fst z *\<^sub>R snd z) ` ?D" by auto have "continuous_on ?D (\z:: real\'N. fst z *\<^sub>R snd z)" by (intro continuous_intros) moreover have "path_connected ?D" by (metis path_connected_Times [OF pc] path_connected_sphere 2) ultimately show ?thesis by (simp add: "*" path_connected_continuous_image) qed ultimately show ?thesis using path_connected_translation by metis qed proposition path_connected_annulus: fixes a :: "'N::euclidean_space" assumes "2 \ DIM('N)" shows "path_connected {x. r1 < norm(x - a) \ norm(x - a) < r2}" "path_connected {x. r1 < norm(x - a) \ norm(x - a) \ r2}" "path_connected {x. r1 \ norm(x - a) \ norm(x - a) < r2}" "path_connected {x. r1 \ norm(x - a) \ norm(x - a) \ r2}" by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms]) proposition connected_annulus: fixes a :: "'N::euclidean_space" assumes "2 \ DIM('N::euclidean_space)" shows "connected {x. r1 < norm(x - a) \ norm(x - a) < r2}" "connected {x. r1 < norm(x - a) \ norm(x - a) \ r2}" "connected {x. r1 \ norm(x - a) \ norm(x - a) < r2}" "connected {x. r1 \ norm(x - a) \ norm(x - a) \ r2}" by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected) subsection\<^marker>\tag unimportant\\Relations between components and path components\ lemma open_connected_component: fixes S :: "'a::real_normed_vector set" assumes "open S" shows "open (connected_component_set S x)" proof (clarsimp simp: open_contains_ball) fix y assume xy: "connected_component S x y" then obtain e where "e>0" "ball y e \ S" using assms connected_component_in openE by blast then show "\e>0. ball y e \ connected_component_set S x" by (metis xy centre_in_ball connected_ball connected_component_eq_eq connected_component_in connected_component_maximal) qed corollary open_components: fixes S :: "'a::real_normed_vector set" shows "\open u; S \ components u\ \ open S" by (simp add: components_iff) (metis open_connected_component) lemma in_closure_connected_component: fixes S :: "'a::real_normed_vector set" assumes x: "x \ S" and S: "open S" shows "x \ closure (connected_component_set S y) \ x \ connected_component_set S y" proof - { assume "x \ closure (connected_component_set S y)" moreover have "x \ connected_component_set S x" using x by simp ultimately have "x \ connected_component_set S y" using S by (meson Compl_disjoint closure_iff_nhds_not_empty connected_component_disjoint disjoint_eq_subset_Compl open_connected_component) } then show ?thesis by (auto simp: closure_def) qed lemma connected_disjoint_Union_open_pick: assumes "pairwise disjnt B" "\S. S \ A \ connected S \ S \ {}" "\S. S \ B \ open S" "\A \ \B" "S \ A" obtains T where "T \ B" "S \ T" "S \ \(B - {T}) = {}" proof - have "S \ \B" "connected S" "S \ {}" using assms \S \ A\ by blast+ then obtain T where "T \ B" "S \ T \ {}" by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute) have 1: "open T" by (simp add: \T \ B\ assms) have 2: "open (\(B-{T}))" using assms by blast have 3: "S \ T \ \(B - {T})" using \S \ \B\ by blast have "T \ \(B - {T}) = {}" using \T \ B\ \pairwise disjnt B\ by (auto simp: pairwise_def disjnt_def) then have 4: "T \ \(B - {T}) \ S = {}" by auto from connectedD [OF \connected S\ 1 2 4 3] have "S \ \(B-{T}) = {}" by (auto simp: Int_commute \S \ T \ {}\) with \T \ B\ have "S \ T" using "3" by auto show ?thesis using \S \ \(B - {T}) = {}\ \S \ T\ \T \ B\ that by auto qed lemma connected_disjoint_Union_open_subset: assumes A: "pairwise disjnt A" and B: "pairwise disjnt B" and SA: "\S. S \ A \ open S \ connected S \ S \ {}" and SB: "\S. S \ B \ open S \ connected S \ S \ {}" and eq [simp]: "\A = \B" shows "A \ B" proof fix S assume "S \ A" obtain T where "T \ B" "S \ T" "S \ \(B - {T}) = {}" using SA SB \S \ A\ connected_disjoint_Union_open_pick [OF B, of A] eq order_refl by blast moreover obtain S' where "S' \ A" "T \ S'" "T \ \(A - {S'}) = {}" using SA SB \T \ B\ connected_disjoint_Union_open_pick [OF A, of B] eq order_refl by blast ultimately have "S' = S" by (metis A Int_subset_iff SA \S \ A\ disjnt_def inf.orderE pairwise_def) with \T \ S'\ have "T \ S" by simp with \S \ T\ have "S = T" by blast with \T \ B\ show "S \ B" by simp qed lemma connected_disjoint_Union_open_unique: assumes A: "pairwise disjnt A" and B: "pairwise disjnt B" and SA: "\S. S \ A \ open S \ connected S \ S \ {}" and SB: "\S. S \ B \ open S \ connected S \ S \ {}" and eq [simp]: "\A = \B" shows "A = B" by (rule subset_antisym; metis connected_disjoint_Union_open_subset assms) proposition components_open_unique: fixes S :: "'a::real_normed_vector set" assumes "pairwise disjnt A" "\A = S" "\X. X \ A \ open X \ connected X \ X \ {}" shows "components S = A" proof - have "open S" using assms by blast show ?thesis proof (rule connected_disjoint_Union_open_unique) show "disjoint (components S)" by (simp add: components_eq disjnt_def pairwise_def) qed (use \open S\ in \simp_all add: assms open_components in_components_connected in_components_nonempty\) qed subsection\<^marker>\tag unimportant\\Existence of unbounded components\ lemma cobounded_unbounded_component: fixes S :: "'a :: euclidean_space set" assumes "bounded (-S)" shows "\x. x \ S \ \ bounded (connected_component_set S x)" proof - obtain i::'a where i: "i \ Basis" using nonempty_Basis by blast obtain B where B: "B>0" "-S \ ball 0 B" using bounded_subset_ballD [OF assms, of 0] by auto then have *: "\x. B \ norm x \ x \ S" by (force simp: ball_def dist_norm) have unbounded_inner: "\ bounded {x. inner i x \ B}" proof (clarsimp simp: bounded_def dist_norm) fix e x show "\y. B \ i \ y \ \ norm (x - y) \ e" using i by (rule_tac x="x + (max B e + 1 + \i \ x\) *\<^sub>R i" in exI) (auto simp: inner_right_distrib) qed have \
: "\x. B \ i \ x \ x \ S" using * Basis_le_norm [OF i] by (metis abs_ge_self inner_commute order_trans) have "{x. B \ i \ x} \ connected_component_set S (B *\<^sub>R i)" by (intro connected_component_maximal) (auto simp: i intro: convex_connected convex_halfspace_ge [of B] \
) then have "\ bounded (connected_component_set S (B *\<^sub>R i))" using bounded_subset unbounded_inner by blast moreover have "B *\<^sub>R i \ S" by (rule *) (simp add: norm_Basis [OF i]) ultimately show ?thesis by blast qed lemma cobounded_unique_unbounded_component: fixes S :: "'a :: euclidean_space set" assumes bs: "bounded (-S)" and "2 \ DIM('a)" and bo: "\ bounded(connected_component_set S x)" "\ bounded(connected_component_set S y)" shows "connected_component_set S x = connected_component_set S y" proof - obtain i::'a where i: "i \ Basis" using nonempty_Basis by blast obtain B where B: "B>0" "-S \ ball 0 B" using bounded_subset_ballD [OF bs, of 0] by auto then have *: "\x. B \ norm x \ x \ S" by (force simp: ball_def dist_norm) obtain x' where x': "connected_component S x x'" "norm x' > B" using bo [unfolded bounded_def dist_norm, simplified, rule_format] by (metis diff_zero norm_minus_commute not_less) obtain y' where y': "connected_component S y y'" "norm y' > B" using bo [unfolded bounded_def dist_norm, simplified, rule_format] by (metis diff_zero norm_minus_commute not_less) have x'y': "connected_component S x' y'" unfolding connected_component_def proof (intro exI conjI) show "connected (- ball 0 B :: 'a set)" using assms by (auto intro: connected_complement_bounded_convex) qed (use x' y' dist_norm * in auto) show ?thesis proof (rule connected_component_eq) show "x \ connected_component_set S y" using x' y' x'y' by (metis (no_types) connected_component_eq_eq connected_component_in mem_Collect_eq) qed qed lemma cobounded_unbounded_components: fixes S :: "'a :: euclidean_space set" shows "bounded (-S) \ \c. c \ components S \ \bounded c" by (metis cobounded_unbounded_component components_def imageI) lemma cobounded_unique_unbounded_components: fixes S :: "'a :: euclidean_space set" shows "\bounded (- S); c \ components S; \ bounded c; c' \ components S; \ bounded c'; 2 \ DIM('a)\ \ c' = c" unfolding components_iff by (metis cobounded_unique_unbounded_component) lemma cobounded_has_bounded_component: fixes S :: "'a :: euclidean_space set" assumes "bounded (- S)" "\ connected S" "2 \ DIM('a)" obtains C where "C \ components S" "bounded C" by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms) subsection\The \inside\ and \outside\ of a Set\ text\<^marker>\tag important\\The inside comprises the points in a bounded connected component of the set's complement. The outside comprises the points in unbounded connected component of the complement.\ definition\<^marker>\tag important\ inside where "inside S \ {x. (x \ S) \ bounded(connected_component_set ( - S) x)}" definition\<^marker>\tag important\ outside where "outside S \ -S \ {x. \ bounded(connected_component_set (- S) x)}" lemma outside: "outside S = {x. \ bounded(connected_component_set (- S) x)}" by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty) lemma inside_no_overlap [simp]: "inside S \ S = {}" by (auto simp: inside_def) lemma outside_no_overlap [simp]: "outside S \ S = {}" by (auto simp: outside_def) lemma inside_Int_outside [simp]: "inside S \ outside S = {}" by (auto simp: inside_def outside_def) lemma inside_Un_outside [simp]: "inside S \ outside S = (- S)" by (auto simp: inside_def outside_def) lemma inside_eq_outside: "inside S = outside S \ S = UNIV" by (auto simp: inside_def outside_def) lemma inside_outside: "inside S = (- (S \ outside S))" by (force simp: inside_def outside) lemma outside_inside: "outside S = (- (S \ inside S))" by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap) lemma union_with_inside: "S \ inside S = - outside S" by (auto simp: inside_outside) (simp add: outside_inside) lemma union_with_outside: "S \ outside S = - inside S" by (simp add: inside_outside) lemma outside_mono: "S \ T \ outside T \ outside S" by (auto simp: outside bounded_subset connected_component_mono) lemma inside_mono: "S \ T \ inside S - T \ inside T" by (auto simp: inside_def bounded_subset connected_component_mono) lemma segment_bound_lemma: fixes u::real assumes "x \ B" "y \ B" "0 \ u" "u \ 1" shows "(1 - u) * x + u * y \ B" proof - obtain dx dy where "dx \ 0" "dy \ 0" "x = B + dx" "y = B + dy" using assms by auto (metis add.commute diff_add_cancel) with \0 \ u\ \u \ 1\ show ?thesis by (simp add: add_increasing2 mult_left_le field_simps) qed lemma cobounded_outside: fixes S :: "'a :: real_normed_vector set" assumes "bounded S" shows "bounded (- outside S)" proof - obtain B where B: "B>0" "S \ ball 0 B" using bounded_subset_ballD [OF assms, of 0] by auto { fix x::'a and C::real assume Bno: "B \ norm x" and C: "0 < C" have "\y. connected_component (- S) x y \ norm y > C" proof (cases "x = 0") case True with B Bno show ?thesis by force next case False have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \ - ball 0 B" proof fix w assume "w \ closed_segment x (((B + C) / norm x) *\<^sub>R x)" then obtain u where w: "w = (1 - u + u * (B + C) / norm x) *\<^sub>R x" "0 \ u" "u \ 1" by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric]) with False B C have "B \ (1 - u) * norm x + u * (B + C)" using segment_bound_lemma [of B "norm x" "B + C" u] Bno by simp with False B C show "w \ - ball 0 B" using distrib_right [of _ _ "norm x"] by (simp add: ball_def w not_less) qed also have "... \ -S" by (simp add: B) finally have "\T. connected T \ T \ - S \ x \ T \ ((B + C) / norm x) *\<^sub>R x \ T" by (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) simp with False B show ?thesis by (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) (simp add: connected_component_def) qed } then show ?thesis apply (simp add: outside_def assms) apply (rule bounded_subset [OF bounded_ball [of 0 B]]) apply (force simp: dist_norm not_less bounded_pos) done qed lemma unbounded_outside: fixes S :: "'a::{real_normed_vector, perfect_space} set" shows "bounded S \ \ bounded(outside S)" using cobounded_imp_unbounded cobounded_outside by blast lemma bounded_inside: fixes S :: "'a::{real_normed_vector, perfect_space} set" shows "bounded S \ bounded(inside S)" by (simp add: bounded_Int cobounded_outside inside_outside) lemma connected_outside: fixes S :: "'a::euclidean_space set" assumes "bounded S" "2 \ DIM('a)" shows "connected(outside S)" apply (clarsimp simp add: connected_iff_connected_component outside) apply (rule_tac S="connected_component_set (- S) x" in connected_component_of_subset) apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq) by (simp add: Collect_mono connected_component_eq) lemma outside_connected_component_lt: "outside S = {x. \B. \y. B < norm(y) \ connected_component (- S) x y}" apply (auto simp: outside bounded_def dist_norm) apply (metis diff_0 norm_minus_cancel not_less) by (metis less_diff_eq norm_minus_commute norm_triangle_ineq2 order.trans pinf(6)) lemma outside_connected_component_le: "outside S = {x. \B. \y. B \ norm(y) \ connected_component (- S) x y}" apply (simp add: outside_connected_component_lt Set.set_eq_iff) by (meson gt_ex leD le_less_linear less_imp_le order.trans) lemma not_outside_connected_component_lt: fixes S :: "'a::euclidean_space set" assumes S: "bounded S" and "2 \ DIM('a)" shows "- (outside S) = {x. \B. \y. B < norm(y) \ \ connected_component (- S) x y}" proof - obtain B::real where B: "0 < B" and Bno: "\x. x \ S \ norm x \ B" using S [simplified bounded_pos] by auto { fix y::'a and z::'a assume yz: "B < norm z" "B < norm y" have "connected_component (- cball 0 B) y z" using assms yz by (force simp: dist_norm intro: connected_componentI [OF _ subset_refl] connected_complement_bounded_convex) then have "connected_component (- S) y z" by (metis connected_component_of_subset Bno Compl_anti_mono mem_cball_0 subset_iff) } note cyz = this show ?thesis apply (auto simp: outside bounded_pos) apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le) by (metis B connected_component_trans cyz not_le) qed lemma not_outside_connected_component_le: fixes S :: "'a::euclidean_space set" assumes S: "bounded S" "2 \ DIM('a)" shows "- (outside S) = {x. \B. \y. B \ norm(y) \ \ connected_component (- S) x y}" apply (auto intro: less_imp_le simp: not_outside_connected_component_lt [OF assms]) by (meson gt_ex less_le_trans) lemma inside_connected_component_lt: fixes S :: "'a::euclidean_space set" assumes S: "bounded S" "2 \ DIM('a)" shows "inside S = {x. (x \ S) \ (\B. \y. B < norm(y) \ \ connected_component (- S) x y)}" by (auto simp: inside_outside not_outside_connected_component_lt [OF assms]) lemma inside_connected_component_le: fixes S :: "'a::euclidean_space set" assumes S: "bounded S" "2 \ DIM('a)" shows "inside S = {x. (x \ S) \ (\B. \y. B \ norm(y) \ \ connected_component (- S) x y)}" by (auto simp: inside_outside not_outside_connected_component_le [OF assms]) lemma inside_subset: assumes "connected U" and "\ bounded U" and "T \ U = - S" shows "inside S \ T" apply (auto simp: inside_def) by (metis bounded_subset [of "connected_component_set (- S) _"] connected_component_maximal Compl_iff Un_iff assms subsetI) lemma frontier_not_empty: fixes S :: "'a :: real_normed_vector set" shows "\S \ {}; S \ UNIV\ \ frontier S \ {}" using connected_Int_frontier [of UNIV S] by auto lemma frontier_eq_empty: fixes S :: "'a :: real_normed_vector set" shows "frontier S = {} \ S = {} \ S = UNIV" using frontier_UNIV frontier_empty frontier_not_empty by blast lemma frontier_of_connected_component_subset: fixes S :: "'a::real_normed_vector set" shows "frontier(connected_component_set S x) \ frontier S" proof - { fix y assume y1: "y \ closure (connected_component_set S x)" and y2: "y \ interior (connected_component_set S x)" have "y \ closure S" using y1 closure_mono connected_component_subset by blast moreover have "z \ interior (connected_component_set S x)" if "0 < e" "ball y e \ interior S" "dist y z < e" for e z proof - have "ball y e \ connected_component_set S y" using connected_component_maximal that interior_subset by (metis centre_in_ball connected_ball subset_trans) then show ?thesis using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior]) by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \0 < e\ y2) qed then have "y \ interior S" using y2 by (force simp: open_contains_ball_eq [OF open_interior]) ultimately have "y \ frontier S" by (auto simp: frontier_def) } then show ?thesis by (auto simp: frontier_def) qed lemma frontier_Union_subset_closure: fixes F :: "'a::real_normed_vector set set" shows "frontier(\F) \ closure(\t \ F. frontier t)" proof - have "\y\F. \y\frontier y. dist y x < e" if "T \ F" "y \ T" "dist y x < e" "x \ interior (\F)" "0 < e" for x y e T proof (cases "x \ T") case True with that show ?thesis by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono) next case False have 1: "closed_segment x y \ T \ {}" using \y \ T\ by blast have 2: "closed_segment x y - T \ {}" using False by blast obtain c where "c \ closed_segment x y" "c \ frontier T" using False connected_Int_frontier [OF connected_segment 1 2] by auto then show ?thesis proof - have "norm (y - x) < e" by (metis dist_norm \dist y x < e\) moreover have "norm (c - x) \ norm (y - x)" by (simp add: \c \ closed_segment x y\ segment_bound(1)) ultimately have "norm (c - x) < e" by linarith then show ?thesis by (metis (no_types) \c \ frontier T\ dist_norm that(1)) qed qed then show ?thesis by (fastforce simp add: frontier_def closure_approachable) qed lemma frontier_Union_subset: fixes F :: "'a::real_normed_vector set set" shows "finite F \ frontier(\F) \ (\t \ F. frontier t)" by (rule order_trans [OF frontier_Union_subset_closure]) (auto simp: closure_subset_eq) lemma frontier_of_components_subset: fixes S :: "'a::real_normed_vector set" shows "C \ components S \ frontier C \ frontier S" by (metis Path_Connected.frontier_of_connected_component_subset components_iff) lemma frontier_of_components_closed_complement: fixes S :: "'a::real_normed_vector set" shows "\closed S; C \ components (- S)\ \ frontier C \ S" using frontier_complement frontier_of_components_subset frontier_subset_eq by blast lemma frontier_minimal_separating_closed: fixes S :: "'a::real_normed_vector set" assumes "closed S" and nconn: "\ connected(- S)" and C: "C \ components (- S)" and conn: "\T. \closed T; T \ S\ \ connected(- T)" shows "frontier C = S" proof (rule ccontr) assume "frontier C \ S" then have "frontier C \ S" using frontier_of_components_closed_complement [OF \closed S\ C] by blast then have "connected(- (frontier C))" by (simp add: conn) have "\ connected(- (frontier C))" unfolding connected_def not_not proof (intro exI conjI) show "open C" using C \closed S\ open_components by blast show "open (- closure C)" by blast show "C \ - closure C \ - frontier C = {}" using closure_subset by blast show "C \ - frontier C \ {}" using C \open C\ components_eq frontier_disjoint_eq by fastforce show "- frontier C \ C \ - closure C" by (simp add: \open C\ closed_Compl frontier_closures) then show "- closure C \ - frontier C \ {}" by (metis (no_types, lifting) C Compl_subset_Compl_iff \frontier C \ S\ compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb) qed then show False using \connected (- frontier C)\ by blast qed lemma connected_component_UNIV [simp]: fixes x :: "'a::real_normed_vector" shows "connected_component_set UNIV x = UNIV" using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV by auto lemma connected_component_eq_UNIV: fixes x :: "'a::real_normed_vector" shows "connected_component_set s x = UNIV \ s = UNIV" using connected_component_in connected_component_UNIV by blast lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}" by (auto simp: components_eq_sing_iff) lemma interior_inside_frontier: fixes S :: "'a::real_normed_vector set" assumes "bounded S" shows "interior S \ inside (frontier S)" proof - { fix x y assume x: "x \ interior S" and y: "y \ S" and cc: "connected_component (- frontier S) x y" have "connected_component_set (- frontier S) x \ frontier S \ {}" proof (rule connected_Int_frontier; simp add: set_eq_iff) show "\u. connected_component (- frontier S) x u \ u \ S" by (meson cc connected_component_in connected_component_refl_eq interior_subset subsetD x) show "\u. connected_component (- frontier S) x u \ u \ S" using y cc by blast qed then have "bounded (connected_component_set (- frontier S) x)" using connected_component_in by auto } then show ?thesis apply (auto simp: inside_def frontier_def) apply (rule classical) apply (rule bounded_subset [OF assms], blast) done qed lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)" by (simp add: inside_def) lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)" using inside_empty inside_Un_outside by blast lemma inside_same_component: "\connected_component (- S) x y; x \ inside S\ \ y \ inside S" using connected_component_eq connected_component_in by (fastforce simp add: inside_def) lemma outside_same_component: "\connected_component (- S) x y; x \ outside S\ \ y \ outside S" using connected_component_eq connected_component_in by (fastforce simp add: outside_def) lemma convex_in_outside: fixes S :: "'a :: {real_normed_vector, perfect_space} set" assumes S: "convex S" and z: "z \ S" shows "z \ outside S" proof (cases "S={}") case True then show ?thesis by simp next case False then obtain a where "a \ S" by blast with z have zna: "z \ a" by auto { assume "bounded (connected_component_set (- S) z)" with bounded_pos_less obtain B where "B>0" and B: "\x. connected_component (- S) z x \ norm x < B" by (metis mem_Collect_eq) define C where "C = (B + 1 + norm z) / norm (z-a)" have "C > 0" using \0 < B\ zna by (simp add: C_def field_split_simps add_strict_increasing) have "\norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\ \ norm z" by (metis add_diff_cancel norm_triangle_ineq3) moreover have "norm (C *\<^sub>R (z-a)) > norm z + B" using zna \B>0\ by (simp add: C_def le_max_iff_disj) ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith { fix u::real assume u: "0\u" "u\1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \ S" then have Cpos: "1 + u * C > 0" by (meson \0 < C\ add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one) then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z" by (simp add: scaleR_add_left [symmetric] field_split_simps) then have False using convexD_alt [OF S \a \ S\ ins, of "1/(u*C + 1)"] \C>0\ \z \ S\ Cpos u by (simp add: * field_split_simps) } note contra = this have "connected_component (- S) z (z + C *\<^sub>R (z-a))" proof (rule connected_componentI [OF connected_segment]) show "closed_segment z (z + C *\<^sub>R (z - a)) \ - S" using contra by (force simp add: closed_segment_def) qed auto then have False using zna B [of "z + C *\<^sub>R (z-a)"] C by (auto simp: field_split_simps max_mult_distrib_right) } then show ?thesis by (auto simp: outside_def z) qed lemma outside_convex: fixes S :: "'a :: {real_normed_vector, perfect_space} set" assumes "convex S" shows "outside S = - S" by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2) lemma outside_singleton [simp]: fixes x :: "'a :: {real_normed_vector, perfect_space}" shows "outside {x} = -{x}" by (auto simp: outside_convex) lemma inside_convex: fixes S :: "'a :: {real_normed_vector, perfect_space} set" shows "convex S \ inside S = {}" by (simp add: inside_outside outside_convex) lemma inside_singleton [simp]: fixes x :: "'a :: {real_normed_vector, perfect_space}" shows "inside {x} = {}" by (auto simp: inside_convex) lemma outside_subset_convex: fixes S :: "'a :: {real_normed_vector, perfect_space} set" shows "\convex T; S \ T\ \ - T \ outside S" using outside_convex outside_mono by blast lemma outside_Un_outside_Un: fixes S :: "'a::real_normed_vector set" assumes "S \ outside(T \ U) = {}" shows "outside(T \ U) \ outside(T \ S)" proof fix x assume x: "x \ outside (T \ U)" have "Y \ - S" if "connected Y" "Y \ - T" "Y \ - U" "x \ Y" "u \ Y" for u Y proof - have "Y \ connected_component_set (- (T \ U)) x" by (simp add: connected_component_maximal that) also have "\ \ outside(T \ U)" by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x) finally have "Y \ outside(T \ U)" . with assms show ?thesis by auto qed with x show "x \ outside (T \ S)" by (simp add: outside_connected_component_lt connected_component_def) meson qed lemma outside_frontier_misses_closure: fixes S :: "'a::real_normed_vector set" assumes "bounded S" shows "outside(frontier S) \ - closure S" - unfolding outside_inside Lattices.boolean_algebra_class.compl_le_compl_iff + unfolding outside_inside boolean_algebra_class.compl_le_compl_iff proof - { assume "interior S \ inside (frontier S)" hence "interior S \ inside (frontier S) = inside (frontier S)" by (simp add: subset_Un_eq) then have "closure S \ frontier S \ inside (frontier S)" using frontier_def by auto } then show "closure S \ frontier S \ inside (frontier S)" using interior_inside_frontier [OF assms] by blast qed lemma outside_frontier_eq_complement_closure: fixes S :: "'a :: {real_normed_vector, perfect_space} set" assumes "bounded S" "convex S" shows "outside(frontier S) = - closure S" by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure outside_subset_convex subset_antisym) lemma inside_frontier_eq_interior: fixes S :: "'a :: {real_normed_vector, perfect_space} set" shows "\bounded S; convex S\ \ inside(frontier S) = interior S" apply (simp add: inside_outside outside_frontier_eq_complement_closure) using closure_subset interior_subset apply (auto simp: frontier_def) done lemma open_inside: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "open (inside S)" proof - { fix x assume x: "x \ inside S" have "open (connected_component_set (- S) x)" using assms open_connected_component by blast then obtain e where e: "e>0" and e: "\y. dist y x < e \ connected_component (- S) x y" using dist_not_less_zero apply (simp add: open_dist) by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x) then have "\e>0. ball x e \ inside S" by (metis e dist_commute inside_same_component mem_ball subsetI x) } then show ?thesis by (simp add: open_contains_ball) qed lemma open_outside: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "open (outside S)" proof - { fix x assume x: "x \ outside S" have "open (connected_component_set (- S) x)" using assms open_connected_component by blast then obtain e where e: "e>0" and e: "\y. dist y x < e \ connected_component (- S) x y" using dist_not_less_zero x by (auto simp add: open_dist outside_def intro: connected_component_refl) then have "\e>0. ball x e \ outside S" by (metis e dist_commute outside_same_component mem_ball subsetI x) } then show ?thesis by (simp add: open_contains_ball) qed lemma closure_inside_subset: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "closure(inside S) \ S \ inside S" by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside) lemma frontier_inside_subset: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "frontier(inside S) \ S" proof - have "closure (inside S) \ - inside S = closure (inside S) - interior (inside S)" by (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside) moreover have "- inside S \ - outside S = S" by (metis (no_types) compl_sup double_compl inside_Un_outside) moreover have "closure (inside S) \ - outside S" by (metis (no_types) assms closure_inside_subset union_with_inside) ultimately have "closure (inside S) - interior (inside S) \ S" by blast then show ?thesis by (simp add: frontier_def open_inside interior_open) qed lemma closure_outside_subset: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "closure(outside S) \ S \ outside S" by (metis assms closed_open closure_minimal inside_outside open_inside sup_ge2) lemma frontier_outside_subset: fixes S :: "'a::real_normed_vector set" assumes "closed S" shows "frontier(outside S) \ S" unfolding frontier_def by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup_aci(1)) lemma inside_complement_unbounded_connected_empty: "\connected (- S); \ bounded (- S)\ \ inside S = {}" using inside_subset by blast lemma inside_bounded_complement_connected_empty: fixes S :: "'a::{real_normed_vector, perfect_space} set" shows "\connected (- S); bounded S\ \ inside S = {}" by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded) lemma inside_inside: assumes "S \ inside T" shows "inside S - T \ inside T" unfolding inside_def proof clarify fix x assume x: "x \ T" "x \ S" and bo: "bounded (connected_component_set (- S) x)" show "bounded (connected_component_set (- T) x)" proof (cases "S \ connected_component_set (- T) x = {}") case True then show ?thesis by (metis bounded_subset [OF bo] compl_le_compl_iff connected_component_idemp connected_component_mono disjoint_eq_subset_Compl double_compl) next case False then obtain y where y: "y \ S" "y \ connected_component_set (- T) x" by (meson disjoint_iff) then have "bounded (connected_component_set (- T) y)" using assms [unfolded inside_def] by blast with y show ?thesis by (metis connected_component_eq) qed qed lemma inside_inside_subset: "inside(inside S) \ S" using inside_inside union_with_outside by fastforce lemma inside_outside_intersect_connected: "\connected T; inside S \ T \ {}; outside S \ T \ {}\ \ S \ T \ {}" apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify) by (metis (no_types, opaque_lifting) Compl_anti_mono connected_component_eq connected_component_maximal contra_subsetD double_compl) lemma outside_bounded_nonempty: fixes S :: "'a :: {real_normed_vector, perfect_space} set" assumes "bounded S" shows "outside S \ {}" by (metis (no_types, lifting) Collect_empty_eq Collect_mem_eq Compl_eq_Diff_UNIV Diff_cancel Diff_disjoint UNIV_I assms ball_eq_empty bounded_diff cobounded_outside convex_ball double_complement order_refl outside_convex outside_def) lemma outside_compact_in_open: fixes S :: "'a :: {real_normed_vector,perfect_space} set" assumes S: "compact S" and T: "open T" and "S \ T" "T \ {}" shows "outside S \ T \ {}" proof - have "outside S \ {}" by (simp add: compact_imp_bounded outside_bounded_nonempty S) with assms obtain a b where a: "a \ outside S" and b: "b \ T" by auto show ?thesis proof (cases "a \ T") case True with a show ?thesis by blast next case False have front: "frontier T \ - S" using \S \ T\ frontier_disjoint_eq T by auto { fix \ assume "path \" and pimg_sbs: "path_image \ - {pathfinish \} \ interior (- T)" and pf: "pathfinish \ \ frontier T" and ps: "pathstart \ = a" define c where "c = pathfinish \" have "c \ -S" unfolding c_def using front pf by blast moreover have "open (-S)" using S compact_imp_closed by blast ultimately obtain \::real where "\ > 0" and \: "cball c \ \ -S" using open_contains_cball[of "-S"] S by blast then obtain d where "d \ T" and d: "dist d c < \" using closure_approachable [of c T] pf unfolding c_def by (metis Diff_iff frontier_def) then have "d \ -S" using \ using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq) have pimg_sbs_cos: "path_image \ \ -S" using \c \ - S\ \S \ T\ c_def interior_subset pimg_sbs by fastforce have "closed_segment c d \ cball c \" by (metis \0 < \\ centre_in_cball closed_segment_subset convex_cball d dist_commute less_eq_real_def mem_cball) with \ have "closed_segment c d \ -S" by blast moreover have con_gcd: "connected (path_image \ \ closed_segment c d)" by (rule connected_Un) (auto simp: c_def \path \\ connected_path_image) ultimately have "connected_component (- S) a d" unfolding connected_component_def using pimg_sbs_cos ps by blast then have "outside S \ T \ {}" using outside_same_component [OF _ a] by (metis IntI \d \ T\ empty_iff) } note * = this have pal: "pathstart (linepath a b) \ closure (- T)" by (auto simp: False closure_def) show ?thesis by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b) qed qed lemma inside_inside_compact_connected: fixes S :: "'a :: euclidean_space set" assumes S: "closed S" and T: "compact T" and "connected T" "S \ inside T" shows "inside S \ inside T" proof (cases "inside T = {}") case True with assms show ?thesis by auto next case False consider "DIM('a) = 1" | "DIM('a) \ 2" using antisym not_less_eq_eq by fastforce then show ?thesis proof cases case 1 then show ?thesis using connected_convex_1_gen assms False inside_convex by blast next case 2 have "bounded S" using assms by (meson bounded_inside bounded_subset compact_imp_bounded) then have coms: "compact S" by (simp add: S compact_eq_bounded_closed) then have bst: "bounded (S \ T)" by (simp add: compact_imp_bounded T) then obtain r where "0 < r" and r: "S \ T \ ball 0 r" using bounded_subset_ballD by blast have outst: "outside S \ outside T \ {}" proof - have "- ball 0 r \ outside S" by (meson convex_ball le_supE outside_subset_convex r) moreover have "- ball 0 r \ outside T" by (meson convex_ball le_supE outside_subset_convex r) ultimately show ?thesis by (metis Compl_subset_Compl_iff Int_subset_iff bounded_ball inf.orderE outside_bounded_nonempty outside_no_overlap) qed have "S \ T = {}" using assms by (metis disjoint_iff_not_equal inside_no_overlap subsetCE) moreover have "outside S \ inside T \ {}" by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open T) ultimately have "inside S \ T = {}" using inside_outside_intersect_connected [OF \connected T\, of S] by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst) then show ?thesis using inside_inside [OF \S \ inside T\] by blast qed qed lemma connected_with_inside: fixes S :: "'a :: real_normed_vector set" assumes S: "closed S" and cons: "connected S" shows "connected(S \ inside S)" proof (cases "S \ inside S = UNIV") case True with assms show ?thesis by auto next case False then obtain b where b: "b \ S" "b \ inside S" by blast have *: "\y T. y \ S \ connected T \ a \ T \ y \ T \ T \ (S \ inside S)" if "a \ S \ inside S" for a using that proof assume "a \ S" then show ?thesis by (rule_tac x=a in exI, rule_tac x="{a}" in exI, simp) next assume a: "a \ inside S" then have ain: "a \ closure (inside S)" by (simp add: closure_def) show ?thesis apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "inside S"]) apply (simp_all add: ain b) subgoal for h apply (rule_tac x="pathfinish h" in exI) apply (simp add: subsetD [OF frontier_inside_subset[OF S]]) apply (rule_tac x="path_image h" in exI) apply (simp add: pathfinish_in_path_image connected_path_image, auto) by (metis Diff_single_insert S frontier_inside_subset insert_iff interior_subset subsetD) done qed show ?thesis apply (simp add: connected_iff_connected_component) apply (clarsimp simp add: connected_component_def dest!: *) subgoal for x y u u' T t' by (rule_tac x="(S \ T \ t')" in exI) (auto intro!: connected_Un cons) done qed text\The proof is virtually the same as that above.\ lemma connected_with_outside: fixes S :: "'a :: real_normed_vector set" assumes S: "closed S" and cons: "connected S" shows "connected(S \ outside S)" proof (cases "S \ outside S = UNIV") case True with assms show ?thesis by auto next case False then obtain b where b: "b \ S" "b \ outside S" by blast have *: "\y T. y \ S \ connected T \ a \ T \ y \ T \ T \ (S \ outside S)" if "a \ (S \ outside S)" for a using that proof assume "a \ S" then show ?thesis by (rule_tac x=a in exI, rule_tac x="{a}" in exI, simp) next assume a: "a \ outside S" then have ain: "a \ closure (outside S)" by (simp add: closure_def) show ?thesis apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "outside S"]) apply (simp_all add: ain b) subgoal for h apply (rule_tac x="pathfinish h" in exI) apply (simp add: subsetD [OF frontier_outside_subset[OF S]]) apply (rule_tac x="path_image h" in exI) apply (simp add: pathfinish_in_path_image connected_path_image, auto) by (metis (no_types, lifting) frontier_outside_subset insertE insert_Diff interior_eq open_outside pathfinish_in_path_image S subsetCE) done qed show ?thesis apply (simp add: connected_iff_connected_component) apply (clarsimp simp add: connected_component_def dest!: *) subgoal for x y u u' T t' by (rule_tac x="(S \ T \ t')" in exI) (auto intro!: connected_Un cons) done qed lemma inside_inside_eq_empty [simp]: fixes S :: "'a :: {real_normed_vector, perfect_space} set" assumes S: "closed S" and cons: "connected S" shows "inside (inside S) = {}" by (metis (no_types) unbounded_outside connected_with_outside [OF assms] bounded_Un inside_complement_unbounded_connected_empty unbounded_outside union_with_outside) lemma inside_in_components: "inside S \ components (- S) \ connected(inside S) \ inside S \ {}" (is "?lhs = ?rhs") proof assume R: ?rhs then have "\x. \x \ S; x \ inside S\ \ \ connected (inside S)" by (simp add: inside_outside) with R show ?lhs unfolding in_components_maximal by (auto intro: inside_same_component connected_componentI) qed (simp add: in_components_maximal) text\The proof is like that above.\ lemma outside_in_components: "outside S \ components (- S) \ connected(outside S) \ outside S \ {}" (is "?lhs = ?rhs") proof assume R: ?rhs then have "\x. \x \ S; x \ outside S\ \ \ connected (outside S)" by (meson disjoint_iff outside_no_overlap) with R show ?lhs unfolding in_components_maximal by (auto intro: outside_same_component connected_componentI) qed (simp add: in_components_maximal) lemma bounded_unique_outside: fixes S :: "'a :: euclidean_space set" assumes "bounded S" "DIM('a) \ 2" shows "(c \ components (- S) \ \ bounded c \ c = outside S)" using assms by (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside) subsection\Condition for an open map's image to contain a ball\ proposition ball_subset_open_map_image: fixes f :: "'a::heine_borel \ 'b :: {real_normed_vector,heine_borel}" assumes contf: "continuous_on (closure S) f" and oint: "open (f ` interior S)" and le_no: "\z. z \ frontier S \ r \ norm(f z - f a)" and "bounded S" "a \ S" "0 < r" shows "ball (f a) r \ f ` S" proof (cases "f ` S = UNIV") case True then show ?thesis by simp next case False then have "closed (frontier (f ` S))" "frontier (f ` S) \ {}" using \a \ S\ by (auto simp: frontier_eq_empty) then obtain w where w: "w \ frontier (f ` S)" and dw_le: "\y. y \ frontier (f ` S) \ norm (f a - w) \ norm (f a - y)" by (auto simp add: dist_norm intro: distance_attains_inf [of "frontier(f ` S)" "f a"]) then obtain \ where \: "\n. \ n \ f ` S" and tendsw: "\ \ w" by (metis Diff_iff frontier_def closure_sequential) then have "\n. \x \ S. \ n = f x" by force then obtain z where zs: "\n. z n \ S" and fz: "\n. \ n = f (z n)" by metis then obtain y K where y: "y \ closure S" and "strict_mono (K :: nat \ nat)" and Klim: "(z \ K) \ y" using \bounded S\ unfolding compact_closure [symmetric] compact_def by (meson closure_subset subset_iff) then have ftendsw: "((\n. f (z n)) \ K) \ w" by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw) have zKs: "\n. (z \ K) n \ S" by (simp add: zs) have fz: "f \ z = \" "(\n. f (z n)) = \" using fz by auto then have "(\ \ K) \ f y" by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially) with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto have rle: "r \ norm (f y - f a)" proof (rule le_no) show "y \ frontier S" using w wy oint by (force simp: imageI image_mono interiorI interior_subset frontier_def y) qed have **: "(b \ (- S) \ {} \ b - (- S) \ {} \ b \ f \ {}) \ (b \ S \ {}) \ b \ f = {} \ b \ S" for b f and S :: "'b set" by blast have \
: "\y. \norm (f a - y) < r; y \ frontier (f ` S)\ \ False" by (metis dw_le norm_minus_commute not_less order_trans rle wy) show ?thesis apply (rule ** [OF connected_Int_frontier [where t = "f`S", OF connected_ball]]) (*such a horrible mess*) using \a \ S\ \0 < r\ by (auto simp: disjoint_iff_not_equal dist_norm dest: \
) qed subsubsection\Special characterizations of classes of functions into and out of R.\ lemma Hausdorff_space_euclidean [simp]: "Hausdorff_space (euclidean :: 'a::metric_space topology)" proof - have "\U V. open U \ open V \ x \ U \ y \ V \ disjnt U V" if "x \ y" for x y :: 'a proof (intro exI conjI) let ?r = "dist x y / 2" have [simp]: "?r > 0" by (simp add: that) show "open (ball x ?r)" "open (ball y ?r)" "x \ (ball x ?r)" "y \ (ball y ?r)" by (auto simp add: that) show "disjnt (ball x ?r) (ball y ?r)" unfolding disjnt_def by (simp add: disjoint_ballI) qed then show ?thesis by (simp add: Hausdorff_space_def) qed proposition embedding_map_into_euclideanreal: assumes "path_connected_space X" shows "embedding_map X euclideanreal f \ continuous_map X euclideanreal f \ inj_on f (topspace X)" proof safe show "continuous_map X euclideanreal f" if "embedding_map X euclideanreal f" using continuous_map_in_subtopology homeomorphic_imp_continuous_map that unfolding embedding_map_def by blast show "inj_on f (topspace X)" if "embedding_map X euclideanreal f" using that homeomorphic_imp_injective_map unfolding embedding_map_def by blast show "embedding_map X euclideanreal f" if cont: "continuous_map X euclideanreal f" and inj: "inj_on f (topspace X)" proof - obtain g where gf: "\x. x \ topspace X \ g (f x) = x" using inv_into_f_f [OF inj] by auto show ?thesis unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def proof (intro exI conjI) show "continuous_map X (top_of_set (f ` topspace X)) f" by (simp add: cont continuous_map_in_subtopology) let ?S = "f ` topspace X" have eq: "{x \ ?S. g x \ U} = f ` U" if "openin X U" for U using openin_subset [OF that] by (auto simp: gf) have 1: "g ` ?S \ topspace X" using eq by blast have "openin (top_of_set ?S) {x \ ?S. g x \ T}" if "openin X T" for T proof - have "T \ topspace X" by (simp add: openin_subset that) have RR: "\x \ ?S \ g -` T. \d>0. \x' \ ?S \ ball x d. g x' \ T" proof (clarsimp simp add: gf) have pcS: "path_connectedin euclidean ?S" using assms cont path_connectedin_continuous_map_image path_connectedin_topspace by blast show "\d>0. \x'\f ` topspace X \ ball (f x) d. g x' \ T" if "x \ T" for x proof - have x: "x \ topspace X" using \T \ topspace X\ \x \ T\ by blast obtain u v d where "0 < d" "u \ topspace X" "v \ topspace X" and sub_fuv: "?S \ {f x - d .. f x + d} \ {f u..f v}" proof (cases "\u \ topspace X. f u < f x") case True then obtain u where u: "u \ topspace X" "f u < f x" .. show ?thesis proof (cases "\v \ topspace X. f x < f v") case True then obtain v where v: "v \ topspace X" "f x < f v" .. show ?thesis proof let ?d = "min (f x - f u) (f v - f x)" show "0 < ?d" by (simp add: \f u < f x\ \f x < f v\) show "f ` topspace X \ {f x - ?d..f x + ?d} \ {f u..f v}" by fastforce qed (auto simp: u v) next case False show ?thesis proof let ?d = "f x - f u" show "0 < ?d" by (simp add: u) show "f ` topspace X \ {f x - ?d..f x + ?d} \ {f u..f x}" using x u False by auto qed (auto simp: x u) qed next case False note no_u = False show ?thesis proof (cases "\v \ topspace X. f x < f v") case True then obtain v where v: "v \ topspace X" "f x < f v" .. show ?thesis proof let ?d = "f v - f x" show "0 < ?d" by (simp add: v) show "f ` topspace X \ {f x - ?d..f x + ?d} \ {f x..f v}" using False by auto qed (auto simp: x v) next case False show ?thesis proof show "f ` topspace X \ {f x - 1..f x + 1} \ {f x..f x}" using False no_u by fastforce qed (auto simp: x) qed qed then obtain h where "pathin X h" "h 0 = u" "h 1 = v" using assms unfolding path_connected_space_def by blast obtain C where "compactin X C" "connectedin X C" "u \ C" "v \ C" proof show "compactin X (h ` {0..1})" using that by (simp add: \pathin X h\ compactin_path_image) show "connectedin X (h ` {0..1})" using \pathin X h\ connectedin_path_image by blast qed (use \h 0 = u\ \h 1 = v\ in auto) have "continuous_map (subtopology euclideanreal (?S \ {f x - d .. f x + d})) (subtopology X C) g" proof (rule continuous_inverse_map) show "compact_space (subtopology X C)" using \compactin X C\ compactin_subspace by blast show "continuous_map (subtopology X C) euclideanreal f" by (simp add: cont continuous_map_from_subtopology) have "{f u .. f v} \ f ` topspace (subtopology X C)" proof (rule connected_contains_Icc) show "connected (f ` topspace (subtopology X C))" using connectedin_continuous_map_image [OF cont] by (simp add: \compactin X C\ \connectedin X C\ compactin_subset_topspace inf_absorb2) show "f u \ f ` topspace (subtopology X C)" by (simp add: \u \ C\ \u \ topspace X\) show "f v \ f ` topspace (subtopology X C)" by (simp add: \v \ C\ \v \ topspace X\) qed then show "f ` topspace X \ {f x - d..f x + d} \ f ` topspace (subtopology X C)" using sub_fuv by blast qed (auto simp: gf) then have contg: "continuous_map (subtopology euclideanreal (?S \ {f x - d .. f x + d})) X g" using continuous_map_in_subtopology by blast have "\e>0. \x \ ?S \ {f x - d .. f x + d} \ ball (f x) e. g x \ T" using openin_continuous_map_preimage [OF contg \openin X T\] x \x \ T\ \0 < d\ unfolding openin_euclidean_subtopology_iff by (force simp: gf dist_commute) then obtain e where "e > 0 \ (\x\f ` topspace X \ {f x - d..f x + d} \ ball (f x) e. g x \ T)" by metis with \0 < d\ have "min d e > 0" "\u. u \ topspace X \ \f x - f u\ < min d e \ u \ T" using dist_real_def gf by force+ then show ?thesis by (metis (full_types) Int_iff dist_real_def image_iff mem_ball gf) qed qed then obtain d where d: "\r. r \ ?S \ g -` T \ d r > 0 \ (\x \ ?S \ ball r (d r). g x \ T)" by metis show ?thesis unfolding openin_subtopology proof (intro exI conjI) show "{x \ ?S. g x \ T} = (\r \ ?S \ g -` T. ball r (d r)) \ f ` topspace X" using d by (auto simp: gf) qed auto qed then show "continuous_map (top_of_set ?S) X g" by (simp add: continuous_map_def gf) qed (auto simp: gf) qed qed subsubsection \An injective function into R is a homeomorphism and so an open map.\ lemma injective_into_1d_eq_homeomorphism: fixes f :: "'a::topological_space \ real" assumes f: "continuous_on S f" and S: "path_connected S" shows "inj_on f S \ (\g. homeomorphism S (f ` S) f g)" proof show "\g. homeomorphism S (f ` S) f g" if "inj_on f S" proof - have "embedding_map (top_of_set S) euclideanreal f" using that embedding_map_into_euclideanreal [of "top_of_set S" f] assms by auto then show ?thesis by (simp add: embedding_map_def) (metis all_closedin_homeomorphic_image f homeomorphism_injective_closed_map that) qed qed (metis homeomorphism_def inj_onI) lemma injective_into_1d_imp_open_map: fixes f :: "'a::topological_space \ real" assumes "continuous_on S f" "path_connected S" "inj_on f S" "openin (subtopology euclidean S) T" shows "openin (subtopology euclidean (f ` S)) (f ` T)" using assms homeomorphism_imp_open_map injective_into_1d_eq_homeomorphism by blast lemma homeomorphism_into_1d: fixes f :: "'a::topological_space \ real" assumes "path_connected S" "continuous_on S f" "f ` S = T" "inj_on f S" shows "\g. homeomorphism S T f g" using assms injective_into_1d_eq_homeomorphism by blast subsection\<^marker>\tag unimportant\ \Rectangular paths\ definition\<^marker>\tag unimportant\ rectpath where "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3) in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)" lemma path_rectpath [simp, intro]: "path (rectpath a b)" by (simp add: Let_def rectpath_def) lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1" by (simp add: rectpath_def Let_def) lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1" by (simp add: rectpath_def Let_def) lemma simple_path_rectpath [simp, intro]: assumes "Re a1 \ Re a3" "Im a1 \ Im a3" shows "simple_path (rectpath a1 a3)" unfolding rectpath_def Let_def using assms by (intro simple_path_join_loop arc_join arc_linepath) (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im) lemma path_image_rectpath: assumes "Re a1 \ Re a3" "Im a1 \ Im a3" shows "path_image (rectpath a1 a3) = {z. Re z \ {Re a1, Re a3} \ Im z \ {Im a1..Im a3}} \ {z. Im z \ {Im a1, Im a3} \ Re z \ {Re a1..Re a3}}" (is "?lhs = ?rhs") proof - define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)" have "?lhs = closed_segment a1 a2 \ closed_segment a2 a3 \ closed_segment a4 a3 \ closed_segment a1 a4" by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute a2_def a4_def Un_assoc) also have "\ = ?rhs" using assms by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl) finally show ?thesis . qed lemma path_image_rectpath_subset_cbox: assumes "Re a \ Re b" "Im a \ Im b" shows "path_image (rectpath a b) \ cbox a b" using assms by (auto simp: path_image_rectpath in_cbox_complex_iff) lemma path_image_rectpath_inter_box: assumes "Re a \ Re b" "Im a \ Im b" shows "path_image (rectpath a b) \ box a b = {}" using assms by (auto simp: path_image_rectpath in_box_complex_iff) lemma path_image_rectpath_cbox_minus_box: assumes "Re a \ Re b" "Im a \ Im b" shows "path_image (rectpath a b) = cbox a b - box a b" using assms by (auto simp: path_image_rectpath in_cbox_complex_iff in_box_complex_iff) end diff --git a/src/HOL/Bit_Operations.thy b/src/HOL/Bit_Operations.thy --- a/src/HOL/Bit_Operations.thy +++ b/src/HOL/Bit_Operations.thy @@ -1,3585 +1,3580 @@ (* Author: Florian Haftmann, TUM *) section \Bit operations in suitable algebraic structures\ theory Bit_Operations imports Presburger Groups_List begin subsection \Abstract bit structures\ class semiring_bits = semiring_parity + assumes bits_induct [case_names stable rec]: \(\a. a div 2 = a \ P a) \ (\a b. P a \ (of_bool b + 2 * a) div 2 = a \ P (of_bool b + 2 * a)) \ P a\ assumes bits_div_0 [simp]: \0 div a = 0\ and bits_div_by_1 [simp]: \a div 1 = a\ and bits_mod_div_trivial [simp]: \a mod b div b = 0\ and even_succ_div_2 [simp]: \even a \ (1 + a) div 2 = a div 2\ and even_mask_div_iff: \even ((2 ^ m - 1) div 2 ^ n) \ 2 ^ n = 0 \ m \ n\ and exp_div_exp_eq: \2 ^ m div 2 ^ n = of_bool (2 ^ m \ 0 \ m \ n) * 2 ^ (m - n)\ and div_exp_eq: \a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)\ and mod_exp_eq: \a mod 2 ^ m mod 2 ^ n = a mod 2 ^ min m n\ and mult_exp_mod_exp_eq: \m \ n \ (a * 2 ^ m) mod (2 ^ n) = (a mod 2 ^ (n - m)) * 2 ^ m\ and div_exp_mod_exp_eq: \a div 2 ^ n mod 2 ^ m = a mod (2 ^ (n + m)) div 2 ^ n\ and even_mult_exp_div_exp_iff: \even (a * 2 ^ m div 2 ^ n) \ m > n \ 2 ^ n = 0 \ (m \ n \ even (a div 2 ^ (n - m)))\ fixes bit :: \'a \ nat \ bool\ assumes bit_iff_odd: \bit a n \ odd (a div 2 ^ n)\ begin text \ Having \<^const>\bit\ as definitional class operation takes into account that specific instances can be implemented differently wrt. code generation. \ lemma bits_div_by_0 [simp]: \a div 0 = 0\ by (metis add_cancel_right_right bits_mod_div_trivial mod_mult_div_eq mult_not_zero) lemma bits_1_div_2 [simp]: \1 div 2 = 0\ using even_succ_div_2 [of 0] by simp lemma bits_1_div_exp [simp]: \1 div 2 ^ n = of_bool (n = 0)\ using div_exp_eq [of 1 1] by (cases n) simp_all lemma even_succ_div_exp [simp]: \(1 + a) div 2 ^ n = a div 2 ^ n\ if \even a\ and \n > 0\ proof (cases n) case 0 with that show ?thesis by simp next case (Suc n) with \even a\ have \(1 + a) div 2 ^ Suc n = a div 2 ^ Suc n\ proof (induction n) case 0 then show ?case by simp next case (Suc n) then show ?case using div_exp_eq [of _ 1 \Suc n\, symmetric] by simp qed with Suc show ?thesis by simp qed lemma even_succ_mod_exp [simp]: \(1 + a) mod 2 ^ n = 1 + (a mod 2 ^ n)\ if \even a\ and \n > 0\ using div_mult_mod_eq [of \1 + a\ \2 ^ n\] that apply simp by (metis local.add.left_commute local.add_left_cancel local.div_mult_mod_eq) lemma bits_mod_by_1 [simp]: \a mod 1 = 0\ using div_mult_mod_eq [of a 1] by simp lemma bits_mod_0 [simp]: \0 mod a = 0\ using div_mult_mod_eq [of 0 a] by simp lemma bits_one_mod_two_eq_one [simp]: \1 mod 2 = 1\ by (simp add: mod2_eq_if) lemma bit_0 [simp]: \bit a 0 \ odd a\ by (simp add: bit_iff_odd) lemma bit_Suc: \bit a (Suc n) \ bit (a div 2) n\ using div_exp_eq [of a 1 n] by (simp add: bit_iff_odd) lemma bit_rec: \bit a n \ (if n = 0 then odd a else bit (a div 2) (n - 1))\ by (cases n) (simp_all add: bit_Suc) lemma bit_0_eq [simp]: \bit 0 = bot\ by (simp add: fun_eq_iff bit_iff_odd) context fixes a assumes stable: \a div 2 = a\ begin lemma bits_stable_imp_add_self: \a + a mod 2 = 0\ proof - have \a div 2 * 2 + a mod 2 = a\ by (fact div_mult_mod_eq) then have \a * 2 + a mod 2 = a\ by (simp add: stable) then show ?thesis by (simp add: mult_2_right ac_simps) qed lemma stable_imp_bit_iff_odd: \bit a n \ odd a\ by (induction n) (simp_all add: stable bit_Suc) end lemma bit_iff_idd_imp_stable: \a div 2 = a\ if \\n. bit a n \ odd a\ using that proof (induction a rule: bits_induct) case (stable a) then show ?case by simp next case (rec a b) from rec.prems [of 1] have [simp]: \b = odd a\ by (simp add: rec.hyps bit_Suc) from rec.hyps have hyp: \(of_bool (odd a) + 2 * a) div 2 = a\ by simp have \bit a n \ odd a\ for n using rec.prems [of \Suc n\] by (simp add: hyp bit_Suc) then have \a div 2 = a\ by (rule rec.IH) then have \of_bool (odd a) + 2 * a = 2 * (a div 2) + of_bool (odd a)\ by (simp add: ac_simps) also have \\ = a\ using mult_div_mod_eq [of 2 a] by (simp add: of_bool_odd_eq_mod_2) finally show ?case using \a div 2 = a\ by (simp add: hyp) qed lemma exp_eq_0_imp_not_bit: \\ bit a n\ if \2 ^ n = 0\ using that by (simp add: bit_iff_odd) lemma bit_eqI: \a = b\ if \\n. 2 ^ n \ 0 \ bit a n \ bit b n\ proof - have \bit a n \ bit b n\ for n proof (cases \2 ^ n = 0\) case True then show ?thesis by (simp add: exp_eq_0_imp_not_bit) next case False then show ?thesis by (rule that) qed then show ?thesis proof (induction a arbitrary: b rule: bits_induct) case (stable a) from stable(2) [of 0] have **: \even b \ even a\ by simp have \b div 2 = b\ proof (rule bit_iff_idd_imp_stable) fix n from stable have *: \bit b n \ bit a n\ by simp also have \bit a n \ odd a\ using stable by (simp add: stable_imp_bit_iff_odd) finally show \bit b n \ odd b\ by (simp add: **) qed from ** have \a mod 2 = b mod 2\ by (simp add: mod2_eq_if) then have \a mod 2 + (a + b) = b mod 2 + (a + b)\ by simp then have \a + a mod 2 + b = b + b mod 2 + a\ by (simp add: ac_simps) with \a div 2 = a\ \b div 2 = b\ show ?case by (simp add: bits_stable_imp_add_self) next case (rec a p) from rec.prems [of 0] have [simp]: \p = odd b\ by simp from rec.hyps have \bit a n \ bit (b div 2) n\ for n using rec.prems [of \Suc n\] by (simp add: bit_Suc) then have \a = b div 2\ by (rule rec.IH) then have \2 * a = 2 * (b div 2)\ by simp then have \b mod 2 + 2 * a = b mod 2 + 2 * (b div 2)\ by simp also have \\ = b\ by (fact mod_mult_div_eq) finally show ?case by (auto simp add: mod2_eq_if) qed qed lemma bit_eq_iff: \a = b \ (\n. bit a n \ bit b n)\ by (auto intro: bit_eqI) named_theorems bit_simps \Simplification rules for \<^const>\bit\\ lemma bit_exp_iff [bit_simps]: \bit (2 ^ m) n \ 2 ^ m \ 0 \ m = n\ by (auto simp add: bit_iff_odd exp_div_exp_eq) lemma bit_1_iff [bit_simps]: \bit 1 n \ 1 \ 0 \ n = 0\ using bit_exp_iff [of 0 n] by simp lemma bit_2_iff [bit_simps]: \bit 2 n \ 2 \ 0 \ n = 1\ using bit_exp_iff [of 1 n] by auto lemma even_bit_succ_iff: \bit (1 + a) n \ bit a n \ n = 0\ if \even a\ using that by (cases \n = 0\) (simp_all add: bit_iff_odd) lemma odd_bit_iff_bit_pred: \bit a n \ bit (a - 1) n \ n = 0\ if \odd a\ proof - from \odd a\ obtain b where \a = 2 * b + 1\ .. moreover have \bit (2 * b) n \ n = 0 \ bit (1 + 2 * b) n\ using even_bit_succ_iff by simp ultimately show ?thesis by (simp add: ac_simps) qed lemma bit_double_iff [bit_simps]: \bit (2 * a) n \ bit a (n - 1) \ n \ 0 \ 2 ^ n \ 0\ using even_mult_exp_div_exp_iff [of a 1 n] by (cases n, auto simp add: bit_iff_odd ac_simps) lemma bit_eq_rec: \a = b \ (even a \ even b) \ a div 2 = b div 2\ (is \?P = ?Q\) proof assume ?P then show ?Q by simp next assume ?Q then have \even a \ even b\ and \a div 2 = b div 2\ by simp_all show ?P proof (rule bit_eqI) fix n show \bit a n \ bit b n\ proof (cases n) case 0 with \even a \ even b\ show ?thesis by simp next case (Suc n) moreover from \a div 2 = b div 2\ have \bit (a div 2) n = bit (b div 2) n\ by simp ultimately show ?thesis by (simp add: bit_Suc) qed qed qed lemma bit_mod_2_iff [simp]: \bit (a mod 2) n \ n = 0 \ odd a\ by (cases a rule: parity_cases) (simp_all add: bit_iff_odd) lemma bit_mask_iff: \bit (2 ^ m - 1) n \ 2 ^ n \ 0 \ n < m\ by (simp add: bit_iff_odd even_mask_div_iff not_le) lemma bit_Numeral1_iff [simp]: \bit (numeral Num.One) n \ n = 0\ by (simp add: bit_rec) lemma exp_add_not_zero_imp: \2 ^ m \ 0\ and \2 ^ n \ 0\ if \2 ^ (m + n) \ 0\ proof - have \\ (2 ^ m = 0 \ 2 ^ n = 0)\ proof (rule notI) assume \2 ^ m = 0 \ 2 ^ n = 0\ then have \2 ^ (m + n) = 0\ by (rule disjE) (simp_all add: power_add) with that show False .. qed then show \2 ^ m \ 0\ and \2 ^ n \ 0\ by simp_all qed lemma bit_disjunctive_add_iff: \bit (a + b) n \ bit a n \ bit b n\ if \\n. \ bit a n \ \ bit b n\ proof (cases \2 ^ n = 0\) case True then show ?thesis by (simp add: exp_eq_0_imp_not_bit) next case False with that show ?thesis proof (induction n arbitrary: a b) case 0 from "0.prems"(1) [of 0] show ?case by auto next case (Suc n) from Suc.prems(1) [of 0] have even: \even a \ even b\ by auto have bit: \\ bit (a div 2) n \ \ bit (b div 2) n\ for n using Suc.prems(1) [of \Suc n\] by (simp add: bit_Suc) from Suc.prems(2) have \2 * 2 ^ n \ 0\ \2 ^ n \ 0\ by (auto simp add: mult_2) have \a + b = (a div 2 * 2 + a mod 2) + (b div 2 * 2 + b mod 2)\ using div_mult_mod_eq [of a 2] div_mult_mod_eq [of b 2] by simp also have \\ = of_bool (odd a \ odd b) + 2 * (a div 2 + b div 2)\ using even by (auto simp add: algebra_simps mod2_eq_if) finally have \bit ((a + b) div 2) n \ bit (a div 2 + b div 2) n\ using \2 * 2 ^ n \ 0\ by simp (simp_all flip: bit_Suc add: bit_double_iff) also have \\ \ bit (a div 2) n \ bit (b div 2) n\ using bit \2 ^ n \ 0\ by (rule Suc.IH) finally show ?case by (simp add: bit_Suc) qed qed lemma exp_add_not_zero_imp_left: \2 ^ m \ 0\ and exp_add_not_zero_imp_right: \2 ^ n \ 0\ if \2 ^ (m + n) \ 0\ proof - have \\ (2 ^ m = 0 \ 2 ^ n = 0)\ proof (rule notI) assume \2 ^ m = 0 \ 2 ^ n = 0\ then have \2 ^ (m + n) = 0\ by (rule disjE) (simp_all add: power_add) with that show False .. qed then show \2 ^ m \ 0\ and \2 ^ n \ 0\ by simp_all qed lemma exp_not_zero_imp_exp_diff_not_zero: \2 ^ (n - m) \ 0\ if \2 ^ n \ 0\ proof (cases \m \ n\) case True moreover define q where \q = n - m\ ultimately have \n = m + q\ by simp with that show ?thesis by (simp add: exp_add_not_zero_imp_right) next case False with that show ?thesis by simp qed end lemma nat_bit_induct [case_names zero even odd]: "P n" if zero: "P 0" and even: "\n. P n \ n > 0 \ P (2 * n)" and odd: "\n. P n \ P (Suc (2 * n))" proof (induction n rule: less_induct) case (less n) show "P n" proof (cases "n = 0") case True with zero show ?thesis by simp next case False with less have hyp: "P (n div 2)" by simp show ?thesis proof (cases "even n") case True then have "n \ 1" by auto with \n \ 0\ have "n div 2 > 0" by simp with \even n\ hyp even [of "n div 2"] show ?thesis by simp next case False with hyp odd [of "n div 2"] show ?thesis by simp qed qed qed instantiation nat :: semiring_bits begin definition bit_nat :: \nat \ nat \ bool\ where \bit_nat m n \ odd (m div 2 ^ n)\ instance proof show \P n\ if stable: \\n. n div 2 = n \ P n\ and rec: \\n b. P n \ (of_bool b + 2 * n) div 2 = n \ P (of_bool b + 2 * n)\ for P and n :: nat proof (induction n rule: nat_bit_induct) case zero from stable [of 0] show ?case by simp next case (even n) with rec [of n False] show ?case by simp next case (odd n) with rec [of n True] show ?case by simp qed show \q mod 2 ^ m mod 2 ^ n = q mod 2 ^ min m n\ for q m n :: nat apply (auto simp add: less_iff_Suc_add power_add mod_mod_cancel split: split_min_lin) apply (metis div_mult2_eq mod_div_trivial mod_eq_self_iff_div_eq_0 mod_mult_self2_is_0 power_commutes) done show \(q * 2 ^ m) mod (2 ^ n) = (q mod 2 ^ (n - m)) * 2 ^ m\ if \m \ n\ for q m n :: nat using that apply (auto simp add: mod_mod_cancel div_mult2_eq power_add mod_mult2_eq le_iff_add split: split_min_lin) done show \even ((2 ^ m - (1::nat)) div 2 ^ n) \ 2 ^ n = (0::nat) \ m \ n\ for m n :: nat using even_mask_div_iff' [where ?'a = nat, of m n] by simp show \even (q * 2 ^ m div 2 ^ n) \ n < m \ (2::nat) ^ n = 0 \ m \ n \ even (q div 2 ^ (n - m))\ for m n q r :: nat apply (auto simp add: not_less power_add ac_simps dest!: le_Suc_ex) apply (metis (full_types) dvd_mult dvd_mult_imp_div dvd_power_iff_le not_less not_less_eq order_refl power_Suc) done qed (auto simp add: div_mult2_eq mod_mult2_eq power_add power_diff bit_nat_def) end lemma int_bit_induct [case_names zero minus even odd]: "P k" if zero_int: "P 0" and minus_int: "P (- 1)" and even_int: "\k. P k \ k \ 0 \ P (k * 2)" and odd_int: "\k. P k \ k \ - 1 \ P (1 + (k * 2))" for k :: int proof (cases "k \ 0") case True define n where "n = nat k" with True have "k = int n" by simp then show "P k" proof (induction n arbitrary: k rule: nat_bit_induct) case zero then show ?case by (simp add: zero_int) next case (even n) have "P (int n * 2)" by (rule even_int) (use even in simp_all) with even show ?case by (simp add: ac_simps) next case (odd n) have "P (1 + (int n * 2))" by (rule odd_int) (use odd in simp_all) with odd show ?case by (simp add: ac_simps) qed next case False define n where "n = nat (- k - 1)" with False have "k = - int n - 1" by simp then show "P k" proof (induction n arbitrary: k rule: nat_bit_induct) case zero then show ?case by (simp add: minus_int) next case (even n) have "P (1 + (- int (Suc n) * 2))" by (rule odd_int) (use even in \simp_all add: algebra_simps\) also have "\ = - int (2 * n) - 1" by (simp add: algebra_simps) finally show ?case using even.prems by simp next case (odd n) have "P (- int (Suc n) * 2)" by (rule even_int) (use odd in \simp_all add: algebra_simps\) also have "\ = - int (Suc (2 * n)) - 1" by (simp add: algebra_simps) finally show ?case using odd.prems by simp qed qed context semiring_bits begin lemma bit_of_bool_iff [bit_simps]: \bit (of_bool b) n \ b \ n = 0\ by (simp add: bit_1_iff) lemma even_of_nat_iff: \even (of_nat n) \ even n\ by (induction n rule: nat_bit_induct) simp_all lemma bit_of_nat_iff [bit_simps]: \bit (of_nat m) n \ (2::'a) ^ n \ 0 \ bit m n\ proof (cases \(2::'a) ^ n = 0\) case True then show ?thesis by (simp add: exp_eq_0_imp_not_bit) next case False then have \bit (of_nat m) n \ bit m n\ proof (induction m arbitrary: n rule: nat_bit_induct) case zero then show ?case by simp next case (even m) then show ?case by (cases n) (auto simp add: bit_double_iff Bit_Operations.bit_double_iff dest: mult_not_zero) next case (odd m) then show ?case by (cases n) (auto simp add: bit_double_iff even_bit_succ_iff Bit_Operations.bit_Suc dest: mult_not_zero) qed with False show ?thesis by simp qed end instantiation int :: semiring_bits begin definition bit_int :: \int \ nat \ bool\ where \bit_int k n \ odd (k div 2 ^ n)\ instance proof show \P k\ if stable: \\k. k div 2 = k \ P k\ and rec: \\k b. P k \ (of_bool b + 2 * k) div 2 = k \ P (of_bool b + 2 * k)\ for P and k :: int proof (induction k rule: int_bit_induct) case zero from stable [of 0] show ?case by simp next case minus from stable [of \- 1\] show ?case by simp next case (even k) with rec [of k False] show ?case by (simp add: ac_simps) next case (odd k) with rec [of k True] show ?case by (simp add: ac_simps) qed show \(2::int) ^ m div 2 ^ n = of_bool ((2::int) ^ m \ 0 \ n \ m) * 2 ^ (m - n)\ for m n :: nat proof (cases \m < n\) case True then have \n = m + (n - m)\ by simp then have \(2::int) ^ m div 2 ^ n = (2::int) ^ m div 2 ^ (m + (n - m))\ by simp also have \\ = (2::int) ^ m div (2 ^ m * 2 ^ (n - m))\ by (simp add: power_add) also have \\ = (2::int) ^ m div 2 ^ m div 2 ^ (n - m)\ by (simp add: zdiv_zmult2_eq) finally show ?thesis using \m < n\ by simp next case False then show ?thesis by (simp add: power_diff) qed show \k mod 2 ^ m mod 2 ^ n = k mod 2 ^ min m n\ for m n :: nat and k :: int using mod_exp_eq [of \nat k\ m n] apply (auto simp add: mod_mod_cancel zdiv_zmult2_eq power_add zmod_zmult2_eq le_iff_add split: split_min_lin) apply (auto simp add: less_iff_Suc_add mod_mod_cancel power_add) apply (simp only: flip: mult.left_commute [of \2 ^ m\]) apply (subst zmod_zmult2_eq) apply simp_all done show \(k * 2 ^ m) mod (2 ^ n) = (k mod 2 ^ (n - m)) * 2 ^ m\ if \m \ n\ for m n :: nat and k :: int using that apply (auto simp add: power_add zmod_zmult2_eq le_iff_add split: split_min_lin) done show \even ((2 ^ m - (1::int)) div 2 ^ n) \ 2 ^ n = (0::int) \ m \ n\ for m n :: nat using even_mask_div_iff' [where ?'a = int, of m n] by simp show \even (k * 2 ^ m div 2 ^ n) \ n < m \ (2::int) ^ n = 0 \ m \ n \ even (k div 2 ^ (n - m))\ for m n :: nat and k l :: int apply (auto simp add: not_less power_add ac_simps dest!: le_Suc_ex) apply (metis Suc_leI dvd_mult dvd_mult_imp_div dvd_power_le dvd_refl power.simps(2)) done qed (auto simp add: zdiv_zmult2_eq zmod_zmult2_eq power_add power_diff not_le bit_int_def) end lemma bit_not_int_iff': \bit (- k - 1) n \ \ bit k n\ for k :: int proof (induction n arbitrary: k) case 0 show ?case by simp next case (Suc n) have \- k - 1 = - (k + 2) + 1\ by simp also have \(- (k + 2) + 1) div 2 = - (k div 2) - 1\ proof (cases \even k\) case True then have \- k div 2 = - (k div 2)\ by rule (simp flip: mult_minus_right) with True show ?thesis by simp next case False have \4 = 2 * (2::int)\ by simp also have \2 * 2 div 2 = (2::int)\ by (simp only: nonzero_mult_div_cancel_left) finally have *: \4 div 2 = (2::int)\ . from False obtain l where k: \k = 2 * l + 1\ .. then have \- k - 2 = 2 * - (l + 2) + 1\ by simp then have \(- k - 2) div 2 + 1 = - (k div 2) - 1\ by (simp flip: mult_minus_right add: *) (simp add: k) with False show ?thesis by simp qed finally have \(- k - 1) div 2 = - (k div 2) - 1\ . with Suc show ?case by (simp add: bit_Suc) qed lemma bit_nat_iff [bit_simps]: \bit (nat k) n \ k \ 0 \ bit k n\ proof (cases \k \ 0\) case True moreover define m where \m = nat k\ ultimately have \k = int m\ by simp then show ?thesis by (simp add: bit_simps) next case False then show ?thesis by simp qed subsection \Bit operations\ class semiring_bit_operations = semiring_bits + fixes "and" :: \'a \ 'a \ 'a\ (infixr \AND\ 64) and or :: \'a \ 'a \ 'a\ (infixr \OR\ 59) and xor :: \'a \ 'a \ 'a\ (infixr \XOR\ 59) and mask :: \nat \ 'a\ and set_bit :: \nat \ 'a \ 'a\ and unset_bit :: \nat \ 'a \ 'a\ and flip_bit :: \nat \ 'a \ 'a\ and push_bit :: \nat \ 'a \ 'a\ and drop_bit :: \nat \ 'a \ 'a\ and take_bit :: \nat \ 'a \ 'a\ assumes bit_and_iff [bit_simps]: \bit (a AND b) n \ bit a n \ bit b n\ and bit_or_iff [bit_simps]: \bit (a OR b) n \ bit a n \ bit b n\ and bit_xor_iff [bit_simps]: \bit (a XOR b) n \ bit a n \ bit b n\ and mask_eq_exp_minus_1: \mask n = 2 ^ n - 1\ and set_bit_eq_or: \set_bit n a = a OR push_bit n 1\ and bit_unset_bit_iff [bit_simps]: \bit (unset_bit m a) n \ bit a n \ m \ n\ and flip_bit_eq_xor: \flip_bit n a = a XOR push_bit n 1\ and push_bit_eq_mult: \push_bit n a = a * 2 ^ n\ and drop_bit_eq_div: \drop_bit n a = a div 2 ^ n\ and take_bit_eq_mod: \take_bit n a = a mod 2 ^ n\ begin text \ We want the bitwise operations to bind slightly weaker than \+\ and \-\. Logically, \<^const>\push_bit\, \<^const>\drop_bit\ and \<^const>\take_bit\ are just aliases; having them as separate operations makes proofs easier, otherwise proof automation would fiddle with concrete expressions \<^term>\2 ^ n\ in a way obfuscating the basic algebraic relationships between those operations. For the sake of code generation operations are specified as definitional class operations, taking into account that specific instances of these can be implemented differently wrt. code generation. \ sublocale "and": semilattice \(AND)\ by standard (auto simp add: bit_eq_iff bit_and_iff) sublocale or: semilattice_neutr \(OR)\ 0 by standard (auto simp add: bit_eq_iff bit_or_iff) sublocale xor: comm_monoid \(XOR)\ 0 by standard (auto simp add: bit_eq_iff bit_xor_iff) lemma even_and_iff: \even (a AND b) \ even a \ even b\ using bit_and_iff [of a b 0] by auto lemma even_or_iff: \even (a OR b) \ even a \ even b\ using bit_or_iff [of a b 0] by auto lemma even_xor_iff: \even (a XOR b) \ (even a \ even b)\ using bit_xor_iff [of a b 0] by auto lemma zero_and_eq [simp]: \0 AND a = 0\ by (simp add: bit_eq_iff bit_and_iff) lemma and_zero_eq [simp]: \a AND 0 = 0\ by (simp add: bit_eq_iff bit_and_iff) lemma one_and_eq: \1 AND a = a mod 2\ by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff) lemma and_one_eq: \a AND 1 = a mod 2\ using one_and_eq [of a] by (simp add: ac_simps) lemma one_or_eq: \1 OR a = a + of_bool (even a)\ by (simp add: bit_eq_iff bit_or_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff) lemma or_one_eq: \a OR 1 = a + of_bool (even a)\ using one_or_eq [of a] by (simp add: ac_simps) lemma one_xor_eq: \1 XOR a = a + of_bool (even a) - of_bool (odd a)\ by (simp add: bit_eq_iff bit_xor_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff odd_bit_iff_bit_pred elim: oddE) lemma xor_one_eq: \a XOR 1 = a + of_bool (even a) - of_bool (odd a)\ using one_xor_eq [of a] by (simp add: ac_simps) lemma bit_iff_odd_drop_bit: \bit a n \ odd (drop_bit n a)\ by (simp add: bit_iff_odd drop_bit_eq_div) lemma even_drop_bit_iff_not_bit: \even (drop_bit n a) \ \ bit a n\ by (simp add: bit_iff_odd_drop_bit) lemma div_push_bit_of_1_eq_drop_bit: \a div push_bit n 1 = drop_bit n a\ by (simp add: push_bit_eq_mult drop_bit_eq_div) lemma bits_ident: "push_bit n (drop_bit n a) + take_bit n a = a" using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div) lemma push_bit_push_bit [simp]: "push_bit m (push_bit n a) = push_bit (m + n) a" by (simp add: push_bit_eq_mult power_add ac_simps) lemma push_bit_0_id [simp]: "push_bit 0 = id" by (simp add: fun_eq_iff push_bit_eq_mult) lemma push_bit_of_0 [simp]: "push_bit n 0 = 0" by (simp add: push_bit_eq_mult) lemma push_bit_of_1: "push_bit n 1 = 2 ^ n" by (simp add: push_bit_eq_mult) lemma push_bit_Suc [simp]: "push_bit (Suc n) a = push_bit n (a * 2)" by (simp add: push_bit_eq_mult ac_simps) lemma push_bit_double: "push_bit n (a * 2) = push_bit n a * 2" by (simp add: push_bit_eq_mult ac_simps) lemma push_bit_add: "push_bit n (a + b) = push_bit n a + push_bit n b" by (simp add: push_bit_eq_mult algebra_simps) lemma push_bit_numeral [simp]: \push_bit (numeral l) (numeral k) = push_bit (pred_numeral l) (numeral (Num.Bit0 k))\ by (simp add: numeral_eq_Suc mult_2_right) (simp add: numeral_Bit0) lemma take_bit_0 [simp]: "take_bit 0 a = 0" by (simp add: take_bit_eq_mod) lemma take_bit_Suc: \take_bit (Suc n) a = take_bit n (a div 2) * 2 + a mod 2\ proof - have \take_bit (Suc n) (a div 2 * 2 + of_bool (odd a)) = take_bit n (a div 2) * 2 + of_bool (odd a)\ using even_succ_mod_exp [of \2 * (a div 2)\ \Suc n\] mult_exp_mod_exp_eq [of 1 \Suc n\ \a div 2\] by (auto simp add: take_bit_eq_mod ac_simps) then show ?thesis using div_mult_mod_eq [of a 2] by (simp add: mod_2_eq_odd) qed lemma take_bit_rec: \take_bit n a = (if n = 0 then 0 else take_bit (n - 1) (a div 2) * 2 + a mod 2)\ by (cases n) (simp_all add: take_bit_Suc) lemma take_bit_Suc_0 [simp]: \take_bit (Suc 0) a = a mod 2\ by (simp add: take_bit_eq_mod) lemma take_bit_of_0 [simp]: "take_bit n 0 = 0" by (simp add: take_bit_eq_mod) lemma take_bit_of_1 [simp]: "take_bit n 1 = of_bool (n > 0)" by (cases n) (simp_all add: take_bit_Suc) lemma drop_bit_of_0 [simp]: "drop_bit n 0 = 0" by (simp add: drop_bit_eq_div) lemma drop_bit_of_1 [simp]: "drop_bit n 1 = of_bool (n = 0)" by (simp add: drop_bit_eq_div) lemma drop_bit_0 [simp]: "drop_bit 0 = id" by (simp add: fun_eq_iff drop_bit_eq_div) lemma drop_bit_Suc: "drop_bit (Suc n) a = drop_bit n (a div 2)" using div_exp_eq [of a 1] by (simp add: drop_bit_eq_div) lemma drop_bit_rec: "drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (a div 2))" by (cases n) (simp_all add: drop_bit_Suc) lemma drop_bit_half: "drop_bit n (a div 2) = drop_bit n a div 2" by (induction n arbitrary: a) (simp_all add: drop_bit_Suc) lemma drop_bit_of_bool [simp]: "drop_bit n (of_bool b) = of_bool (n = 0 \ b)" by (cases n) simp_all lemma even_take_bit_eq [simp]: \even (take_bit n a) \ n = 0 \ even a\ by (simp add: take_bit_rec [of n a]) lemma take_bit_take_bit [simp]: "take_bit m (take_bit n a) = take_bit (min m n) a" by (simp add: take_bit_eq_mod mod_exp_eq ac_simps) lemma drop_bit_drop_bit [simp]: "drop_bit m (drop_bit n a) = drop_bit (m + n) a" by (simp add: drop_bit_eq_div power_add div_exp_eq ac_simps) lemma push_bit_take_bit: "push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)" apply (simp add: push_bit_eq_mult take_bit_eq_mod power_add ac_simps) using mult_exp_mod_exp_eq [of m \m + n\ a] apply (simp add: ac_simps power_add) done lemma take_bit_push_bit: "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)" proof (cases "m \ n") case True then show ?thesis apply (simp add:) apply (simp_all add: push_bit_eq_mult take_bit_eq_mod) apply (auto dest!: le_Suc_ex simp add: power_add ac_simps) using mult_exp_mod_exp_eq [of m m \a * 2 ^ n\ for n] apply (simp add: ac_simps) done next case False then show ?thesis using push_bit_take_bit [of n "m - n" a] by simp qed lemma take_bit_drop_bit: "take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)" by (simp add: drop_bit_eq_div take_bit_eq_mod ac_simps div_exp_mod_exp_eq) lemma drop_bit_take_bit: "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)" proof (cases "m \ n") case True then show ?thesis using take_bit_drop_bit [of "n - m" m a] by simp next case False then obtain q where \m = n + q\ by (auto simp add: not_le dest: less_imp_Suc_add) then have \drop_bit m (take_bit n a) = 0\ using div_exp_eq [of \a mod 2 ^ n\ n q] by (simp add: take_bit_eq_mod drop_bit_eq_div) with False show ?thesis by simp qed lemma even_push_bit_iff [simp]: \even (push_bit n a) \ n \ 0 \ even a\ by (simp add: push_bit_eq_mult) auto lemma bit_push_bit_iff [bit_simps]: \bit (push_bit m a) n \ m \ n \ 2 ^ n \ 0 \ bit a (n - m)\ by (auto simp add: bit_iff_odd push_bit_eq_mult even_mult_exp_div_exp_iff) lemma bit_drop_bit_eq [bit_simps]: \bit (drop_bit n a) = bit a \ (+) n\ by (simp add: bit_iff_odd fun_eq_iff ac_simps flip: drop_bit_eq_div) lemma bit_take_bit_iff [bit_simps]: \bit (take_bit m a) n \ n < m \ bit a n\ by (simp add: bit_iff_odd drop_bit_take_bit not_le flip: drop_bit_eq_div) lemma stable_imp_drop_bit_eq: \drop_bit n a = a\ if \a div 2 = a\ by (induction n) (simp_all add: that drop_bit_Suc) lemma stable_imp_take_bit_eq: \take_bit n a = (if even a then 0 else 2 ^ n - 1)\ if \a div 2 = a\ proof (rule bit_eqI) fix m assume \2 ^ m \ 0\ with that show \bit (take_bit n a) m \ bit (if even a then 0 else 2 ^ n - 1) m\ by (simp add: bit_take_bit_iff bit_mask_iff stable_imp_bit_iff_odd) qed lemma exp_dvdE: assumes \2 ^ n dvd a\ obtains b where \a = push_bit n b\ proof - from assms obtain b where \a = 2 ^ n * b\ .. then have \a = push_bit n b\ by (simp add: push_bit_eq_mult ac_simps) with that show thesis . qed lemma take_bit_eq_0_iff: \take_bit n a = 0 \ 2 ^ n dvd a\ (is \?P \ ?Q\) proof assume ?P then show ?Q by (simp add: take_bit_eq_mod mod_0_imp_dvd) next assume ?Q then obtain b where \a = push_bit n b\ by (rule exp_dvdE) then show ?P by (simp add: take_bit_push_bit) qed lemma take_bit_tightened: \take_bit m a = take_bit m b\ if \take_bit n a = take_bit n b\ and \m \ n\ proof - from that have \take_bit m (take_bit n a) = take_bit m (take_bit n b)\ by simp then have \take_bit (min m n) a = take_bit (min m n) b\ by simp with that show ?thesis by (simp add: min_def) qed lemma take_bit_eq_self_iff_drop_bit_eq_0: \take_bit n a = a \ drop_bit n a = 0\ (is \?P \ ?Q\) proof assume ?P show ?Q proof (rule bit_eqI) fix m from \?P\ have \a = take_bit n a\ .. also have \\ bit (take_bit n a) (n + m)\ unfolding bit_simps by (simp add: bit_simps) finally show \bit (drop_bit n a) m \ bit 0 m\ by (simp add: bit_simps) qed next assume ?Q show ?P proof (rule bit_eqI) fix m from \?Q\ have \\ bit (drop_bit n a) (m - n)\ by simp then have \ \ bit a (n + (m - n))\ by (simp add: bit_simps) then show \bit (take_bit n a) m \ bit a m\ by (cases \m < n\) (auto simp add: bit_simps) qed qed lemma drop_bit_exp_eq: \drop_bit m (2 ^ n) = of_bool (m \ n \ 2 ^ n \ 0) * 2 ^ (n - m)\ by (rule bit_eqI) (auto simp add: bit_simps) lemma take_bit_and [simp]: \take_bit n (a AND b) = take_bit n a AND take_bit n b\ by (auto simp add: bit_eq_iff bit_take_bit_iff bit_and_iff) lemma take_bit_or [simp]: \take_bit n (a OR b) = take_bit n a OR take_bit n b\ by (auto simp add: bit_eq_iff bit_take_bit_iff bit_or_iff) lemma take_bit_xor [simp]: \take_bit n (a XOR b) = take_bit n a XOR take_bit n b\ by (auto simp add: bit_eq_iff bit_take_bit_iff bit_xor_iff) lemma push_bit_and [simp]: \push_bit n (a AND b) = push_bit n a AND push_bit n b\ by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_and_iff) lemma push_bit_or [simp]: \push_bit n (a OR b) = push_bit n a OR push_bit n b\ by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_or_iff) lemma push_bit_xor [simp]: \push_bit n (a XOR b) = push_bit n a XOR push_bit n b\ by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_xor_iff) lemma drop_bit_and [simp]: \drop_bit n (a AND b) = drop_bit n a AND drop_bit n b\ by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_and_iff) lemma drop_bit_or [simp]: \drop_bit n (a OR b) = drop_bit n a OR drop_bit n b\ by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_or_iff) lemma drop_bit_xor [simp]: \drop_bit n (a XOR b) = drop_bit n a XOR drop_bit n b\ by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_xor_iff) lemma bit_mask_iff [bit_simps]: \bit (mask m) n \ 2 ^ n \ 0 \ n < m\ by (simp add: mask_eq_exp_minus_1 bit_mask_iff) lemma even_mask_iff: \even (mask n) \ n = 0\ using bit_mask_iff [of n 0] by auto lemma mask_0 [simp]: \mask 0 = 0\ by (simp add: mask_eq_exp_minus_1) lemma mask_Suc_0 [simp]: \mask (Suc 0) = 1\ by (simp add: mask_eq_exp_minus_1 add_implies_diff sym) lemma mask_Suc_exp: \mask (Suc n) = 2 ^ n OR mask n\ by (rule bit_eqI) (auto simp add: bit_or_iff bit_mask_iff bit_exp_iff not_less le_less_Suc_eq) lemma mask_Suc_double: \mask (Suc n) = 1 OR 2 * mask n\ proof (rule bit_eqI) fix q assume \2 ^ q \ 0\ show \bit (mask (Suc n)) q \ bit (1 OR 2 * mask n) q\ by (cases q) (simp_all add: even_mask_iff even_or_iff bit_or_iff bit_mask_iff bit_exp_iff bit_double_iff not_less le_less_Suc_eq bit_1_iff, auto simp add: mult_2) qed lemma mask_numeral: \mask (numeral n) = 1 + 2 * mask (pred_numeral n)\ by (simp add: numeral_eq_Suc mask_Suc_double one_or_eq ac_simps) lemma take_bit_mask [simp]: \take_bit m (mask n) = mask (min m n)\ by (rule bit_eqI) (simp add: bit_simps) lemma take_bit_eq_mask: \take_bit n a = a AND mask n\ by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_and_iff bit_mask_iff) lemma or_eq_0_iff: \a OR b = 0 \ a = 0 \ b = 0\ by (auto simp add: bit_eq_iff bit_or_iff) lemma disjunctive_add: \a + b = a OR b\ if \\n. \ bit a n \ \ bit b n\ by (rule bit_eqI) (use that in \simp add: bit_disjunctive_add_iff bit_or_iff\) lemma bit_iff_and_drop_bit_eq_1: \bit a n \ drop_bit n a AND 1 = 1\ by (simp add: bit_iff_odd_drop_bit and_one_eq odd_iff_mod_2_eq_one) lemma bit_iff_and_push_bit_not_eq_0: \bit a n \ a AND push_bit n 1 \ 0\ apply (cases \2 ^ n = 0\) apply (simp_all add: push_bit_of_1 bit_eq_iff bit_and_iff bit_push_bit_iff exp_eq_0_imp_not_bit) apply (simp_all add: bit_exp_iff) done lemmas set_bit_def = set_bit_eq_or lemma bit_set_bit_iff [bit_simps]: \bit (set_bit m a) n \ bit a n \ (m = n \ 2 ^ n \ 0)\ by (auto simp add: set_bit_def push_bit_of_1 bit_or_iff bit_exp_iff) lemma even_set_bit_iff: \even (set_bit m a) \ even a \ m \ 0\ using bit_set_bit_iff [of m a 0] by auto lemma even_unset_bit_iff: \even (unset_bit m a) \ even a \ m = 0\ using bit_unset_bit_iff [of m a 0] by auto lemma and_exp_eq_0_iff_not_bit: \a AND 2 ^ n = 0 \ \ bit a n\ (is \?P \ ?Q\) proof assume ?Q then show ?P by (auto intro: bit_eqI simp add: bit_simps) next assume ?P show ?Q proof (rule notI) assume \bit a n\ then have \a AND 2 ^ n = 2 ^ n\ by (auto intro: bit_eqI simp add: bit_simps) with \?P\ show False using \bit a n\ exp_eq_0_imp_not_bit by auto qed qed lemmas flip_bit_def = flip_bit_eq_xor lemma bit_flip_bit_iff [bit_simps]: \bit (flip_bit m a) n \ (m = n \ \ bit a n) \ 2 ^ n \ 0\ by (auto simp add: flip_bit_def push_bit_of_1 bit_xor_iff bit_exp_iff exp_eq_0_imp_not_bit) lemma even_flip_bit_iff: \even (flip_bit m a) \ \ (even a \ m = 0)\ using bit_flip_bit_iff [of m a 0] by auto lemma set_bit_0 [simp]: \set_bit 0 a = 1 + 2 * (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ then show \bit (set_bit 0 a) m = bit (1 + 2 * (a div 2)) m\ by (simp add: bit_set_bit_iff bit_double_iff even_bit_succ_iff) (cases m, simp_all add: bit_Suc) qed lemma set_bit_Suc: \set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ show \bit (set_bit (Suc n) a) m \ bit (a mod 2 + 2 * set_bit n (a div 2)) m\ proof (cases m) case 0 then show ?thesis by (simp add: even_set_bit_iff) next case (Suc m) with * have \2 ^ m \ 0\ using mult_2 by auto show ?thesis by (cases a rule: parity_cases) (simp_all add: bit_set_bit_iff bit_double_iff even_bit_succ_iff *, simp_all add: Suc \2 ^ m \ 0\ bit_Suc) qed qed lemma unset_bit_0 [simp]: \unset_bit 0 a = 2 * (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ then show \bit (unset_bit 0 a) m = bit (2 * (a div 2)) m\ by (simp add: bit_unset_bit_iff bit_double_iff) (cases m, simp_all add: bit_Suc) qed lemma unset_bit_Suc: \unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ then show \bit (unset_bit (Suc n) a) m \ bit (a mod 2 + 2 * unset_bit n (a div 2)) m\ proof (cases m) case 0 then show ?thesis by (simp add: even_unset_bit_iff) next case (Suc m) show ?thesis by (cases a rule: parity_cases) (simp_all add: bit_unset_bit_iff bit_double_iff even_bit_succ_iff *, simp_all add: Suc bit_Suc) qed qed lemma flip_bit_0 [simp]: \flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ then show \bit (flip_bit 0 a) m = bit (of_bool (even a) + 2 * (a div 2)) m\ by (simp add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff) (cases m, simp_all add: bit_Suc) qed lemma flip_bit_Suc: \flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ show \bit (flip_bit (Suc n) a) m \ bit (a mod 2 + 2 * flip_bit n (a div 2)) m\ proof (cases m) case 0 then show ?thesis by (simp add: even_flip_bit_iff) next case (Suc m) with * have \2 ^ m \ 0\ using mult_2 by auto show ?thesis by (cases a rule: parity_cases) (simp_all add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff, simp_all add: Suc \2 ^ m \ 0\ bit_Suc) qed qed lemma flip_bit_eq_if: \flip_bit n a = (if bit a n then unset_bit else set_bit) n a\ by (rule bit_eqI) (auto simp add: bit_set_bit_iff bit_unset_bit_iff bit_flip_bit_iff) lemma take_bit_set_bit_eq: \take_bit n (set_bit m a) = (if n \ m then take_bit n a else set_bit m (take_bit n a))\ by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_set_bit_iff) lemma take_bit_unset_bit_eq: \take_bit n (unset_bit m a) = (if n \ m then take_bit n a else unset_bit m (take_bit n a))\ by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_unset_bit_iff) lemma take_bit_flip_bit_eq: \take_bit n (flip_bit m a) = (if n \ m then take_bit n a else flip_bit m (take_bit n a))\ by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_flip_bit_iff) end class ring_bit_operations = semiring_bit_operations + ring_parity + fixes not :: \'a \ 'a\ (\NOT\) assumes bit_not_iff [bit_simps]: \\n. bit (NOT a) n \ 2 ^ n \ 0 \ \ bit a n\ assumes minus_eq_not_minus_1: \- a = NOT (a - 1)\ begin text \ For the sake of code generation \<^const>\not\ is specified as definitional class operation. Note that \<^const>\not\ has no sensible definition for unlimited but only positive bit strings (type \<^typ>\nat\). \ lemma bits_minus_1_mod_2_eq [simp]: \(- 1) mod 2 = 1\ by (simp add: mod_2_eq_odd) lemma not_eq_complement: \NOT a = - a - 1\ using minus_eq_not_minus_1 [of \a + 1\] by simp lemma minus_eq_not_plus_1: \- a = NOT a + 1\ using not_eq_complement [of a] by simp lemma bit_minus_iff [bit_simps]: \bit (- a) n \ 2 ^ n \ 0 \ \ bit (a - 1) n\ by (simp add: minus_eq_not_minus_1 bit_not_iff) lemma even_not_iff [simp]: \even (NOT a) \ odd a\ using bit_not_iff [of a 0] by auto lemma bit_not_exp_iff [bit_simps]: \bit (NOT (2 ^ m)) n \ 2 ^ n \ 0 \ n \ m\ by (auto simp add: bit_not_iff bit_exp_iff) lemma bit_minus_1_iff [simp]: \bit (- 1) n \ 2 ^ n \ 0\ by (simp add: bit_minus_iff) lemma bit_minus_exp_iff [bit_simps]: \bit (- (2 ^ m)) n \ 2 ^ n \ 0 \ n \ m\ by (auto simp add: bit_simps simp flip: mask_eq_exp_minus_1) lemma bit_minus_2_iff [simp]: \bit (- 2) n \ 2 ^ n \ 0 \ n > 0\ by (simp add: bit_minus_iff bit_1_iff) lemma not_one [simp]: \NOT 1 = - 2\ by (simp add: bit_eq_iff bit_not_iff) (simp add: bit_1_iff) sublocale "and": semilattice_neutr \(AND)\ \- 1\ by standard (rule bit_eqI, simp add: bit_and_iff) -sublocale bit: boolean_algebra \(AND)\ \(OR)\ NOT 0 \- 1\ - rewrites \bit.xor = (XOR)\ -proof - - interpret bit: boolean_algebra \(AND)\ \(OR)\ NOT 0 \- 1\ - by standard (auto simp add: bit_and_iff bit_or_iff bit_not_iff intro: bit_eqI) - show \boolean_algebra (AND) (OR) NOT 0 (- 1)\ - by standard - show \boolean_algebra.xor (AND) (OR) NOT = (XOR)\ - by (rule ext, rule ext, rule bit_eqI) - (auto simp add: bit.xor_def bit_and_iff bit_or_iff bit_xor_iff bit_not_iff) -qed +sublocale bit: abstract_boolean_algebra \(AND)\ \(OR)\ NOT 0 \- 1\ + by standard (auto simp add: bit_and_iff bit_or_iff bit_not_iff intro: bit_eqI) + +sublocale bit: abstract_boolean_algebra_sym_diff \(AND)\ \(OR)\ NOT 0 \- 1\ \(XOR)\ + apply standard + apply (rule bit_eqI) + apply (auto simp add: bit_simps) + done lemma and_eq_not_not_or: \a AND b = NOT (NOT a OR NOT b)\ by simp lemma or_eq_not_not_and: \a OR b = NOT (NOT a AND NOT b)\ by simp lemma not_add_distrib: \NOT (a + b) = NOT a - b\ by (simp add: not_eq_complement algebra_simps) lemma not_diff_distrib: \NOT (a - b) = NOT a + b\ using not_add_distrib [of a \- b\] by simp lemma and_eq_minus_1_iff: \a AND b = - 1 \ a = - 1 \ b = - 1\ proof assume \a = - 1 \ b = - 1\ then show \a AND b = - 1\ by simp next assume \a AND b = - 1\ have *: \bit a n\ \bit b n\ if \2 ^ n \ 0\ for n proof - from \a AND b = - 1\ have \bit (a AND b) n = bit (- 1) n\ by (simp add: bit_eq_iff) then show \bit a n\ \bit b n\ using that by (simp_all add: bit_and_iff) qed have \a = - 1\ by (rule bit_eqI) (simp add: *) moreover have \b = - 1\ by (rule bit_eqI) (simp add: *) ultimately show \a = - 1 \ b = - 1\ by simp qed lemma disjunctive_diff: \a - b = a AND NOT b\ if \\n. bit b n \ bit a n\ proof - have \NOT a + b = NOT a OR b\ by (rule disjunctive_add) (auto simp add: bit_not_iff dest: that) then have \NOT (NOT a + b) = NOT (NOT a OR b)\ by simp then show ?thesis by (simp add: not_add_distrib) qed lemma push_bit_minus: \push_bit n (- a) = - push_bit n a\ by (simp add: push_bit_eq_mult) lemma take_bit_not_take_bit: \take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)\ by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff) lemma take_bit_not_iff: \take_bit n (NOT a) = take_bit n (NOT b) \ take_bit n a = take_bit n b\ apply (simp add: bit_eq_iff) apply (simp add: bit_not_iff bit_take_bit_iff bit_exp_iff) apply (use exp_eq_0_imp_not_bit in blast) done lemma take_bit_not_eq_mask_diff: \take_bit n (NOT a) = mask n - take_bit n a\ proof - have \take_bit n (NOT a) = take_bit n (NOT (take_bit n a))\ by (simp add: take_bit_not_take_bit) also have \\ = mask n AND NOT (take_bit n a)\ by (simp add: take_bit_eq_mask ac_simps) also have \\ = mask n - take_bit n a\ by (subst disjunctive_diff) (auto simp add: bit_take_bit_iff bit_mask_iff exp_eq_0_imp_not_bit) finally show ?thesis by simp qed lemma mask_eq_take_bit_minus_one: \mask n = take_bit n (- 1)\ by (simp add: bit_eq_iff bit_mask_iff bit_take_bit_iff conj_commute) lemma take_bit_minus_one_eq_mask: \take_bit n (- 1) = mask n\ by (simp add: mask_eq_take_bit_minus_one) lemma minus_exp_eq_not_mask: \- (2 ^ n) = NOT (mask n)\ by (rule bit_eqI) (simp add: bit_minus_iff bit_not_iff flip: mask_eq_exp_minus_1) lemma push_bit_minus_one_eq_not_mask: \push_bit n (- 1) = NOT (mask n)\ by (simp add: push_bit_eq_mult minus_exp_eq_not_mask) lemma take_bit_not_mask_eq_0: \take_bit m (NOT (mask n)) = 0\ if \n \ m\ by (rule bit_eqI) (use that in \simp add: bit_take_bit_iff bit_not_iff bit_mask_iff\) lemma unset_bit_eq_and_not: \unset_bit n a = a AND NOT (push_bit n 1)\ by (rule bit_eqI) (auto simp add: bit_simps) lemmas unset_bit_def = unset_bit_eq_and_not end subsection \Instance \<^typ>\int\\ instantiation int :: ring_bit_operations begin definition not_int :: \int \ int\ where \not_int k = - k - 1\ lemma not_int_rec: \NOT k = of_bool (even k) + 2 * NOT (k div 2)\ for k :: int by (auto simp add: not_int_def elim: oddE) lemma even_not_iff_int: \even (NOT k) \ odd k\ for k :: int by (simp add: not_int_def) lemma not_int_div_2: \NOT k div 2 = NOT (k div 2)\ for k :: int by (cases k) (simp_all add: not_int_def divide_int_def nat_add_distrib) lemma bit_not_int_iff [bit_simps]: \bit (NOT k) n \ \ bit k n\ for k :: int by (simp add: bit_not_int_iff' not_int_def) function and_int :: \int \ int \ int\ where \(k::int) AND l = (if k \ {0, - 1} \ l \ {0, - 1} then - of_bool (odd k \ odd l) else of_bool (odd k \ odd l) + 2 * ((k div 2) AND (l div 2)))\ by auto termination proof (relation \measure (\(k, l). nat (\k\ + \l\))\) show \wf (measure (\(k, l). nat (\k\ + \l\)))\ by simp show \((k div 2, l div 2), k, l) \ measure (\(k, l). nat (\k\ + \l\))\ if \\ (k \ {0, - 1} \ l \ {0, - 1})\ for k l proof - have less_eq: \\k div 2\ \ \k\\ for k :: int by (cases k) (simp_all add: divide_int_def nat_add_distrib) have less: \\k div 2\ < \k\\ if \k \ {0, - 1}\ for k :: int proof (cases k) case (nonneg n) with that show ?thesis by (simp add: int_div_less_self) next case (neg n) with that have \n \ 0\ by simp then have \n div 2 < n\ by (simp add: div_less_iff_less_mult) with neg that show ?thesis by (simp add: divide_int_def nat_add_distrib) qed from that have *: \k \ {0, - 1} \ l \ {0, - 1}\ by simp then have \0 < \k\ + \l\\ by auto moreover from * have \\k div 2\ + \l div 2\ < \k\ + \l\\ proof assume \k \ {0, - 1}\ then have \\k div 2\ < \k\\ by (rule less) with less_eq [of l] show ?thesis by auto next assume \l \ {0, - 1}\ then have \\l div 2\ < \l\\ by (rule less) with less_eq [of k] show ?thesis by auto qed ultimately show ?thesis by simp qed qed declare and_int.simps [simp del] lemma and_int_rec: \k AND l = of_bool (odd k \ odd l) + 2 * ((k div 2) AND (l div 2))\ for k l :: int proof (cases \k \ {0, - 1} \ l \ {0, - 1}\) case True then show ?thesis by auto (simp_all add: and_int.simps) next case False then show ?thesis by (auto simp add: ac_simps and_int.simps [of k l]) qed lemma bit_and_int_iff: \bit (k AND l) n \ bit k n \ bit l n\ for k l :: int proof (induction n arbitrary: k l) case 0 then show ?case by (simp add: and_int_rec [of k l]) next case (Suc n) then show ?case by (simp add: and_int_rec [of k l] bit_Suc) qed lemma even_and_iff_int: \even (k AND l) \ even k \ even l\ for k l :: int using bit_and_int_iff [of k l 0] by auto definition or_int :: \int \ int \ int\ where \k OR l = NOT (NOT k AND NOT l)\ for k l :: int lemma or_int_rec: \k OR l = of_bool (odd k \ odd l) + 2 * ((k div 2) OR (l div 2))\ for k l :: int using and_int_rec [of \NOT k\ \NOT l\] by (simp add: or_int_def even_not_iff_int not_int_div_2) (simp_all add: not_int_def) lemma bit_or_int_iff: \bit (k OR l) n \ bit k n \ bit l n\ for k l :: int by (simp add: or_int_def bit_not_int_iff bit_and_int_iff) definition xor_int :: \int \ int \ int\ where \k XOR l = k AND NOT l OR NOT k AND l\ for k l :: int lemma xor_int_rec: \k XOR l = of_bool (odd k \ odd l) + 2 * ((k div 2) XOR (l div 2))\ for k l :: int by (simp add: xor_int_def or_int_rec [of \k AND NOT l\ \NOT k AND l\] even_and_iff_int even_not_iff_int) (simp add: and_int_rec [of \NOT k\ \l\] and_int_rec [of \k\ \NOT l\] not_int_div_2) lemma bit_xor_int_iff: \bit (k XOR l) n \ bit k n \ bit l n\ for k l :: int by (auto simp add: xor_int_def bit_or_int_iff bit_and_int_iff bit_not_int_iff) definition mask_int :: \nat \ int\ where \mask n = (2 :: int) ^ n - 1\ definition push_bit_int :: \nat \ int \ int\ where \push_bit_int n k = k * 2 ^ n\ definition drop_bit_int :: \nat \ int \ int\ where \drop_bit_int n k = k div 2 ^ n\ definition take_bit_int :: \nat \ int \ int\ where \take_bit_int n k = k mod 2 ^ n\ definition set_bit_int :: \nat \ int \ int\ where \set_bit n k = k OR push_bit n 1\ for k :: int definition unset_bit_int :: \nat \ int \ int\ where \unset_bit n k = k AND NOT (push_bit n 1)\ for k :: int definition flip_bit_int :: \nat \ int \ int\ where \flip_bit n k = k XOR push_bit n 1\ for k :: int instance proof fix k l :: int and m n :: nat show \- k = NOT (k - 1)\ by (simp add: not_int_def) show \bit (k AND l) n \ bit k n \ bit l n\ by (fact bit_and_int_iff) show \bit (k OR l) n \ bit k n \ bit l n\ by (fact bit_or_int_iff) show \bit (k XOR l) n \ bit k n \ bit l n\ by (fact bit_xor_int_iff) show \bit (unset_bit m k) n \ bit k n \ m \ n\ proof - have \unset_bit m k = k AND NOT (push_bit m 1)\ by (simp add: unset_bit_int_def) also have \NOT (push_bit m 1 :: int) = - (push_bit m 1 + 1)\ by (simp add: not_int_def) finally show ?thesis by (simp only: bit_simps bit_and_int_iff) (auto simp add: bit_simps bit_not_int_iff' push_bit_int_def) qed qed (simp_all add: bit_not_int_iff mask_int_def set_bit_int_def flip_bit_int_def push_bit_int_def drop_bit_int_def take_bit_int_def) end lemma bit_push_bit_iff_int: \bit (push_bit m k) n \ m \ n \ bit k (n - m)\ for k :: int by (auto simp add: bit_push_bit_iff) lemma take_bit_nonnegative [simp]: \take_bit n k \ 0\ for k :: int by (simp add: take_bit_eq_mod) lemma not_take_bit_negative [simp]: \\ take_bit n k < 0\ for k :: int by (simp add: not_less) lemma take_bit_int_less_exp [simp]: \take_bit n k < 2 ^ n\ for k :: int by (simp add: take_bit_eq_mod) lemma take_bit_int_eq_self_iff: \take_bit n k = k \ 0 \ k \ k < 2 ^ n\ (is \?P \ ?Q\) for k :: int proof assume ?P moreover note take_bit_int_less_exp [of n k] take_bit_nonnegative [of n k] ultimately show ?Q by simp next assume ?Q then show ?P by (simp add: take_bit_eq_mod) qed lemma take_bit_int_eq_self: \take_bit n k = k\ if \0 \ k\ \k < 2 ^ n\ for k :: int using that by (simp add: take_bit_int_eq_self_iff) lemma mask_half_int: \mask n div 2 = (mask (n - 1) :: int)\ by (cases n) (simp_all add: mask_eq_exp_minus_1 algebra_simps) lemma mask_nonnegative_int [simp]: \mask n \ (0::int)\ by (simp add: mask_eq_exp_minus_1) lemma not_mask_negative_int [simp]: \\ mask n < (0::int)\ by (simp add: not_less) lemma not_nonnegative_int_iff [simp]: \NOT k \ 0 \ k < 0\ for k :: int by (simp add: not_int_def) lemma not_negative_int_iff [simp]: \NOT k < 0 \ k \ 0\ for k :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less not_le) lemma and_nonnegative_int_iff [simp]: \k AND l \ 0 \ k \ 0 \ l \ 0\ for k l :: int proof (induction k arbitrary: l rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even k) then show ?case using and_int_rec [of \k * 2\ l] by (simp add: pos_imp_zdiv_nonneg_iff zero_le_mult_iff) next case (odd k) from odd have \0 \ k AND l div 2 \ 0 \ k \ 0 \ l div 2\ by simp then have \0 \ (1 + k * 2) div 2 AND l div 2 \ 0 \ (1 + k * 2) div 2 \ 0 \ l div 2\ by simp with and_int_rec [of \1 + k * 2\ l] show ?case by (auto simp add: zero_le_mult_iff not_le) qed lemma and_negative_int_iff [simp]: \k AND l < 0 \ k < 0 \ l < 0\ for k l :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less) lemma and_less_eq: \k AND l \ k\ if \l < 0\ for k l :: int using that proof (induction k arbitrary: l rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even k) from even.IH [of \l div 2\] even.hyps even.prems show ?case by (simp add: and_int_rec [of _ l]) next case (odd k) from odd.IH [of \l div 2\] odd.hyps odd.prems show ?case by (simp add: and_int_rec [of _ l]) linarith qed lemma or_nonnegative_int_iff [simp]: \k OR l \ 0 \ k \ 0 \ l \ 0\ for k l :: int by (simp only: or_eq_not_not_and not_nonnegative_int_iff) simp lemma or_negative_int_iff [simp]: \k OR l < 0 \ k < 0 \ l < 0\ for k l :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less) lemma or_greater_eq: \k OR l \ k\ if \l \ 0\ for k l :: int using that proof (induction k arbitrary: l rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even k) from even.IH [of \l div 2\] even.hyps even.prems show ?case by (simp add: or_int_rec [of _ l]) linarith next case (odd k) from odd.IH [of \l div 2\] odd.hyps odd.prems show ?case by (simp add: or_int_rec [of _ l]) qed lemma xor_nonnegative_int_iff [simp]: \k XOR l \ 0 \ (k \ 0 \ l \ 0)\ for k l :: int by (simp only: bit.xor_def or_nonnegative_int_iff) auto lemma xor_negative_int_iff [simp]: \k XOR l < 0 \ (k < 0) \ (l < 0)\ for k l :: int by (subst Not_eq_iff [symmetric]) (auto simp add: not_less) lemma OR_upper: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ \x < 2 ^ n\ \y < 2 ^ n\ shows \x OR y < 2 ^ n\ using assms proof (induction x arbitrary: y n rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even x) from even.IH [of \n - 1\ \y div 2\] even.prems even.hyps show ?case by (cases n) (auto simp add: or_int_rec [of \_ * 2\] elim: oddE) next case (odd x) from odd.IH [of \n - 1\ \y div 2\] odd.prems odd.hyps show ?case by (cases n) (auto simp add: or_int_rec [of \1 + _ * 2\], linarith) qed lemma XOR_upper: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ \x < 2 ^ n\ \y < 2 ^ n\ shows \x XOR y < 2 ^ n\ using assms proof (induction x arbitrary: y n rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even x) from even.IH [of \n - 1\ \y div 2\] even.prems even.hyps show ?case by (cases n) (auto simp add: xor_int_rec [of \_ * 2\] elim: oddE) next case (odd x) from odd.IH [of \n - 1\ \y div 2\] odd.prems odd.hyps show ?case by (cases n) (auto simp add: xor_int_rec [of \1 + _ * 2\]) qed lemma AND_lower [simp]: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ shows \0 \ x AND y\ using assms by simp lemma OR_lower [simp]: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ \0 \ y\ shows \0 \ x OR y\ using assms by simp lemma XOR_lower [simp]: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ \0 \ y\ shows \0 \ x XOR y\ using assms by simp lemma AND_upper1 [simp]: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ x\ shows \x AND y \ x\ using assms proof (induction x arbitrary: y rule: int_bit_induct) case (odd k) then have \k AND y div 2 \ k\ by simp then show ?case by (simp add: and_int_rec [of \1 + _ * 2\]) qed (simp_all add: and_int_rec [of \_ * 2\]) lemmas AND_upper1' [simp] = order_trans [OF AND_upper1] \<^marker>\contributor \Stefan Berghofer\\ lemmas AND_upper1'' [simp] = order_le_less_trans [OF AND_upper1] \<^marker>\contributor \Stefan Berghofer\\ lemma AND_upper2 [simp]: \<^marker>\contributor \Stefan Berghofer\\ fixes x y :: int assumes \0 \ y\ shows \x AND y \ y\ using assms AND_upper1 [of y x] by (simp add: ac_simps) lemmas AND_upper2' [simp] = order_trans [OF AND_upper2] \<^marker>\contributor \Stefan Berghofer\\ lemmas AND_upper2'' [simp] = order_le_less_trans [OF AND_upper2] \<^marker>\contributor \Stefan Berghofer\\ lemma plus_and_or: \(x AND y) + (x OR y) = x + y\ for x y :: int proof (induction x arbitrary: y rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even x) from even.IH [of \y div 2\] show ?case by (auto simp add: and_int_rec [of _ y] or_int_rec [of _ y] elim: oddE) next case (odd x) from odd.IH [of \y div 2\] show ?case by (auto simp add: and_int_rec [of _ y] or_int_rec [of _ y] elim: oddE) qed lemma push_bit_minus_one: "push_bit n (- 1 :: int) = - (2 ^ n)" by (simp add: push_bit_eq_mult) lemma minus_1_div_exp_eq_int: \- 1 div (2 :: int) ^ n = - 1\ by (induction n) (use div_exp_eq [symmetric, of \- 1 :: int\ 1] in \simp_all add: ac_simps\) lemma drop_bit_minus_one [simp]: \drop_bit n (- 1 :: int) = - 1\ by (simp add: drop_bit_eq_div minus_1_div_exp_eq_int) lemma take_bit_Suc_from_most: \take_bit (Suc n) k = 2 ^ n * of_bool (bit k n) + take_bit n k\ for k :: int by (simp only: take_bit_eq_mod power_Suc2) (simp_all add: bit_iff_odd odd_iff_mod_2_eq_one zmod_zmult2_eq) lemma take_bit_minus: \take_bit n (- take_bit n k) = take_bit n (- k)\ for k :: int by (simp add: take_bit_eq_mod mod_minus_eq) lemma take_bit_diff: \take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)\ for k l :: int by (simp add: take_bit_eq_mod mod_diff_eq) lemma bit_imp_take_bit_positive: \0 < take_bit m k\ if \n < m\ and \bit k n\ for k :: int proof (rule ccontr) assume \\ 0 < take_bit m k\ then have \take_bit m k = 0\ by (auto simp add: not_less intro: order_antisym) then have \bit (take_bit m k) n = bit 0 n\ by simp with that show False by (simp add: bit_take_bit_iff) qed lemma take_bit_mult: \take_bit n (take_bit n k * take_bit n l) = take_bit n (k * l)\ for k l :: int by (simp add: take_bit_eq_mod mod_mult_eq) lemma (in ring_1) of_nat_nat_take_bit_eq [simp]: \of_nat (nat (take_bit n k)) = of_int (take_bit n k)\ by simp lemma take_bit_minus_small_eq: \take_bit n (- k) = 2 ^ n - k\ if \0 < k\ \k \ 2 ^ n\ for k :: int proof - define m where \m = nat k\ with that have \k = int m\ and \0 < m\ and \m \ 2 ^ n\ by simp_all have \(2 ^ n - m) mod 2 ^ n = 2 ^ n - m\ using \0 < m\ by simp then have \int ((2 ^ n - m) mod 2 ^ n) = int (2 ^ n - m)\ by simp then have \(2 ^ n - int m) mod 2 ^ n = 2 ^ n - int m\ using \m \ 2 ^ n\ by (simp only: of_nat_mod of_nat_diff) simp with \k = int m\ have \(2 ^ n - k) mod 2 ^ n = 2 ^ n - k\ by simp then show ?thesis by (simp add: take_bit_eq_mod) qed lemma drop_bit_push_bit_int: \drop_bit m (push_bit n k) = drop_bit (m - n) (push_bit (n - m) k)\ for k :: int by (cases \m \ n\) (auto simp add: mult.left_commute [of _ \2 ^ n\] mult.commute [of _ \2 ^ n\] mult.assoc mult.commute [of k] drop_bit_eq_div push_bit_eq_mult not_le power_add dest!: le_Suc_ex less_imp_Suc_add) lemma push_bit_nonnegative_int_iff [simp]: \push_bit n k \ 0 \ k \ 0\ for k :: int by (simp add: push_bit_eq_mult zero_le_mult_iff power_le_zero_eq) lemma push_bit_negative_int_iff [simp]: \push_bit n k < 0 \ k < 0\ for k :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less) lemma drop_bit_nonnegative_int_iff [simp]: \drop_bit n k \ 0 \ k \ 0\ for k :: int by (induction n) (auto simp add: drop_bit_Suc drop_bit_half) lemma drop_bit_negative_int_iff [simp]: \drop_bit n k < 0 \ k < 0\ for k :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less) lemma set_bit_nonnegative_int_iff [simp]: \set_bit n k \ 0 \ k \ 0\ for k :: int by (simp add: set_bit_def) lemma set_bit_negative_int_iff [simp]: \set_bit n k < 0 \ k < 0\ for k :: int by (simp add: set_bit_def) lemma unset_bit_nonnegative_int_iff [simp]: \unset_bit n k \ 0 \ k \ 0\ for k :: int by (simp add: unset_bit_def) lemma unset_bit_negative_int_iff [simp]: \unset_bit n k < 0 \ k < 0\ for k :: int by (simp add: unset_bit_def) lemma flip_bit_nonnegative_int_iff [simp]: \flip_bit n k \ 0 \ k \ 0\ for k :: int by (simp add: flip_bit_def) lemma flip_bit_negative_int_iff [simp]: \flip_bit n k < 0 \ k < 0\ for k :: int by (simp add: flip_bit_def) lemma set_bit_greater_eq: \set_bit n k \ k\ for k :: int by (simp add: set_bit_def or_greater_eq) lemma unset_bit_less_eq: \unset_bit n k \ k\ for k :: int by (simp add: unset_bit_def and_less_eq) lemma set_bit_eq: \set_bit n k = k + of_bool (\ bit k n) * 2 ^ n\ for k :: int proof (rule bit_eqI) fix m show \bit (set_bit n k) m \ bit (k + of_bool (\ bit k n) * 2 ^ n) m\ proof (cases \m = n\) case True then show ?thesis apply (simp add: bit_set_bit_iff) apply (simp add: bit_iff_odd div_plus_div_distrib_dvd_right) done next case False then show ?thesis apply (clarsimp simp add: bit_set_bit_iff) apply (subst disjunctive_add) apply (clarsimp simp add: bit_exp_iff) apply (clarsimp simp add: bit_or_iff bit_exp_iff) done qed qed lemma unset_bit_eq: \unset_bit n k = k - of_bool (bit k n) * 2 ^ n\ for k :: int proof (rule bit_eqI) fix m show \bit (unset_bit n k) m \ bit (k - of_bool (bit k n) * 2 ^ n) m\ proof (cases \m = n\) case True then show ?thesis apply (simp add: bit_unset_bit_iff) apply (simp add: bit_iff_odd) using div_plus_div_distrib_dvd_right [of \2 ^ n\ \- (2 ^ n)\ k] apply (simp add: dvd_neg_div) done next case False then show ?thesis apply (clarsimp simp add: bit_unset_bit_iff) apply (subst disjunctive_diff) apply (clarsimp simp add: bit_exp_iff) apply (clarsimp simp add: bit_and_iff bit_not_iff bit_exp_iff) done qed qed lemma and_int_unfold [code]: \k AND l = (if k = 0 \ l = 0 then 0 else if k = - 1 then l else if l = - 1 then k else (k mod 2) * (l mod 2) + 2 * ((k div 2) AND (l div 2)))\ for k l :: int by (auto simp add: and_int_rec [of k l] zmult_eq_1_iff elim: oddE) lemma or_int_unfold [code]: \k OR l = (if k = - 1 \ l = - 1 then - 1 else if k = 0 then l else if l = 0 then k else max (k mod 2) (l mod 2) + 2 * ((k div 2) OR (l div 2)))\ for k l :: int by (auto simp add: or_int_rec [of k l] elim: oddE) lemma xor_int_unfold [code]: \k XOR l = (if k = - 1 then NOT l else if l = - 1 then NOT k else if k = 0 then l else if l = 0 then k else \k mod 2 - l mod 2\ + 2 * ((k div 2) XOR (l div 2)))\ for k l :: int by (auto simp add: xor_int_rec [of k l] not_int_def elim!: oddE) subsection \Instance \<^typ>\nat\\ instantiation nat :: semiring_bit_operations begin definition and_nat :: \nat \ nat \ nat\ where \m AND n = nat (int m AND int n)\ for m n :: nat definition or_nat :: \nat \ nat \ nat\ where \m OR n = nat (int m OR int n)\ for m n :: nat definition xor_nat :: \nat \ nat \ nat\ where \m XOR n = nat (int m XOR int n)\ for m n :: nat definition mask_nat :: \nat \ nat\ where \mask n = (2 :: nat) ^ n - 1\ definition push_bit_nat :: \nat \ nat \ nat\ where \push_bit_nat n m = m * 2 ^ n\ definition drop_bit_nat :: \nat \ nat \ nat\ where \drop_bit_nat n m = m div 2 ^ n\ definition take_bit_nat :: \nat \ nat \ nat\ where \take_bit_nat n m = m mod 2 ^ n\ definition set_bit_nat :: \nat \ nat \ nat\ where \set_bit m n = n OR push_bit m 1\ for m n :: nat definition unset_bit_nat :: \nat \ nat \ nat\ where \unset_bit m n = nat (unset_bit m (int n))\ for m n :: nat definition flip_bit_nat :: \nat \ nat \ nat\ where \flip_bit m n = n XOR push_bit m 1\ for m n :: nat instance proof fix m n q :: nat show \bit (m AND n) q \ bit m q \ bit n q\ by (simp add: and_nat_def bit_simps) show \bit (m OR n) q \ bit m q \ bit n q\ by (simp add: or_nat_def bit_simps) show \bit (m XOR n) q \ bit m q \ bit n q\ by (simp add: xor_nat_def bit_simps) show \bit (unset_bit m n) q \ bit n q \ m \ q\ by (simp add: unset_bit_nat_def bit_simps) qed (simp_all add: mask_nat_def set_bit_nat_def flip_bit_nat_def push_bit_nat_def drop_bit_nat_def take_bit_nat_def) end lemma take_bit_nat_less_exp [simp]: \take_bit n m < 2 ^ n\ for n m ::nat by (simp add: take_bit_eq_mod) lemma take_bit_nat_eq_self_iff: \take_bit n m = m \ m < 2 ^ n\ (is \?P \ ?Q\) for n m :: nat proof assume ?P moreover note take_bit_nat_less_exp [of n m] ultimately show ?Q by simp next assume ?Q then show ?P by (simp add: take_bit_eq_mod) qed lemma take_bit_nat_eq_self: \take_bit n m = m\ if \m < 2 ^ n\ for m n :: nat using that by (simp add: take_bit_nat_eq_self_iff) lemma take_bit_nat_less_eq_self [simp]: \take_bit n m \ m\ for n m :: nat by (simp add: take_bit_eq_mod) lemma take_bit_nat_less_self_iff: \take_bit n m < m \ 2 ^ n \ m\ (is \?P \ ?Q\) for m n :: nat proof assume ?P then have \take_bit n m \ m\ by simp then show \?Q\ by (simp add: take_bit_nat_eq_self_iff) next have \take_bit n m < 2 ^ n\ by (fact take_bit_nat_less_exp) also assume ?Q finally show ?P . qed lemma bit_push_bit_iff_nat: \bit (push_bit m q) n \ m \ n \ bit q (n - m)\ for q :: nat by (auto simp add: bit_push_bit_iff) lemma and_nat_rec: \m AND n = of_bool (odd m \ odd n) + 2 * ((m div 2) AND (n div 2))\ for m n :: nat apply (simp add: and_nat_def and_int_rec [of \int m\ \int n\] zdiv_int nat_add_distrib nat_mult_distrib) apply (subst nat_add_distrib) apply auto done lemma or_nat_rec: \m OR n = of_bool (odd m \ odd n) + 2 * ((m div 2) OR (n div 2))\ for m n :: nat apply (simp add: or_nat_def or_int_rec [of \int m\ \int n\] zdiv_int nat_add_distrib nat_mult_distrib) apply (subst nat_add_distrib) apply auto done lemma xor_nat_rec: \m XOR n = of_bool (odd m \ odd n) + 2 * ((m div 2) XOR (n div 2))\ for m n :: nat apply (simp add: xor_nat_def xor_int_rec [of \int m\ \int n\] zdiv_int nat_add_distrib nat_mult_distrib) apply (subst nat_add_distrib) apply auto done lemma Suc_0_and_eq [simp]: \Suc 0 AND n = n mod 2\ using one_and_eq [of n] by simp lemma and_Suc_0_eq [simp]: \n AND Suc 0 = n mod 2\ using and_one_eq [of n] by simp lemma Suc_0_or_eq: \Suc 0 OR n = n + of_bool (even n)\ using one_or_eq [of n] by simp lemma or_Suc_0_eq: \n OR Suc 0 = n + of_bool (even n)\ using or_one_eq [of n] by simp lemma Suc_0_xor_eq: \Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\ using one_xor_eq [of n] by simp lemma xor_Suc_0_eq: \n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\ using xor_one_eq [of n] by simp lemma and_nat_unfold [code]: \m AND n = (if m = 0 \ n = 0 then 0 else (m mod 2) * (n mod 2) + 2 * ((m div 2) AND (n div 2)))\ for m n :: nat by (auto simp add: and_nat_rec [of m n] elim: oddE) lemma or_nat_unfold [code]: \m OR n = (if m = 0 then n else if n = 0 then m else max (m mod 2) (n mod 2) + 2 * ((m div 2) OR (n div 2)))\ for m n :: nat by (auto simp add: or_nat_rec [of m n] elim: oddE) lemma xor_nat_unfold [code]: \m XOR n = (if m = 0 then n else if n = 0 then m else (m mod 2 + n mod 2) mod 2 + 2 * ((m div 2) XOR (n div 2)))\ for m n :: nat by (auto simp add: xor_nat_rec [of m n] elim!: oddE) lemma [code]: \unset_bit 0 m = 2 * (m div 2)\ \unset_bit (Suc n) m = m mod 2 + 2 * unset_bit n (m div 2)\ by (simp_all add: unset_bit_Suc) subsection \Common algebraic structure\ class unique_euclidean_semiring_with_bit_operations = unique_euclidean_semiring_with_nat + semiring_bit_operations begin lemma take_bit_of_exp [simp]: \take_bit m (2 ^ n) = of_bool (n < m) * 2 ^ n\ by (simp add: take_bit_eq_mod exp_mod_exp) lemma take_bit_of_2 [simp]: \take_bit n 2 = of_bool (2 \ n) * 2\ using take_bit_of_exp [of n 1] by simp lemma take_bit_of_mask: \take_bit m (2 ^ n - 1) = 2 ^ min m n - 1\ by (simp add: take_bit_eq_mod mask_mod_exp) lemma push_bit_eq_0_iff [simp]: "push_bit n a = 0 \ a = 0" by (simp add: push_bit_eq_mult) lemma take_bit_add: "take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)" by (simp add: take_bit_eq_mod mod_simps) lemma take_bit_of_1_eq_0_iff [simp]: "take_bit n 1 = 0 \ n = 0" by (simp add: take_bit_eq_mod) lemma take_bit_Suc_1 [simp]: \take_bit (Suc n) 1 = 1\ by (simp add: take_bit_Suc) lemma take_bit_Suc_bit0 [simp]: \take_bit (Suc n) (numeral (Num.Bit0 k)) = take_bit n (numeral k) * 2\ by (simp add: take_bit_Suc numeral_Bit0_div_2) lemma take_bit_Suc_bit1 [simp]: \take_bit (Suc n) (numeral (Num.Bit1 k)) = take_bit n (numeral k) * 2 + 1\ by (simp add: take_bit_Suc numeral_Bit1_div_2 mod_2_eq_odd) lemma take_bit_numeral_1 [simp]: \take_bit (numeral l) 1 = 1\ by (simp add: take_bit_rec [of \numeral l\ 1]) lemma take_bit_numeral_bit0 [simp]: \take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2\ by (simp add: take_bit_rec numeral_Bit0_div_2) lemma take_bit_numeral_bit1 [simp]: \take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1\ by (simp add: take_bit_rec numeral_Bit1_div_2 mod_2_eq_odd) lemma drop_bit_Suc_bit0 [simp]: \drop_bit (Suc n) (numeral (Num.Bit0 k)) = drop_bit n (numeral k)\ by (simp add: drop_bit_Suc numeral_Bit0_div_2) lemma drop_bit_Suc_bit1 [simp]: \drop_bit (Suc n) (numeral (Num.Bit1 k)) = drop_bit n (numeral k)\ by (simp add: drop_bit_Suc numeral_Bit1_div_2) lemma drop_bit_numeral_bit0 [simp]: \drop_bit (numeral l) (numeral (Num.Bit0 k)) = drop_bit (pred_numeral l) (numeral k)\ by (simp add: drop_bit_rec numeral_Bit0_div_2) lemma drop_bit_numeral_bit1 [simp]: \drop_bit (numeral l) (numeral (Num.Bit1 k)) = drop_bit (pred_numeral l) (numeral k)\ by (simp add: drop_bit_rec numeral_Bit1_div_2) lemma drop_bit_of_nat: "drop_bit n (of_nat m) = of_nat (drop_bit n m)" by (simp add: drop_bit_eq_div Bit_Operations.drop_bit_eq_div of_nat_div [of m "2 ^ n"]) lemma bit_of_nat_iff_bit [bit_simps]: \bit (of_nat m) n \ bit m n\ proof - have \even (m div 2 ^ n) \ even (of_nat (m div 2 ^ n))\ by simp also have \of_nat (m div 2 ^ n) = of_nat m div of_nat (2 ^ n)\ by (simp add: of_nat_div) finally show ?thesis by (simp add: bit_iff_odd semiring_bits_class.bit_iff_odd) qed lemma of_nat_drop_bit: \of_nat (drop_bit m n) = drop_bit m (of_nat n)\ by (simp add: drop_bit_eq_div Bit_Operations.drop_bit_eq_div of_nat_div) lemma bit_push_bit_iff_of_nat_iff [bit_simps]: \bit (push_bit m (of_nat r)) n \ m \ n \ bit (of_nat r) (n - m)\ by (auto simp add: bit_push_bit_iff) lemma take_bit_sum: "take_bit n a = (\k = 0..k = 0..k = Suc 0..k = Suc 0..k = 0..More properties\ lemma take_bit_eq_mask_iff: \take_bit n k = mask n \ take_bit n (k + 1) = 0\ (is \?P \ ?Q\) for k :: int proof assume ?P then have \take_bit n (take_bit n k + take_bit n 1) = 0\ by (simp add: mask_eq_exp_minus_1 take_bit_eq_0_iff) then show ?Q by (simp only: take_bit_add) next assume ?Q then have \take_bit n (k + 1) - 1 = - 1\ by simp then have \take_bit n (take_bit n (k + 1) - 1) = take_bit n (- 1)\ by simp moreover have \take_bit n (take_bit n (k + 1) - 1) = take_bit n k\ by (simp add: take_bit_eq_mod mod_simps) ultimately show ?P by (simp add: take_bit_minus_one_eq_mask) qed lemma take_bit_eq_mask_iff_exp_dvd: \take_bit n k = mask n \ 2 ^ n dvd k + 1\ for k :: int by (simp add: take_bit_eq_mask_iff flip: take_bit_eq_0_iff) context ring_bit_operations begin lemma even_of_int_iff: \even (of_int k) \ even k\ by (induction k rule: int_bit_induct) simp_all lemma bit_of_int_iff [bit_simps]: \bit (of_int k) n \ (2::'a) ^ n \ 0 \ bit k n\ proof (cases \(2::'a) ^ n = 0\) case True then show ?thesis by (simp add: exp_eq_0_imp_not_bit) next case False then have \bit (of_int k) n \ bit k n\ proof (induction k arbitrary: n rule: int_bit_induct) case zero then show ?case by simp next case minus then show ?case by simp next case (even k) then show ?case using bit_double_iff [of \of_int k\ n] Bit_Operations.bit_double_iff [of k n] by (cases n) (auto simp add: ac_simps dest: mult_not_zero) next case (odd k) then show ?case using bit_double_iff [of \of_int k\ n] by (cases n) (auto simp add: ac_simps bit_double_iff even_bit_succ_iff Bit_Operations.bit_Suc dest: mult_not_zero) qed with False show ?thesis by simp qed lemma push_bit_of_int: \push_bit n (of_int k) = of_int (push_bit n k)\ by (simp add: push_bit_eq_mult Bit_Operations.push_bit_eq_mult) lemma of_int_push_bit: \of_int (push_bit n k) = push_bit n (of_int k)\ by (simp add: push_bit_eq_mult Bit_Operations.push_bit_eq_mult) lemma take_bit_of_int: \take_bit n (of_int k) = of_int (take_bit n k)\ by (rule bit_eqI) (simp add: bit_take_bit_iff Bit_Operations.bit_take_bit_iff bit_of_int_iff) lemma of_int_take_bit: \of_int (take_bit n k) = take_bit n (of_int k)\ by (rule bit_eqI) (simp add: bit_take_bit_iff Bit_Operations.bit_take_bit_iff bit_of_int_iff) lemma of_int_not_eq: \of_int (NOT k) = NOT (of_int k)\ by (rule bit_eqI) (simp add: bit_not_iff Bit_Operations.bit_not_iff bit_of_int_iff) lemma of_int_and_eq: \of_int (k AND l) = of_int k AND of_int l\ by (rule bit_eqI) (simp add: bit_of_int_iff bit_and_iff Bit_Operations.bit_and_iff) lemma of_int_or_eq: \of_int (k OR l) = of_int k OR of_int l\ by (rule bit_eqI) (simp add: bit_of_int_iff bit_or_iff Bit_Operations.bit_or_iff) lemma of_int_xor_eq: \of_int (k XOR l) = of_int k XOR of_int l\ by (rule bit_eqI) (simp add: bit_of_int_iff bit_xor_iff Bit_Operations.bit_xor_iff) lemma of_int_mask_eq: \of_int (mask n) = mask n\ by (induction n) (simp_all add: mask_Suc_double Bit_Operations.mask_Suc_double of_int_or_eq) end lemma take_bit_incr_eq: \take_bit n (k + 1) = 1 + take_bit n k\ if \take_bit n k \ 2 ^ n - 1\ for k :: int proof - from that have \2 ^ n \ k mod 2 ^ n + 1\ by (simp add: take_bit_eq_mod) moreover have \k mod 2 ^ n < 2 ^ n\ by simp ultimately have *: \k mod 2 ^ n + 1 < 2 ^ n\ by linarith have \(k + 1) mod 2 ^ n = (k mod 2 ^ n + 1) mod 2 ^ n\ by (simp add: mod_simps) also have \\ = k mod 2 ^ n + 1\ using * by (simp add: zmod_trivial_iff) finally have \(k + 1) mod 2 ^ n = k mod 2 ^ n + 1\ . then show ?thesis by (simp add: take_bit_eq_mod) qed lemma take_bit_decr_eq: \take_bit n (k - 1) = take_bit n k - 1\ if \take_bit n k \ 0\ for k :: int proof - from that have \k mod 2 ^ n \ 0\ by (simp add: take_bit_eq_mod) moreover have \k mod 2 ^ n \ 0\ \k mod 2 ^ n < 2 ^ n\ by simp_all ultimately have *: \k mod 2 ^ n > 0\ by linarith have \(k - 1) mod 2 ^ n = (k mod 2 ^ n - 1) mod 2 ^ n\ by (simp add: mod_simps) also have \\ = k mod 2 ^ n - 1\ by (simp add: zmod_trivial_iff) (use \k mod 2 ^ n < 2 ^ n\ * in linarith) finally have \(k - 1) mod 2 ^ n = k mod 2 ^ n - 1\ . then show ?thesis by (simp add: take_bit_eq_mod) qed lemma take_bit_int_greater_eq: \k + 2 ^ n \ take_bit n k\ if \k < 0\ for k :: int proof - have \k + 2 ^ n \ take_bit n (k + 2 ^ n)\ proof (cases \k > - (2 ^ n)\) case False then have \k + 2 ^ n \ 0\ by simp also note take_bit_nonnegative finally show ?thesis . next case True with that have \0 \ k + 2 ^ n\ and \k + 2 ^ n < 2 ^ n\ by simp_all then show ?thesis by (simp only: take_bit_eq_mod mod_pos_pos_trivial) qed then show ?thesis by (simp add: take_bit_eq_mod) qed lemma take_bit_int_less_eq: \take_bit n k \ k - 2 ^ n\ if \2 ^ n \ k\ and \n > 0\ for k :: int using that zmod_le_nonneg_dividend [of \k - 2 ^ n\ \2 ^ n\] by (simp add: take_bit_eq_mod) lemma take_bit_int_less_eq_self_iff: \take_bit n k \ k \ 0 \ k\ (is \?P \ ?Q\) for k :: int proof assume ?P show ?Q proof (rule ccontr) assume \\ 0 \ k\ then have \k < 0\ by simp with \?P\ have \take_bit n k < 0\ by (rule le_less_trans) then show False by simp qed next assume ?Q then show ?P by (simp add: take_bit_eq_mod zmod_le_nonneg_dividend) qed lemma take_bit_int_less_self_iff: \take_bit n k < k \ 2 ^ n \ k\ for k :: int by (auto simp add: less_le take_bit_int_less_eq_self_iff take_bit_int_eq_self_iff intro: order_trans [of 0 \2 ^ n\ k]) lemma take_bit_int_greater_self_iff: \k < take_bit n k \ k < 0\ for k :: int using take_bit_int_less_eq_self_iff [of n k] by auto lemma take_bit_int_greater_eq_self_iff: \k \ take_bit n k \ k < 2 ^ n\ for k :: int by (auto simp add: le_less take_bit_int_greater_self_iff take_bit_int_eq_self_iff dest: sym not_sym intro: less_trans [of k 0 \2 ^ n\]) lemma minus_numeral_inc_eq: \- numeral (Num.inc n) = NOT (numeral n :: int)\ by (simp add: not_int_def sub_inc_One_eq add_One) lemma sub_one_eq_not_neg: \Num.sub n num.One = NOT (- numeral n :: int)\ by (simp add: not_int_def) lemma bit_numeral_int_iff [bit_simps]: \bit (numeral m :: int) n \ bit (numeral m :: nat) n\ using bit_of_nat_iff_bit [of \numeral m\ n] by simp lemma bit_minus_int_iff [bit_simps]: \bit (- k) n \ \ bit (k - 1) n\ for k :: int using bit_not_int_iff' [of \k - 1\] by simp lemma bit_numeral_int_simps [simp]: \bit (1 :: int) (numeral n) \ bit (0 :: int) (pred_numeral n)\ \bit (numeral (num.Bit0 w) :: int) (numeral n) \ bit (numeral w :: int) (pred_numeral n)\ \bit (numeral (num.Bit1 w) :: int) (numeral n) \ bit (numeral w :: int) (pred_numeral n)\ \bit (numeral (Num.BitM w) :: int) (numeral n) \ \ bit (- numeral w :: int) (pred_numeral n)\ \bit (- numeral (num.Bit0 w) :: int) (numeral n) \ bit (- numeral w :: int) (pred_numeral n)\ \bit (- numeral (num.Bit1 w) :: int) (numeral n) \ \ bit (numeral w :: int) (pred_numeral n)\ \bit (- numeral (Num.BitM w) :: int) (numeral n) \ bit (- (numeral w) :: int) (pred_numeral n)\ by (simp_all add: bit_1_iff numeral_eq_Suc bit_Suc add_One sub_inc_One_eq bit_minus_int_iff) lemma bit_numeral_Bit0_Suc_iff [simp]: \bit (numeral (Num.Bit0 m) :: int) (Suc n) \ bit (numeral m :: int) n\ by (simp add: bit_Suc) lemma bit_numeral_Bit1_Suc_iff [simp]: \bit (numeral (Num.Bit1 m) :: int) (Suc n) \ bit (numeral m :: int) n\ by (simp add: bit_Suc) lemma int_not_numerals [simp]: \NOT (numeral (Num.Bit0 n) :: int) = - numeral (Num.Bit1 n)\ \NOT (numeral (Num.Bit1 n) :: int) = - numeral (Num.inc (num.Bit1 n))\ \NOT (numeral (Num.BitM n) :: int) = - numeral (num.Bit0 n)\ \NOT (- numeral (Num.Bit0 n) :: int) = numeral (Num.BitM n)\ \NOT (- numeral (Num.Bit1 n) :: int) = numeral (Num.Bit0 n)\ by (simp_all add: not_int_def add_One inc_BitM_eq) text \FIXME: The rule sets below are very large (24 rules for each operator). Is there a simpler way to do this?\ context begin private lemma eqI: \k = l\ if num: \\n. bit k (numeral n) \ bit l (numeral n)\ and even: \even k \ even l\ for k l :: int proof (rule bit_eqI) fix n show \bit k n \ bit l n\ proof (cases n) case 0 with even show ?thesis by simp next case (Suc n) with num [of \num_of_nat (Suc n)\] show ?thesis by (simp only: numeral_num_of_nat) qed qed lemma int_and_numerals [simp]: \numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND numeral y)\ \numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (2 :: int) * (numeral x AND numeral y)\ \numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND numeral y)\ \numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x AND numeral y)\ \numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND - numeral y)\ \numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (2 :: int) * (numeral x AND - numeral (y + Num.One))\ \numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND - numeral y)\ \numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x AND - numeral (y + Num.One))\ \- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (2 :: int) * (- numeral x AND numeral y)\ \- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (2 :: int) * (- numeral x AND numeral y)\ \- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (2 :: int) * (- numeral (x + Num.One) AND numeral y)\ \- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) AND numeral y)\ \- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x AND - numeral y)\ \- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (2 :: int) * (- numeral x AND - numeral (y + Num.One))\ \- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (- numeral (x + Num.One) AND - numeral y)\ \- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) AND - numeral (y + Num.One))\ \(1::int) AND numeral (Num.Bit0 y) = 0\ \(1::int) AND numeral (Num.Bit1 y) = 1\ \(1::int) AND - numeral (Num.Bit0 y) = 0\ \(1::int) AND - numeral (Num.Bit1 y) = 1\ \numeral (Num.Bit0 x) AND (1::int) = 0\ \numeral (Num.Bit1 x) AND (1::int) = 1\ \- numeral (Num.Bit0 x) AND (1::int) = 0\ \- numeral (Num.Bit1 x) AND (1::int) = 1\ by (auto simp add: bit_and_iff bit_minus_iff even_and_iff bit_double_iff even_bit_succ_iff add_One sub_inc_One_eq intro: eqI) lemma int_or_numerals [simp]: \numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (2 :: int) * (numeral x OR numeral y)\ \numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR numeral y)\ \numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x OR numeral y)\ \numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR numeral y)\ \numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (2 :: int) * (numeral x OR - numeral y)\ \numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR - numeral (y + Num.One))\ \numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x OR - numeral y)\ \numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR - numeral (y + Num.One))\ \- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (2 :: int) * (- numeral x OR numeral y)\ \- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x OR numeral y)\ \- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR numeral y)\ \- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR numeral y)\ \- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x OR - numeral y)\ \- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x OR - numeral (y + Num.One))\ \- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR - numeral y)\ \- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR - numeral (y + Num.One))\ \(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)\ \(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)\ \(1::int) OR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)\ \(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)\ \numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)\ \numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)\ \- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)\ \- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)\ by (auto simp add: bit_or_iff bit_minus_iff even_or_iff bit_double_iff even_bit_succ_iff add_One sub_inc_One_eq sub_BitM_One_eq intro: eqI) lemma int_xor_numerals [simp]: \numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (2 :: int) * (numeral x XOR numeral y)\ \numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x XOR numeral y)\ \numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x XOR numeral y)\ \numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (2 :: int) * (numeral x XOR numeral y)\ \numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (2 :: int) * (numeral x XOR - numeral y)\ \numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x XOR - numeral (y + Num.One))\ \numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x XOR - numeral y)\ \numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (2 :: int) * (numeral x XOR - numeral (y + Num.One))\ \- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (2 :: int) * (- numeral x XOR numeral y)\ \- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x XOR numeral y)\ \- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) XOR numeral y)\ \- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (2 :: int) * (- numeral (x + Num.One) XOR numeral y)\ \- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x XOR - numeral y)\ \- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x XOR - numeral (y + Num.One))\ \- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) XOR - numeral y)\ \- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (2 :: int) * (- numeral (x + Num.One) XOR - numeral (y + Num.One))\ \(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)\ \(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)\ \(1::int) XOR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)\ \(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))\ \numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)\ \numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)\ \- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)\ \- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))\ by (auto simp add: bit_xor_iff bit_minus_iff even_xor_iff bit_double_iff even_bit_succ_iff add_One sub_inc_One_eq sub_BitM_One_eq intro: eqI) end context semiring_bit_operations begin lemma push_bit_of_nat: \push_bit n (of_nat m) = of_nat (push_bit n m)\ by (simp add: push_bit_eq_mult Bit_Operations.push_bit_eq_mult) lemma of_nat_push_bit: \of_nat (push_bit m n) = push_bit m (of_nat n)\ by (simp add: push_bit_eq_mult Bit_Operations.push_bit_eq_mult) lemma take_bit_of_nat: \take_bit n (of_nat m) = of_nat (take_bit n m)\ by (rule bit_eqI) (simp add: bit_take_bit_iff Bit_Operations.bit_take_bit_iff bit_of_nat_iff) lemma of_nat_take_bit: \of_nat (take_bit n m) = take_bit n (of_nat m)\ by (rule bit_eqI) (simp add: bit_take_bit_iff Bit_Operations.bit_take_bit_iff bit_of_nat_iff) end lemma push_bit_nat_eq: \push_bit n (nat k) = nat (push_bit n k)\ by (cases \k \ 0\) (simp_all add: push_bit_eq_mult nat_mult_distrib not_le mult_nonneg_nonpos2) lemma drop_bit_nat_eq: \drop_bit n (nat k) = nat (drop_bit n k)\ apply (cases \k \ 0\) apply (simp_all add: drop_bit_eq_div nat_div_distrib nat_power_eq not_le) apply (simp add: divide_int_def) done lemma take_bit_nat_eq: \take_bit n (nat k) = nat (take_bit n k)\ if \k \ 0\ using that by (simp add: take_bit_eq_mod nat_mod_distrib nat_power_eq) lemma nat_take_bit_eq: \nat (take_bit n k) = take_bit n (nat k)\ if \k \ 0\ using that by (simp add: take_bit_eq_mod nat_mod_distrib nat_power_eq) lemma not_exp_less_eq_0_int [simp]: \\ 2 ^ n \ (0::int)\ by (simp add: power_le_zero_eq) lemma half_nonnegative_int_iff [simp]: \k div 2 \ 0 \ k \ 0\ for k :: int proof (cases \k \ 0\) case True then show ?thesis by (auto simp add: divide_int_def sgn_1_pos) next case False then show ?thesis by (auto simp add: divide_int_def not_le elim!: evenE) qed lemma half_negative_int_iff [simp]: \k div 2 < 0 \ k < 0\ for k :: int by (subst Not_eq_iff [symmetric]) (simp add: not_less) lemma push_bit_of_Suc_0 [simp]: "push_bit n (Suc 0) = 2 ^ n" using push_bit_of_1 [where ?'a = nat] by simp lemma take_bit_of_Suc_0 [simp]: "take_bit n (Suc 0) = of_bool (0 < n)" using take_bit_of_1 [where ?'a = nat] by simp lemma drop_bit_of_Suc_0 [simp]: "drop_bit n (Suc 0) = of_bool (n = 0)" using drop_bit_of_1 [where ?'a = nat] by simp lemma int_bit_bound: fixes k :: int obtains n where \\m. n \ m \ bit k m \ bit k n\ and \n > 0 \ bit k (n - 1) \ bit k n\ proof - obtain q where *: \\m. q \ m \ bit k m \ bit k q\ proof (cases \k \ 0\) case True moreover from power_gt_expt [of 2 \nat k\] have \nat k < 2 ^ nat k\ by simp then have \int (nat k) < int (2 ^ nat k)\ by (simp only: of_nat_less_iff) ultimately have *: \k div 2 ^ nat k = 0\ by simp show thesis proof (rule that [of \nat k\]) fix m assume \nat k \ m\ then show \bit k m \ bit k (nat k)\ by (auto simp add: * bit_iff_odd power_add zdiv_zmult2_eq dest!: le_Suc_ex) qed next case False moreover from power_gt_expt [of 2 \nat (- k)\] have \nat (- k) < 2 ^ nat (- k)\ by simp then have \int (nat (- k)) < int (2 ^ nat (- k))\ by (simp only: of_nat_less_iff) ultimately have \- k div - (2 ^ nat (- k)) = - 1\ by (subst div_pos_neg_trivial) simp_all then have *: \k div 2 ^ nat (- k) = - 1\ by simp show thesis proof (rule that [of \nat (- k)\]) fix m assume \nat (- k) \ m\ then show \bit k m \ bit k (nat (- k))\ by (auto simp add: * bit_iff_odd power_add zdiv_zmult2_eq minus_1_div_exp_eq_int dest!: le_Suc_ex) qed qed show thesis proof (cases \\m. bit k m \ bit k q\) case True then have \bit k 0 \ bit k q\ by blast with True that [of 0] show thesis by simp next case False then obtain r where **: \bit k r \ bit k q\ by blast have \r < q\ by (rule ccontr) (use * [of r] ** in simp) define N where \N = {n. n < q \ bit k n \ bit k q}\ moreover have \finite N\ \r \ N\ using ** N_def \r < q\ by auto moreover define n where \n = Suc (Max N)\ ultimately have \\m. n \ m \ bit k m \ bit k n\ apply auto apply (metis (full_types, lifting) "*" Max_ge_iff Suc_n_not_le_n \finite N\ all_not_in_conv mem_Collect_eq not_le) apply (metis "*" Max_ge Suc_n_not_le_n \finite N\ linorder_not_less mem_Collect_eq) apply (metis "*" Max_ge Suc_n_not_le_n \finite N\ linorder_not_less mem_Collect_eq) apply (metis (full_types, lifting) "*" Max_ge_iff Suc_n_not_le_n \finite N\ all_not_in_conv mem_Collect_eq not_le) done have \bit k (Max N) \ bit k n\ by (metis (mono_tags, lifting) "*" Max_in N_def \\m. n \ m \ bit k m = bit k n\ \finite N\ \r \ N\ empty_iff le_cases mem_Collect_eq) show thesis apply (rule that [of n]) using \\m. n \ m \ bit k m = bit k n\ apply blast using \bit k (Max N) \ bit k n\ n_def by auto qed qed context semiring_bit_operations begin lemma of_nat_and_eq: \of_nat (m AND n) = of_nat m AND of_nat n\ by (rule bit_eqI) (simp add: bit_of_nat_iff bit_and_iff Bit_Operations.bit_and_iff) lemma of_nat_or_eq: \of_nat (m OR n) = of_nat m OR of_nat n\ by (rule bit_eqI) (simp add: bit_of_nat_iff bit_or_iff Bit_Operations.bit_or_iff) lemma of_nat_xor_eq: \of_nat (m XOR n) = of_nat m XOR of_nat n\ by (rule bit_eqI) (simp add: bit_of_nat_iff bit_xor_iff Bit_Operations.bit_xor_iff) end context ring_bit_operations begin lemma of_nat_mask_eq: \of_nat (mask n) = mask n\ by (induction n) (simp_all add: mask_Suc_double Bit_Operations.mask_Suc_double of_nat_or_eq) end lemma Suc_mask_eq_exp: \Suc (mask n) = 2 ^ n\ by (simp add: mask_eq_exp_minus_1) lemma less_eq_mask: \n \ mask n\ by (simp add: mask_eq_exp_minus_1 le_diff_conv2) (metis Suc_mask_eq_exp diff_Suc_1 diff_le_diff_pow diff_zero le_refl not_less_eq_eq power_0) lemma less_mask: \n < mask n\ if \Suc 0 < n\ proof - define m where \m = n - 2\ with that have *: \n = m + 2\ by simp have \Suc (Suc (Suc m)) < 4 * 2 ^ m\ by (induction m) simp_all then have \Suc (m + 2) < Suc (mask (m + 2))\ by (simp add: Suc_mask_eq_exp) then have \m + 2 < mask (m + 2)\ by (simp add: less_le) with * show ?thesis by simp qed subsection \Bit concatenation\ definition concat_bit :: \nat \ int \ int \ int\ where \concat_bit n k l = take_bit n k OR push_bit n l\ lemma bit_concat_bit_iff [bit_simps]: \bit (concat_bit m k l) n \ n < m \ bit k n \ m \ n \ bit l (n - m)\ by (simp add: concat_bit_def bit_or_iff bit_and_iff bit_take_bit_iff bit_push_bit_iff ac_simps) lemma concat_bit_eq: \concat_bit n k l = take_bit n k + push_bit n l\ by (simp add: concat_bit_def take_bit_eq_mask bit_and_iff bit_mask_iff bit_push_bit_iff disjunctive_add) lemma concat_bit_0 [simp]: \concat_bit 0 k l = l\ by (simp add: concat_bit_def) lemma concat_bit_Suc: \concat_bit (Suc n) k l = k mod 2 + 2 * concat_bit n (k div 2) l\ by (simp add: concat_bit_eq take_bit_Suc push_bit_double) lemma concat_bit_of_zero_1 [simp]: \concat_bit n 0 l = push_bit n l\ by (simp add: concat_bit_def) lemma concat_bit_of_zero_2 [simp]: \concat_bit n k 0 = take_bit n k\ by (simp add: concat_bit_def take_bit_eq_mask) lemma concat_bit_nonnegative_iff [simp]: \concat_bit n k l \ 0 \ l \ 0\ by (simp add: concat_bit_def) lemma concat_bit_negative_iff [simp]: \concat_bit n k l < 0 \ l < 0\ by (simp add: concat_bit_def) lemma concat_bit_assoc: \concat_bit n k (concat_bit m l r) = concat_bit (m + n) (concat_bit n k l) r\ by (rule bit_eqI) (auto simp add: bit_concat_bit_iff ac_simps) lemma concat_bit_assoc_sym: \concat_bit m (concat_bit n k l) r = concat_bit (min m n) k (concat_bit (m - n) l r)\ by (rule bit_eqI) (auto simp add: bit_concat_bit_iff ac_simps min_def) lemma concat_bit_eq_iff: \concat_bit n k l = concat_bit n r s \ take_bit n k = take_bit n r \ l = s\ (is \?P \ ?Q\) proof assume ?Q then show ?P by (simp add: concat_bit_def) next assume ?P then have *: \bit (concat_bit n k l) m = bit (concat_bit n r s) m\ for m by (simp add: bit_eq_iff) have \take_bit n k = take_bit n r\ proof (rule bit_eqI) fix m from * [of m] show \bit (take_bit n k) m \ bit (take_bit n r) m\ by (auto simp add: bit_take_bit_iff bit_concat_bit_iff) qed moreover have \push_bit n l = push_bit n s\ proof (rule bit_eqI) fix m from * [of m] show \bit (push_bit n l) m \ bit (push_bit n s) m\ by (auto simp add: bit_push_bit_iff bit_concat_bit_iff) qed then have \l = s\ by (simp add: push_bit_eq_mult) ultimately show ?Q by (simp add: concat_bit_def) qed lemma take_bit_concat_bit_eq: \take_bit m (concat_bit n k l) = concat_bit (min m n) k (take_bit (m - n) l)\ by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_concat_bit_iff min_def) lemma concat_bit_take_bit_eq: \concat_bit n (take_bit n b) = concat_bit n b\ by (simp add: concat_bit_def [abs_def]) subsection \Taking bits with sign propagation\ context ring_bit_operations begin definition signed_take_bit :: \nat \ 'a \ 'a\ where \signed_take_bit n a = take_bit n a OR (of_bool (bit a n) * NOT (mask n))\ lemma signed_take_bit_eq_if_positive: \signed_take_bit n a = take_bit n a\ if \\ bit a n\ using that by (simp add: signed_take_bit_def) lemma signed_take_bit_eq_if_negative: \signed_take_bit n a = take_bit n a OR NOT (mask n)\ if \bit a n\ using that by (simp add: signed_take_bit_def) lemma even_signed_take_bit_iff: \even (signed_take_bit m a) \ even a\ by (auto simp add: signed_take_bit_def even_or_iff even_mask_iff bit_double_iff) lemma bit_signed_take_bit_iff [bit_simps]: \bit (signed_take_bit m a) n \ 2 ^ n \ 0 \ bit a (min m n)\ by (simp add: signed_take_bit_def bit_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff min_def not_le) (use exp_eq_0_imp_not_bit in blast) lemma signed_take_bit_0 [simp]: \signed_take_bit 0 a = - (a mod 2)\ by (simp add: signed_take_bit_def odd_iff_mod_2_eq_one) lemma signed_take_bit_Suc: \signed_take_bit (Suc n) a = a mod 2 + 2 * signed_take_bit n (a div 2)\ proof (rule bit_eqI) fix m assume *: \2 ^ m \ 0\ show \bit (signed_take_bit (Suc n) a) m \ bit (a mod 2 + 2 * signed_take_bit n (a div 2)) m\ proof (cases m) case 0 then show ?thesis by (simp add: even_signed_take_bit_iff) next case (Suc m) with * have \2 ^ m \ 0\ by (metis mult_not_zero power_Suc) with Suc show ?thesis by (simp add: bit_signed_take_bit_iff mod2_eq_if bit_double_iff even_bit_succ_iff ac_simps flip: bit_Suc) qed qed lemma signed_take_bit_of_0 [simp]: \signed_take_bit n 0 = 0\ by (simp add: signed_take_bit_def) lemma signed_take_bit_of_minus_1 [simp]: \signed_take_bit n (- 1) = - 1\ by (simp add: signed_take_bit_def take_bit_minus_one_eq_mask mask_eq_exp_minus_1) lemma signed_take_bit_Suc_1 [simp]: \signed_take_bit (Suc n) 1 = 1\ by (simp add: signed_take_bit_Suc) lemma signed_take_bit_rec: \signed_take_bit n a = (if n = 0 then - (a mod 2) else a mod 2 + 2 * signed_take_bit (n - 1) (a div 2))\ by (cases n) (simp_all add: signed_take_bit_Suc) lemma signed_take_bit_eq_iff_take_bit_eq: \signed_take_bit n a = signed_take_bit n b \ take_bit (Suc n) a = take_bit (Suc n) b\ proof - have \bit (signed_take_bit n a) = bit (signed_take_bit n b) \ bit (take_bit (Suc n) a) = bit (take_bit (Suc n) b)\ by (simp add: fun_eq_iff bit_signed_take_bit_iff bit_take_bit_iff not_le less_Suc_eq_le min_def) (use exp_eq_0_imp_not_bit in fastforce) then show ?thesis by (simp add: bit_eq_iff fun_eq_iff) qed lemma signed_take_bit_signed_take_bit [simp]: \signed_take_bit m (signed_take_bit n a) = signed_take_bit (min m n) a\ proof (rule bit_eqI) fix q show \bit (signed_take_bit m (signed_take_bit n a)) q \ bit (signed_take_bit (min m n) a) q\ by (simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff) (use le_Suc_ex exp_add_not_zero_imp in blast) qed lemma signed_take_bit_take_bit: \signed_take_bit m (take_bit n a) = (if n \ m then take_bit n else signed_take_bit m) a\ by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff) lemma take_bit_signed_take_bit: \take_bit m (signed_take_bit n a) = take_bit m a\ if \m \ Suc n\ using that by (rule le_SucE; intro bit_eqI) (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def less_Suc_eq) end text \Modulus centered around 0\ lemma signed_take_bit_eq_concat_bit: \signed_take_bit n k = concat_bit n k (- of_bool (bit k n))\ by (simp add: concat_bit_def signed_take_bit_def push_bit_minus_one_eq_not_mask) lemma signed_take_bit_add: \signed_take_bit n (signed_take_bit n k + signed_take_bit n l) = signed_take_bit n (k + l)\ for k l :: int proof - have \take_bit (Suc n) (take_bit (Suc n) (signed_take_bit n k) + take_bit (Suc n) (signed_take_bit n l)) = take_bit (Suc n) (k + l)\ by (simp add: take_bit_signed_take_bit take_bit_add) then show ?thesis by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_add) qed lemma signed_take_bit_diff: \signed_take_bit n (signed_take_bit n k - signed_take_bit n l) = signed_take_bit n (k - l)\ for k l :: int proof - have \take_bit (Suc n) (take_bit (Suc n) (signed_take_bit n k) - take_bit (Suc n) (signed_take_bit n l)) = take_bit (Suc n) (k - l)\ by (simp add: take_bit_signed_take_bit take_bit_diff) then show ?thesis by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_diff) qed lemma signed_take_bit_minus: \signed_take_bit n (- signed_take_bit n k) = signed_take_bit n (- k)\ for k :: int proof - have \take_bit (Suc n) (- take_bit (Suc n) (signed_take_bit n k)) = take_bit (Suc n) (- k)\ by (simp add: take_bit_signed_take_bit take_bit_minus) then show ?thesis by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_minus) qed lemma signed_take_bit_mult: \signed_take_bit n (signed_take_bit n k * signed_take_bit n l) = signed_take_bit n (k * l)\ for k l :: int proof - have \take_bit (Suc n) (take_bit (Suc n) (signed_take_bit n k) * take_bit (Suc n) (signed_take_bit n l)) = take_bit (Suc n) (k * l)\ by (simp add: take_bit_signed_take_bit take_bit_mult) then show ?thesis by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_mult) qed lemma signed_take_bit_eq_take_bit_minus: \signed_take_bit n k = take_bit (Suc n) k - 2 ^ Suc n * of_bool (bit k n)\ for k :: int proof (cases \bit k n\) case True have \signed_take_bit n k = take_bit (Suc n) k OR NOT (mask (Suc n))\ by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff less_Suc_eq True) then have \signed_take_bit n k = take_bit (Suc n) k + NOT (mask (Suc n))\ by (simp add: disjunctive_add bit_take_bit_iff bit_not_iff bit_mask_iff) with True show ?thesis by (simp flip: minus_exp_eq_not_mask) next case False show ?thesis by (rule bit_eqI) (simp add: False bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq) qed lemma signed_take_bit_eq_take_bit_shift: \signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n\ for k :: int proof - have *: \take_bit n k OR 2 ^ n = take_bit n k + 2 ^ n\ by (simp add: disjunctive_add bit_exp_iff bit_take_bit_iff) have \take_bit n k - 2 ^ n = take_bit n k + NOT (mask n)\ by (simp add: minus_exp_eq_not_mask) also have \\ = take_bit n k OR NOT (mask n)\ by (rule disjunctive_add) (simp add: bit_exp_iff bit_take_bit_iff bit_not_iff bit_mask_iff) finally have **: \take_bit n k - 2 ^ n = take_bit n k OR NOT (mask n)\ . have \take_bit (Suc n) (k + 2 ^ n) = take_bit (Suc n) (take_bit (Suc n) k + take_bit (Suc n) (2 ^ n))\ by (simp only: take_bit_add) also have \take_bit (Suc n) k = 2 ^ n * of_bool (bit k n) + take_bit n k\ by (simp add: take_bit_Suc_from_most) finally have \take_bit (Suc n) (k + 2 ^ n) = take_bit (Suc n) (2 ^ (n + of_bool (bit k n)) + take_bit n k)\ by (simp add: ac_simps) also have \2 ^ (n + of_bool (bit k n)) + take_bit n k = 2 ^ (n + of_bool (bit k n)) OR take_bit n k\ by (rule disjunctive_add) (auto simp add: disjunctive_add bit_take_bit_iff bit_double_iff bit_exp_iff) finally show ?thesis using * ** by (simp add: signed_take_bit_def concat_bit_Suc min_def ac_simps) qed lemma signed_take_bit_nonnegative_iff [simp]: \0 \ signed_take_bit n k \ \ bit k n\ for k :: int by (simp add: signed_take_bit_def not_less concat_bit_def) lemma signed_take_bit_negative_iff [simp]: \signed_take_bit n k < 0 \ bit k n\ for k :: int by (simp add: signed_take_bit_def not_less concat_bit_def) lemma signed_take_bit_int_greater_eq_minus_exp [simp]: \- (2 ^ n) \ signed_take_bit n k\ for k :: int by (simp add: signed_take_bit_eq_take_bit_shift) lemma signed_take_bit_int_less_exp [simp]: \signed_take_bit n k < 2 ^ n\ for k :: int using take_bit_int_less_exp [of \Suc n\] by (simp add: signed_take_bit_eq_take_bit_shift) lemma signed_take_bit_int_eq_self_iff: \signed_take_bit n k = k \ - (2 ^ n) \ k \ k < 2 ^ n\ for k :: int by (auto simp add: signed_take_bit_eq_take_bit_shift take_bit_int_eq_self_iff algebra_simps) lemma signed_take_bit_int_eq_self: \signed_take_bit n k = k\ if \- (2 ^ n) \ k\ \k < 2 ^ n\ for k :: int using that by (simp add: signed_take_bit_int_eq_self_iff) lemma signed_take_bit_int_less_eq_self_iff: \signed_take_bit n k \ k \ - (2 ^ n) \ k\ for k :: int by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_less_eq_self_iff algebra_simps) linarith lemma signed_take_bit_int_less_self_iff: \signed_take_bit n k < k \ 2 ^ n \ k\ for k :: int by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_less_self_iff algebra_simps) lemma signed_take_bit_int_greater_self_iff: \k < signed_take_bit n k \ k < - (2 ^ n)\ for k :: int by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_greater_self_iff algebra_simps) linarith lemma signed_take_bit_int_greater_eq_self_iff: \k \ signed_take_bit n k \ k < 2 ^ n\ for k :: int by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_greater_eq_self_iff algebra_simps) lemma signed_take_bit_int_greater_eq: \k + 2 ^ Suc n \ signed_take_bit n k\ if \k < - (2 ^ n)\ for k :: int using that take_bit_int_greater_eq [of \k + 2 ^ n\ \Suc n\] by (simp add: signed_take_bit_eq_take_bit_shift) lemma signed_take_bit_int_less_eq: \signed_take_bit n k \ k - 2 ^ Suc n\ if \k \ 2 ^ n\ for k :: int using that take_bit_int_less_eq [of \Suc n\ \k + 2 ^ n\] by (simp add: signed_take_bit_eq_take_bit_shift) lemma signed_take_bit_Suc_bit0 [simp]: \signed_take_bit (Suc n) (numeral (Num.Bit0 k)) = signed_take_bit n (numeral k) * (2 :: int)\ by (simp add: signed_take_bit_Suc) lemma signed_take_bit_Suc_bit1 [simp]: \signed_take_bit (Suc n) (numeral (Num.Bit1 k)) = signed_take_bit n (numeral k) * 2 + (1 :: int)\ by (simp add: signed_take_bit_Suc) lemma signed_take_bit_Suc_minus_bit0 [simp]: \signed_take_bit (Suc n) (- numeral (Num.Bit0 k)) = signed_take_bit n (- numeral k) * (2 :: int)\ by (simp add: signed_take_bit_Suc) lemma signed_take_bit_Suc_minus_bit1 [simp]: \signed_take_bit (Suc n) (- numeral (Num.Bit1 k)) = signed_take_bit n (- numeral k - 1) * 2 + (1 :: int)\ by (simp add: signed_take_bit_Suc) lemma signed_take_bit_numeral_bit0 [simp]: \signed_take_bit (numeral l) (numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (numeral k) * (2 :: int)\ by (simp add: signed_take_bit_rec) lemma signed_take_bit_numeral_bit1 [simp]: \signed_take_bit (numeral l) (numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2 + (1 :: int)\ by (simp add: signed_take_bit_rec) lemma signed_take_bit_numeral_minus_bit0 [simp]: \signed_take_bit (numeral l) (- numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (- numeral k) * (2 :: int)\ by (simp add: signed_take_bit_rec) lemma signed_take_bit_numeral_minus_bit1 [simp]: \signed_take_bit (numeral l) (- numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (- numeral k - 1) * 2 + (1 :: int)\ by (simp add: signed_take_bit_rec) lemma signed_take_bit_code [code]: \signed_take_bit n a = (let l = take_bit (Suc n) a in if bit l n then l + push_bit (Suc n) (- 1) else l)\ proof - have *: \take_bit (Suc n) a + push_bit n (- 2) = take_bit (Suc n) a OR NOT (mask (Suc n))\ by (auto simp add: bit_take_bit_iff bit_push_bit_iff bit_not_iff bit_mask_iff disjunctive_add simp flip: push_bit_minus_one_eq_not_mask) show ?thesis by (rule bit_eqI) (auto simp add: Let_def * bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq bit_not_iff bit_mask_iff bit_or_iff) qed subsection \Horner sums\ context semiring_bit_operations begin lemma horner_sum_bit_eq_take_bit: \horner_sum of_bool 2 (map (bit a) [0.. proof (induction a arbitrary: n rule: bits_induct) case (stable a) moreover have \bit a = (\_. odd a)\ using stable by (simp add: stable_imp_bit_iff_odd fun_eq_iff) moreover have \{q. q < n} = {0.. by auto ultimately show ?case by (simp add: stable_imp_take_bit_eq horner_sum_eq_sum mask_eq_sum_exp) next case (rec a b) show ?case proof (cases n) case 0 then show ?thesis by simp next case (Suc m) have \map (bit (of_bool b + 2 * a)) [0.. by (simp only: upt_conv_Cons) simp also have \\ = b # map (bit a) [0.. by (simp only: flip: map_Suc_upt) (simp add: bit_Suc rec.hyps) finally show ?thesis using Suc rec.IH [of m] by (simp add: take_bit_Suc rec.hyps) (simp_all add: ac_simps mod_2_eq_odd) qed qed end context unique_euclidean_semiring_with_bit_operations begin lemma bit_horner_sum_bit_iff [bit_simps]: \bit (horner_sum of_bool 2 bs) n \ n < length bs \ bs ! n\ proof (induction bs arbitrary: n) case Nil then show ?case by simp next case (Cons b bs) show ?case proof (cases n) case 0 then show ?thesis by simp next case (Suc m) with bit_rec [of _ n] Cons.prems Cons.IH [of m] show ?thesis by simp qed qed lemma take_bit_horner_sum_bit_eq: \take_bit n (horner_sum of_bool 2 bs) = horner_sum of_bool 2 (take n bs)\ by (auto simp add: bit_eq_iff bit_take_bit_iff bit_horner_sum_bit_iff) end lemma horner_sum_of_bool_2_less: \(horner_sum of_bool 2 bs :: int) < 2 ^ length bs\ proof - have \(\n = 0.. (\n = 0.. by (rule sum_mono) simp also have \\ = 2 ^ length bs - 1\ by (induction bs) simp_all finally show ?thesis by (simp add: horner_sum_eq_sum) qed subsection \Symbolic computations on numeral expressions\ \<^marker>\contributor \Andreas Lochbihler\\ fun and_num :: \num \ num \ num option\ where \and_num num.One num.One = Some num.One\ | \and_num num.One (num.Bit0 n) = None\ | \and_num num.One (num.Bit1 n) = Some num.One\ | \and_num (num.Bit0 m) num.One = None\ | \and_num (num.Bit0 m) (num.Bit0 n) = map_option num.Bit0 (and_num m n)\ | \and_num (num.Bit0 m) (num.Bit1 n) = map_option num.Bit0 (and_num m n)\ | \and_num (num.Bit1 m) num.One = Some num.One\ | \and_num (num.Bit1 m) (num.Bit0 n) = map_option num.Bit0 (and_num m n)\ | \and_num (num.Bit1 m) (num.Bit1 n) = (case and_num m n of None \ Some num.One | Some n' \ Some (num.Bit1 n'))\ fun and_not_num :: \num \ num \ num option\ where \and_not_num num.One num.One = None\ | \and_not_num num.One (num.Bit0 n) = Some num.One\ | \and_not_num num.One (num.Bit1 n) = None\ | \and_not_num (num.Bit0 m) num.One = Some (num.Bit0 m)\ | \and_not_num (num.Bit0 m) (num.Bit0 n) = map_option num.Bit0 (and_not_num m n)\ | \and_not_num (num.Bit0 m) (num.Bit1 n) = map_option num.Bit0 (and_not_num m n)\ | \and_not_num (num.Bit1 m) num.One = Some (num.Bit0 m)\ | \and_not_num (num.Bit1 m) (num.Bit0 n) = (case and_not_num m n of None \ Some num.One | Some n' \ Some (num.Bit1 n'))\ | \and_not_num (num.Bit1 m) (num.Bit1 n) = map_option num.Bit0 (and_not_num m n)\ fun or_num :: \num \ num \ num\ where \or_num num.One num.One = num.One\ | \or_num num.One (num.Bit0 n) = num.Bit1 n\ | \or_num num.One (num.Bit1 n) = num.Bit1 n\ | \or_num (num.Bit0 m) num.One = num.Bit1 m\ | \or_num (num.Bit0 m) (num.Bit0 n) = num.Bit0 (or_num m n)\ | \or_num (num.Bit0 m) (num.Bit1 n) = num.Bit1 (or_num m n)\ | \or_num (num.Bit1 m) num.One = num.Bit1 m\ | \or_num (num.Bit1 m) (num.Bit0 n) = num.Bit1 (or_num m n)\ | \or_num (num.Bit1 m) (num.Bit1 n) = num.Bit1 (or_num m n)\ fun or_not_num_neg :: \num \ num \ num\ where \or_not_num_neg num.One num.One = num.One\ | \or_not_num_neg num.One (num.Bit0 m) = num.Bit1 m\ | \or_not_num_neg num.One (num.Bit1 m) = num.Bit1 m\ | \or_not_num_neg (num.Bit0 n) num.One = num.Bit0 num.One\ | \or_not_num_neg (num.Bit0 n) (num.Bit0 m) = Num.BitM (or_not_num_neg n m)\ | \or_not_num_neg (num.Bit0 n) (num.Bit1 m) = num.Bit0 (or_not_num_neg n m)\ | \or_not_num_neg (num.Bit1 n) num.One = num.One\ | \or_not_num_neg (num.Bit1 n) (num.Bit0 m) = Num.BitM (or_not_num_neg n m)\ | \or_not_num_neg (num.Bit1 n) (num.Bit1 m) = Num.BitM (or_not_num_neg n m)\ fun xor_num :: \num \ num \ num option\ where \xor_num num.One num.One = None\ | \xor_num num.One (num.Bit0 n) = Some (num.Bit1 n)\ | \xor_num num.One (num.Bit1 n) = Some (num.Bit0 n)\ | \xor_num (num.Bit0 m) num.One = Some (num.Bit1 m)\ | \xor_num (num.Bit0 m) (num.Bit0 n) = map_option num.Bit0 (xor_num m n)\ | \xor_num (num.Bit0 m) (num.Bit1 n) = Some (case xor_num m n of None \ num.One | Some n' \ num.Bit1 n')\ | \xor_num (num.Bit1 m) num.One = Some (num.Bit0 m)\ | \xor_num (num.Bit1 m) (num.Bit0 n) = Some (case xor_num m n of None \ num.One | Some n' \ num.Bit1 n')\ | \xor_num (num.Bit1 m) (num.Bit1 n) = map_option num.Bit0 (xor_num m n)\ lemma int_numeral_and_num: \numeral m AND numeral n = (case and_num m n of None \ 0 :: int | Some n' \ numeral n')\ by (induction m n rule: and_num.induct) (simp_all split: option.split) lemma and_num_eq_None_iff: \and_num m n = None \ numeral m AND numeral n = (0::int)\ by (simp add: int_numeral_and_num split: option.split) lemma and_num_eq_Some_iff: \and_num m n = Some q \ numeral m AND numeral n = (numeral q :: int)\ by (simp add: int_numeral_and_num split: option.split) lemma int_numeral_and_not_num: \numeral m AND NOT (numeral n) = (case and_not_num m n of None \ 0 :: int | Some n' \ numeral n')\ by (induction m n rule: and_not_num.induct) (simp_all add: add_One BitM_inc_eq not_int_def split: option.split) lemma int_numeral_not_and_num: \NOT (numeral m) AND numeral n = (case and_not_num n m of None \ 0 :: int | Some n' \ numeral n')\ using int_numeral_and_not_num [of n m] by (simp add: ac_simps) lemma and_not_num_eq_None_iff: \and_not_num m n = None \ numeral m AND NOT (numeral n) = (0::int)\ by (simp add: int_numeral_and_not_num split: option.split) lemma and_not_num_eq_Some_iff: \and_not_num m n = Some q \ numeral m AND NOT (numeral n) = (numeral q :: int)\ by (simp add: int_numeral_and_not_num split: option.split) lemma int_numeral_or_num: \numeral m OR numeral n = (numeral (or_num m n) :: int)\ by (induction m n rule: or_num.induct) simp_all lemma numeral_or_num_eq: \numeral (or_num m n) = (numeral m OR numeral n :: int)\ by (simp add: int_numeral_or_num) lemma int_numeral_or_not_num_neg: \numeral m OR NOT (numeral n :: int) = - numeral (or_not_num_neg m n)\ by (induction m n rule: or_not_num_neg.induct) (simp_all add: add_One BitM_inc_eq not_int_def) lemma int_numeral_not_or_num_neg: \NOT (numeral m) OR (numeral n :: int) = - numeral (or_not_num_neg n m)\ using int_numeral_or_not_num_neg [of n m] by (simp add: ac_simps) lemma numeral_or_not_num_eq: \numeral (or_not_num_neg m n) = - (numeral m OR NOT (numeral n :: int))\ using int_numeral_or_not_num_neg [of m n] by simp lemma int_numeral_xor_num: \numeral m XOR numeral n = (case xor_num m n of None \ 0 :: int | Some n' \ numeral n')\ by (induction m n rule: xor_num.induct) (simp_all split: option.split) lemma xor_num_eq_None_iff: \xor_num m n = None \ numeral m XOR numeral n = (0::int)\ by (simp add: int_numeral_xor_num split: option.split) lemma xor_num_eq_Some_iff: \xor_num m n = Some q \ numeral m XOR numeral n = (numeral q :: int)\ by (simp add: int_numeral_xor_num split: option.split) subsection \Key ideas of bit operations\ text \ When formalizing bit operations, it is tempting to represent bit values as explicit lists over a binary type. This however is a bad idea, mainly due to the inherent ambiguities in representation concerning repeating leading bits. Hence this approach avoids such explicit lists altogether following an algebraic path: \<^item> Bit values are represented by numeric types: idealized unbounded bit values can be represented by type \<^typ>\int\, bounded bit values by quotient types over \<^typ>\int\. \<^item> (A special case are idealized unbounded bit values ending in @{term [source] 0} which can be represented by type \<^typ>\nat\ but only support a restricted set of operations). \<^item> From this idea follows that \<^item> multiplication by \<^term>\2 :: int\ is a bit shift to the left and \<^item> division by \<^term>\2 :: int\ is a bit shift to the right. \<^item> Concerning bounded bit values, iterated shifts to the left may result in eliminating all bits by shifting them all beyond the boundary. The property \<^prop>\(2 :: int) ^ n \ 0\ represents that \<^term>\n\ is \<^emph>\not\ beyond that boundary. \<^item> The projection on a single bit is then @{thm bit_iff_odd [where ?'a = int, no_vars]}. \<^item> This leads to the most fundamental properties of bit values: \<^item> Equality rule: @{thm bit_eqI [where ?'a = int, no_vars]} \<^item> Induction rule: @{thm bits_induct [where ?'a = int, no_vars]} \<^item> Typical operations are characterized as follows: \<^item> Singleton \<^term>\n\th bit: \<^term>\(2 :: int) ^ n\ \<^item> Bit mask upto bit \<^term>\n\: @{thm mask_eq_exp_minus_1 [where ?'a = int, no_vars]} \<^item> Left shift: @{thm push_bit_eq_mult [where ?'a = int, no_vars]} \<^item> Right shift: @{thm drop_bit_eq_div [where ?'a = int, no_vars]} \<^item> Truncation: @{thm take_bit_eq_mod [where ?'a = int, no_vars]} \<^item> Negation: @{thm bit_not_iff [where ?'a = int, no_vars]} \<^item> And: @{thm bit_and_iff [where ?'a = int, no_vars]} \<^item> Or: @{thm bit_or_iff [where ?'a = int, no_vars]} \<^item> Xor: @{thm bit_xor_iff [where ?'a = int, no_vars]} \<^item> Set a single bit: @{thm set_bit_def [where ?'a = int, no_vars]} \<^item> Unset a single bit: @{thm unset_bit_def [where ?'a = int, no_vars]} \<^item> Flip a single bit: @{thm flip_bit_def [where ?'a = int, no_vars]} \<^item> Signed truncation, or modulus centered around \<^term>\0::int\: @{thm signed_take_bit_def [no_vars]} \<^item> Bit concatenation: @{thm concat_bit_def [no_vars]} \<^item> (Bounded) conversion from and to a list of bits: @{thm horner_sum_bit_eq_take_bit [where ?'a = int, no_vars]} \ -find_theorems \(_ AND _) * _ = _\ - no_notation "and" (infixr \AND\ 64) and or (infixr \OR\ 59) and xor (infixr \XOR\ 59) bundle bit_operations_syntax begin notation "and" (infixr \AND\ 64) and or (infixr \OR\ 59) and xor (infixr \XOR\ 59) end end diff --git a/src/HOL/Boolean_Algebra.thy b/src/HOL/Boolean_Algebras.thy rename from src/HOL/Boolean_Algebra.thy rename to src/HOL/Boolean_Algebras.thy --- a/src/HOL/Boolean_Algebra.thy +++ b/src/HOL/Boolean_Algebras.thy @@ -1,296 +1,573 @@ -(* Title: HOL/Boolean_Algebra.thy +(* Title: HOL/Boolean_Algebras.thy Author: Brian Huffman + Author: Florian Haftmann *) -section \Abstract boolean Algebras\ +section \Boolean Algebras\ -theory Boolean_Algebra +theory Boolean_Algebras imports Lattices begin -locale boolean_algebra = conj: abel_semigroup "(\<^bold>\)" + disj: abel_semigroup "(\<^bold>\)" - for conj :: "'a \ 'a \ 'a" (infixr "\<^bold>\" 70) - and disj :: "'a \ 'a \ 'a" (infixr "\<^bold>\" 65) + - fixes compl :: "'a \ 'a" ("\<^bold>- _" [81] 80) - and zero :: "'a" ("\<^bold>0") - and one :: "'a" ("\<^bold>1") - assumes conj_disj_distrib: "x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)" - and disj_conj_distrib: "x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)" - and conj_one_right: "x \<^bold>\ \<^bold>1 = x" - and disj_zero_right: "x \<^bold>\ \<^bold>0 = x" - and conj_cancel_right [simp]: "x \<^bold>\ \<^bold>- x = \<^bold>0" - and disj_cancel_right [simp]: "x \<^bold>\ \<^bold>- x = \<^bold>1" +subsection \Abstract boolean algebra\ + +locale abstract_boolean_algebra = conj: abel_semigroup \(\<^bold>\)\ + disj: abel_semigroup \(\<^bold>\)\ + for conj :: \'a \ 'a \ 'a\ (infixr \\<^bold>\\ 70) + and disj :: \'a \ 'a \ 'a\ (infixr \\<^bold>\\ 65) + + fixes compl :: \'a \ 'a\ (\\<^bold>- _\ [81] 80) + and zero :: \'a\ (\\<^bold>0\) + and one :: \'a\ (\\<^bold>1\) + assumes conj_disj_distrib: \x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)\ + and disj_conj_distrib: \x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)\ + and conj_one_right: \x \<^bold>\ \<^bold>1 = x\ + and disj_zero_right: \x \<^bold>\ \<^bold>0 = x\ + and conj_cancel_right [simp]: \x \<^bold>\ \<^bold>- x = \<^bold>0\ + and disj_cancel_right [simp]: \x \<^bold>\ \<^bold>- x = \<^bold>1\ begin -sublocale conj: semilattice_neutr "(\<^bold>\)" "\<^bold>1" +sublocale conj: semilattice_neutr \(\<^bold>\)\ \\<^bold>1\ proof - show "x \<^bold>\ \<^bold>1 = x" for x + show "x \<^bold>\ \<^bold>1 = x" for x by (fact conj_one_right) show "x \<^bold>\ x = x" for x proof - have "x \<^bold>\ x = (x \<^bold>\ x) \<^bold>\ \<^bold>0" by (simp add: disj_zero_right) also have "\ = (x \<^bold>\ x) \<^bold>\ (x \<^bold>\ \<^bold>- x)" by simp also have "\ = x \<^bold>\ (x \<^bold>\ \<^bold>- x)" by (simp only: conj_disj_distrib) also have "\ = x \<^bold>\ \<^bold>1" by simp also have "\ = x" by (simp add: conj_one_right) finally show ?thesis . qed qed -sublocale disj: semilattice_neutr "(\<^bold>\)" "\<^bold>0" +sublocale disj: semilattice_neutr \(\<^bold>\)\ \\<^bold>0\ proof show "x \<^bold>\ \<^bold>0 = x" for x by (fact disj_zero_right) show "x \<^bold>\ x = x" for x proof - have "x \<^bold>\ x = (x \<^bold>\ x) \<^bold>\ \<^bold>1" by simp also have "\ = (x \<^bold>\ x) \<^bold>\ (x \<^bold>\ \<^bold>- x)" by simp also have "\ = x \<^bold>\ (x \<^bold>\ \<^bold>- x)" by (simp only: disj_conj_distrib) also have "\ = x \<^bold>\ \<^bold>0" by simp also have "\ = x" by (simp add: disj_zero_right) finally show ?thesis . qed qed -subsection \Complement\ +subsubsection \Complement\ lemma complement_unique: assumes 1: "a \<^bold>\ x = \<^bold>0" assumes 2: "a \<^bold>\ x = \<^bold>1" assumes 3: "a \<^bold>\ y = \<^bold>0" assumes 4: "a \<^bold>\ y = \<^bold>1" shows "x = y" proof - from 1 3 have "(a \<^bold>\ x) \<^bold>\ (x \<^bold>\ y) = (a \<^bold>\ y) \<^bold>\ (x \<^bold>\ y)" by simp then have "(x \<^bold>\ a) \<^bold>\ (x \<^bold>\ y) = (y \<^bold>\ a) \<^bold>\ (y \<^bold>\ x)" by (simp add: ac_simps) then have "x \<^bold>\ (a \<^bold>\ y) = y \<^bold>\ (a \<^bold>\ x)" by (simp add: conj_disj_distrib) with 2 4 have "x \<^bold>\ \<^bold>1 = y \<^bold>\ \<^bold>1" by simp then show "x = y" by simp qed lemma compl_unique: "x \<^bold>\ y = \<^bold>0 \ x \<^bold>\ y = \<^bold>1 \ \<^bold>- x = y" by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) lemma double_compl [simp]: "\<^bold>- (\<^bold>- x) = x" proof (rule compl_unique) show "\<^bold>- x \<^bold>\ x = \<^bold>0" by (simp only: conj_cancel_right conj.commute) show "\<^bold>- x \<^bold>\ x = \<^bold>1" by (simp only: disj_cancel_right disj.commute) qed lemma compl_eq_compl_iff [simp]: \\<^bold>- x = \<^bold>- y \ x = y\ (is \?P \ ?Q\) proof assume \?Q\ then show ?P by simp next assume \?P\ then have \\<^bold>- (\<^bold>- x) = \<^bold>- (\<^bold>- y)\ by simp then show ?Q by simp qed -subsection \Conjunction\ +subsubsection \Conjunction\ lemma conj_zero_right [simp]: "x \<^bold>\ \<^bold>0 = \<^bold>0" using conj.left_idem conj_cancel_right by fastforce lemma compl_one [simp]: "\<^bold>- \<^bold>1 = \<^bold>0" by (rule compl_unique [OF conj_zero_right disj_zero_right]) lemma conj_zero_left [simp]: "\<^bold>0 \<^bold>\ x = \<^bold>0" by (subst conj.commute) (rule conj_zero_right) lemma conj_cancel_left [simp]: "\<^bold>- x \<^bold>\ x = \<^bold>0" by (subst conj.commute) (rule conj_cancel_right) lemma conj_disj_distrib2: "(y \<^bold>\ z) \<^bold>\ x = (y \<^bold>\ x) \<^bold>\ (z \<^bold>\ x)" by (simp only: conj.commute conj_disj_distrib) lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2 -lemma conj_assoc: "(x \<^bold>\ y) \<^bold>\ z = x \<^bold>\ (y \<^bold>\ z)" - by (fact ac_simps) - -lemma conj_commute: "x \<^bold>\ y = y \<^bold>\ x" - by (fact ac_simps) - -lemmas conj_left_commute = conj.left_commute -lemmas conj_ac = conj.assoc conj.commute conj.left_commute -lemma conj_one_left: "\<^bold>1 \<^bold>\ x = x" - by (fact conj.left_neutral) - -lemma conj_left_absorb: "x \<^bold>\ (x \<^bold>\ y) = x \<^bold>\ y" - by (fact conj.left_idem) +subsubsection \Disjunction\ -lemma conj_absorb: "x \<^bold>\ x = x" - by (fact conj.idem) +context +begin - -subsection \Disjunction\ - -interpretation dual: boolean_algebra "(\<^bold>\)" "(\<^bold>\)" compl \\<^bold>1\ \\<^bold>0\ +interpretation dual: abstract_boolean_algebra \(\<^bold>\)\ \(\<^bold>\)\ compl \\<^bold>1\ \\<^bold>0\ apply standard apply (rule disj_conj_distrib) apply (rule conj_disj_distrib) apply simp_all done +lemma disj_one_right [simp]: "x \<^bold>\ \<^bold>1 = \<^bold>1" + by (fact dual.conj_zero_right) + lemma compl_zero [simp]: "\<^bold>- \<^bold>0 = \<^bold>1" by (fact dual.compl_one) -lemma disj_one_right [simp]: "x \<^bold>\ \<^bold>1 = \<^bold>1" - by (fact dual.conj_zero_right) - lemma disj_one_left [simp]: "\<^bold>1 \<^bold>\ x = \<^bold>1" by (fact dual.conj_zero_left) lemma disj_cancel_left [simp]: "\<^bold>- x \<^bold>\ x = \<^bold>1" - by (rule dual.conj_cancel_left) + by (fact dual.conj_cancel_left) lemma disj_conj_distrib2: "(y \<^bold>\ z) \<^bold>\ x = (y \<^bold>\ x) \<^bold>\ (z \<^bold>\ x)" - by (rule dual.conj_disj_distrib2) + by (fact dual.conj_disj_distrib2) lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2 -lemma disj_assoc: "(x \<^bold>\ y) \<^bold>\ z = x \<^bold>\ (y \<^bold>\ z)" - by (fact ac_simps) - -lemma disj_commute: "x \<^bold>\ y = y \<^bold>\ x" - by (fact ac_simps) - -lemmas disj_left_commute = disj.left_commute - -lemmas disj_ac = disj.assoc disj.commute disj.left_commute - -lemma disj_zero_left: "\<^bold>0 \<^bold>\ x = x" - by (fact disj.left_neutral) - -lemma disj_left_absorb: "x \<^bold>\ (x \<^bold>\ y) = x \<^bold>\ y" - by (fact disj.left_idem) - -lemma disj_absorb: "x \<^bold>\ x = x" - by (fact disj.idem) +end -subsection \De Morgan's Laws\ +subsubsection \De Morgan's Laws\ lemma de_Morgan_conj [simp]: "\<^bold>- (x \<^bold>\ y) = \<^bold>- x \<^bold>\ \<^bold>- y" proof (rule compl_unique) have "(x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y) = ((x \<^bold>\ y) \<^bold>\ \<^bold>- x) \<^bold>\ ((x \<^bold>\ y) \<^bold>\ \<^bold>- y)" by (rule conj_disj_distrib) also have "\ = (y \<^bold>\ (x \<^bold>\ \<^bold>- x)) \<^bold>\ (x \<^bold>\ (y \<^bold>\ \<^bold>- y))" - by (simp only: conj_ac) + by (simp only: ac_simps) finally show "(x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y) = \<^bold>0" by (simp only: conj_cancel_right conj_zero_right disj_zero_right) next have "(x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y) = (x \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y)) \<^bold>\ (y \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y))" by (rule disj_conj_distrib2) also have "\ = (\<^bold>- y \<^bold>\ (x \<^bold>\ \<^bold>- x)) \<^bold>\ (\<^bold>- x \<^bold>\ (y \<^bold>\ \<^bold>- y))" - by (simp only: disj_ac) + by (simp only: ac_simps) finally show "(x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y) = \<^bold>1" by (simp only: disj_cancel_right disj_one_right conj_one_right) qed +context +begin + +interpretation dual: abstract_boolean_algebra \(\<^bold>\)\ \(\<^bold>\)\ compl \\<^bold>1\ \\<^bold>0\ + apply standard + apply (rule disj_conj_distrib) + apply (rule conj_disj_distrib) + apply simp_all + done + lemma de_Morgan_disj [simp]: "\<^bold>- (x \<^bold>\ y) = \<^bold>- x \<^bold>\ \<^bold>- y" - using dual.boolean_algebra_axioms by (rule boolean_algebra.de_Morgan_conj) + by (fact dual.de_Morgan_conj) + +end + +end subsection \Symmetric Difference\ -definition xor :: "'a \ 'a \ 'a" (infixr "\<^bold>\" 65) - where "x \<^bold>\ y = (x \<^bold>\ \<^bold>- y) \<^bold>\ (\<^bold>- x \<^bold>\ y)" +locale abstract_boolean_algebra_sym_diff = abstract_boolean_algebra + + fixes xor :: \'a \ 'a \ 'a\ (infixr \\<^bold>\\ 65) + assumes xor_def : \x \<^bold>\ y = (x \<^bold>\ \<^bold>- y) \<^bold>\ (\<^bold>- x \<^bold>\ y)\ +begin sublocale xor: comm_monoid xor \\<^bold>0\ proof fix x y z :: 'a let ?t = "(x \<^bold>\ y \<^bold>\ z) \<^bold>\ (x \<^bold>\ \<^bold>- y \<^bold>\ \<^bold>- z) \<^bold>\ (\<^bold>- x \<^bold>\ y \<^bold>\ \<^bold>- z) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y \<^bold>\ z)" have "?t \<^bold>\ (z \<^bold>\ x \<^bold>\ \<^bold>- x) \<^bold>\ (z \<^bold>\ y \<^bold>\ \<^bold>- y) = ?t \<^bold>\ (x \<^bold>\ y \<^bold>\ \<^bold>- y) \<^bold>\ (x \<^bold>\ z \<^bold>\ \<^bold>- z)" by (simp only: conj_cancel_right conj_zero_right) then show "(x \<^bold>\ y) \<^bold>\ z = x \<^bold>\ (y \<^bold>\ z)" by (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) - (simp only: conj_disj_distribs conj_ac disj_ac) + (simp only: conj_disj_distribs conj_ac ac_simps) show "x \<^bold>\ y = y \<^bold>\ x" - by (simp only: xor_def conj_commute disj_commute) + by (simp only: xor_def ac_simps) show "x \<^bold>\ \<^bold>0 = x" by (simp add: xor_def) qed -lemmas xor_assoc = xor.assoc -lemmas xor_commute = xor.commute -lemmas xor_left_commute = xor.left_commute - -lemmas xor_ac = xor.assoc xor.commute xor.left_commute - -lemma xor_def2: "x \<^bold>\ y = (x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y)" - using conj.commute conj_disj_distrib2 disj.commute xor_def by auto - -lemma xor_zero_right: "x \<^bold>\ \<^bold>0 = x" - by (fact xor.comm_neutral) - -lemma xor_zero_left: "\<^bold>0 \<^bold>\ x = x" - by (fact xor.left_neutral) +lemma xor_def2: + \x \<^bold>\ y = (x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y)\ +proof - + note xor_def [of x y] + also have \x \<^bold>\ \<^bold>- y \<^bold>\ \<^bold>- x \<^bold>\ y = ((x \<^bold>\ \<^bold>- x) \<^bold>\ (\<^bold>- y \<^bold>\ \<^bold>- x)) \<^bold>\ (x \<^bold>\ y) \<^bold>\ (\<^bold>- y \<^bold>\ y)\ + by (simp add: ac_simps disj_conj_distribs) + also have \\ = (x \<^bold>\ y) \<^bold>\ (\<^bold>- x \<^bold>\ \<^bold>- y)\ + by (simp add: ac_simps) + finally show ?thesis . +qed lemma xor_one_right [simp]: "x \<^bold>\ \<^bold>1 = \<^bold>- x" - by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) + by (simp only: xor_def compl_one conj_zero_right conj_one_right disj.left_neutral) lemma xor_one_left [simp]: "\<^bold>1 \<^bold>\ x = \<^bold>- x" - by (subst xor_commute) (rule xor_one_right) + using xor_one_right [of x] by (simp add: ac_simps) lemma xor_self [simp]: "x \<^bold>\ x = \<^bold>0" by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) lemma xor_left_self [simp]: "x \<^bold>\ (x \<^bold>\ y) = y" - by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) + by (simp only: xor.assoc [symmetric] xor_self xor.left_neutral) lemma xor_compl_left [simp]: "\<^bold>- x \<^bold>\ y = \<^bold>- (x \<^bold>\ y)" by (simp add: ac_simps flip: xor_one_left) lemma xor_compl_right [simp]: "x \<^bold>\ \<^bold>- y = \<^bold>- (x \<^bold>\ y)" - using xor_commute xor_compl_left by auto + using xor.commute xor_compl_left by auto -lemma xor_cancel_right: "x \<^bold>\ \<^bold>- x = \<^bold>1" +lemma xor_cancel_right [simp]: "x \<^bold>\ \<^bold>- x = \<^bold>1" by (simp only: xor_compl_right xor_self compl_zero) -lemma xor_cancel_left: "\<^bold>- x \<^bold>\ x = \<^bold>1" +lemma xor_cancel_left [simp]: "\<^bold>- x \<^bold>\ x = \<^bold>1" by (simp only: xor_compl_left xor_self compl_zero) lemma conj_xor_distrib: "x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)" proof - have *: "(x \<^bold>\ y \<^bold>\ \<^bold>- z) \<^bold>\ (x \<^bold>\ \<^bold>- y \<^bold>\ z) = (y \<^bold>\ x \<^bold>\ \<^bold>- x) \<^bold>\ (z \<^bold>\ x \<^bold>\ \<^bold>- x) \<^bold>\ (x \<^bold>\ y \<^bold>\ \<^bold>- z) \<^bold>\ (x \<^bold>\ \<^bold>- y \<^bold>\ z)" - by (simp only: conj_cancel_right conj_zero_right disj_zero_left) + by (simp only: conj_cancel_right conj_zero_right disj.left_neutral) then show "x \<^bold>\ (y \<^bold>\ z) = (x \<^bold>\ y) \<^bold>\ (x \<^bold>\ z)" by (simp (no_asm_use) only: xor_def de_Morgan_disj de_Morgan_conj double_compl - conj_disj_distribs conj_ac disj_ac) + conj_disj_distribs ac_simps) qed lemma conj_xor_distrib2: "(y \<^bold>\ z) \<^bold>\ x = (y \<^bold>\ x) \<^bold>\ (z \<^bold>\ x)" by (simp add: conj.commute conj_xor_distrib) lemmas conj_xor_distribs = conj_xor_distrib conj_xor_distrib2 end + +subsection \Type classes\ + +class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus + + assumes inf_compl_bot: \x \ - x = \\ + and sup_compl_top: \x \ - x = \\ + assumes diff_eq: \x - y = x \ - y\ +begin + +sublocale boolean_algebra: abstract_boolean_algebra \(\)\ \(\)\ uminus \ \ + apply standard + apply (rule inf_sup_distrib1) + apply (rule sup_inf_distrib1) + apply (simp_all add: ac_simps inf_compl_bot sup_compl_top) + done + +lemma compl_inf_bot: "- x \ x = \" + by (fact boolean_algebra.conj_cancel_left) + +lemma compl_sup_top: "- x \ x = \" + by (fact boolean_algebra.disj_cancel_left) + +lemma compl_unique: + assumes "x \ y = \" + and "x \ y = \" + shows "- x = y" + using assms by (rule boolean_algebra.compl_unique) + +lemma double_compl: "- (- x) = x" + by (fact boolean_algebra.double_compl) + +lemma compl_eq_compl_iff: "- x = - y \ x = y" + by (fact boolean_algebra.compl_eq_compl_iff) + +lemma compl_bot_eq: "- \ = \" + by (fact boolean_algebra.compl_zero) + +lemma compl_top_eq: "- \ = \" + by (fact boolean_algebra.compl_one) + +lemma compl_inf: "- (x \ y) = - x \ - y" + by (fact boolean_algebra.de_Morgan_conj) + +lemma compl_sup: "- (x \ y) = - x \ - y" + by (fact boolean_algebra.de_Morgan_disj) + +lemma compl_mono: + assumes "x \ y" + shows "- y \ - x" +proof - + from assms have "x \ y = y" by (simp only: le_iff_sup) + then have "- (x \ y) = - y" by simp + then have "- x \ - y = - y" by simp + then have "- y \ - x = - y" by (simp only: inf_commute) + then show ?thesis by (simp only: le_iff_inf) +qed + +lemma compl_le_compl_iff [simp]: "- x \ - y \ y \ x" + by (auto dest: compl_mono) + +lemma compl_le_swap1: + assumes "y \ - x" + shows "x \ -y" +proof - + from assms have "- (- x) \ - y" by (simp only: compl_le_compl_iff) + then show ?thesis by simp +qed + +lemma compl_le_swap2: + assumes "- y \ x" + shows "- x \ y" +proof - + from assms have "- x \ - (- y)" by (simp only: compl_le_compl_iff) + then show ?thesis by simp +qed + +lemma compl_less_compl_iff [simp]: "- x < - y \ y < x" + by (auto simp add: less_le) + +lemma compl_less_swap1: + assumes "y < - x" + shows "x < - y" +proof - + from assms have "- (- x) < - y" by (simp only: compl_less_compl_iff) + then show ?thesis by simp +qed + +lemma compl_less_swap2: + assumes "- y < x" + shows "- x < y" +proof - + from assms have "- x < - (- y)" + by (simp only: compl_less_compl_iff) + then show ?thesis by simp +qed + +lemma sup_cancel_left1: \x \ a \ (- x \ b) = \\ + by (simp add: ac_simps) + +lemma sup_cancel_left2: \- x \ a \ (x \ b) = \\ + by (simp add: ac_simps) + +lemma inf_cancel_left1: \x \ a \ (- x \ b) = \\ + by (simp add: ac_simps) + +lemma inf_cancel_left2: \- x \ a \ (x \ b) = \\ + by (simp add: ac_simps) + +lemma sup_compl_top_left1 [simp]: \- x \ (x \ y) = \\ + by (simp add: sup_assoc [symmetric]) + +lemma sup_compl_top_left2 [simp]: \x \ (- x \ y) = \\ + using sup_compl_top_left1 [of "- x" y] by simp + +lemma inf_compl_bot_left1 [simp]: \- x \ (x \ y) = \\ + by (simp add: inf_assoc [symmetric]) + +lemma inf_compl_bot_left2 [simp]: \x \ (- x \ y) = \\ + using inf_compl_bot_left1 [of "- x" y] by simp + +lemma inf_compl_bot_right [simp]: \x \ (y \ - x) = \\ + by (subst inf_left_commute) simp + end + + +subsection \Lattice on \<^typ>\bool\\ + +instantiation bool :: boolean_algebra +begin + +definition bool_Compl_def [simp]: "uminus = Not" + +definition bool_diff_def [simp]: "A - B \ A \ \ B" + +definition [simp]: "P \ Q \ P \ Q" + +definition [simp]: "P \ Q \ P \ Q" + +instance by standard auto + +end + +lemma sup_boolI1: "P \ P \ Q" + by simp + +lemma sup_boolI2: "Q \ P \ Q" + by simp + +lemma sup_boolE: "P \ Q \ (P \ R) \ (Q \ R) \ R" + by auto + +instance "fun" :: (type, boolean_algebra) boolean_algebra + by standard (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+ + + +subsection \Lattice on unary and binary predicates\ + +lemma inf1I: "A x \ B x \ (A \ B) x" + by (simp add: inf_fun_def) + +lemma inf2I: "A x y \ B x y \ (A \ B) x y" + by (simp add: inf_fun_def) + +lemma inf1E: "(A \ B) x \ (A x \ B x \ P) \ P" + by (simp add: inf_fun_def) + +lemma inf2E: "(A \ B) x y \ (A x y \ B x y \ P) \ P" + by (simp add: inf_fun_def) + +lemma inf1D1: "(A \ B) x \ A x" + by (rule inf1E) + +lemma inf2D1: "(A \ B) x y \ A x y" + by (rule inf2E) + +lemma inf1D2: "(A \ B) x \ B x" + by (rule inf1E) + +lemma inf2D2: "(A \ B) x y \ B x y" + by (rule inf2E) + +lemma sup1I1: "A x \ (A \ B) x" + by (simp add: sup_fun_def) + +lemma sup2I1: "A x y \ (A \ B) x y" + by (simp add: sup_fun_def) + +lemma sup1I2: "B x \ (A \ B) x" + by (simp add: sup_fun_def) + +lemma sup2I2: "B x y \ (A \ B) x y" + by (simp add: sup_fun_def) + +lemma sup1E: "(A \ B) x \ (A x \ P) \ (B x \ P) \ P" + by (simp add: sup_fun_def) iprover + +lemma sup2E: "(A \ B) x y \ (A x y \ P) \ (B x y \ P) \ P" + by (simp add: sup_fun_def) iprover + +text \ \<^medskip> Classical introduction rule: no commitment to \A\ vs \B\.\ + +lemma sup1CI: "(\ B x \ A x) \ (A \ B) x" + by (auto simp add: sup_fun_def) + +lemma sup2CI: "(\ B x y \ A x y) \ (A \ B) x y" + by (auto simp add: sup_fun_def) + + +subsection \Simproc setup\ + +locale boolean_algebra_cancel +begin + +lemma sup1: "(A::'a::semilattice_sup) \ sup k a \ sup A b \ sup k (sup a b)" + by (simp only: ac_simps) + +lemma sup2: "(B::'a::semilattice_sup) \ sup k b \ sup a B \ sup k (sup a b)" + by (simp only: ac_simps) + +lemma sup0: "(a::'a::bounded_semilattice_sup_bot) \ sup a bot" + by simp + +lemma inf1: "(A::'a::semilattice_inf) \ inf k a \ inf A b \ inf k (inf a b)" + by (simp only: ac_simps) + +lemma inf2: "(B::'a::semilattice_inf) \ inf k b \ inf a B \ inf k (inf a b)" + by (simp only: ac_simps) + +lemma inf0: "(a::'a::bounded_semilattice_inf_top) \ inf a top" + by simp + +end + +ML_file \Tools/boolean_algebra_cancel.ML\ + +simproc_setup boolean_algebra_cancel_sup ("sup a b::'a::boolean_algebra") = + \fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_sup_conv\ + +simproc_setup boolean_algebra_cancel_inf ("inf a b::'a::boolean_algebra") = + \fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_inf_conv\ + + +context boolean_algebra +begin + +lemma shunt1: "(x \ y \ z) \ (x \ -y \ z)" +proof + assume "x \ y \ z" + hence "-y \ (x \ y) \ -y \ z" + using sup.mono by blast + hence "-y \ x \ -y \ z" + by (simp add: sup_inf_distrib1) + thus "x \ -y \ z" + by simp +next + assume "x \ -y \ z" + hence "x \ y \ (-y \ z) \ y" + using inf_mono by auto + thus "x \ y \ z" + using inf.boundedE inf_sup_distrib2 by auto +qed + +lemma shunt2: "(x \ -y \ z) \ (x \ y \ z)" + by (simp add: shunt1) + +lemma inf_shunt: "(x \ y = \) \ (x \ - y)" + by (simp add: order.eq_iff shunt1) + +lemma sup_shunt: "(x \ y = \) \ (- x \ y)" + using inf_shunt [of \- x\ \- y\, symmetric] + by (simp flip: compl_sup compl_top_eq) + +lemma diff_shunt_var: "(x - y = \) \ (x \ y)" + by (simp add: diff_eq inf_shunt) + +lemma sup_neg_inf: + \p \ q \ r \ p \ -q \ r\ (is \?P \ ?Q\) +proof + assume ?P + then have \p \ - q \ (q \ r) \ - q\ + by (rule inf_mono) simp + then show ?Q + by (simp add: inf_sup_distrib2) +next + assume ?Q + then have \p \ - q \ q \ r \ q\ + by (rule sup_mono) simp + then show ?P + by (simp add: sup_inf_distrib ac_simps) +qed + +end + +end diff --git a/src/HOL/Fun.thy b/src/HOL/Fun.thy --- a/src/HOL/Fun.thy +++ b/src/HOL/Fun.thy @@ -1,1069 +1,1069 @@ (* Title: HOL/Fun.thy Author: Tobias Nipkow, Cambridge University Computer Laboratory Author: Andrei Popescu, TU Muenchen Copyright 1994, 2012 *) section \Notions about functions\ theory Fun imports Set keywords "functor" :: thy_goal_defn begin lemma apply_inverse: "f x = u \ (\x. P x \ g (f x) = x) \ P x \ x = g u" by auto text \Uniqueness, so NOT the axiom of choice.\ lemma uniq_choice: "\x. \!y. Q x y \ \f. \x. Q x (f x)" by (force intro: theI') lemma b_uniq_choice: "\x\S. \!y. Q x y \ \f. \x\S. Q x (f x)" by (force intro: theI') subsection \The Identity Function \id\\ definition id :: "'a \ 'a" where "id = (\x. x)" lemma id_apply [simp]: "id x = x" by (simp add: id_def) lemma image_id [simp]: "image id = id" by (simp add: id_def fun_eq_iff) lemma vimage_id [simp]: "vimage id = id" by (simp add: id_def fun_eq_iff) lemma eq_id_iff: "(\x. f x = x) \ f = id" by auto code_printing constant id \ (Haskell) "id" subsection \The Composition Operator \f \ g\\ definition comp :: "('b \ 'c) \ ('a \ 'b) \ 'a \ 'c" (infixl "\" 55) where "f \ g = (\x. f (g x))" notation (ASCII) comp (infixl "o" 55) lemma comp_apply [simp]: "(f \ g) x = f (g x)" by (simp add: comp_def) lemma comp_assoc: "(f \ g) \ h = f \ (g \ h)" by (simp add: fun_eq_iff) lemma id_comp [simp]: "id \ g = g" by (simp add: fun_eq_iff) lemma comp_id [simp]: "f \ id = f" by (simp add: fun_eq_iff) lemma comp_eq_dest: "a \ b = c \ d \ a (b v) = c (d v)" by (simp add: fun_eq_iff) lemma comp_eq_elim: "a \ b = c \ d \ ((\v. a (b v) = c (d v)) \ R) \ R" by (simp add: fun_eq_iff) lemma comp_eq_dest_lhs: "a \ b = c \ a (b v) = c v" by clarsimp lemma comp_eq_id_dest: "a \ b = id \ c \ a (b v) = c v" by clarsimp lemma image_comp: "f ` (g ` r) = (f \ g) ` r" by auto lemma vimage_comp: "f -` (g -` x) = (g \ f) -` x" by auto lemma image_eq_imp_comp: "f ` A = g ` B \ (h \ f) ` A = (h \ g) ` B" by (auto simp: comp_def elim!: equalityE) lemma image_bind: "f ` (Set.bind A g) = Set.bind A ((`) f \ g)" by (auto simp add: Set.bind_def) lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \ f)" by (auto simp add: Set.bind_def) lemma (in group_add) minus_comp_minus [simp]: "uminus \ uminus = id" by (simp add: fun_eq_iff) -lemma (in Lattices.boolean_algebra) minus_comp_minus [simp]: "uminus \ uminus = id" +lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus \ uminus = id" by (simp add: fun_eq_iff) code_printing constant comp \ (SML) infixl 5 "o" and (Haskell) infixr 9 "." subsection \The Forward Composition Operator \fcomp\\ definition fcomp :: "('a \ 'b) \ ('b \ 'c) \ 'a \ 'c" (infixl "\>" 60) where "f \> g = (\x. g (f x))" lemma fcomp_apply [simp]: "(f \> g) x = g (f x)" by (simp add: fcomp_def) lemma fcomp_assoc: "(f \> g) \> h = f \> (g \> h)" by (simp add: fcomp_def) lemma id_fcomp [simp]: "id \> g = g" by (simp add: fcomp_def) lemma fcomp_id [simp]: "f \> id = f" by (simp add: fcomp_def) lemma fcomp_comp: "fcomp f g = comp g f" by (simp add: ext) code_printing constant fcomp \ (Eval) infixl 1 "#>" no_notation fcomp (infixl "\>" 60) subsection \Mapping functions\ definition map_fun :: "('c \ 'a) \ ('b \ 'd) \ ('a \ 'b) \ 'c \ 'd" where "map_fun f g h = g \ h \ f" lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))" by (simp add: map_fun_def) subsection \Injectivity and Bijectivity\ definition inj_on :: "('a \ 'b) \ 'a set \ bool" \ \injective\ where "inj_on f A \ (\x\A. \y\A. f x = f y \ x = y)" definition bij_betw :: "('a \ 'b) \ 'a set \ 'b set \ bool" \ \bijective\ where "bij_betw f A B \ inj_on f A \ f ` A = B" text \ A common special case: functions injective, surjective or bijective over the entire domain type. \ abbreviation inj :: "('a \ 'b) \ bool" where "inj f \ inj_on f UNIV" abbreviation surj :: "('a \ 'b) \ bool" where "surj f \ range f = UNIV" translations \ \The negated case:\ "\ CONST surj f" \ "CONST range f \ CONST UNIV" abbreviation bij :: "('a \ 'b) \ bool" where "bij f \ bij_betw f UNIV UNIV" lemma inj_def: "inj f \ (\x y. f x = f y \ x = y)" unfolding inj_on_def by blast lemma injI: "(\x y. f x = f y \ x = y) \ inj f" unfolding inj_def by blast theorem range_ex1_eq: "inj f \ b \ range f \ (\!x. b = f x)" unfolding inj_def by blast lemma injD: "inj f \ f x = f y \ x = y" by (simp add: inj_def) lemma inj_on_eq_iff: "inj_on f A \ x \ A \ y \ A \ f x = f y \ x = y" by (auto simp: inj_on_def) lemma inj_on_cong: "(\a. a \ A \ f a = g a) \ inj_on f A \ inj_on g A" by (auto simp: inj_on_def) lemma inj_on_strict_subset: "inj_on f B \ A \ B \ f ` A \ f ` B" unfolding inj_on_def by blast lemma inj_compose: "inj f \ inj g \ inj (f \ g)" by (simp add: inj_def) lemma inj_fun: "inj f \ inj (\x y. f x)" by (simp add: inj_def fun_eq_iff) lemma inj_eq: "inj f \ f x = f y \ x = y" by (simp add: inj_on_eq_iff) lemma inj_on_iff_Uniq: "inj_on f A \ (\x\A. \\<^sub>\\<^sub>1y. y\A \ f x = f y)" by (auto simp: Uniq_def inj_on_def) lemma inj_on_id[simp]: "inj_on id A" by (simp add: inj_on_def) lemma inj_on_id2[simp]: "inj_on (\x. x) A" by (simp add: inj_on_def) lemma inj_on_Int: "inj_on f A \ inj_on f B \ inj_on f (A \ B)" unfolding inj_on_def by blast lemma surj_id: "surj id" by simp lemma bij_id[simp]: "bij id" by (simp add: bij_betw_def) lemma bij_uminus: "bij (uminus :: 'a \ 'a::ab_group_add)" unfolding bij_betw_def inj_on_def by (force intro: minus_minus [symmetric]) lemma bij_betwE: "bij_betw f A B \ \a\A. f a \ B" unfolding bij_betw_def by auto lemma inj_onI [intro?]: "(\x y. x \ A \ y \ A \ f x = f y \ x = y) \ inj_on f A" by (simp add: inj_on_def) lemma inj_on_inverseI: "(\x. x \ A \ g (f x) = x) \ inj_on f A" by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) lemma inj_onD: "inj_on f A \ f x = f y \ x \ A \ y \ A \ x = y" unfolding inj_on_def by blast lemma inj_on_subset: assumes "inj_on f A" and "B \ A" shows "inj_on f B" proof (rule inj_onI) fix a b assume "a \ B" and "b \ B" with assms have "a \ A" and "b \ A" by auto moreover assume "f a = f b" ultimately show "a = b" using assms by (auto dest: inj_onD) qed lemma comp_inj_on: "inj_on f A \ inj_on g (f ` A) \ inj_on (g \ f) A" by (simp add: comp_def inj_on_def) lemma inj_on_imageI: "inj_on (g \ f) A \ inj_on g (f ` A)" by (auto simp add: inj_on_def) lemma inj_on_image_iff: "\x\A. \y\A. g (f x) = g (f y) \ g x = g y \ inj_on f A \ inj_on g (f ` A) \ inj_on g A" unfolding inj_on_def by blast lemma inj_on_contraD: "inj_on f A \ x \ y \ x \ A \ y \ A \ f x \ f y" unfolding inj_on_def by blast lemma inj_singleton [simp]: "inj_on (\x. {x}) A" by (simp add: inj_on_def) lemma inj_on_empty[iff]: "inj_on f {}" by (simp add: inj_on_def) lemma subset_inj_on: "inj_on f B \ A \ B \ inj_on f A" unfolding inj_on_def by blast lemma inj_on_Un: "inj_on f (A \ B) \ inj_on f A \ inj_on f B \ f ` (A - B) \ f ` (B - A) = {}" unfolding inj_on_def by (blast intro: sym) lemma inj_on_insert [iff]: "inj_on f (insert a A) \ inj_on f A \ f a \ f ` (A - {a})" unfolding inj_on_def by (blast intro: sym) lemma inj_on_diff: "inj_on f A \ inj_on f (A - B)" unfolding inj_on_def by blast lemma comp_inj_on_iff: "inj_on f A \ inj_on f' (f ` A) \ inj_on (f' \ f) A" by (auto simp: comp_inj_on inj_on_def) lemma inj_on_imageI2: "inj_on (f' \ f) A \ inj_on f A" by (auto simp: comp_inj_on inj_on_def) lemma inj_img_insertE: assumes "inj_on f A" assumes "x \ B" and "insert x B = f ` A" obtains x' A' where "x' \ A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'" proof - from assms have "x \ f ` A" by auto then obtain x' where *: "x' \ A" "x = f x'" by auto then have A: "A = insert x' (A - {x'})" by auto with assms * have B: "B = f ` (A - {x'})" by (auto dest: inj_on_contraD) have "x' \ A - {x'}" by simp from this A \x = f x'\ B show ?thesis .. qed lemma linorder_inj_onI: fixes A :: "'a::order set" assumes ne: "\x y. \x < y; x\A; y\A\ \ f x \ f y" and lin: "\x y. \x\A; y\A\ \ x\y \ y\x" shows "inj_on f A" proof (rule inj_onI) fix x y assume eq: "f x = f y" and "x\A" "y\A" then show "x = y" using lin [of x y] ne by (force simp: dual_order.order_iff_strict) qed lemma linorder_injI: assumes "\x y::'a::linorder. x < y \ f x \ f y" shows "inj f" \ \Courtesy of Stephan Merz\ using assms by (auto intro: linorder_inj_onI linear) lemma inj_on_image_Pow: "inj_on f A \inj_on (image f) (Pow A)" unfolding Pow_def inj_on_def by blast lemma bij_betw_image_Pow: "bij_betw f A B \ bij_betw (image f) (Pow A) (Pow B)" by (auto simp add: bij_betw_def inj_on_image_Pow image_Pow_surj) lemma surj_def: "surj f \ (\y. \x. y = f x)" by auto lemma surjI: assumes "\x. g (f x) = x" shows "surj g" using assms [symmetric] by auto lemma surjD: "surj f \ \x. y = f x" by (simp add: surj_def) lemma surjE: "surj f \ (\x. y = f x \ C) \ C" by (simp add: surj_def) blast lemma comp_surj: "surj f \ surj g \ surj (g \ f)" using image_comp [of g f UNIV] by simp lemma bij_betw_imageI: "inj_on f A \ f ` A = B \ bij_betw f A B" unfolding bij_betw_def by clarify lemma bij_betw_imp_surj_on: "bij_betw f A B \ f ` A = B" unfolding bij_betw_def by clarify lemma bij_betw_imp_surj: "bij_betw f A UNIV \ surj f" unfolding bij_betw_def by auto lemma bij_betw_empty1: "bij_betw f {} A \ A = {}" unfolding bij_betw_def by blast lemma bij_betw_empty2: "bij_betw f A {} \ A = {}" unfolding bij_betw_def by blast lemma inj_on_imp_bij_betw: "inj_on f A \ bij_betw f A (f ` A)" unfolding bij_betw_def by simp lemma bij_betw_apply: "\bij_betw f A B; a \ A\ \ f a \ B" unfolding bij_betw_def by auto lemma bij_def: "bij f \ inj f \ surj f" by (rule bij_betw_def) lemma bijI: "inj f \ surj f \ bij f" by (rule bij_betw_imageI) lemma bij_is_inj: "bij f \ inj f" by (simp add: bij_def) lemma bij_is_surj: "bij f \ surj f" by (simp add: bij_def) lemma bij_betw_imp_inj_on: "bij_betw f A B \ inj_on f A" by (simp add: bij_betw_def) lemma bij_betw_trans: "bij_betw f A B \ bij_betw g B C \ bij_betw (g \ f) A C" by (auto simp add:bij_betw_def comp_inj_on) lemma bij_comp: "bij f \ bij g \ bij (g \ f)" by (rule bij_betw_trans) lemma bij_betw_comp_iff: "bij_betw f A A' \ bij_betw f' A' A'' \ bij_betw (f' \ f) A A''" by (auto simp add: bij_betw_def inj_on_def) lemma bij_betw_comp_iff2: assumes bij: "bij_betw f' A' A''" and img: "f ` A \ A'" shows "bij_betw f A A' \ bij_betw (f' \ f) A A''" using assms proof (auto simp add: bij_betw_comp_iff) assume *: "bij_betw (f' \ f) A A''" then show "bij_betw f A A'" using img proof (auto simp add: bij_betw_def) assume "inj_on (f' \ f) A" then show "inj_on f A" using inj_on_imageI2 by blast next fix a' assume **: "a' \ A'" with bij have "f' a' \ A''" unfolding bij_betw_def by auto with * obtain a where 1: "a \ A \ f' (f a) = f' a'" unfolding bij_betw_def by force with img have "f a \ A'" by auto with bij ** 1 have "f a = a'" unfolding bij_betw_def inj_on_def by auto with 1 show "a' \ f ` A" by auto qed qed lemma bij_betw_inv: assumes "bij_betw f A B" shows "\g. bij_betw g B A" proof - have i: "inj_on f A" and s: "f ` A = B" using assms by (auto simp: bij_betw_def) let ?P = "\b a. a \ A \ f a = b" let ?g = "\b. The (?P b)" have g: "?g b = a" if P: "?P b a" for a b proof - from that s have ex1: "\a. ?P b a" by blast then have uex1: "\!a. ?P b a" by (blast dest:inj_onD[OF i]) then show ?thesis using the1_equality[OF uex1, OF P] P by simp qed have "inj_on ?g B" proof (rule inj_onI) fix x y assume "x \ B" "y \ B" "?g x = ?g y" from s \x \ B\ obtain a1 where a1: "?P x a1" by blast from s \y \ B\ obtain a2 where a2: "?P y a2" by blast from g [OF a1] a1 g [OF a2] a2 \?g x = ?g y\ show "x = y" by simp qed moreover have "?g ` B = A" proof (auto simp: image_def) fix b assume "b \ B" with s obtain a where P: "?P b a" by blast with g[OF P] show "?g b \ A" by auto next fix a assume "a \ A" with s obtain b where P: "?P b a" by blast with s have "b \ B" by blast with g[OF P] show "\b\B. a = ?g b" by blast qed ultimately show ?thesis by (auto simp: bij_betw_def) qed lemma bij_betw_cong: "(\a. a \ A \ f a = g a) \ bij_betw f A A' = bij_betw g A A'" unfolding bij_betw_def inj_on_def by safe force+ (* somewhat slow *) lemma bij_betw_id[intro, simp]: "bij_betw id A A" unfolding bij_betw_def id_def by auto lemma bij_betw_id_iff: "bij_betw id A B \ A = B" by (auto simp add: bij_betw_def) lemma bij_betw_combine: "bij_betw f A B \ bij_betw f C D \ B \ D = {} \ bij_betw f (A \ C) (B \ D)" unfolding bij_betw_def inj_on_Un image_Un by auto lemma bij_betw_subset: "bij_betw f A A' \ B \ A \ f ` B = B' \ bij_betw f B B'" by (auto simp add: bij_betw_def inj_on_def) lemma bij_pointE: assumes "bij f" obtains x where "y = f x" and "\x'. y = f x' \ x' = x" proof - from assms have "inj f" by (rule bij_is_inj) moreover from assms have "surj f" by (rule bij_is_surj) then have "y \ range f" by simp ultimately have "\!x. y = f x" by (simp add: range_ex1_eq) with that show thesis by blast qed lemma bij_iff: \<^marker>\contributor \Amine Chaieb\\ \bij f \ (\x. \!y. f y = x)\ (is \?P \ ?Q\) proof assume ?P then have \inj f\ \surj f\ by (simp_all add: bij_def) show ?Q proof fix y from \surj f\ obtain x where \y = f x\ by (auto simp add: surj_def) with \inj f\ show \\!x. f x = y\ by (auto simp add: inj_def) qed next assume ?Q then have \inj f\ by (auto simp add: inj_def) moreover have \\x. y = f x\ for y proof - from \?Q\ obtain x where \f x = y\ by blast then have \y = f x\ by simp then show ?thesis .. qed then have \surj f\ by (auto simp add: surj_def) ultimately show ?P by (rule bijI) qed lemma bij_betw_partition: \bij_betw f A B\ if \bij_betw f (A \ C) (B \ D)\ \bij_betw f C D\ \A \ C = {}\ \B \ D = {}\ proof - from that have \inj_on f (A \ C)\ \inj_on f C\ \f ` (A \ C) = B \ D\ \f ` C = D\ by (simp_all add: bij_betw_def) then have \inj_on f A\ and \f ` (A - C) \ f ` (C - A) = {}\ by (simp_all add: inj_on_Un) with \A \ C = {}\ have \f ` A \ f ` C = {}\ by auto with \f ` (A \ C) = B \ D\ \f ` C = D\ \B \ D = {}\ have \f ` A = B\ by blast with \inj_on f A\ show ?thesis by (simp add: bij_betw_def) qed lemma surj_image_vimage_eq: "surj f \ f ` (f -` A) = A" by simp lemma surj_vimage_empty: assumes "surj f" shows "f -` A = {} \ A = {}" using surj_image_vimage_eq [OF \surj f\, of A] by (intro iffI) fastforce+ lemma inj_vimage_image_eq: "inj f \ f -` (f ` A) = A" unfolding inj_def by blast lemma vimage_subsetD: "surj f \ f -` B \ A \ B \ f ` A" by (blast intro: sym) lemma vimage_subsetI: "inj f \ B \ f ` A \ f -` B \ A" unfolding inj_def by blast lemma vimage_subset_eq: "bij f \ f -` B \ A \ B \ f ` A" unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD) lemma inj_on_image_eq_iff: "inj_on f C \ A \ C \ B \ C \ f ` A = f ` B \ A = B" by (fastforce simp: inj_on_def) lemma inj_on_Un_image_eq_iff: "inj_on f (A \ B) \ f ` A = f ` B \ A = B" by (erule inj_on_image_eq_iff) simp_all lemma inj_on_image_Int: "inj_on f C \ A \ C \ B \ C \ f ` (A \ B) = f ` A \ f ` B" unfolding inj_on_def by blast lemma inj_on_image_set_diff: "inj_on f C \ A - B \ C \ B \ C \ f ` (A - B) = f ` A - f ` B" unfolding inj_on_def by blast lemma image_Int: "inj f \ f ` (A \ B) = f ` A \ f ` B" unfolding inj_def by blast lemma image_set_diff: "inj f \ f ` (A - B) = f ` A - f ` B" unfolding inj_def by blast lemma inj_on_image_mem_iff: "inj_on f B \ a \ B \ A \ B \ f a \ f ` A \ a \ A" by (auto simp: inj_on_def) lemma inj_image_mem_iff: "inj f \ f a \ f ` A \ a \ A" by (blast dest: injD) lemma inj_image_subset_iff: "inj f \ f ` A \ f ` B \ A \ B" by (blast dest: injD) lemma inj_image_eq_iff: "inj f \ f ` A = f ` B \ A = B" by (blast dest: injD) lemma surj_Compl_image_subset: "surj f \ - (f ` A) \ f ` (- A)" by auto lemma inj_image_Compl_subset: "inj f \ f ` (- A) \ - (f ` A)" by (auto simp: inj_def) lemma bij_image_Compl_eq: "bij f \ f ` (- A) = - (f ` A)" by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI) lemma inj_vimage_singleton: "inj f \ f -` {a} \ {THE x. f x = a}" \ \The inverse image of a singleton under an injective function is included in a singleton.\ by (simp add: inj_def) (blast intro: the_equality [symmetric]) lemma inj_on_vimage_singleton: "inj_on f A \ f -` {a} \ A \ {THE x. x \ A \ f x = a}" by (auto simp add: inj_on_def intro: the_equality [symmetric]) lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" by (auto intro!: inj_onI) lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \ inj_on f A" by (auto intro!: inj_onI dest: strict_mono_eq) lemma bij_betw_byWitness: assumes left: "\a \ A. f' (f a) = a" and right: "\a' \ A'. f (f' a') = a'" and "f ` A \ A'" and img2: "f' ` A' \ A" shows "bij_betw f A A'" using assms unfolding bij_betw_def inj_on_def proof safe fix a b assume "a \ A" "b \ A" with left have "a = f' (f a) \ b = f' (f b)" by simp moreover assume "f a = f b" ultimately show "a = b" by simp next fix a' assume *: "a' \ A'" with img2 have "f' a' \ A" by blast moreover from * right have "a' = f (f' a')" by simp ultimately show "a' \ f ` A" by blast qed corollary notIn_Un_bij_betw: assumes "b \ A" and "f b \ A'" and "bij_betw f A A'" shows "bij_betw f (A \ {b}) (A' \ {f b})" proof - have "bij_betw f {b} {f b}" unfolding bij_betw_def inj_on_def by simp with assms show ?thesis using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast qed lemma notIn_Un_bij_betw3: assumes "b \ A" and "f b \ A'" shows "bij_betw f A A' = bij_betw f (A \ {b}) (A' \ {f b})" proof assume "bij_betw f A A'" then show "bij_betw f (A \ {b}) (A' \ {f b})" using assms notIn_Un_bij_betw [of b A f A'] by blast next assume *: "bij_betw f (A \ {b}) (A' \ {f b})" have "f ` A = A'" proof auto fix a assume **: "a \ A" then have "f a \ A' \ {f b}" using * unfolding bij_betw_def by blast moreover have False if "f a = f b" proof - have "a = b" using * ** that unfolding bij_betw_def inj_on_def by blast with \b \ A\ ** show ?thesis by blast qed ultimately show "f a \ A'" by blast next fix a' assume **: "a' \ A'" then have "a' \ f ` (A \ {b})" using * by (auto simp add: bij_betw_def) then obtain a where 1: "a \ A \ {b} \ f a = a'" by blast moreover have False if "a = b" using 1 ** \f b \ A'\ that by blast ultimately have "a \ A" by blast with 1 show "a' \ f ` A" by blast qed then show "bij_betw f A A'" using * bij_betw_subset[of f "A \ {b}" _ A] by blast qed lemma inj_on_disjoint_Un: assumes "inj_on f A" and "inj_on g B" and "f ` A \ g ` B = {}" shows "inj_on (\x. if x \ A then f x else g x) (A \ B)" using assms by (simp add: inj_on_def disjoint_iff) (blast) lemma bij_betw_disjoint_Un: assumes "bij_betw f A C" and "bij_betw g B D" and "A \ B = {}" and "C \ D = {}" shows "bij_betw (\x. if x \ A then f x else g x) (A \ B) (C \ D)" using assms by (auto simp: inj_on_disjoint_Un bij_betw_def) lemma involuntory_imp_bij: \bij f\ if \\x. f (f x) = x\ proof (rule bijI) from that show \surj f\ by (rule surjI) show \inj f\ proof (rule injI) fix x y assume \f x = f y\ then have \f (f x) = f (f y)\ by simp then show \x = y\ by (simp add: that) qed qed subsubsection \Important examples\ context cancel_semigroup_add begin lemma inj_on_add [simp]: "inj_on ((+) a) A" by (rule inj_onI) simp lemma inj_add_left: \inj ((+) a)\ by simp lemma inj_on_add' [simp]: "inj_on (\b. b + a) A" by (rule inj_onI) simp lemma bij_betw_add [simp]: "bij_betw ((+) a) A B \ (+) a ` A = B" by (simp add: bij_betw_def) end context ab_group_add begin lemma surj_plus [simp]: "surj ((+) a)" by (auto intro!: range_eqI [of b "(+) a" "b - a" for b]) (simp add: algebra_simps) lemma inj_diff_right [simp]: \inj (\b. b - a)\ proof - have \inj ((+) (- a))\ by (fact inj_add_left) also have \(+) (- a) = (\b. b - a)\ by (simp add: fun_eq_iff) finally show ?thesis . qed lemma surj_diff_right [simp]: "surj (\x. x - a)" using surj_plus [of "- a"] by (simp cong: image_cong_simp) lemma translation_Compl: "(+) a ` (- t) = - ((+) a ` t)" proof (rule set_eqI) fix b show "b \ (+) a ` (- t) \ b \ - (+) a ` t" by (auto simp: image_iff algebra_simps intro!: bexI [of _ "b - a"]) qed lemma translation_subtract_Compl: "(\x. x - a) ` (- t) = - ((\x. x - a) ` t)" using translation_Compl [of "- a" t] by (simp cong: image_cong_simp) lemma translation_diff: "(+) a ` (s - t) = ((+) a ` s) - ((+) a ` t)" by auto lemma translation_subtract_diff: "(\x. x - a) ` (s - t) = ((\x. x - a) ` s) - ((\x. x - a) ` t)" using translation_diff [of "- a"] by (simp cong: image_cong_simp) lemma translation_Int: "(+) a ` (s \ t) = ((+) a ` s) \ ((+) a ` t)" by auto lemma translation_subtract_Int: "(\x. x - a) ` (s \ t) = ((\x. x - a) ` s) \ ((\x. x - a) ` t)" using translation_Int [of " -a"] by (simp cong: image_cong_simp) end subsection \Function Updating\ definition fun_upd :: "('a \ 'b) \ 'a \ 'b \ ('a \ 'b)" where "fun_upd f a b = (\x. if x = a then b else f x)" nonterminal updbinds and updbind syntax "_updbind" :: "'a \ 'a \ updbind" ("(2_ :=/ _)") "" :: "updbind \ updbinds" ("_") "_updbinds":: "updbind \ updbinds \ updbinds" ("_,/ _") "_Update" :: "'a \ updbinds \ 'a" ("_/'((_)')" [1000, 0] 900) translations "_Update f (_updbinds b bs)" \ "_Update (_Update f b) bs" "f(x:=y)" \ "CONST fun_upd f x y" (* Hint: to define the sum of two functions (or maps), use case_sum. A nice infix syntax could be defined by notation case_sum (infixr "'(+')"80) *) lemma fun_upd_idem_iff: "f(x:=y) = f \ f x = y" unfolding fun_upd_def apply safe apply (erule subst) apply (rule_tac [2] ext) apply auto done lemma fun_upd_idem: "f x = y \ f(x := y) = f" by (simp only: fun_upd_idem_iff) lemma fun_upd_triv [iff]: "f(x := f x) = f" by (simp only: fun_upd_idem) lemma fun_upd_apply [simp]: "(f(x := y)) z = (if z = x then y else f z)" by (simp add: fun_upd_def) (* fun_upd_apply supersedes these two, but they are useful if fun_upd_apply is intentionally removed from the simpset *) lemma fun_upd_same: "(f(x := y)) x = y" by simp lemma fun_upd_other: "z \ x \ (f(x := y)) z = f z" by simp lemma fun_upd_upd [simp]: "f(x := y, x := z) = f(x := z)" by (simp add: fun_eq_iff) lemma fun_upd_twist: "a \ c \ (m(a := b))(c := d) = (m(c := d))(a := b)" by auto lemma inj_on_fun_updI: "inj_on f A \ y \ f ` A \ inj_on (f(x := y)) A" by (auto simp: inj_on_def) lemma fun_upd_image: "f(x := y) ` A = (if x \ A then insert y (f ` (A - {x})) else f ` A)" by auto lemma fun_upd_comp: "f \ (g(x := y)) = (f \ g)(x := f y)" by auto lemma fun_upd_eqD: "f(x := y) = g(x := z) \ y = z" by (simp add: fun_eq_iff split: if_split_asm) subsection \\override_on\\ definition override_on :: "('a \ 'b) \ ('a \ 'b) \ 'a set \ 'a \ 'b" where "override_on f g A = (\a. if a \ A then g a else f a)" lemma override_on_emptyset[simp]: "override_on f g {} = f" by (simp add: override_on_def) lemma override_on_apply_notin[simp]: "a \ A \ (override_on f g A) a = f a" by (simp add: override_on_def) lemma override_on_apply_in[simp]: "a \ A \ (override_on f g A) a = g a" by (simp add: override_on_def) lemma override_on_insert: "override_on f g (insert x X) = (override_on f g X)(x:=g x)" by (simp add: override_on_def fun_eq_iff) lemma override_on_insert': "override_on f g (insert x X) = (override_on (f(x:=g x)) g X)" by (simp add: override_on_def fun_eq_iff) subsection \Inversion of injective functions\ definition the_inv_into :: "'a set \ ('a \ 'b) \ ('b \ 'a)" where "the_inv_into A f = (\x. THE y. y \ A \ f y = x)" lemma the_inv_into_f_f: "inj_on f A \ x \ A \ the_inv_into A f (f x) = x" unfolding the_inv_into_def inj_on_def by blast lemma f_the_inv_into_f: "inj_on f A \ y \ f ` A \ f (the_inv_into A f y) = y" unfolding the_inv_into_def by (rule the1I2; blast dest: inj_onD) lemma f_the_inv_into_f_bij_betw: "bij_betw f A B \ (bij_betw f A B \ x \ B) \ f (the_inv_into A f x) = x" unfolding bij_betw_def by (blast intro: f_the_inv_into_f) lemma the_inv_into_into: "inj_on f A \ x \ f ` A \ A \ B \ the_inv_into A f x \ B" unfolding the_inv_into_def by (rule the1I2; blast dest: inj_onD) lemma the_inv_into_onto [simp]: "inj_on f A \ the_inv_into A f ` (f ` A) = A" by (fast intro: the_inv_into_into the_inv_into_f_f [symmetric]) lemma the_inv_into_f_eq: "inj_on f A \ f x = y \ x \ A \ the_inv_into A f y = x" by (force simp add: the_inv_into_f_f) lemma the_inv_into_comp: "inj_on f (g ` A) \ inj_on g A \ x \ f ` g ` A \ the_inv_into A (f \ g) x = (the_inv_into A g \ the_inv_into (g ` A) f) x" apply (rule the_inv_into_f_eq) apply (fast intro: comp_inj_on) apply (simp add: f_the_inv_into_f the_inv_into_into) apply (simp add: the_inv_into_into) done lemma inj_on_the_inv_into: "inj_on f A \ inj_on (the_inv_into A f) (f ` A)" by (auto intro: inj_onI simp: the_inv_into_f_f) lemma bij_betw_the_inv_into: "bij_betw f A B \ bij_betw (the_inv_into A f) B A" by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) lemma bij_betw_iff_bijections: "bij_betw f A B \ (\g. (\x \ A. f x \ B \ g(f x) = x) \ (\y \ B. g y \ A \ f(g y) = y))" (is "?lhs = ?rhs") proof assume L: ?lhs then show ?rhs apply (rule_tac x="the_inv_into A f" in exI) apply (auto simp: bij_betw_def f_the_inv_into_f the_inv_into_f_f the_inv_into_into) done qed (force intro: bij_betw_byWitness) abbreviation the_inv :: "('a \ 'b) \ ('b \ 'a)" where "the_inv f \ the_inv_into UNIV f" lemma the_inv_f_f: "the_inv f (f x) = x" if "inj f" using that UNIV_I by (rule the_inv_into_f_f) subsection \Cantor's Paradox\ theorem Cantors_paradox: "\f. f ` A = Pow A" proof assume "\f. f ` A = Pow A" then obtain f where f: "f ` A = Pow A" .. let ?X = "{a \ A. a \ f a}" have "?X \ Pow A" by blast then have "?X \ f ` A" by (simp only: f) then obtain x where "x \ A" and "f x = ?X" by blast then show False by blast qed subsection \Monotonic functions over a set\ definition "mono_on f A \ \r s. r \ A \ s \ A \ r \ s \ f r \ f s" lemma mono_onI: "(\r s. r \ A \ s \ A \ r \ s \ f r \ f s) \ mono_on f A" unfolding mono_on_def by simp lemma mono_onD: "\mono_on f A; r \ A; s \ A; r \ s\ \ f r \ f s" unfolding mono_on_def by simp lemma mono_imp_mono_on: "mono f \ mono_on f A" unfolding mono_def mono_on_def by auto lemma mono_on_subset: "mono_on f A \ B \ A \ mono_on f B" unfolding mono_on_def by auto definition "strict_mono_on f A \ \r s. r \ A \ s \ A \ r < s \ f r < f s" lemma strict_mono_onI: "(\r s. r \ A \ s \ A \ r < s \ f r < f s) \ strict_mono_on f A" unfolding strict_mono_on_def by simp lemma strict_mono_onD: "\strict_mono_on f A; r \ A; s \ A; r < s\ \ f r < f s" unfolding strict_mono_on_def by simp lemma mono_on_greaterD: assumes "mono_on g A" "x \ A" "y \ A" "g x > (g (y::_::linorder) :: _ :: linorder)" shows "x > y" proof (rule ccontr) assume "\x > y" hence "x \ y" by (simp add: not_less) from assms(1-3) and this have "g x \ g y" by (rule mono_onD) with assms(4) show False by simp qed lemma strict_mono_inv: fixes f :: "('a::linorder) \ ('b::linorder)" assumes "strict_mono f" and "surj f" and inv: "\x. g (f x) = x" shows "strict_mono g" proof fix x y :: 'b assume "x < y" from \surj f\ obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast with \x < y\ and \strict_mono f\ have "x' < y'" by (simp add: strict_mono_less) with inv show "g x < g y" by simp qed lemma strict_mono_on_imp_inj_on: assumes "strict_mono_on (f :: (_ :: linorder) \ (_ :: preorder)) A" shows "inj_on f A" proof (rule inj_onI) fix x y assume "x \ A" "y \ A" "f x = f y" thus "x = y" by (cases x y rule: linorder_cases) (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x]) qed lemma strict_mono_on_leD: assumes "strict_mono_on (f :: (_ :: linorder) \ _ :: preorder) A" "x \ A" "y \ A" "x \ y" shows "f x \ f y" proof (insert le_less_linear[of y x], elim disjE) assume "x < y" with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all thus ?thesis by (rule less_imp_le) qed (insert assms, simp) lemma strict_mono_on_eqD: fixes f :: "(_ :: linorder) \ (_ :: preorder)" assumes "strict_mono_on f A" "f x = f y" "x \ A" "y \ A" shows "y = x" using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD) lemma strict_mono_on_imp_mono_on: "strict_mono_on (f :: (_ :: linorder) \ _ :: preorder) A \ mono_on f A" by (rule mono_onI, rule strict_mono_on_leD) subsection \Setup\ subsubsection \Proof tools\ text \Simplify terms of the form \f(\,x:=y,\,x:=z,\)\ to \f(\,x:=z,\)\\ simproc_setup fun_upd2 ("f(v := w, x := y)") = \fn _ => let fun gen_fun_upd NONE T _ _ = NONE | gen_fun_upd (SOME f) T x y = SOME (Const (\<^const_name>\fun_upd\, T) $ f $ x $ y) fun dest_fun_T1 (Type (_, T :: Ts)) = T fun find_double (t as Const (\<^const_name>\fun_upd\,T) $ f $ x $ y) = let fun find (Const (\<^const_name>\fun_upd\,T) $ g $ v $ w) = if v aconv x then SOME g else gen_fun_upd (find g) T v w | find t = NONE in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end val ss = simpset_of \<^context> fun proc ctxt ct = let val t = Thm.term_of ct in (case find_double t of (T, NONE) => NONE | (T, SOME rhs) => SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) (fn _ => resolve_tac ctxt [eq_reflection] 1 THEN resolve_tac ctxt @{thms ext} 1 THEN simp_tac (put_simpset ss ctxt) 1))) end in proc end \ subsubsection \Functorial structure of types\ ML_file \Tools/functor.ML\ functor map_fun: map_fun by (simp_all add: fun_eq_iff) functor vimage by (simp_all add: fun_eq_iff vimage_comp) text \Legacy theorem names\ lemmas o_def = comp_def lemmas o_apply = comp_apply lemmas o_assoc = comp_assoc [symmetric] lemmas id_o = id_comp lemmas o_id = comp_id lemmas o_eq_dest = comp_eq_dest lemmas o_eq_elim = comp_eq_elim lemmas o_eq_dest_lhs = comp_eq_dest_lhs lemmas o_eq_id_dest = comp_eq_id_dest end diff --git a/src/HOL/Hilbert_Choice.thy b/src/HOL/Hilbert_Choice.thy --- a/src/HOL/Hilbert_Choice.thy +++ b/src/HOL/Hilbert_Choice.thy @@ -1,1266 +1,1254 @@ (* Title: HOL/Hilbert_Choice.thy Author: Lawrence C Paulson, Tobias Nipkow Author: Viorel Preoteasa (Results about complete distributive lattices) Copyright 2001 University of Cambridge *) section \Hilbert's Epsilon-Operator and the Axiom of Choice\ theory Hilbert_Choice imports Wellfounded keywords "specification" :: thy_goal_defn begin subsection \Hilbert's epsilon\ axiomatization Eps :: "('a \ bool) \ 'a" where someI: "P x \ P (Eps P)" syntax (epsilon) "_Eps" :: "pttrn \ bool \ 'a" ("(3\_./ _)" [0, 10] 10) syntax (input) "_Eps" :: "pttrn \ bool \ 'a" ("(3@ _./ _)" [0, 10] 10) syntax "_Eps" :: "pttrn \ bool \ 'a" ("(3SOME _./ _)" [0, 10] 10) translations "SOME x. P" \ "CONST Eps (\x. P)" print_translation \ [(\<^const_syntax>\Eps\, fn _ => fn [Abs abs] => let val (x, t) = Syntax_Trans.atomic_abs_tr' abs in Syntax.const \<^syntax_const>\_Eps\ $ x $ t end)] \ \ \to avoid eta-contraction of body\ definition inv_into :: "'a set \ ('a \ 'b) \ ('b \ 'a)" where "inv_into A f = (\x. SOME y. y \ A \ f y = x)" lemma inv_into_def2: "inv_into A f x = (SOME y. y \ A \ f y = x)" by(simp add: inv_into_def) abbreviation inv :: "('a \ 'b) \ ('b \ 'a)" where "inv \ inv_into UNIV" subsection \Hilbert's Epsilon-operator\ lemma Eps_cong: assumes "\x. P x = Q x" shows "Eps P = Eps Q" using ext[of P Q, OF assms] by simp text \ Easier to use than \someI\ if the witness comes from an existential formula. \ lemma someI_ex [elim?]: "\x. P x \ P (SOME x. P x)" by (elim exE someI) lemma some_eq_imp: assumes "Eps P = a" "P b" shows "P a" using assms someI_ex by force text \ Easier to use than \someI\ because the conclusion has only one occurrence of \<^term>\P\. \ lemma someI2: "P a \ (\x. P x \ Q x) \ Q (SOME x. P x)" by (blast intro: someI) text \ Easier to use than \someI2\ if the witness comes from an existential formula. \ lemma someI2_ex: "\a. P a \ (\x. P x \ Q x) \ Q (SOME x. P x)" by (blast intro: someI2) lemma someI2_bex: "\a\A. P a \ (\x. x \ A \ P x \ Q x) \ Q (SOME x. x \ A \ P x)" by (blast intro: someI2) lemma some_equality [intro]: "P a \ (\x. P x \ x = a) \ (SOME x. P x) = a" by (blast intro: someI2) lemma some1_equality: "\!x. P x \ P a \ (SOME x. P x) = a" by blast lemma some_eq_ex: "P (SOME x. P x) \ (\x. P x)" by (blast intro: someI) lemma some_in_eq: "(SOME x. x \ A) \ A \ A \ {}" unfolding ex_in_conv[symmetric] by (rule some_eq_ex) lemma some_eq_trivial [simp]: "(SOME y. y = x) = x" by (rule some_equality) (rule refl) lemma some_sym_eq_trivial [simp]: "(SOME y. x = y) = x" by (iprover intro: some_equality) subsection \Axiom of Choice, Proved Using the Description Operator\ lemma choice: "\x. \y. Q x y \ \f. \x. Q x (f x)" by (fast elim: someI) lemma bchoice: "\x\S. \y. Q x y \ \f. \x\S. Q x (f x)" by (fast elim: someI) lemma choice_iff: "(\x. \y. Q x y) \ (\f. \x. Q x (f x))" by (fast elim: someI) lemma choice_iff': "(\x. P x \ (\y. Q x y)) \ (\f. \x. P x \ Q x (f x))" by (fast elim: someI) lemma bchoice_iff: "(\x\S. \y. Q x y) \ (\f. \x\S. Q x (f x))" by (fast elim: someI) lemma bchoice_iff': "(\x\S. P x \ (\y. Q x y)) \ (\f. \x\S. P x \ Q x (f x))" by (fast elim: someI) lemma dependent_nat_choice: assumes 1: "\x. P 0 x" and 2: "\x n. P n x \ \y. P (Suc n) y \ Q n x y" shows "\f. \n. P n (f n) \ Q n (f n) (f (Suc n))" proof (intro exI allI conjI) fix n define f where "f = rec_nat (SOME x. P 0 x) (\n x. SOME y. P (Suc n) y \ Q n x y)" then have "P 0 (f 0)" "\n. P n (f n) \ P (Suc n) (f (Suc n)) \ Q n (f n) (f (Suc n))" using someI_ex[OF 1] someI_ex[OF 2] by simp_all then show "P n (f n)" "Q n (f n) (f (Suc n))" by (induct n) auto qed lemma finite_subset_Union: assumes "finite A" "A \ \\" obtains \ where "finite \" "\ \ \" "A \ \\" proof - have "\x\A. \B\\. x\B" using assms by blast then obtain f where f: "\x. x \ A \ f x \ \ \ x \ f x" by (auto simp add: bchoice_iff Bex_def) show thesis proof show "finite (f ` A)" using assms by auto qed (use f in auto) qed subsection \Function Inverse\ lemma inv_def: "inv f = (\y. SOME x. f x = y)" by (simp add: inv_into_def) lemma inv_into_into: "x \ f ` A \ inv_into A f x \ A" by (simp add: inv_into_def) (fast intro: someI2) lemma inv_identity [simp]: "inv (\a. a) = (\a. a)" by (simp add: inv_def) lemma inv_id [simp]: "inv id = id" by (simp add: id_def) lemma inv_into_f_f [simp]: "inj_on f A \ x \ A \ inv_into A f (f x) = x" by (simp add: inv_into_def inj_on_def) (blast intro: someI2) lemma inv_f_f: "inj f \ inv f (f x) = x" by simp lemma f_inv_into_f: "y \ f`A \ f (inv_into A f y) = y" by (simp add: inv_into_def) (fast intro: someI2) lemma inv_into_f_eq: "inj_on f A \ x \ A \ f x = y \ inv_into A f y = x" by (erule subst) (fast intro: inv_into_f_f) lemma inv_f_eq: "inj f \ f x = y \ inv f y = x" by (simp add:inv_into_f_eq) lemma inj_imp_inv_eq: "inj f \ \x. f (g x) = x \ inv f = g" by (blast intro: inv_into_f_eq) text \But is it useful?\ lemma inj_transfer: assumes inj: "inj f" and minor: "\y. y \ range f \ P (inv f y)" shows "P x" proof - have "f x \ range f" by auto then have "P(inv f (f x))" by (rule minor) then show "P x" by (simp add: inv_into_f_f [OF inj]) qed lemma inj_iff: "inj f \ inv f \ f = id" by (simp add: o_def fun_eq_iff) (blast intro: inj_on_inverseI inv_into_f_f) lemma inv_o_cancel[simp]: "inj f \ inv f \ f = id" by (simp add: inj_iff) lemma o_inv_o_cancel[simp]: "inj f \ g \ inv f \ f = g" by (simp add: comp_assoc) lemma inv_into_image_cancel[simp]: "inj_on f A \ S \ A \ inv_into A f ` f ` S = S" by (fastforce simp: image_def) lemma inj_imp_surj_inv: "inj f \ surj (inv f)" by (blast intro!: surjI inv_into_f_f) lemma surj_f_inv_f: "surj f \ f (inv f y) = y" by (simp add: f_inv_into_f) lemma bij_inv_eq_iff: "bij p \ x = inv p y \ p x = y" using surj_f_inv_f[of p] by (auto simp add: bij_def) lemma inv_into_injective: assumes eq: "inv_into A f x = inv_into A f y" and x: "x \ f`A" and y: "y \ f`A" shows "x = y" proof - from eq have "f (inv_into A f x) = f (inv_into A f y)" by simp with x y show ?thesis by (simp add: f_inv_into_f) qed lemma inj_on_inv_into: "B \ f`A \ inj_on (inv_into A f) B" by (blast intro: inj_onI dest: inv_into_injective injD) lemma inj_imp_bij_betw_inv: "inj f \ bij_betw (inv f) (f ` M) M" by (simp add: bij_betw_def image_subsetI inj_on_inv_into) lemma bij_betw_inv_into: "bij_betw f A B \ bij_betw (inv_into A f) B A" by (auto simp add: bij_betw_def inj_on_inv_into) lemma surj_imp_inj_inv: "surj f \ inj (inv f)" by (simp add: inj_on_inv_into) lemma surj_iff: "surj f \ f \ inv f = id" by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a]) lemma surj_iff_all: "surj f \ (\x. f (inv f x) = x)" by (simp add: o_def surj_iff fun_eq_iff) lemma surj_imp_inv_eq: assumes "surj f" and gf: "\x. g (f x) = x" shows "inv f = g" proof (rule ext) fix x have "g (f (inv f x)) = inv f x" by (rule gf) then show "inv f x = g x" by (simp add: surj_f_inv_f \surj f\) qed lemma bij_imp_bij_inv: "bij f \ bij (inv f)" by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv) lemma inv_equality: "(\x. g (f x) = x) \ (\y. f (g y) = y) \ inv f = g" by (rule ext) (auto simp add: inv_into_def) lemma inv_inv_eq: "bij f \ inv (inv f) = f" by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) text \ \bij (inv f)\ implies little about \f\. Consider \f :: bool \ bool\ such that \f True = f False = True\. Then it ia consistent with axiom \someI\ that \inv f\ could be any function at all, including the identity function. If \inv f = id\ then \inv f\ is a bijection, but \inj f\, \surj f\ and \inv (inv f) = f\ all fail. \ lemma inv_into_comp: "inj_on f (g ` A) \ inj_on g A \ x \ f ` g ` A \ inv_into A (f \ g) x = (inv_into A g \ inv_into (g ` A) f) x" by (auto simp: f_inv_into_f inv_into_into intro: inv_into_f_eq comp_inj_on) lemma o_inv_distrib: "bij f \ bij g \ inv (f \ g) = inv g \ inv f" by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) lemma image_f_inv_f: "surj f \ f ` (inv f ` A) = A" by (simp add: surj_f_inv_f image_comp comp_def) lemma image_inv_f_f: "inj f \ inv f ` (f ` A) = A" by simp lemma bij_image_Collect_eq: assumes "bij f" shows "f ` Collect P = {y. P (inv f y)}" proof show "f ` Collect P \ {y. P (inv f y)}" using assms by (force simp add: bij_is_inj) show "{y. P (inv f y)} \ f ` Collect P" using assms by (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric]) qed lemma bij_vimage_eq_inv_image: assumes "bij f" shows "f -` A = inv f ` A" proof show "f -` A \ inv f ` A" using assms by (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric]) show "inv f ` A \ f -` A" using assms by (auto simp add: bij_is_surj [THEN surj_f_inv_f]) qed lemma inv_fn_o_fn_is_id: fixes f::"'a \ 'a" assumes "bij f" shows "((inv f)^^n) o (f^^n) = (\x. x)" proof - have "((inv f)^^n)((f^^n) x) = x" for x n proof (induction n) case (Suc n) have *: "(inv f) (f y) = y" for y by (simp add: assms bij_is_inj) have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))" by (simp add: funpow_swap1) also have "... = (inv f^^n) ((f^^n) x)" using * by auto also have "... = x" using Suc.IH by auto finally show ?case by simp qed (auto) then show ?thesis unfolding o_def by blast qed lemma fn_o_inv_fn_is_id: fixes f::"'a \ 'a" assumes "bij f" shows "(f^^n) o ((inv f)^^n) = (\x. x)" proof - have "(f^^n) (((inv f)^^n) x) = x" for x n proof (induction n) case (Suc n) have *: "f(inv f y) = y" for y using bij_inv_eq_iff[OF assms] by auto have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))" by (simp add: funpow_swap1) also have "... = (f^^n) ((inv f^^n) x)" using * by auto also have "... = x" using Suc.IH by auto finally show ?case by simp qed (auto) then show ?thesis unfolding o_def by blast qed lemma inv_fn: fixes f::"'a \ 'a" assumes "bij f" shows "inv (f^^n) = ((inv f)^^n)" proof - have "inv (f^^n) x = ((inv f)^^n) x" for x proof (rule inv_into_f_eq) show "inj (f ^^ n)" by (simp add: inj_fn[OF bij_is_inj [OF assms]]) show "(f ^^ n) ((inv f ^^ n) x) = x" using fn_o_inv_fn_is_id[OF assms, THEN fun_cong] by force qed auto then show ?thesis by auto qed lemma funpow_inj_finite: \<^marker>\contributor \Lars Noschinski\\ assumes \inj p\ \finite {y. \n. y = (p ^^ n) x}\ obtains n where \n > 0\ \(p ^^ n) x = x\ proof - have \infinite (UNIV :: nat set)\ by simp moreover have \{y. \n. y = (p ^^ n) x} = (\n. (p ^^ n) x) ` UNIV\ by auto with assms have \finite \\ by simp ultimately have "\n \ UNIV. \ finite {m \ UNIV. (p ^^ m) x = (p ^^ n) x}" by (rule pigeonhole_infinite) then obtain n where "infinite {m. (p ^^ m) x = (p ^^ n) x}" by auto then have "infinite ({m. (p ^^ m) x = (p ^^ n) x} - {n})" by auto then have "({m. (p ^^ m) x = (p ^^ n) x} - {n}) \ {}" by (auto simp add: subset_singleton_iff) then obtain m where m: "(p ^^ m) x = (p ^^ n) x" "m \ n" by auto { fix m n assume "(p ^^ n) x = (p ^^ m) x" "m < n" have "(p ^^ (n - m)) x = inv (p ^^ m) ((p ^^ m) ((p ^^ (n - m)) x))" using \inj p\ by (simp add: inv_f_f) also have "((p ^^ m) ((p ^^ (n - m)) x)) = (p ^^ n) x" using \m < n\ funpow_add [of m \n - m\ p] by simp also have "inv (p ^^ m) \ = x" using \inj p\ by (simp add: \(p ^^ n) x = _\) finally have "(p ^^ (n - m)) x = x" "0 < n - m" using \m < n\ by auto } note general = this show thesis proof (cases m n rule: linorder_cases) case less then have \n - m > 0\ \(p ^^ (n - m)) x = x\ using general [of n m] m by simp_all then show thesis by (blast intro: that) next case equal then show thesis using m by simp next case greater then have \m - n > 0\ \(p ^^ (m - n)) x = x\ using general [of m n] m by simp_all then show thesis by (blast intro: that) qed qed lemma mono_inv: fixes f::"'a::linorder \ 'b::linorder" assumes "mono f" "bij f" shows "mono (inv f)" proof fix x y::'b assume "x \ y" from \bij f\ obtain a b where x: "x = f a" and y: "y = f b" by(fastforce simp: bij_def surj_def) show "inv f x \ inv f y" proof (rule le_cases) assume "a \ b" thus ?thesis using \bij f\ x y by(simp add: bij_def inv_f_f) next assume "b \ a" hence "f b \ f a" by(rule monoD[OF \mono f\]) hence "y \ x" using x y by simp hence "x = y" using \x \ y\ by auto thus ?thesis by simp qed qed lemma strict_mono_inv_on_range: fixes f :: "'a::linorder \ 'b::order" assumes "strict_mono f" shows "strict_mono_on (inv f) (range f)" proof (clarsimp simp: strict_mono_on_def) fix x y assume "f x < f y" then show "inv f (f x) < inv f (f y)" using assms strict_mono_imp_inj_on strict_mono_less by fastforce qed lemma mono_bij_Inf: fixes f :: "'a::complete_linorder \ 'b::complete_linorder" assumes "mono f" "bij f" shows "f (Inf A) = Inf (f`A)" proof - have "surj f" using \bij f\ by (auto simp: bij_betw_def) have *: "(inv f) (Inf (f`A)) \ Inf ((inv f)`(f`A))" using mono_Inf[OF mono_inv[OF assms], of "f`A"] by simp have "Inf (f`A) \ f (Inf ((inv f)`(f`A)))" using monoD[OF \mono f\ *] by(simp add: surj_f_inv_f[OF \surj f\]) also have "... = f(Inf A)" using assms by (simp add: bij_is_inj) finally show ?thesis using mono_Inf[OF assms(1), of A] by auto qed lemma finite_fun_UNIVD1: assumes fin: "finite (UNIV :: ('a \ 'b) set)" and card: "card (UNIV :: 'b set) \ Suc 0" shows "finite (UNIV :: 'a set)" proof - let ?UNIV_b = "UNIV :: 'b set" from fin have "finite ?UNIV_b" by (rule finite_fun_UNIVD2) with card have "card ?UNIV_b \ Suc (Suc 0)" by (cases "card ?UNIV_b") (auto simp: card_eq_0_iff) then have "card ?UNIV_b = Suc (Suc (card ?UNIV_b - Suc (Suc 0)))" by simp then obtain b1 b2 :: 'b where b1b2: "b1 \ b2" by (auto simp: card_Suc_eq) from fin have fin': "finite (range (\f :: 'a \ 'b. inv f b1))" by (rule finite_imageI) have "UNIV = range (\f :: 'a \ 'b. inv f b1)" proof (rule UNIV_eq_I) fix x :: 'a from b1b2 have "x = inv (\y. if y = x then b1 else b2) b1" by (simp add: inv_into_def) then show "x \ range (\f::'a \ 'b. inv f b1)" by blast qed with fin' show ?thesis by simp qed text \ Every infinite set contains a countable subset. More precisely we show that a set \S\ is infinite if and only if there exists an injective function from the naturals into \S\. The ``only if'' direction is harder because it requires the construction of a sequence of pairwise different elements of an infinite set \S\. The idea is to construct a sequence of non-empty and infinite subsets of \S\ obtained by successively removing elements of \S\. \ lemma infinite_countable_subset: assumes inf: "\ finite S" shows "\f::nat \ 'a. inj f \ range f \ S" \ \Courtesy of Stephan Merz\ proof - define Sseq where "Sseq = rec_nat S (\n T. T - {SOME e. e \ T})" define pick where "pick n = (SOME e. e \ Sseq n)" for n have *: "Sseq n \ S" "\ finite (Sseq n)" for n by (induct n) (auto simp: Sseq_def inf) then have **: "\n. pick n \ Sseq n" unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex) with * have "range pick \ S" by auto moreover have "pick n \ pick (n + Suc m)" for m n proof - have "pick n \ Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def) with ** show ?thesis by auto qed then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add) ultimately show ?thesis by blast qed lemma infinite_iff_countable_subset: "\ finite S \ (\f::nat \ 'a. inj f \ range f \ S)" \ \Courtesy of Stephan Merz\ using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto lemma image_inv_into_cancel: assumes surj: "f`A = A'" and sub: "B' \ A'" shows "f `((inv_into A f)`B') = B'" using assms proof (auto simp: f_inv_into_f) let ?f' = "inv_into A f" fix a' assume *: "a' \ B'" with sub have "a' \ A'" by auto with surj have "a' = f (?f' a')" by (auto simp: f_inv_into_f) with * show "a' \ f ` (?f' ` B')" by blast qed lemma inv_into_inv_into_eq: assumes "bij_betw f A A'" and a: "a \ A" shows "inv_into A' (inv_into A f) a = f a" proof - let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'" from assms have *: "bij_betw ?f' A' A" by (auto simp: bij_betw_inv_into) with a obtain a' where a': "a' \ A'" "?f' a' = a" unfolding bij_betw_def by force with a * have "?f'' a = a'" by (auto simp: f_inv_into_f bij_betw_def) moreover from assms a' have "f a = a'" by (auto simp: bij_betw_def) ultimately show "?f'' a = f a" by simp qed lemma inj_on_iff_surj: assumes "A \ {}" shows "(\f. inj_on f A \ f ` A \ A') \ (\g. g ` A' = A)" proof safe fix f assume inj: "inj_on f A" and incl: "f ` A \ A'" let ?phi = "\a' a. a \ A \ f a = a'" let ?csi = "\a. a \ A" let ?g = "\a'. if a' \ f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)" have "?g ` A' = A" proof show "?g ` A' \ A" proof clarify fix a' assume *: "a' \ A'" show "?g a' \ A" proof (cases "a' \ f ` A") case True then obtain a where "?phi a' a" by blast then have "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast with True show ?thesis by auto next case False with assms have "?csi (SOME a. ?csi a)" using someI_ex[of ?csi] by blast with False show ?thesis by auto qed qed next show "A \ ?g ` A'" proof - have "?g (f a) = a \ f a \ A'" if a: "a \ A" for a proof - let ?b = "SOME aa. ?phi (f a) aa" from a have "?phi (f a) a" by auto then have *: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast then have "?g (f a) = ?b" using a by auto moreover from inj * a have "a = ?b" by (auto simp add: inj_on_def) ultimately have "?g(f a) = a" by simp with incl a show ?thesis by auto qed then show ?thesis by force qed qed then show "\g. g ` A' = A" by blast next fix g let ?f = "inv_into A' g" have "inj_on ?f (g ` A')" by (auto simp: inj_on_inv_into) moreover have "?f (g a') \ A'" if a': "a' \ A'" for a' proof - let ?phi = "\ b'. b' \ A' \ g b' = g a'" from a' have "?phi a'" by auto then have "?phi (SOME b'. ?phi b')" using someI[of ?phi] by blast then show ?thesis by (auto simp: inv_into_def) qed ultimately show "\f. inj_on f (g ` A') \ f ` g ` A' \ A'" by auto qed lemma Ex_inj_on_UNION_Sigma: "\f. (inj_on f (\i \ I. A i) \ f ` (\i \ I. A i) \ (SIGMA i : I. A i))" proof let ?phi = "\a i. i \ I \ a \ A i" let ?sm = "\a. SOME i. ?phi a i" let ?f = "\a. (?sm a, a)" have "inj_on ?f (\i \ I. A i)" by (auto simp: inj_on_def) moreover have "?sm a \ I \ a \ A(?sm a)" if "i \ I" and "a \ A i" for i a using that someI[of "?phi a" i] by auto then have "?f ` (\i \ I. A i) \ (SIGMA i : I. A i)" by auto ultimately show "inj_on ?f (\i \ I. A i) \ ?f ` (\i \ I. A i) \ (SIGMA i : I. A i)" by auto qed lemma inv_unique_comp: assumes fg: "f \ g = id" and gf: "g \ f = id" shows "inv f = g" using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff) lemma subset_image_inj: "S \ f ` T \ (\U. U \ T \ inj_on f U \ S = f ` U)" proof safe show "\U\T. inj_on f U \ S = f ` U" if "S \ f ` T" proof - from that [unfolded subset_image_iff subset_iff] obtain g where g: "\x. x \ S \ g x \ T \ x = f (g x)" by (auto simp add: image_iff Bex_def choice_iff') show ?thesis proof (intro exI conjI) show "g ` S \ T" by (simp add: g image_subsetI) show "inj_on f (g ` S)" using g by (auto simp: inj_on_def) show "S = f ` (g ` S)" using g image_subset_iff by auto qed qed qed blast subsection \Other Consequences of Hilbert's Epsilon\ text \Hilbert's Epsilon and the \<^term>\split\ Operator\ text \Looping simprule!\ lemma split_paired_Eps: "(SOME x. P x) = (SOME (a, b). P (a, b))" by simp lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))" by (simp add: split_def) lemma Eps_case_prod_eq [simp]: "(SOME (x', y'). x = x' \ y = y') = (x, y)" by blast text \A relation is wellfounded iff it has no infinite descending chain.\ lemma wf_iff_no_infinite_down_chain: "wf r \ (\f. \i. (f (Suc i), f i) \ r)" (is "_ \ \ ?ex") proof assume "wf r" show "\ ?ex" proof assume ?ex then obtain f where f: "(f (Suc i), f i) \ r" for i by blast from \wf r\ have minimal: "x \ Q \ \z\Q. \y. (y, z) \ r \ y \ Q" for x Q by (auto simp: wf_eq_minimal) let ?Q = "{w. \i. w = f i}" fix n have "f n \ ?Q" by blast from minimal [OF this] obtain j where "(y, f j) \ r \ y \ ?Q" for y by blast with this [OF \(f (Suc j), f j) \ r\] have "f (Suc j) \ ?Q" by simp then show False by blast qed next assume "\ ?ex" then show "wf r" proof (rule contrapos_np) assume "\ wf r" then obtain Q x where x: "x \ Q" and rec: "z \ Q \ \y. (y, z) \ r \ y \ Q" for z by (auto simp add: wf_eq_minimal) obtain descend :: "nat \ 'a" where descend_0: "descend 0 = x" and descend_Suc: "descend (Suc n) = (SOME y. y \ Q \ (y, descend n) \ r)" for n by (rule that [of "rec_nat x (\_ rec. (SOME y. y \ Q \ (y, rec) \ r))"]) simp_all have descend_Q: "descend n \ Q" for n proof (induct n) case 0 with x show ?case by (simp only: descend_0) next case Suc then show ?case by (simp only: descend_Suc) (rule someI2_ex; use rec in blast) qed have "(descend (Suc i), descend i) \ r" for i by (simp only: descend_Suc) (rule someI2_ex; use descend_Q rec in blast) then show "\f. \i. (f (Suc i), f i) \ r" by blast qed qed lemma wf_no_infinite_down_chainE: assumes "wf r" obtains k where "(f (Suc k), f k) \ r" using assms wf_iff_no_infinite_down_chain[of r] by blast text \A dynamically-scoped fact for TFL\ lemma tfl_some: "\P x. P x \ P (Eps P)" by (blast intro: someI) subsection \An aside: bounded accessible part\ text \Finite monotone eventually stable sequences\ lemma finite_mono_remains_stable_implies_strict_prefix: fixes f :: "nat \ 'a::order" assumes S: "finite (range f)" "mono f" and eq: "\n. f n = f (Suc n) \ f (Suc n) = f (Suc (Suc n))" shows "\N. (\n\N. \m\N. m < n \ f m < f n) \ (\n\N. f N = f n)" using assms proof - have "\n. f n = f (Suc n)" proof (rule ccontr) assume "\ ?thesis" then have "\n. f n \ f (Suc n)" by auto with \mono f\ have "\n. f n < f (Suc n)" by (auto simp: le_less mono_iff_le_Suc) with lift_Suc_mono_less_iff[of f] have *: "\n m. n < m \ f n < f m" by auto have "inj f" proof (intro injI) fix x y assume "f x = f y" then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *) qed with \finite (range f)\ have "finite (UNIV::nat set)" by (rule finite_imageD) then show False by simp qed then obtain n where n: "f n = f (Suc n)" .. define N where "N = (LEAST n. f n = f (Suc n))" have N: "f N = f (Suc N)" unfolding N_def using n by (rule LeastI) show ?thesis proof (intro exI[of _ N] conjI allI impI) fix n assume "N \ n" then have "\m. N \ m \ m \ n \ f m = f N" proof (induct rule: dec_induct) case base then show ?case by simp next case (step n) then show ?case using eq [rule_format, of "n - 1"] N by (cases n) (auto simp add: le_Suc_eq) qed from this[of n] \N \ n\ show "f N = f n" by auto next fix n m :: nat assume "m < n" "n \ N" then show "f m < f n" proof (induct rule: less_Suc_induct) case (1 i) then have "i < N" by simp then have "f i \ f (Suc i)" unfolding N_def by (rule not_less_Least) with \mono f\ show ?case by (simp add: mono_iff_le_Suc less_le) next case 2 then show ?case by simp qed qed qed lemma finite_mono_strict_prefix_implies_finite_fixpoint: fixes f :: "nat \ 'a set" assumes S: "\i. f i \ S" "finite S" and ex: "\N. (\n\N. \m\N. m < n \ f m \ f n) \ (\n\N. f N = f n)" shows "f (card S) = (\n. f n)" proof - from ex obtain N where inj: "\n m. n \ N \ m \ N \ m < n \ f m \ f n" and eq: "\n\N. f N = f n" by atomize auto have "i \ N \ i \ card (f i)" for i proof (induct i) case 0 then show ?case by simp next case (Suc i) with inj [of "Suc i" i] have "(f i) \ (f (Suc i))" by auto moreover have "finite (f (Suc i))" using S by (rule finite_subset) ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono) with Suc inj show ?case by auto qed then have "N \ card (f N)" by simp also have "\ \ card S" using S by (intro card_mono) finally have \
: "f (card S) = f N" using eq by auto moreover have "\ (range f) \ f N" proof clarify fix x n assume "x \ f n" with eq inj [of N] show "x \ f N" by (cases "n < N") (auto simp: not_less) qed ultimately show ?thesis by auto qed subsection \More on injections, bijections, and inverses\ locale bijection = fixes f :: "'a \ 'a" assumes bij: "bij f" begin lemma bij_inv: "bij (inv f)" using bij by (rule bij_imp_bij_inv) lemma surj [simp]: "surj f" using bij by (rule bij_is_surj) lemma inj: "inj f" using bij by (rule bij_is_inj) lemma surj_inv [simp]: "surj (inv f)" using inj by (rule inj_imp_surj_inv) lemma inj_inv: "inj (inv f)" using surj by (rule surj_imp_inj_inv) lemma eqI: "f a = f b \ a = b" using inj by (rule injD) lemma eq_iff [simp]: "f a = f b \ a = b" by (auto intro: eqI) lemma eq_invI: "inv f a = inv f b \ a = b" using inj_inv by (rule injD) lemma eq_inv_iff [simp]: "inv f a = inv f b \ a = b" by (auto intro: eq_invI) lemma inv_left [simp]: "inv f (f a) = a" using inj by (simp add: inv_f_eq) lemma inv_comp_left [simp]: "inv f \ f = id" by (simp add: fun_eq_iff) lemma inv_right [simp]: "f (inv f a) = a" using surj by (simp add: surj_f_inv_f) lemma inv_comp_right [simp]: "f \ inv f = id" by (simp add: fun_eq_iff) lemma inv_left_eq_iff [simp]: "inv f a = b \ f b = a" by auto lemma inv_right_eq_iff [simp]: "b = inv f a \ f b = a" by auto end lemma infinite_imp_bij_betw: assumes infinite: "\ finite A" shows "\h. bij_betw h A (A - {a})" proof (cases "a \ A") case False then have "A - {a} = A" by blast then show ?thesis using bij_betw_id[of A] by auto next case True with infinite have "\ finite (A - {a})" by auto with infinite_iff_countable_subset[of "A - {a}"] obtain f :: "nat \ 'a" where "inj f" and f: "f ` UNIV \ A - {a}" by blast define g where "g n = (if n = 0 then a else f (Suc n))" for n define A' where "A' = g ` UNIV" have *: "\y. f y \ a" using f by blast have 3: "inj_on g UNIV \ g ` UNIV \ A \ a \ g ` UNIV" using \inj f\ f * unfolding inj_on_def g_def by (auto simp add: True image_subset_iff) then have 4: "bij_betw g UNIV A' \ a \ A' \ A' \ A" using inj_on_imp_bij_betw[of g] by (auto simp: A'_def) then have 5: "bij_betw (inv g) A' UNIV" by (auto simp add: bij_betw_inv_into) from 3 obtain n where n: "g n = a" by auto have 6: "bij_betw g (UNIV - {n}) (A' - {a})" by (rule bij_betw_subset) (use 3 4 n in \auto simp: image_set_diff A'_def\) define v where "v m = (if m < n then m else Suc m)" for m have "m < n \ m = n" if "\k. k < n \ m \ Suc k" for m using that [of "m-1"] by auto then have 7: "bij_betw v UNIV (UNIV - {n})" unfolding bij_betw_def inj_on_def v_def by auto define h' where "h' = g \ v \ (inv g)" with 5 6 7 have 8: "bij_betw h' A' (A' - {a})" by (auto simp add: bij_betw_trans) define h where "h b = (if b \ A' then h' b else b)" for b with 8 have "bij_betw h A' (A' - {a})" using bij_betw_cong[of A' h] by auto moreover have "\b \ A - A'. h b = b" by (auto simp: h_def) then have "bij_betw h (A - A') (A - A')" using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto moreover from 4 have "(A' \ (A - A') = {} \ A' \ (A - A') = A) \ ((A' - {a}) \ (A - A') = {} \ (A' - {a}) \ (A - A') = A - {a})" by blast ultimately have "bij_betw h A (A - {a})" using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp then show ?thesis by blast qed lemma infinite_imp_bij_betw2: assumes "\ finite A" shows "\h. bij_betw h A (A \ {a})" proof (cases "a \ A") case True then have "A \ {a} = A" by blast then show ?thesis using bij_betw_id[of A] by auto next case False let ?A' = "A \ {a}" from False have "A = ?A' - {a}" by blast moreover from assms have "\ finite ?A'" by auto ultimately obtain f where "bij_betw f ?A' A" using infinite_imp_bij_betw[of ?A' a] by auto then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into) then show ?thesis by auto qed lemma bij_betw_inv_into_left: "bij_betw f A A' \ a \ A \ inv_into A f (f a) = a" unfolding bij_betw_def by clarify (rule inv_into_f_f) lemma bij_betw_inv_into_right: "bij_betw f A A' \ a' \ A' \ f (inv_into A f a') = a'" unfolding bij_betw_def using f_inv_into_f by force lemma bij_betw_inv_into_subset: "bij_betw f A A' \ B \ A \ f ` B = B' \ bij_betw (inv_into A f) B' B" by (auto simp: bij_betw_def intro: inj_on_inv_into) subsection \Specification package -- Hilbertized version\ lemma exE_some: "Ex P \ c \ Eps P \ P c" by (simp only: someI_ex) ML_file \Tools/choice_specification.ML\ subsection \Complete Distributive Lattices -- Properties depending on Hilbert Choice\ context complete_distrib_lattice begin lemma Sup_Inf: "\ (Inf ` A) = \ (Sup ` {f ` A |f. \B\A. f B \ B})" proof (rule order.antisym) show "\ (Inf ` A) \ \ (Sup ` {f ` A |f. \B\A. f B \ B})" using Inf_lower2 Sup_upper by (fastforce simp add: intro: Sup_least INF_greatest) next show "\ (Sup ` {f ` A |f. \B\A. f B \ B}) \ \ (Inf ` A)" proof (simp add: Inf_Sup, rule SUP_least, simp, safe) fix f assume "\Y. (\f. Y = f ` A \ (\Y\A. f Y \ Y)) \ f Y \ Y" then have B: "\ F . (\ Y \ A . F Y \ Y) \ \ Z \ A . f (F ` A) = F Z" by auto show "\(f ` {f ` A |f. \Y\A. f Y \ Y}) \ \(Inf ` A)" proof (cases "\ Z \ A . \(f ` {f ` A |f. \Y\A. f Y \ Y}) \ Inf Z") case True from this obtain Z where [simp]: "Z \ A" and A: "\(f ` {f ` A |f. \Y\A. f Y \ Y}) \ Inf Z" by blast have B: "... \ \(Inf ` A)" by (simp add: SUP_upper) from A and B show ?thesis by simp next case False then have X: "\ Z . Z \ A \ \ x . x \ Z \ \ \(f ` {f ` A |f. \Y\A. f Y \ Y}) \ x" using Inf_greatest by blast define F where "F = (\ Z . SOME x . x \ Z \ \ \(f ` {f ` A |f. \Y\A. f Y \ Y}) \ x)" have C: "\Y. Y \ A \ F Y \ Y" using X by (simp add: F_def, rule someI2_ex, auto) have E: "\Y. Y \ A \ \ \(f ` {f ` A |f. \Y\A. f Y \ Y}) \ F Y" using X by (simp add: F_def, rule someI2_ex, auto) from C and B obtain Z where D: "Z \ A " and Y: "f (F ` A) = F Z" by blast from E and D have W: "\ \(f ` {f ` A |f. \Y\A. f Y \ Y}) \ F Z" by simp have "\(f ` {f ` A |f. \Y\A. f Y \ Y}) \ f (F ` A)" using C by (blast intro: INF_lower) with W Y show ?thesis by simp qed qed qed lemma dual_complete_distrib_lattice: "class.complete_distrib_lattice Sup Inf sup (\) (>) inf \ \" by (simp add: class.complete_distrib_lattice.intro [OF dual_complete_lattice] class.complete_distrib_lattice_axioms_def Sup_Inf) lemma sup_Inf: "a \ \B = \((\) a ` B)" proof (rule order.antisym) show "a \ \B \ \((\) a ` B)" using Inf_lower sup.mono by (fastforce intro: INF_greatest) next have "\((\) a ` B) \ \(Sup ` {{f {a}, f B} |f. f {a} = a \ f B \ B})" by (rule INF_greatest, auto simp add: INF_lower) also have "... = \(Inf ` {{a}, B})" by (unfold Sup_Inf, simp) finally show "\((\) a ` B) \ a \ \B" by simp qed lemma inf_Sup: "a \ \B = \((\) a ` B)" using dual_complete_distrib_lattice by (rule complete_distrib_lattice.sup_Inf) lemma INF_SUP: "(\y. \x. P x y) = (\f. \x. P (f x) x)" proof (rule order.antisym) show "(SUP x. INF y. P (x y) y) \ (INF y. SUP x. P x y)" by (rule SUP_least, rule INF_greatest, rule SUP_upper2, simp_all, rule INF_lower2, simp, blast) next have "(INF y. SUP x. ((P x y))) \ Inf (Sup ` {{P x y | x . True} | y . True })" (is "?A \ ?B") proof (rule INF_greatest, clarsimp) fix y have "?A \ (SUP x. P x y)" by (rule INF_lower, simp) also have "... \ Sup {uu. \x. uu = P x y}" by (simp add: full_SetCompr_eq) finally show "?A \ Sup {uu. \x. uu = P x y}" by simp qed also have "... \ (SUP x. INF y. P (x y) y)" proof (subst Inf_Sup, rule SUP_least, clarsimp) fix f assume A: "\Y. (\y. Y = {uu. \x. uu = P x y}) \ f Y \ Y" have " \(f ` {uu. \y. uu = {uu. \x. uu = P x y}}) \ (\y. P (SOME x. f {P x y |x. True} = P x y) y)" proof (rule INF_greatest, clarsimp) fix y have "(INF x\{uu. \y. uu = {uu. \x. uu = P x y}}. f x) \ f {uu. \x. uu = P x y}" by (rule INF_lower, blast) also have "... \ P (SOME x. f {uu . \x. uu = P x y} = P x y) y" by (rule someI2_ex) (use A in auto) finally show "\(f ` {uu. \y. uu = {uu. \x. uu = P x y}}) \ P (SOME x. f {uu. \x. uu = P x y} = P x y) y" by simp qed also have "... \ (SUP x. INF y. P (x y) y)" by (rule SUP_upper, simp) finally show "\(f ` {uu. \y. uu = {uu. \x. uu = P x y}}) \ (\x. \y. P (x y) y)" by simp qed finally show "(INF y. SUP x. P x y) \ (SUP x. INF y. P (x y) y)" by simp qed lemma INF_SUP_set: "(\B\A. \(g ` B)) = (\B\{f ` A |f. \C\A. f C \ C}. \(g ` B))" (is "_ = (\B\?F. _)") proof (rule order.antisym) have "\ ((g \ f) ` A) \ \ (g ` B)" if "\B. B \ A \ f B \ B" "B \ A" for f B using that by (auto intro: SUP_upper2 INF_lower2) then show "(\x\?F. \a\x. g a) \ (\x\A. \a\x. g a)" by (auto intro!: SUP_least INF_greatest simp add: image_comp) next show "(\x\A. \a\x. g a) \ (\x\?F. \a\x. g a)" proof (cases "{} \ A") case True then show ?thesis by (rule INF_lower2) simp_all next case False {fix x have "(\x\A. \x\x. g x) \ (\u. if x \ A then if u \ x then g u else \ else \)" proof (cases "x \ A") case True then show ?thesis by (intro INF_lower2 SUP_least SUP_upper2) auto qed auto } then have "(\Y\A. \a\Y. g a) \ (\Y. \y. if Y \ A then if y \ Y then g y else \ else \)" by (rule INF_greatest) also have "... = (\x. \Y. if Y \ A then if x Y \ Y then g (x Y) else \ else \)" by (simp only: INF_SUP) also have "... \ (\x\?F. \a\x. g a)" proof (rule SUP_least) show "(\B. if B \ A then if x B \ B then g (x B) else \ else \) \ (\x\?F. \x\x. g x)" for x proof - define G where "G \ \Y. if x Y \ Y then x Y else (SOME x. x \Y)" have "\Y\A. G Y \ Y" using False some_in_eq G_def by auto then have A: "G ` A \ ?F" by blast show "(\Y. if Y \ A then if x Y \ Y then g (x Y) else \ else \) \ (\x\?F. \x\x. g x)" by (fastforce simp: G_def intro: SUP_upper2 [OF A] INF_greatest INF_lower2) qed qed finally show ?thesis by simp qed qed lemma SUP_INF: "(\y. \x. P x y) = (\x. \y. P (x y) y)" using dual_complete_distrib_lattice by (rule complete_distrib_lattice.INF_SUP) lemma SUP_INF_set: "(\x\A. \ (g ` x)) = (\x\{f ` A |f. \Y\A. f Y \ Y}. \ (g ` x))" using dual_complete_distrib_lattice by (rule complete_distrib_lattice.INF_SUP_set) end (*properties of the former complete_distrib_lattice*) context complete_distrib_lattice begin lemma sup_INF: "a \ (\b\B. f b) = (\b\B. a \ f b)" by (simp add: sup_Inf image_comp) lemma inf_SUP: "a \ (\b\B. f b) = (\b\B. a \ f b)" by (simp add: inf_Sup image_comp) lemma Inf_sup: "\B \ a = (\b\B. b \ a)" by (simp add: sup_Inf sup_commute) lemma Sup_inf: "\B \ a = (\b\B. b \ a)" by (simp add: inf_Sup inf_commute) lemma INF_sup: "(\b\B. f b) \ a = (\b\B. f b \ a)" by (simp add: sup_INF sup_commute) lemma SUP_inf: "(\b\B. f b) \ a = (\b\B. f b \ a)" by (simp add: inf_SUP inf_commute) lemma Inf_sup_eq_top_iff: "(\B \ a = \) \ (\b\B. b \ a = \)" by (simp only: Inf_sup INF_top_conv) lemma Sup_inf_eq_bot_iff: "(\B \ a = \) \ (\b\B. b \ a = \)" by (simp only: Sup_inf SUP_bot_conv) lemma INF_sup_distrib2: "(\a\A. f a) \ (\b\B. g b) = (\a\A. \b\B. f a \ g b)" by (subst INF_commute) (simp add: sup_INF INF_sup) lemma SUP_inf_distrib2: "(\a\A. f a) \ (\b\B. g b) = (\a\A. \b\B. f a \ g b)" by (subst SUP_commute) (simp add: inf_SUP SUP_inf) end -context complete_boolean_algebra -begin - -lemma dual_complete_boolean_algebra: - "class.complete_boolean_algebra Sup Inf sup (\) (>) inf \ \ (\x y. x \ - y) uminus" - by (rule class.complete_boolean_algebra.intro, - rule dual_complete_distrib_lattice, - rule dual_boolean_algebra) -end - - - instantiation set :: (type) complete_distrib_lattice begin instance proof (standard, clarsimp) fix A :: "(('a set) set) set" fix x::'a assume A: "\\\A. \X\\. x \ X" define F where "F \ \Y. SOME X. Y \ A \ X \ Y \ x \ X" have "(\S \ F ` A. x \ S)" using A unfolding F_def by (fastforce intro: someI2_ex) moreover have "\Y\A. F Y \ Y" using A unfolding F_def by (fastforce intro: someI2_ex) then have "\f. F ` A = f ` A \ (\Y\A. f Y \ Y)" by blast ultimately show "\X. (\f. X = f ` A \ (\Y\A. f Y \ Y)) \ (\S\X. x \ S)" by auto qed end instance set :: (type) complete_boolean_algebra .. instantiation "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice begin instance by standard (simp add: le_fun_def INF_SUP_set image_comp) end instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. context complete_linorder begin subclass complete_distrib_lattice proof (standard, rule ccontr) fix A :: "'a set set" let ?F = "{f ` A |f. \Y\A. f Y \ Y}" assume "\ \(Sup ` A) \ \(Inf ` ?F)" then have C: "\(Sup ` A) > \(Inf ` ?F)" by (simp add: not_le) show False proof (cases "\ z . \(Sup ` A) > z \ z > \(Inf ` ?F)") case True then obtain z where A: "z < \(Sup ` A)" and X: "z > \(Inf ` ?F)" by blast then have B: "\Y. Y \ A \ \k \Y . z < k" using local.less_Sup_iff by(force dest: less_INF_D) define G where "G \ \Y. SOME k . k \ Y \ z < k" have E: "\Y. Y \ A \ G Y \ Y" using B unfolding G_def by (fastforce intro: someI2_ex) have "z \ Inf (G ` A)" proof (rule INF_greatest) show "\Y. Y \ A \ z \ G Y" using B unfolding G_def by (fastforce intro: someI2_ex) qed also have "... \ \(Inf ` ?F)" by (rule SUP_upper) (use E in blast) finally have "z \ \(Inf ` ?F)" by simp with X show ?thesis using local.not_less by blast next case False have B: "\Y. Y \ A \ \ k \Y . \(Inf ` ?F) < k" using C local.less_Sup_iff by(force dest: less_INF_D) define G where "G \ \ Y . SOME k . k \ Y \ \(Inf ` ?F) < k" have E: "\Y. Y \ A \ G Y \ Y" using B unfolding G_def by (fastforce intro: someI2_ex) have "\Y. Y \ A \ \(Sup ` A) \ G Y" using B False local.leI unfolding G_def by (fastforce intro: someI2_ex) then have "\(Sup ` A) \ Inf (G ` A)" by (simp add: local.INF_greatest) also have "Inf (G ` A) \ \(Inf ` ?F)" by (rule SUP_upper) (use E in blast) finally have "\(Sup ` A) \ \(Inf ` ?F)" by simp with C show ?thesis using not_less by blast qed qed end end diff --git a/src/HOL/Lattices.thy b/src/HOL/Lattices.thy --- a/src/HOL/Lattices.thy +++ b/src/HOL/Lattices.thy @@ -1,997 +1,724 @@ (* Title: HOL/Lattices.thy Author: Tobias Nipkow *) section \Abstract lattices\ theory Lattices imports Groups begin subsection \Abstract semilattice\ text \ These locales provide a basic structure for interpretation into bigger structures; extensions require careful thinking, otherwise undesired effects may occur due to interpretation. \ locale semilattice = abel_semigroup + assumes idem [simp]: "a \<^bold>* a = a" begin lemma left_idem [simp]: "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b" by (simp add: assoc [symmetric]) lemma right_idem [simp]: "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b" by (simp add: assoc) end locale semilattice_neutr = semilattice + comm_monoid locale semilattice_order = semilattice + fixes less_eq :: "'a \ 'a \ bool" (infix "\<^bold>\" 50) and less :: "'a \ 'a \ bool" (infix "\<^bold><" 50) assumes order_iff: "a \<^bold>\ b \ a = a \<^bold>* b" and strict_order_iff: "a \<^bold>< b \ a = a \<^bold>* b \ a \ b" begin lemma orderI: "a = a \<^bold>* b \ a \<^bold>\ b" by (simp add: order_iff) lemma orderE: assumes "a \<^bold>\ b" obtains "a = a \<^bold>* b" using assms by (unfold order_iff) sublocale ordering less_eq less proof show "a \<^bold>< b \ a \<^bold>\ b \ a \ b" for a b by (simp add: order_iff strict_order_iff) next show "a \<^bold>\ a" for a by (simp add: order_iff) next fix a b assume "a \<^bold>\ b" "b \<^bold>\ a" then have "a = a \<^bold>* b" "a \<^bold>* b = b" by (simp_all add: order_iff commute) then show "a = b" by simp next fix a b c assume "a \<^bold>\ b" "b \<^bold>\ c" then have "a = a \<^bold>* b" "b = b \<^bold>* c" by (simp_all add: order_iff commute) then have "a = a \<^bold>* (b \<^bold>* c)" by simp then have "a = (a \<^bold>* b) \<^bold>* c" by (simp add: assoc) with \a = a \<^bold>* b\ [symmetric] have "a = a \<^bold>* c" by simp then show "a \<^bold>\ c" by (rule orderI) qed lemma cobounded1 [simp]: "a \<^bold>* b \<^bold>\ a" by (simp add: order_iff commute) lemma cobounded2 [simp]: "a \<^bold>* b \<^bold>\ b" by (simp add: order_iff) lemma boundedI: assumes "a \<^bold>\ b" and "a \<^bold>\ c" shows "a \<^bold>\ b \<^bold>* c" proof (rule orderI) from assms obtain "a \<^bold>* b = a" and "a \<^bold>* c = a" by (auto elim!: orderE) then show "a = a \<^bold>* (b \<^bold>* c)" by (simp add: assoc [symmetric]) qed lemma boundedE: assumes "a \<^bold>\ b \<^bold>* c" obtains "a \<^bold>\ b" and "a \<^bold>\ c" using assms by (blast intro: trans cobounded1 cobounded2) lemma bounded_iff [simp]: "a \<^bold>\ b \<^bold>* c \ a \<^bold>\ b \ a \<^bold>\ c" by (blast intro: boundedI elim: boundedE) lemma strict_boundedE: assumes "a \<^bold>< b \<^bold>* c" obtains "a \<^bold>< b" and "a \<^bold>< c" using assms by (auto simp add: commute strict_iff_order elim: orderE intro!: that)+ lemma coboundedI1: "a \<^bold>\ c \ a \<^bold>* b \<^bold>\ c" by (rule trans) auto lemma coboundedI2: "b \<^bold>\ c \ a \<^bold>* b \<^bold>\ c" by (rule trans) auto lemma strict_coboundedI1: "a \<^bold>< c \ a \<^bold>* b \<^bold>< c" using irrefl by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order elim: strict_boundedE) lemma strict_coboundedI2: "b \<^bold>< c \ a \<^bold>* b \<^bold>< c" using strict_coboundedI1 [of b c a] by (simp add: commute) lemma mono: "a \<^bold>\ c \ b \<^bold>\ d \ a \<^bold>* b \<^bold>\ c \<^bold>* d" by (blast intro: boundedI coboundedI1 coboundedI2) lemma absorb1: "a \<^bold>\ b \ a \<^bold>* b = a" by (rule antisym) (auto simp: refl) lemma absorb2: "b \<^bold>\ a \ a \<^bold>* b = b" by (rule antisym) (auto simp: refl) lemma absorb3: "a \<^bold>< b \ a \<^bold>* b = a" by (rule absorb1) (rule strict_implies_order) lemma absorb4: "b \<^bold>< a \ a \<^bold>* b = b" by (rule absorb2) (rule strict_implies_order) lemma absorb_iff1: "a \<^bold>\ b \ a \<^bold>* b = a" using order_iff by auto lemma absorb_iff2: "b \<^bold>\ a \ a \<^bold>* b = b" using order_iff by (auto simp add: commute) end locale semilattice_neutr_order = semilattice_neutr + semilattice_order begin sublocale ordering_top less_eq less "\<^bold>1" by standard (simp add: order_iff) lemma eq_neutr_iff [simp]: \a \<^bold>* b = \<^bold>1 \ a = \<^bold>1 \ b = \<^bold>1\ by (simp add: eq_iff) lemma neutr_eq_iff [simp]: \\<^bold>1 = a \<^bold>* b \ a = \<^bold>1 \ b = \<^bold>1\ by (simp add: eq_iff) end text \Interpretations for boolean operators\ interpretation conj: semilattice_neutr \(\)\ True by standard auto interpretation disj: semilattice_neutr \(\)\ False by standard auto declare conj_assoc [ac_simps del] disj_assoc [ac_simps del] \ \already simp by default\ subsection \Syntactic infimum and supremum operations\ class inf = fixes inf :: "'a \ 'a \ 'a" (infixl "\" 70) class sup = fixes sup :: "'a \ 'a \ 'a" (infixl "\" 65) subsection \Concrete lattices\ class semilattice_inf = order + inf + assumes inf_le1 [simp]: "x \ y \ x" and inf_le2 [simp]: "x \ y \ y" and inf_greatest: "x \ y \ x \ z \ x \ y \ z" class semilattice_sup = order + sup + assumes sup_ge1 [simp]: "x \ x \ y" and sup_ge2 [simp]: "y \ x \ y" and sup_least: "y \ x \ z \ x \ y \ z \ x" begin text \Dual lattice.\ lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater" by (rule class.semilattice_inf.intro, rule dual_order) (unfold_locales, simp_all add: sup_least) end class lattice = semilattice_inf + semilattice_sup subsubsection \Intro and elim rules\ context semilattice_inf begin lemma le_infI1: "a \ x \ a \ b \ x" by (rule order_trans) auto lemma le_infI2: "b \ x \ a \ b \ x" by (rule order_trans) auto lemma le_infI: "x \ a \ x \ b \ x \ a \ b" by (fact inf_greatest) (* FIXME: duplicate lemma *) lemma le_infE: "x \ a \ b \ (x \ a \ x \ b \ P) \ P" by (blast intro: order_trans inf_le1 inf_le2) lemma le_inf_iff: "x \ y \ z \ x \ y \ x \ z" by (blast intro: le_infI elim: le_infE) lemma le_iff_inf: "x \ y \ x \ y = x" by (auto intro: le_infI1 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_inf_iff) lemma inf_mono: "a \ c \ b \ d \ a \ b \ c \ d" by (fast intro: inf_greatest le_infI1 le_infI2) lemma mono_inf: "mono f \ f (A \ B) \ f A \ f B" for f :: "'a \ 'b::semilattice_inf" by (auto simp add: mono_def intro: Lattices.inf_greatest) end context semilattice_sup begin lemma le_supI1: "x \ a \ x \ a \ b" by (rule order_trans) auto lemma le_supI2: "x \ b \ x \ a \ b" by (rule order_trans) auto lemma le_supI: "a \ x \ b \ x \ a \ b \ x" by (fact sup_least) (* FIXME: duplicate lemma *) lemma le_supE: "a \ b \ x \ (a \ x \ b \ x \ P) \ P" by (blast intro: order_trans sup_ge1 sup_ge2) lemma le_sup_iff: "x \ y \ z \ x \ z \ y \ z" by (blast intro: le_supI elim: le_supE) lemma le_iff_sup: "x \ y \ x \ y = y" by (auto intro: le_supI2 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_sup_iff) lemma sup_mono: "a \ c \ b \ d \ a \ b \ c \ d" by (fast intro: sup_least le_supI1 le_supI2) lemma mono_sup: "mono f \ f A \ f B \ f (A \ B)" for f :: "'a \ 'b::semilattice_sup" by (auto simp add: mono_def intro: Lattices.sup_least) end subsubsection \Equational laws\ context semilattice_inf begin sublocale inf: semilattice inf proof fix a b c show "(a \ b) \ c = a \ (b \ c)" by (rule order.antisym) (auto intro: le_infI1 le_infI2 simp add: le_inf_iff) show "a \ b = b \ a" by (rule order.antisym) (auto simp add: le_inf_iff) show "a \ a = a" by (rule order.antisym) (auto simp add: le_inf_iff) qed sublocale inf: semilattice_order inf less_eq less by standard (auto simp add: le_iff_inf less_le) lemma inf_assoc: "(x \ y) \ z = x \ (y \ z)" by (fact inf.assoc) lemma inf_commute: "(x \ y) = (y \ x)" by (fact inf.commute) lemma inf_left_commute: "x \ (y \ z) = y \ (x \ z)" by (fact inf.left_commute) lemma inf_idem: "x \ x = x" by (fact inf.idem) (* already simp *) lemma inf_left_idem: "x \ (x \ y) = x \ y" by (fact inf.left_idem) (* already simp *) lemma inf_right_idem: "(x \ y) \ y = x \ y" by (fact inf.right_idem) (* already simp *) lemma inf_absorb1: "x \ y \ x \ y = x" by (rule order.antisym) auto lemma inf_absorb2: "y \ x \ x \ y = y" by (rule order.antisym) auto lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem end context semilattice_sup begin sublocale sup: semilattice sup proof fix a b c show "(a \ b) \ c = a \ (b \ c)" by (rule order.antisym) (auto intro: le_supI1 le_supI2 simp add: le_sup_iff) show "a \ b = b \ a" by (rule order.antisym) (auto simp add: le_sup_iff) show "a \ a = a" by (rule order.antisym) (auto simp add: le_sup_iff) qed sublocale sup: semilattice_order sup greater_eq greater by standard (auto simp add: le_iff_sup sup.commute less_le) lemma sup_assoc: "(x \ y) \ z = x \ (y \ z)" by (fact sup.assoc) lemma sup_commute: "(x \ y) = (y \ x)" by (fact sup.commute) lemma sup_left_commute: "x \ (y \ z) = y \ (x \ z)" by (fact sup.left_commute) lemma sup_idem: "x \ x = x" by (fact sup.idem) (* already simp *) lemma sup_left_idem [simp]: "x \ (x \ y) = x \ y" by (fact sup.left_idem) lemma sup_absorb1: "y \ x \ x \ y = x" by (rule order.antisym) auto lemma sup_absorb2: "x \ y \ x \ y = y" by (rule order.antisym) auto lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem end context lattice begin lemma dual_lattice: "class.lattice sup (\) (>) inf" by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order) (unfold_locales, auto) lemma inf_sup_absorb [simp]: "x \ (x \ y) = x" by (blast intro: order.antisym inf_le1 inf_greatest sup_ge1) lemma sup_inf_absorb [simp]: "x \ (x \ y) = x" by (blast intro: order.antisym sup_ge1 sup_least inf_le1) lemmas inf_sup_aci = inf_aci sup_aci lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2 text \Towards distributivity.\ lemma distrib_sup_le: "x \ (y \ z) \ (x \ y) \ (x \ z)" by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2) lemma distrib_inf_le: "(x \ y) \ (x \ z) \ x \ (y \ z)" by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2) text \If you have one of them, you have them all.\ lemma distrib_imp1: assumes distrib: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)" shows "x \ (y \ z) = (x \ y) \ (x \ z)" proof- have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)" by simp also have "\ = x \ (z \ (x \ y))" by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb) also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)" by (simp add: inf_commute) also have "\ = (x \ y) \ (x \ z)" by(simp add:distrib) finally show ?thesis . qed lemma distrib_imp2: assumes distrib: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)" shows "x \ (y \ z) = (x \ y) \ (x \ z)" proof- have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)" by simp also have "\ = x \ (z \ (x \ y))" by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb) also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)" by (simp add: sup_commute) also have "\ = (x \ y) \ (x \ z)" by (simp add:distrib) finally show ?thesis . qed end subsubsection \Strict order\ context semilattice_inf begin lemma less_infI1: "a < x \ a \ b < x" by (auto simp add: less_le inf_absorb1 intro: le_infI1) lemma less_infI2: "b < x \ a \ b < x" by (auto simp add: less_le inf_absorb2 intro: le_infI2) end context semilattice_sup begin lemma less_supI1: "x < a \ x < a \ b" using dual_semilattice by (rule semilattice_inf.less_infI1) lemma less_supI2: "x < b \ x < a \ b" using dual_semilattice by (rule semilattice_inf.less_infI2) end subsection \Distributive lattices\ class distrib_lattice = lattice + assumes sup_inf_distrib1: "x \ (y \ z) = (x \ y) \ (x \ z)" context distrib_lattice begin lemma sup_inf_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" by (simp add: sup_commute sup_inf_distrib1) lemma inf_sup_distrib1: "x \ (y \ z) = (x \ y) \ (x \ z)" by (rule distrib_imp2 [OF sup_inf_distrib1]) lemma inf_sup_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" by (simp add: inf_commute inf_sup_distrib1) lemma dual_distrib_lattice: "class.distrib_lattice sup (\) (>) inf" by (rule class.distrib_lattice.intro, rule dual_lattice) (unfold_locales, fact inf_sup_distrib1) lemmas sup_inf_distrib = sup_inf_distrib1 sup_inf_distrib2 lemmas inf_sup_distrib = inf_sup_distrib1 inf_sup_distrib2 lemmas distrib = sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2 end -subsection \Bounded lattices and boolean algebras\ +subsection \Bounded lattices\ class bounded_semilattice_inf_top = semilattice_inf + order_top begin sublocale inf_top: semilattice_neutr inf top + inf_top: semilattice_neutr_order inf top less_eq less proof show "x \ \ = x" for x by (rule inf_absorb1) simp qed lemma inf_top_left: "\ \ x = x" by (fact inf_top.left_neutral) lemma inf_top_right: "x \ \ = x" by (fact inf_top.right_neutral) lemma inf_eq_top_iff: "x \ y = \ \ x = \ \ y = \" by (fact inf_top.eq_neutr_iff) lemma top_eq_inf_iff: "\ = x \ y \ x = \ \ y = \" by (fact inf_top.neutr_eq_iff) end class bounded_semilattice_sup_bot = semilattice_sup + order_bot begin sublocale sup_bot: semilattice_neutr sup bot + sup_bot: semilattice_neutr_order sup bot greater_eq greater proof show "x \ \ = x" for x by (rule sup_absorb1) simp qed lemma sup_bot_left: "\ \ x = x" by (fact sup_bot.left_neutral) lemma sup_bot_right: "x \ \ = x" by (fact sup_bot.right_neutral) lemma sup_eq_bot_iff: "x \ y = \ \ x = \ \ y = \" by (fact sup_bot.eq_neutr_iff) lemma bot_eq_sup_iff: "\ = x \ y \ x = \ \ y = \" by (fact sup_bot.neutr_eq_iff) end class bounded_lattice_bot = lattice + order_bot begin subclass bounded_semilattice_sup_bot .. lemma inf_bot_left [simp]: "\ \ x = \" by (rule inf_absorb1) simp lemma inf_bot_right [simp]: "x \ \ = \" by (rule inf_absorb2) simp end class bounded_lattice_top = lattice + order_top begin subclass bounded_semilattice_inf_top .. lemma sup_top_left [simp]: "\ \ x = \" by (rule sup_absorb1) simp lemma sup_top_right [simp]: "x \ \ = \" by (rule sup_absorb2) simp end class bounded_lattice = lattice + order_bot + order_top begin subclass bounded_lattice_bot .. subclass bounded_lattice_top .. lemma dual_bounded_lattice: "class.bounded_lattice sup greater_eq greater inf \ \" by unfold_locales (auto simp add: less_le_not_le) end -class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus + - assumes inf_compl_bot: "x \ - x = \" - and sup_compl_top: "x \ - x = \" - assumes diff_eq: "x - y = x \ - y" -begin - -lemma dual_boolean_algebra: - "class.boolean_algebra (\x y. x \ - y) uminus sup greater_eq greater inf \ \" - by (rule class.boolean_algebra.intro, - rule dual_bounded_lattice, - rule dual_distrib_lattice) - (unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq) - -lemma compl_inf_bot [simp]: "- x \ x = \" - by (simp add: inf_commute inf_compl_bot) - -lemma compl_sup_top [simp]: "- x \ x = \" - by (simp add: sup_commute sup_compl_top) - -lemma compl_unique: - assumes "x \ y = \" - and "x \ y = \" - shows "- x = y" -proof - - have "(x \ - x) \ (- x \ y) = (x \ y) \ (- x \ y)" - using inf_compl_bot assms(1) by simp - then have "(- x \ x) \ (- x \ y) = (y \ x) \ (y \ - x)" - by (simp add: inf_commute) - then have "- x \ (x \ y) = y \ (x \ - x)" - by (simp add: inf_sup_distrib1) - then have "- x \ \ = y \ \" - using sup_compl_top assms(2) by simp - then show "- x = y" by simp -qed - -lemma double_compl [simp]: "- (- x) = x" - using compl_inf_bot compl_sup_top by (rule compl_unique) - -lemma compl_eq_compl_iff [simp]: "- x = - y \ x = y" -proof - assume "- x = - y" - then have "- (- x) = - (- y)" by (rule arg_cong) - then show "x = y" by simp -next - assume "x = y" - then show "- x = - y" by simp -qed - -lemma compl_bot_eq [simp]: "- \ = \" -proof - - from sup_compl_top have "\ \ - \ = \" . - then show ?thesis by simp -qed - -lemma compl_top_eq [simp]: "- \ = \" -proof - - from inf_compl_bot have "\ \ - \ = \" . - then show ?thesis by simp -qed - -lemma compl_inf [simp]: "- (x \ y) = - x \ - y" -proof (rule compl_unique) - have "(x \ y) \ (- x \ - y) = (y \ (x \ - x)) \ (x \ (y \ - y))" - by (simp only: inf_sup_distrib inf_aci) - then show "(x \ y) \ (- x \ - y) = \" - by (simp add: inf_compl_bot) -next - have "(x \ y) \ (- x \ - y) = (- y \ (x \ - x)) \ (- x \ (y \ - y))" - by (simp only: sup_inf_distrib sup_aci) - then show "(x \ y) \ (- x \ - y) = \" - by (simp add: sup_compl_top) -qed - -lemma compl_sup [simp]: "- (x \ y) = - x \ - y" - using dual_boolean_algebra - by (rule boolean_algebra.compl_inf) - -lemma compl_mono: - assumes "x \ y" - shows "- y \ - x" -proof - - from assms have "x \ y = y" by (simp only: le_iff_sup) - then have "- (x \ y) = - y" by simp - then have "- x \ - y = - y" by simp - then have "- y \ - x = - y" by (simp only: inf_commute) - then show ?thesis by (simp only: le_iff_inf) -qed - -lemma compl_le_compl_iff [simp]: "- x \ - y \ y \ x" - by (auto dest: compl_mono) - -lemma compl_le_swap1: - assumes "y \ - x" - shows "x \ -y" -proof - - from assms have "- (- x) \ - y" by (simp only: compl_le_compl_iff) - then show ?thesis by simp -qed - -lemma compl_le_swap2: - assumes "- y \ x" - shows "- x \ y" -proof - - from assms have "- x \ - (- y)" by (simp only: compl_le_compl_iff) - then show ?thesis by simp -qed - -lemma compl_less_compl_iff [simp]: "- x < - y \ y < x" - by (auto simp add: less_le) - -lemma compl_less_swap1: - assumes "y < - x" - shows "x < - y" -proof - - from assms have "- (- x) < - y" by (simp only: compl_less_compl_iff) - then show ?thesis by simp -qed - -lemma compl_less_swap2: - assumes "- y < x" - shows "- x < y" -proof - - from assms have "- x < - (- y)" - by (simp only: compl_less_compl_iff) - then show ?thesis by simp -qed - -lemma sup_cancel_left1: "sup (sup x a) (sup (- x) b) = top" - by (simp add: ac_simps sup_compl_top) - -lemma sup_cancel_left2: "sup (sup (- x) a) (sup x b) = top" - by (simp add: ac_simps sup_compl_top) - -lemma inf_cancel_left1: "inf (inf x a) (inf (- x) b) = bot" - by (simp add: ac_simps inf_compl_bot) - -lemma inf_cancel_left2: "inf (inf (- x) a) (inf x b) = bot" - by (simp add: ac_simps inf_compl_bot) - -declare inf_compl_bot [simp] - and sup_compl_top [simp] - -lemma sup_compl_top_left1 [simp]: "sup (- x) (sup x y) = top" - by (simp add: sup_assoc[symmetric]) - -lemma sup_compl_top_left2 [simp]: "sup x (sup (- x) y) = top" - using sup_compl_top_left1[of "- x" y] by simp - -lemma inf_compl_bot_left1 [simp]: "inf (- x) (inf x y) = bot" - by (simp add: inf_assoc[symmetric]) - -lemma inf_compl_bot_left2 [simp]: "inf x (inf (- x) y) = bot" - using inf_compl_bot_left1[of "- x" y] by simp - -lemma inf_compl_bot_right [simp]: "inf x (inf y (- x)) = bot" - by (subst inf_left_commute) simp - -end - -locale boolean_algebra_cancel -begin - -lemma sup1: "(A::'a::semilattice_sup) \ sup k a \ sup A b \ sup k (sup a b)" - by (simp only: ac_simps) - -lemma sup2: "(B::'a::semilattice_sup) \ sup k b \ sup a B \ sup k (sup a b)" - by (simp only: ac_simps) - -lemma sup0: "(a::'a::bounded_semilattice_sup_bot) \ sup a bot" - by simp - -lemma inf1: "(A::'a::semilattice_inf) \ inf k a \ inf A b \ inf k (inf a b)" - by (simp only: ac_simps) - -lemma inf2: "(B::'a::semilattice_inf) \ inf k b \ inf a B \ inf k (inf a b)" - by (simp only: ac_simps) - -lemma inf0: "(a::'a::bounded_semilattice_inf_top) \ inf a top" - by simp - -end - -ML_file \Tools/boolean_algebra_cancel.ML\ - -simproc_setup boolean_algebra_cancel_sup ("sup a b::'a::boolean_algebra") = - \fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_sup_conv\ - -simproc_setup boolean_algebra_cancel_inf ("inf a b::'a::boolean_algebra") = - \fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_inf_conv\ - subsection \\min/max\ as special case of lattice\ context linorder begin sublocale min: semilattice_order min less_eq less + max: semilattice_order max greater_eq greater by standard (auto simp add: min_def max_def) declare min.absorb1 [simp] min.absorb2 [simp] min.absorb3 [simp] min.absorb4 [simp] max.absorb1 [simp] max.absorb2 [simp] max.absorb3 [simp] max.absorb4 [simp] lemma min_le_iff_disj: "min x y \ z \ x \ z \ y \ z" unfolding min_def using linear by (auto intro: order_trans) lemma le_max_iff_disj: "z \ max x y \ z \ x \ z \ y" unfolding max_def using linear by (auto intro: order_trans) lemma min_less_iff_disj: "min x y < z \ x < z \ y < z" unfolding min_def le_less using less_linear by (auto intro: less_trans) lemma less_max_iff_disj: "z < max x y \ z < x \ z < y" unfolding max_def le_less using less_linear by (auto intro: less_trans) lemma min_less_iff_conj [simp]: "z < min x y \ z < x \ z < y" unfolding min_def le_less using less_linear by (auto intro: less_trans) lemma max_less_iff_conj [simp]: "max x y < z \ x < z \ y < z" unfolding max_def le_less using less_linear by (auto intro: less_trans) lemma min_max_distrib1: "min (max b c) a = max (min b a) (min c a)" by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym) lemma min_max_distrib2: "min a (max b c) = max (min a b) (min a c)" by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym) lemma max_min_distrib1: "max (min b c) a = min (max b a) (max c a)" by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym) lemma max_min_distrib2: "max a (min b c) = min (max a b) (max a c)" by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym) lemmas min_max_distribs = min_max_distrib1 min_max_distrib2 max_min_distrib1 max_min_distrib2 lemma split_min [no_atp]: "P (min i j) \ (i \ j \ P i) \ (\ i \ j \ P j)" by (simp add: min_def) lemma split_max [no_atp]: "P (max i j) \ (i \ j \ P j) \ (\ i \ j \ P i)" by (simp add: max_def) lemma split_min_lin [no_atp]: \P (min a b) \ (b = a \ P a) \ (a < b \ P a) \ (b < a \ P b)\ by (cases a b rule: linorder_cases) auto lemma split_max_lin [no_atp]: \P (max a b) \ (b = a \ P a) \ (a < b \ P b) \ (b < a \ P a)\ by (cases a b rule: linorder_cases) auto lemma min_of_mono: "mono f \ min (f m) (f n) = f (min m n)" for f :: "'a \ 'b::linorder" by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym) lemma max_of_mono: "mono f \ max (f m) (f n) = f (max m n)" for f :: "'a \ 'b::linorder" by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym) end lemma max_of_antimono: "antimono f \ max (f x) (f y) = f (min x y)" and min_of_antimono: "antimono f \ min (f x) (f y) = f (max x y)" for f::"'a::linorder \ 'b::linorder" by (auto simp: antimono_def Orderings.max_def min_def intro!: antisym) lemma inf_min: "inf = (min :: 'a::{semilattice_inf,linorder} \ 'a \ 'a)" by (auto intro: antisym simp add: min_def fun_eq_iff) lemma sup_max: "sup = (max :: 'a::{semilattice_sup,linorder} \ 'a \ 'a)" by (auto intro: antisym simp add: max_def fun_eq_iff) subsection \Uniqueness of inf and sup\ lemma (in semilattice_inf) inf_unique: fixes f (infixl "\" 70) assumes le1: "\x y. x \ y \ x" and le2: "\x y. x \ y \ y" and greatest: "\x y z. x \ y \ x \ z \ x \ y \ z" shows "x \ y = x \ y" proof (rule order.antisym) show "x \ y \ x \ y" by (rule le_infI) (rule le1, rule le2) have leI: "\x y z. x \ y \ x \ z \ x \ y \ z" by (blast intro: greatest) show "x \ y \ x \ y" by (rule leI) simp_all qed lemma (in semilattice_sup) sup_unique: fixes f (infixl "\" 70) assumes ge1 [simp]: "\x y. x \ x \ y" and ge2: "\x y. y \ x \ y" and least: "\x y z. y \ x \ z \ x \ y \ z \ x" shows "x \ y = x \ y" proof (rule order.antisym) show "x \ y \ x \ y" by (rule le_supI) (rule ge1, rule ge2) have leI: "\x y z. x \ z \ y \ z \ x \ y \ z" by (blast intro: least) show "x \ y \ x \ y" by (rule leI) simp_all qed -subsection \Lattice on \<^typ>\bool\\ - -instantiation bool :: boolean_algebra -begin - -definition bool_Compl_def [simp]: "uminus = Not" - -definition bool_diff_def [simp]: "A - B \ A \ \ B" - -definition [simp]: "P \ Q \ P \ Q" - -definition [simp]: "P \ Q \ P \ Q" - -instance by standard auto - -end - -lemma sup_boolI1: "P \ P \ Q" - by simp - -lemma sup_boolI2: "Q \ P \ Q" - by simp - -lemma sup_boolE: "P \ Q \ (P \ R) \ (Q \ R) \ R" - by auto - - subsection \Lattice on \<^typ>\_ \ _\\ instantiation "fun" :: (type, semilattice_sup) semilattice_sup begin definition "f \ g = (\x. f x \ g x)" lemma sup_apply [simp, code]: "(f \ g) x = f x \ g x" by (simp add: sup_fun_def) instance by standard (simp_all add: le_fun_def) end instantiation "fun" :: (type, semilattice_inf) semilattice_inf begin definition "f \ g = (\x. f x \ g x)" lemma inf_apply [simp, code]: "(f \ g) x = f x \ g x" by (simp add: inf_fun_def) instance by standard (simp_all add: le_fun_def) end instance "fun" :: (type, lattice) lattice .. instance "fun" :: (type, distrib_lattice) distrib_lattice by standard (rule ext, simp add: sup_inf_distrib1) instance "fun" :: (type, bounded_lattice) bounded_lattice .. instantiation "fun" :: (type, uminus) uminus begin definition fun_Compl_def: "- A = (\x. - A x)" lemma uminus_apply [simp, code]: "(- A) x = - (A x)" by (simp add: fun_Compl_def) instance .. end instantiation "fun" :: (type, minus) minus begin definition fun_diff_def: "A - B = (\x. A x - B x)" lemma minus_apply [simp, code]: "(A - B) x = A x - B x" by (simp add: fun_diff_def) instance .. end -instance "fun" :: (type, boolean_algebra) boolean_algebra - by standard (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+ - - -subsection \Lattice on unary and binary predicates\ - -lemma inf1I: "A x \ B x \ (A \ B) x" - by (simp add: inf_fun_def) - -lemma inf2I: "A x y \ B x y \ (A \ B) x y" - by (simp add: inf_fun_def) - -lemma inf1E: "(A \ B) x \ (A x \ B x \ P) \ P" - by (simp add: inf_fun_def) - -lemma inf2E: "(A \ B) x y \ (A x y \ B x y \ P) \ P" - by (simp add: inf_fun_def) - -lemma inf1D1: "(A \ B) x \ A x" - by (rule inf1E) - -lemma inf2D1: "(A \ B) x y \ A x y" - by (rule inf2E) - -lemma inf1D2: "(A \ B) x \ B x" - by (rule inf1E) - -lemma inf2D2: "(A \ B) x y \ B x y" - by (rule inf2E) - -lemma sup1I1: "A x \ (A \ B) x" - by (simp add: sup_fun_def) - -lemma sup2I1: "A x y \ (A \ B) x y" - by (simp add: sup_fun_def) - -lemma sup1I2: "B x \ (A \ B) x" - by (simp add: sup_fun_def) - -lemma sup2I2: "B x y \ (A \ B) x y" - by (simp add: sup_fun_def) - -lemma sup1E: "(A \ B) x \ (A x \ P) \ (B x \ P) \ P" - by (simp add: sup_fun_def) iprover - -lemma sup2E: "(A \ B) x y \ (A x y \ P) \ (B x y \ P) \ P" - by (simp add: sup_fun_def) iprover - -text \ \<^medskip> Classical introduction rule: no commitment to \A\ vs \B\.\ - -lemma sup1CI: "(\ B x \ A x) \ (A \ B) x" - by (auto simp add: sup_fun_def) - -lemma sup2CI: "(\ B x y \ A x y) \ (A \ B) x y" - by (auto simp add: sup_fun_def) - end diff --git a/src/HOL/Set.thy b/src/HOL/Set.thy --- a/src/HOL/Set.thy +++ b/src/HOL/Set.thy @@ -1,2041 +1,2041 @@ (* Title: HOL/Set.thy Author: Tobias Nipkow Author: Lawrence C Paulson Author: Markus Wenzel *) section \Set theory for higher-order logic\ theory Set - imports Lattices Boolean_Algebra + imports Lattices Boolean_Algebras begin subsection \Sets as predicates\ typedecl 'a set axiomatization Collect :: "('a \ bool) \ 'a set" \ \comprehension\ and member :: "'a \ 'a set \ bool" \ \membership\ where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a" and Collect_mem_eq [simp]: "Collect (\x. member x A) = A" notation member ("'(\')") and member ("(_/ \ _)" [51, 51] 50) abbreviation not_member where "not_member x A \ \ (x \ A)" \ \non-membership\ notation not_member ("'(\')") and not_member ("(_/ \ _)" [51, 51] 50) notation (ASCII) member ("'(:')") and member ("(_/ : _)" [51, 51] 50) and not_member ("'(~:')") and not_member ("(_/ ~: _)" [51, 51] 50) text \Set comprehensions\ syntax "_Coll" :: "pttrn \ bool \ 'a set" ("(1{_./ _})") translations "{x. P}" \ "CONST Collect (\x. P)" syntax (ASCII) "_Collect" :: "pttrn \ 'a set \ bool \ 'a set" ("(1{(_/: _)./ _})") syntax "_Collect" :: "pttrn \ 'a set \ bool \ 'a set" ("(1{(_/ \ _)./ _})") translations "{p:A. P}" \ "CONST Collect (\p. p \ A \ P)" lemma CollectI: "P a \ a \ {x. P x}" by simp lemma CollectD: "a \ {x. P x} \ P a" by simp lemma Collect_cong: "(\x. P x = Q x) \ {x. P x} = {x. Q x}" by simp text \ Simproc for pulling \x = t\ in \{x. \ \ x = t \ \}\ to the front (and similarly for \t = x\): \ simproc_setup defined_Collect ("{x. P x \ Q x}") = \ fn _ => Quantifier1.rearrange_Collect (fn ctxt => resolve_tac ctxt @{thms Collect_cong} 1 THEN resolve_tac ctxt @{thms iffI} 1 THEN ALLGOALS (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE}, DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})])) \ lemmas CollectE = CollectD [elim_format] lemma set_eqI: assumes "\x. x \ A \ x \ B" shows "A = B" proof - from assms have "{x. x \ A} = {x. x \ B}" by simp then show ?thesis by simp qed lemma set_eq_iff: "A = B \ (\x. x \ A \ x \ B)" by (auto intro:set_eqI) lemma Collect_eqI: assumes "\x. P x = Q x" shows "Collect P = Collect Q" using assms by (auto intro: set_eqI) text \Lifting of predicate class instances\ instantiation set :: (type) boolean_algebra begin definition less_eq_set where "A \ B \ (\x. member x A) \ (\x. member x B)" definition less_set where "A < B \ (\x. member x A) < (\x. member x B)" definition inf_set where "A \ B = Collect ((\x. member x A) \ (\x. member x B))" definition sup_set where "A \ B = Collect ((\x. member x A) \ (\x. member x B))" definition bot_set where "\ = Collect \" definition top_set where "\ = Collect \" definition uminus_set where "- A = Collect (- (\x. member x A))" definition minus_set where "A - B = Collect ((\x. member x A) - (\x. member x B))" instance by standard (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def bot_set_def top_set_def uminus_set_def minus_set_def less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply) end text \Set enumerations\ abbreviation empty :: "'a set" ("{}") where "{} \ bot" definition insert :: "'a \ 'a set \ 'a set" where insert_compr: "insert a B = {x. x = a \ x \ B}" syntax "_Finset" :: "args \ 'a set" ("{(_)}") translations "{x, xs}" \ "CONST insert x {xs}" "{x}" \ "CONST insert x {}" subsection \Subsets and bounded quantifiers\ abbreviation subset :: "'a set \ 'a set \ bool" where "subset \ less" abbreviation subset_eq :: "'a set \ 'a set \ bool" where "subset_eq \ less_eq" notation subset ("'(\')") and subset ("(_/ \ _)" [51, 51] 50) and subset_eq ("'(\')") and subset_eq ("(_/ \ _)" [51, 51] 50) abbreviation (input) supset :: "'a set \ 'a set \ bool" where "supset \ greater" abbreviation (input) supset_eq :: "'a set \ 'a set \ bool" where "supset_eq \ greater_eq" notation supset ("'(\')") and supset ("(_/ \ _)" [51, 51] 50) and supset_eq ("'(\')") and supset_eq ("(_/ \ _)" [51, 51] 50) notation (ASCII output) subset ("'(<')") and subset ("(_/ < _)" [51, 51] 50) and subset_eq ("'(<=')") and subset_eq ("(_/ <= _)" [51, 51] 50) definition Ball :: "'a set \ ('a \ bool) \ bool" where "Ball A P \ (\x. x \ A \ P x)" \ \bounded universal quantifiers\ definition Bex :: "'a set \ ('a \ bool) \ bool" where "Bex A P \ (\x. x \ A \ P x)" \ \bounded existential quantifiers\ syntax (ASCII) "_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3ALL (_/:_)./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3EX (_/:_)./ _)" [0, 0, 10] 10) "_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3EX! (_/:_)./ _)" [0, 0, 10] 10) "_Bleast" :: "id \ 'a set \ bool \ 'a" ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10) syntax (input) "_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3! (_/:_)./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3? (_/:_)./ _)" [0, 0, 10] 10) "_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3?! (_/:_)./ _)" [0, 0, 10] 10) syntax "_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3\(_/\_)./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3\(_/\_)./ _)" [0, 0, 10] 10) "_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3\!(_/\_)./ _)" [0, 0, 10] 10) "_Bleast" :: "id \ 'a set \ bool \ 'a" ("(3LEAST(_/\_)./ _)" [0, 0, 10] 10) translations "\x\A. P" \ "CONST Ball A (\x. P)" "\x\A. P" \ "CONST Bex A (\x. P)" "\!x\A. P" \ "\!x. x \ A \ P" "LEAST x:A. P" \ "LEAST x. x \ A \ P" syntax (ASCII output) "_setlessAll" :: "[idt, 'a, bool] \ bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] \ bool" ("(3EX _<_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] \ bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] \ bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) "_setleEx1" :: "[idt, 'a, bool] \ bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) syntax "_setlessAll" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleEx1" :: "[idt, 'a, bool] \ bool" ("(3\!_\_./ _)" [0, 0, 10] 10) translations "\A\B. P" \ "\A. A \ B \ P" "\A\B. P" \ "\A. A \ B \ P" "\A\B. P" \ "\A. A \ B \ P" "\A\B. P" \ "\A. A \ B \ P" "\!A\B. P" \ "\!A. A \ B \ P" print_translation \ let val All_binder = Mixfix.binder_name \<^const_syntax>\All\; val Ex_binder = Mixfix.binder_name \<^const_syntax>\Ex\; val impl = \<^const_syntax>\HOL.implies\; val conj = \<^const_syntax>\HOL.conj\; val sbset = \<^const_syntax>\subset\; val sbset_eq = \<^const_syntax>\subset_eq\; val trans = [((All_binder, impl, sbset), \<^syntax_const>\_setlessAll\), ((All_binder, impl, sbset_eq), \<^syntax_const>\_setleAll\), ((Ex_binder, conj, sbset), \<^syntax_const>\_setlessEx\), ((Ex_binder, conj, sbset_eq), \<^syntax_const>\_setleEx\)]; fun mk v (v', T) c n P = if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P else raise Match; fun tr' q = (q, fn _ => (fn [Const (\<^syntax_const>\_bound\, _) $ Free (v, Type (\<^type_name>\set\, _)), Const (c, _) $ (Const (d, _) $ (Const (\<^syntax_const>\_bound\, _) $ Free (v', T)) $ n) $ P] => (case AList.lookup (=) trans (q, c, d) of NONE => raise Match | SOME l => mk v (v', T) l n P) | _ => raise Match)); in [tr' All_binder, tr' Ex_binder] end \ text \ \<^medskip> Translate between \{e | x1\xn. P}\ and \{u. \x1\xn. u = e \ P}\; \{y. \x1\xn. y = e \ P}\ is only translated if \[0..n] \ bvs e\. \ syntax "_Setcompr" :: "'a \ idts \ bool \ 'a set" ("(1{_ |/_./ _})") parse_translation \ let val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", \<^const_syntax>\Ex\)); fun nvars (Const (\<^syntax_const>\_idts\, _) $ _ $ idts) = nvars idts + 1 | nvars _ = 1; fun setcompr_tr ctxt [e, idts, b] = let val eq = Syntax.const \<^const_syntax>\HOL.eq\ $ Bound (nvars idts) $ e; val P = Syntax.const \<^const_syntax>\HOL.conj\ $ eq $ b; val exP = ex_tr ctxt [idts, P]; in Syntax.const \<^const_syntax>\Collect\ $ absdummy dummyT exP end; in [(\<^syntax_const>\_Setcompr\, setcompr_tr)] end \ print_translation \ [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\Ball\ \<^syntax_const>\_Ball\, Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\Bex\ \<^syntax_const>\_Bex\] \ \ \to avoid eta-contraction of body\ print_translation \ let val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (\<^const_syntax>\Ex\, "DUMMY")); fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] = let fun check (Const (\<^const_syntax>\Ex\, _) $ Abs (_, _, P), n) = check (P, n + 1) | check (Const (\<^const_syntax>\HOL.conj\, _) $ (Const (\<^const_syntax>\HOL.eq\, _) $ Bound m $ e) $ P, n) = n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, [])) | check _ = false; fun tr' (_ $ abs) = let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs] in Syntax.const \<^syntax_const>\_Setcompr\ $ e $ idts $ Q end; in if check (P, 0) then tr' P else let val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs; val M = Syntax.const \<^syntax_const>\_Coll\ $ x $ t; in case t of Const (\<^const_syntax>\HOL.conj\, _) $ (Const (\<^const_syntax>\Set.member\, _) $ (Const (\<^syntax_const>\_bound\, _) $ Free (yN, _)) $ A) $ P => if xN = yN then Syntax.const \<^syntax_const>\_Collect\ $ x $ A $ P else M | _ => M end end; in [(\<^const_syntax>\Collect\, setcompr_tr')] end \ simproc_setup defined_Bex ("\x\A. P x \ Q x") = \ fn _ => Quantifier1.rearrange_Bex (fn ctxt => unfold_tac ctxt @{thms Bex_def}) \ simproc_setup defined_All ("\x\A. P x \ Q x") = \ fn _ => Quantifier1.rearrange_Ball (fn ctxt => unfold_tac ctxt @{thms Ball_def}) \ lemma ballI [intro!]: "(\x. x \ A \ P x) \ \x\A. P x" by (simp add: Ball_def) lemmas strip = impI allI ballI lemma bspec [dest?]: "\x\A. P x \ x \ A \ P x" by (simp add: Ball_def) text \Gives better instantiation for bound:\ setup \ map_theory_claset (fn ctxt => ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt')) \ ML \ structure Simpdata = struct open Simpdata; val mksimps_pairs = [(\<^const_name>\Ball\, @{thms bspec})] @ mksimps_pairs; end; open Simpdata; \ declaration \fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\ lemma ballE [elim]: "\x\A. P x \ (P x \ Q) \ (x \ A \ Q) \ Q" unfolding Ball_def by blast lemma bexI [intro]: "P x \ x \ A \ \x\A. P x" \ \Normally the best argument order: \P x\ constrains the choice of \x \ A\.\ unfolding Bex_def by blast lemma rev_bexI [intro?]: "x \ A \ P x \ \x\A. P x" \ \The best argument order when there is only one \x \ A\.\ unfolding Bex_def by blast lemma bexCI: "(\x\A. \ P x \ P a) \ a \ A \ \x\A. P x" unfolding Bex_def by blast lemma bexE [elim!]: "\x\A. P x \ (\x. x \ A \ P x \ Q) \ Q" unfolding Bex_def by blast lemma ball_triv [simp]: "(\x\A. P) \ ((\x. x \ A) \ P)" \ \trivial rewrite rule.\ by (simp add: Ball_def) lemma bex_triv [simp]: "(\x\A. P) \ ((\x. x \ A) \ P)" \ \Dual form for existentials.\ by (simp add: Bex_def) lemma bex_triv_one_point1 [simp]: "(\x\A. x = a) \ a \ A" by blast lemma bex_triv_one_point2 [simp]: "(\x\A. a = x) \ a \ A" by blast lemma bex_one_point1 [simp]: "(\x\A. x = a \ P x) \ a \ A \ P a" by blast lemma bex_one_point2 [simp]: "(\x\A. a = x \ P x) \ a \ A \ P a" by blast lemma ball_one_point1 [simp]: "(\x\A. x = a \ P x) \ (a \ A \ P a)" by blast lemma ball_one_point2 [simp]: "(\x\A. a = x \ P x) \ (a \ A \ P a)" by blast lemma ball_conj_distrib: "(\x\A. P x \ Q x) \ (\x\A. P x) \ (\x\A. Q x)" by blast lemma bex_disj_distrib: "(\x\A. P x \ Q x) \ (\x\A. P x) \ (\x\A. Q x)" by blast text \Congruence rules\ lemma ball_cong: "\ A = B; \x. x \ B \ P x \ Q x \ \ (\x\A. P x) \ (\x\B. Q x)" by (simp add: Ball_def) lemma ball_cong_simp [cong]: "\ A = B; \x. x \ B =simp=> P x \ Q x \ \ (\x\A. P x) \ (\x\B. Q x)" by (simp add: simp_implies_def Ball_def) lemma bex_cong: "\ A = B; \x. x \ B \ P x \ Q x \ \ (\x\A. P x) \ (\x\B. Q x)" by (simp add: Bex_def cong: conj_cong) lemma bex_cong_simp [cong]: "\ A = B; \x. x \ B =simp=> P x \ Q x \ \ (\x\A. P x) \ (\x\B. Q x)" by (simp add: simp_implies_def Bex_def cong: conj_cong) lemma bex1_def: "(\!x\X. P x) \ (\x\X. P x) \ (\x\X. \y\X. P x \ P y \ x = y)" by auto subsection \Basic operations\ subsubsection \Subsets\ lemma subsetI [intro!]: "(\x. x \ A \ x \ B) \ A \ B" by (simp add: less_eq_set_def le_fun_def) text \ \<^medskip> Map the type \'a set \ anything\ to just \'a\; for overloading constants whose first argument has type \'a set\. \ lemma subsetD [elim, intro?]: "A \ B \ c \ A \ c \ B" by (simp add: less_eq_set_def le_fun_def) \ \Rule in Modus Ponens style.\ lemma rev_subsetD [intro?,no_atp]: "c \ A \ A \ B \ c \ B" \ \The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\ by (rule subsetD) lemma subsetCE [elim,no_atp]: "A \ B \ (c \ A \ P) \ (c \ B \ P) \ P" \ \Classical elimination rule.\ by (auto simp add: less_eq_set_def le_fun_def) lemma subset_eq: "A \ B \ (\x\A. x \ B)" by blast lemma contra_subsetD [no_atp]: "A \ B \ c \ B \ c \ A" by blast lemma subset_refl: "A \ A" by (fact order_refl) (* already [iff] *) lemma subset_trans: "A \ B \ B \ C \ A \ C" by (fact order_trans) lemma subset_not_subset_eq [code]: "A \ B \ A \ B \ \ B \ A" by (fact less_le_not_le) lemma eq_mem_trans: "a = b \ b \ A \ a \ A" by simp lemmas basic_trans_rules [trans] = order_trans_rules rev_subsetD subsetD eq_mem_trans subsubsection \Equality\ lemma subset_antisym [intro!]: "A \ B \ B \ A \ A = B" \ \Anti-symmetry of the subset relation.\ by (iprover intro: set_eqI subsetD) text \\<^medskip> Equality rules from ZF set theory -- are they appropriate here?\ lemma equalityD1: "A = B \ A \ B" by simp lemma equalityD2: "A = B \ B \ A" by simp text \ \<^medskip> Be careful when adding this to the claset as \subset_empty\ is in the simpset: \<^prop>\A = {}\ goes to \<^prop>\{} \ A\ and \<^prop>\A \ {}\ and then back to \<^prop>\A = {}\! \ lemma equalityE: "A = B \ (A \ B \ B \ A \ P) \ P" by simp lemma equalityCE [elim]: "A = B \ (c \ A \ c \ B \ P) \ (c \ A \ c \ B \ P) \ P" by blast lemma eqset_imp_iff: "A = B \ x \ A \ x \ B" by simp lemma eqelem_imp_iff: "x = y \ x \ A \ y \ A" by simp subsubsection \The empty set\ lemma empty_def: "{} = {x. False}" by (simp add: bot_set_def bot_fun_def) lemma empty_iff [simp]: "c \ {} \ False" by (simp add: empty_def) lemma emptyE [elim!]: "a \ {} \ P" by simp lemma empty_subsetI [iff]: "{} \ A" \ \One effect is to delete the ASSUMPTION \<^prop>\{} \ A\\ by blast lemma equals0I: "(\y. y \ A \ False) \ A = {}" by blast lemma equals0D: "A = {} \ a \ A" \ \Use for reasoning about disjointness: \A \ B = {}\\ by blast lemma ball_empty [simp]: "Ball {} P \ True" by (simp add: Ball_def) lemma bex_empty [simp]: "Bex {} P \ False" by (simp add: Bex_def) subsubsection \The universal set -- UNIV\ abbreviation UNIV :: "'a set" where "UNIV \ top" lemma UNIV_def: "UNIV = {x. True}" by (simp add: top_set_def top_fun_def) lemma UNIV_I [simp]: "x \ UNIV" by (simp add: UNIV_def) declare UNIV_I [intro] \ \unsafe makes it less likely to cause problems\ lemma UNIV_witness [intro?]: "\x. x \ UNIV" by simp lemma subset_UNIV: "A \ UNIV" by (fact top_greatest) (* already simp *) text \ \<^medskip> Eta-contracting these two rules (to remove \P\) causes them to be ignored because of their interaction with congruence rules. \ lemma ball_UNIV [simp]: "Ball UNIV P \ All P" by (simp add: Ball_def) lemma bex_UNIV [simp]: "Bex UNIV P \ Ex P" by (simp add: Bex_def) lemma UNIV_eq_I: "(\x. x \ A) \ UNIV = A" by auto lemma UNIV_not_empty [iff]: "UNIV \ {}" by (blast elim: equalityE) lemma empty_not_UNIV[simp]: "{} \ UNIV" by blast subsubsection \The Powerset operator -- Pow\ definition Pow :: "'a set \ 'a set set" where Pow_def: "Pow A = {B. B \ A}" lemma Pow_iff [iff]: "A \ Pow B \ A \ B" by (simp add: Pow_def) lemma PowI: "A \ B \ A \ Pow B" by (simp add: Pow_def) lemma PowD: "A \ Pow B \ A \ B" by (simp add: Pow_def) lemma Pow_bottom: "{} \ Pow B" by simp lemma Pow_top: "A \ Pow A" by simp lemma Pow_not_empty: "Pow A \ {}" using Pow_top by blast subsubsection \Set complement\ lemma Compl_iff [simp]: "c \ - A \ c \ A" by (simp add: fun_Compl_def uminus_set_def) lemma ComplI [intro!]: "(c \ A \ False) \ c \ - A" by (simp add: fun_Compl_def uminus_set_def) blast text \ \<^medskip> This form, with negated conclusion, works well with the Classical prover. Negated assumptions behave like formulae on the right side of the notional turnstile \dots \ lemma ComplD [dest!]: "c \ - A \ c \ A" by simp lemmas ComplE = ComplD [elim_format] lemma Compl_eq: "- A = {x. \ x \ A}" by blast subsubsection \Binary intersection\ abbreviation inter :: "'a set \ 'a set \ 'a set" (infixl "\" 70) where "(\) \ inf" notation (ASCII) inter (infixl "Int" 70) lemma Int_def: "A \ B = {x. x \ A \ x \ B}" by (simp add: inf_set_def inf_fun_def) lemma Int_iff [simp]: "c \ A \ B \ c \ A \ c \ B" unfolding Int_def by blast lemma IntI [intro!]: "c \ A \ c \ B \ c \ A \ B" by simp lemma IntD1: "c \ A \ B \ c \ A" by simp lemma IntD2: "c \ A \ B \ c \ B" by simp lemma IntE [elim!]: "c \ A \ B \ (c \ A \ c \ B \ P) \ P" by simp lemma mono_Int: "mono f \ f (A \ B) \ f A \ f B" by (fact mono_inf) subsubsection \Binary union\ abbreviation union :: "'a set \ 'a set \ 'a set" (infixl "\" 65) where "union \ sup" notation (ASCII) union (infixl "Un" 65) lemma Un_def: "A \ B = {x. x \ A \ x \ B}" by (simp add: sup_set_def sup_fun_def) lemma Un_iff [simp]: "c \ A \ B \ c \ A \ c \ B" unfolding Un_def by blast lemma UnI1 [elim?]: "c \ A \ c \ A \ B" by simp lemma UnI2 [elim?]: "c \ B \ c \ A \ B" by simp text \\<^medskip> Classical introduction rule: no commitment to \A\ vs. \B\.\ lemma UnCI [intro!]: "(c \ B \ c \ A) \ c \ A \ B" by auto lemma UnE [elim!]: "c \ A \ B \ (c \ A \ P) \ (c \ B \ P) \ P" unfolding Un_def by blast lemma insert_def: "insert a B = {x. x = a} \ B" by (simp add: insert_compr Un_def) lemma mono_Un: "mono f \ f A \ f B \ f (A \ B)" by (fact mono_sup) subsubsection \Set difference\ lemma Diff_iff [simp]: "c \ A - B \ c \ A \ c \ B" by (simp add: minus_set_def fun_diff_def) lemma DiffI [intro!]: "c \ A \ c \ B \ c \ A - B" by simp lemma DiffD1: "c \ A - B \ c \ A" by simp lemma DiffD2: "c \ A - B \ c \ B \ P" by simp lemma DiffE [elim!]: "c \ A - B \ (c \ A \ c \ B \ P) \ P" by simp lemma set_diff_eq: "A - B = {x. x \ A \ x \ B}" by blast lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)" by blast subsubsection \Augmenting a set -- \<^const>\insert\\ lemma insert_iff [simp]: "a \ insert b A \ a = b \ a \ A" unfolding insert_def by blast lemma insertI1: "a \ insert a B" by simp lemma insertI2: "a \ B \ a \ insert b B" by simp lemma insertE [elim!]: "a \ insert b A \ (a = b \ P) \ (a \ A \ P) \ P" unfolding insert_def by blast lemma insertCI [intro!]: "(a \ B \ a = b) \ a \ insert b B" \ \Classical introduction rule.\ by auto lemma subset_insert_iff: "A \ insert x B \ (if x \ A then A - {x} \ B else A \ B)" by auto lemma set_insert: assumes "x \ A" obtains B where "A = insert x B" and "x \ B" proof show "A = insert x (A - {x})" using assms by blast show "x \ A - {x}" by blast qed lemma insert_ident: "x \ A \ x \ B \ insert x A = insert x B \ A = B" by auto lemma insert_eq_iff: assumes "a \ A" "b \ B" shows "insert a A = insert b B \ (if a = b then A = B else \C. A = insert b C \ b \ C \ B = insert a C \ a \ C)" (is "?L \ ?R") proof show ?R if ?L proof (cases "a = b") case True with assms \?L\ show ?R by (simp add: insert_ident) next case False let ?C = "A - {b}" have "A = insert b ?C \ b \ ?C \ B = insert a ?C \ a \ ?C" using assms \?L\ \a \ b\ by auto then show ?R using \a \ b\ by auto qed show ?L if ?R using that by (auto split: if_splits) qed lemma insert_UNIV: "insert x UNIV = UNIV" by auto subsubsection \Singletons, using insert\ lemma singletonI [intro!]: "a \ {a}" \ \Redundant? But unlike \insertCI\, it proves the subgoal immediately!\ by (rule insertI1) lemma singletonD [dest!]: "b \ {a} \ b = a" by blast lemmas singletonE = singletonD [elim_format] lemma singleton_iff: "b \ {a} \ b = a" by blast lemma singleton_inject [dest!]: "{a} = {b} \ a = b" by blast lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \ a = b \ A \ {b}" by blast lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \ a = b \ A \ {b}" by blast lemma subset_singletonD: "A \ {x} \ A = {} \ A = {x}" by fast lemma subset_singleton_iff: "X \ {a} \ X = {} \ X = {a}" by blast lemma subset_singleton_iff_Uniq: "(\a. A \ {a}) \ (\\<^sub>\\<^sub>1x. x \ A)" unfolding Uniq_def by blast lemma singleton_conv [simp]: "{x. x = a} = {a}" by blast lemma singleton_conv2 [simp]: "{x. a = x} = {a}" by blast lemma Diff_single_insert: "A - {x} \ B \ A \ insert x B" by blast lemma subset_Diff_insert: "A \ B - insert x C \ A \ B - C \ x \ A" by blast lemma doubleton_eq_iff: "{a, b} = {c, d} \ a = c \ b = d \ a = d \ b = c" by (blast elim: equalityE) lemma Un_singleton_iff: "A \ B = {x} \ A = {} \ B = {x} \ A = {x} \ B = {} \ A = {x} \ B = {x}" by auto lemma singleton_Un_iff: "{x} = A \ B \ A = {} \ B = {x} \ A = {x} \ B = {} \ A = {x} \ B = {x}" by auto subsubsection \Image of a set under a function\ text \Frequently \b\ does not have the syntactic form of \f x\.\ definition image :: "('a \ 'b) \ 'a set \ 'b set" (infixr "`" 90) where "f ` A = {y. \x\A. y = f x}" lemma image_eqI [simp, intro]: "b = f x \ x \ A \ b \ f ` A" unfolding image_def by blast lemma imageI: "x \ A \ f x \ f ` A" by (rule image_eqI) (rule refl) lemma rev_image_eqI: "x \ A \ b = f x \ b \ f ` A" \ \This version's more effective when we already have the required \x\.\ by (rule image_eqI) lemma imageE [elim!]: assumes "b \ (\x. f x) ` A" \ \The eta-expansion gives variable-name preservation.\ obtains x where "b = f x" and "x \ A" using assms unfolding image_def by blast lemma Compr_image_eq: "{x \ f ` A. P x} = f ` {x \ A. P (f x)}" by auto lemma image_Un: "f ` (A \ B) = f ` A \ f ` B" by blast lemma image_iff: "z \ f ` A \ (\x\A. z = f x)" by blast lemma image_subsetI: "(\x. x \ A \ f x \ B) \ f ` A \ B" \ \Replaces the three steps \subsetI\, \imageE\, \hypsubst\, but breaks too many existing proofs.\ by blast lemma image_subset_iff: "f ` A \ B \ (\x\A. f x \ B)" \ \This rewrite rule would confuse users if made default.\ by blast lemma subset_imageE: assumes "B \ f ` A" obtains C where "C \ A" and "B = f ` C" proof - from assms have "B = f ` {a \ A. f a \ B}" by fast moreover have "{a \ A. f a \ B} \ A" by blast ultimately show thesis by (blast intro: that) qed lemma subset_image_iff: "B \ f ` A \ (\AA\A. B = f ` AA)" by (blast elim: subset_imageE) lemma image_ident [simp]: "(\x. x) ` Y = Y" by blast lemma image_empty [simp]: "f ` {} = {}" by blast lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)" by blast lemma image_constant: "x \ A \ (\x. c) ` A = {c}" by auto lemma image_constant_conv: "(\x. c) ` A = (if A = {} then {} else {c})" by auto lemma image_image: "f ` (g ` A) = (\x. f (g x)) ` A" by blast lemma insert_image [simp]: "x \ A \ insert (f x) (f ` A) = f ` A" by blast lemma image_is_empty [iff]: "f ` A = {} \ A = {}" by blast lemma empty_is_image [iff]: "{} = f ` A \ A = {}" by blast lemma image_Collect: "f ` {x. P x} = {f x | x. P x}" \ \NOT suitable as a default simp rule: the RHS isn't simpler than the LHS, with its implicit quantifier and conjunction. Also image enjoys better equational properties than does the RHS.\ by blast lemma if_image_distrib [simp]: "(\x. if P x then f x else g x) ` S = f ` (S \ {x. P x}) \ g ` (S \ {x. \ P x})" by auto lemma image_cong: "f ` M = g ` N" if "M = N" "\x. x \ N \ f x = g x" using that by (simp add: image_def) lemma image_cong_simp [cong]: "f ` M = g ` N" if "M = N" "\x. x \ N =simp=> f x = g x" using that image_cong [of M N f g] by (simp add: simp_implies_def) lemma image_Int_subset: "f ` (A \ B) \ f ` A \ f ` B" by blast lemma image_diff_subset: "f ` A - f ` B \ f ` (A - B)" by blast lemma Setcompr_eq_image: "{f x |x. x \ A} = f ` A" by blast lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}" by auto lemma ball_imageD: "\x\f ` A. P x \ \x\A. P (f x)" by simp lemma bex_imageD: "\x\f ` A. P x \ \x\A. P (f x)" by auto lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S" by auto text \\<^medskip> Range of a function -- just an abbreviation for image!\ abbreviation range :: "('a \ 'b) \ 'b set" \ \of function\ where "range f \ f ` UNIV" lemma range_eqI: "b = f x \ b \ range f" by simp lemma rangeI: "f x \ range f" by simp lemma rangeE [elim?]: "b \ range (\x. f x) \ (\x. b = f x \ P) \ P" by (rule imageE) lemma full_SetCompr_eq: "{u. \x. u = f x} = range f" by auto lemma range_composition: "range (\x. f (g x)) = f ` range g" by auto lemma range_constant [simp]: "range (\_. x) = {x}" by (simp add: image_constant) lemma range_eq_singletonD: "range f = {a} \ f x = a" by auto subsubsection \Some rules with \if\\ text \Elimination of \{x. \ \ x = t \ \}\.\ lemma Collect_conv_if: "{x. x = a \ P x} = (if P a then {a} else {})" by auto lemma Collect_conv_if2: "{x. a = x \ P x} = (if P a then {a} else {})" by auto text \ Rewrite rules for boolean case-splitting: faster than \if_split [split]\. \ lemma if_split_eq1: "(if Q then x else y) = b \ (Q \ x = b) \ (\ Q \ y = b)" by (rule if_split) lemma if_split_eq2: "a = (if Q then x else y) \ (Q \ a = x) \ (\ Q \ a = y)" by (rule if_split) text \ Split ifs on either side of the membership relation. Not for \[simp]\ -- can cause goals to blow up! \ lemma if_split_mem1: "(if Q then x else y) \ b \ (Q \ x \ b) \ (\ Q \ y \ b)" by (rule if_split) lemma if_split_mem2: "(a \ (if Q then x else y)) \ (Q \ a \ x) \ (\ Q \ a \ y)" by (rule if_split [where P = "\S. a \ S"]) lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2 (*Would like to add these, but the existing code only searches for the outer-level constant, which in this case is just Set.member; we instead need to use term-nets to associate patterns with rules. Also, if a rule fails to apply, then the formula should be kept. [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]), ("Int", [IntD1,IntD2]), ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] *) subsection \Further operations and lemmas\ subsubsection \The ``proper subset'' relation\ lemma psubsetI [intro!]: "A \ B \ A \ B \ A \ B" unfolding less_le by blast lemma psubsetE [elim!]: "A \ B \ (A \ B \ \ B \ A \ R) \ R" unfolding less_le by blast lemma psubset_insert_iff: "A \ insert x B \ (if x \ B then A \ B else if x \ A then A - {x} \ B else A \ B)" by (auto simp add: less_le subset_insert_iff) lemma psubset_eq: "A \ B \ A \ B \ A \ B" by (simp only: less_le) lemma psubset_imp_subset: "A \ B \ A \ B" by (simp add: psubset_eq) lemma psubset_trans: "A \ B \ B \ C \ A \ C" unfolding less_le by (auto dest: subset_antisym) lemma psubsetD: "A \ B \ c \ A \ c \ B" unfolding less_le by (auto dest: subsetD) lemma psubset_subset_trans: "A \ B \ B \ C \ A \ C" by (auto simp add: psubset_eq) lemma subset_psubset_trans: "A \ B \ B \ C \ A \ C" by (auto simp add: psubset_eq) lemma psubset_imp_ex_mem: "A \ B \ \b. b \ B - A" unfolding less_le by blast lemma atomize_ball: "(\x. x \ A \ P x) \ Trueprop (\x\A. P x)" by (simp only: Ball_def atomize_all atomize_imp) lemmas [symmetric, rulify] = atomize_ball and [symmetric, defn] = atomize_ball lemma image_Pow_mono: "f ` A \ B \ image f ` Pow A \ Pow B" by blast lemma image_Pow_surj: "f ` A = B \ image f ` Pow A = Pow B" by (blast elim: subset_imageE) subsubsection \Derived rules involving subsets.\ text \\insert\.\ lemma subset_insertI: "B \ insert a B" by (rule subsetI) (erule insertI2) lemma subset_insertI2: "A \ B \ A \ insert b B" by blast lemma subset_insert: "x \ A \ A \ insert x B \ A \ B" by blast text \\<^medskip> Finite Union -- the least upper bound of two sets.\ lemma Un_upper1: "A \ A \ B" by (fact sup_ge1) lemma Un_upper2: "B \ A \ B" by (fact sup_ge2) lemma Un_least: "A \ C \ B \ C \ A \ B \ C" by (fact sup_least) text \\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\ lemma Int_lower1: "A \ B \ A" by (fact inf_le1) lemma Int_lower2: "A \ B \ B" by (fact inf_le2) lemma Int_greatest: "C \ A \ C \ B \ C \ A \ B" by (fact inf_greatest) text \\<^medskip> Set difference.\ lemma Diff_subset[simp]: "A - B \ A" by blast lemma Diff_subset_conv: "A - B \ C \ A \ B \ C" by blast subsubsection \Equalities involving union, intersection, inclusion, etc.\ text \\{}\.\ lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" \ \supersedes \Collect_False_empty\\ by auto lemma subset_empty [simp]: "A \ {} \ A = {}" by (fact bot_unique) lemma not_psubset_empty [iff]: "\ (A < {})" by (fact not_less_bot) (* FIXME: already simp *) lemma Collect_subset [simp]: "{x\A. P x} \ A" by auto lemma Collect_empty_eq [simp]: "Collect P = {} \ (\x. \ P x)" by blast lemma empty_Collect_eq [simp]: "{} = Collect P \ (\x. \ P x)" by blast lemma Collect_neg_eq: "{x. \ P x} = - {x. P x}" by blast lemma Collect_disj_eq: "{x. P x \ Q x} = {x. P x} \ {x. Q x}" by blast lemma Collect_imp_eq: "{x. P x \ Q x} = - {x. P x} \ {x. Q x}" by blast lemma Collect_conj_eq: "{x. P x \ Q x} = {x. P x} \ {x. Q x}" by blast lemma Collect_mono_iff: "Collect P \ Collect Q \ (\x. P x \ Q x)" by blast text \\<^medskip> \insert\.\ lemma insert_is_Un: "insert a A = {a} \ A" \ \NOT SUITABLE FOR REWRITING since \{a} \ insert a {}\\ by blast lemma insert_not_empty [simp]: "insert a A \ {}" and empty_not_insert [simp]: "{} \ insert a A" by blast+ lemma insert_absorb: "a \ A \ insert a A = A" \ \\[simp]\ causes recursive calls when there are nested inserts\ \ \with \<^emph>\quadratic\ running time\ by blast lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" by blast lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" by blast lemma insert_subset [simp]: "insert x A \ B \ x \ B \ A \ B" by blast lemma mk_disjoint_insert: "a \ A \ \B. A = insert a B \ a \ B" \ \use new \B\ rather than \A - {a}\ to avoid infinite unfolding\ by (rule exI [where x = "A - {a}"]) blast lemma insert_Collect: "insert a (Collect P) = {u. u \ a \ P u}" by auto lemma insert_inter_insert [simp]: "insert a A \ insert a B = insert a (A \ B)" by blast lemma insert_disjoint [simp]: "insert a A \ B = {} \ a \ B \ A \ B = {}" "{} = insert a A \ B \ a \ B \ {} = A \ B" by auto lemma disjoint_insert [simp]: "B \ insert a A = {} \ a \ B \ B \ A = {}" "{} = A \ insert b B \ b \ A \ {} = A \ B" by auto text \\<^medskip> \Int\\ lemma Int_absorb: "A \ A = A" by (fact inf_idem) (* already simp *) lemma Int_left_absorb: "A \ (A \ B) = A \ B" by (fact inf_left_idem) lemma Int_commute: "A \ B = B \ A" by (fact inf_commute) lemma Int_left_commute: "A \ (B \ C) = B \ (A \ C)" by (fact inf_left_commute) lemma Int_assoc: "(A \ B) \ C = A \ (B \ C)" by (fact inf_assoc) lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute \ \Intersection is an AC-operator\ lemma Int_absorb1: "B \ A \ A \ B = B" by (fact inf_absorb2) lemma Int_absorb2: "A \ B \ A \ B = A" by (fact inf_absorb1) lemma Int_empty_left: "{} \ B = {}" by (fact inf_bot_left) (* already simp *) lemma Int_empty_right: "A \ {} = {}" by (fact inf_bot_right) (* already simp *) lemma disjoint_eq_subset_Compl: "A \ B = {} \ A \ - B" by blast lemma disjoint_iff: "A \ B = {} \ (\x. x\A \ x \ B)" by blast lemma disjoint_iff_not_equal: "A \ B = {} \ (\x\A. \y\B. x \ y)" by blast lemma Int_UNIV_left: "UNIV \ B = B" by (fact inf_top_left) (* already simp *) lemma Int_UNIV_right: "A \ UNIV = A" by (fact inf_top_right) (* already simp *) lemma Int_Un_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)" by (fact inf_sup_distrib1) lemma Int_Un_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)" by (fact inf_sup_distrib2) lemma Int_UNIV [simp]: "A \ B = UNIV \ A = UNIV \ B = UNIV" by (fact inf_eq_top_iff) (* already simp *) lemma Int_subset_iff [simp]: "C \ A \ B \ C \ A \ C \ B" by (fact le_inf_iff) lemma Int_Collect: "x \ A \ {x. P x} \ x \ A \ P x" by blast text \\<^medskip> \Un\.\ lemma Un_absorb: "A \ A = A" by (fact sup_idem) (* already simp *) lemma Un_left_absorb: "A \ (A \ B) = A \ B" by (fact sup_left_idem) lemma Un_commute: "A \ B = B \ A" by (fact sup_commute) lemma Un_left_commute: "A \ (B \ C) = B \ (A \ C)" by (fact sup_left_commute) lemma Un_assoc: "(A \ B) \ C = A \ (B \ C)" by (fact sup_assoc) lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute \ \Union is an AC-operator\ lemma Un_absorb1: "A \ B \ A \ B = B" by (fact sup_absorb2) lemma Un_absorb2: "B \ A \ A \ B = A" by (fact sup_absorb1) lemma Un_empty_left: "{} \ B = B" by (fact sup_bot_left) (* already simp *) lemma Un_empty_right: "A \ {} = A" by (fact sup_bot_right) (* already simp *) lemma Un_UNIV_left: "UNIV \ B = UNIV" by (fact sup_top_left) (* already simp *) lemma Un_UNIV_right: "A \ UNIV = UNIV" by (fact sup_top_right) (* already simp *) lemma Un_insert_left [simp]: "(insert a B) \ C = insert a (B \ C)" by blast lemma Un_insert_right [simp]: "A \ (insert a B) = insert a (A \ B)" by blast lemma Int_insert_left: "(insert a B) \ C = (if a \ C then insert a (B \ C) else B \ C)" by auto lemma Int_insert_left_if0 [simp]: "a \ C \ (insert a B) \ C = B \ C" by auto lemma Int_insert_left_if1 [simp]: "a \ C \ (insert a B) \ C = insert a (B \ C)" by auto lemma Int_insert_right: "A \ (insert a B) = (if a \ A then insert a (A \ B) else A \ B)" by auto lemma Int_insert_right_if0 [simp]: "a \ A \ A \ (insert a B) = A \ B" by auto lemma Int_insert_right_if1 [simp]: "a \ A \ A \ (insert a B) = insert a (A \ B)" by auto lemma Un_Int_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)" by (fact sup_inf_distrib1) lemma Un_Int_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)" by (fact sup_inf_distrib2) lemma Un_Int_crazy: "(A \ B) \ (B \ C) \ (C \ A) = (A \ B) \ (B \ C) \ (C \ A)" by blast lemma subset_Un_eq: "A \ B \ A \ B = B" by (fact le_iff_sup) lemma Un_empty [iff]: "A \ B = {} \ A = {} \ B = {}" by (fact sup_eq_bot_iff) (* FIXME: already simp *) lemma Un_subset_iff [simp]: "A \ B \ C \ A \ C \ B \ C" by (fact le_sup_iff) lemma Un_Diff_Int: "(A - B) \ (A \ B) = A" by blast lemma Diff_Int2: "A \ C - B \ C = A \ C - B" by blast lemma subset_UnE: assumes "C \ A \ B" obtains A' B' where "A' \ A" "B' \ B" "C = A' \ B'" proof show "C \ A \ A" "C \ B \ B" "C = (C \ A) \ (C \ B)" using assms by blast+ qed lemma Un_Int_eq [simp]: "(S \ T) \ S = S" "(S \ T) \ T = T" "S \ (S \ T) = S" "T \ (S \ T) = T" by auto lemma Int_Un_eq [simp]: "(S \ T) \ S = S" "(S \ T) \ T = T" "S \ (S \ T) = S" "T \ (S \ T) = T" by auto text \\<^medskip> Set complement\ lemma Compl_disjoint [simp]: "A \ - A = {}" by (fact inf_compl_bot) lemma Compl_disjoint2 [simp]: "- A \ A = {}" by (fact compl_inf_bot) lemma Compl_partition: "A \ - A = UNIV" by (fact sup_compl_top) lemma Compl_partition2: "- A \ A = UNIV" by (fact compl_sup_top) lemma double_complement: "- (-A) = A" for A :: "'a set" by (fact double_compl) (* already simp *) lemma Compl_Un: "- (A \ B) = (- A) \ (- B)" by (fact compl_sup) (* already simp *) lemma Compl_Int: "- (A \ B) = (- A) \ (- B)" by (fact compl_inf) (* already simp *) lemma subset_Compl_self_eq: "A \ - A \ A = {}" by blast lemma Un_Int_assoc_eq: "(A \ B) \ C = A \ (B \ C) \ C \ A" \ \Halmos, Naive Set Theory, page 16.\ by blast lemma Compl_UNIV_eq: "- UNIV = {}" by (fact compl_top_eq) (* already simp *) lemma Compl_empty_eq: "- {} = UNIV" by (fact compl_bot_eq) (* already simp *) lemma Compl_subset_Compl_iff [iff]: "- A \ - B \ B \ A" by (fact compl_le_compl_iff) (* FIXME: already simp *) lemma Compl_eq_Compl_iff [iff]: "- A = - B \ A = B" for A B :: "'a set" by (fact compl_eq_compl_iff) (* FIXME: already simp *) lemma Compl_insert: "- insert x A = (- A) - {x}" by blast text \\<^medskip> Bounded quantifiers. The following are not added to the default simpset because (a) they duplicate the body and (b) there are no similar rules for \Int\. \ lemma ball_Un: "(\x \ A \ B. P x) \ (\x\A. P x) \ (\x\B. P x)" by blast lemma bex_Un: "(\x \ A \ B. P x) \ (\x\A. P x) \ (\x\B. P x)" by blast text \\<^medskip> Set difference.\ lemma Diff_eq: "A - B = A \ (- B)" by blast lemma Diff_eq_empty_iff [simp]: "A - B = {} \ A \ B" by blast lemma Diff_cancel [simp]: "A - A = {}" by blast lemma Diff_idemp [simp]: "(A - B) - B = A - B" for A B :: "'a set" by blast lemma Diff_triv: "A \ B = {} \ A - B = A" by (blast elim: equalityE) lemma empty_Diff [simp]: "{} - A = {}" by blast lemma Diff_empty [simp]: "A - {} = A" by blast lemma Diff_UNIV [simp]: "A - UNIV = {}" by blast lemma Diff_insert0 [simp]: "x \ A \ A - insert x B = A - B" by blast lemma Diff_insert: "A - insert a B = A - B - {a}" \ \NOT SUITABLE FOR REWRITING since \{a} \ insert a 0\\ by blast lemma Diff_insert2: "A - insert a B = A - {a} - B" \ \NOT SUITABLE FOR REWRITING since \{a} \ insert a 0\\ by blast lemma insert_Diff_if: "insert x A - B = (if x \ B then A - B else insert x (A - B))" by auto lemma insert_Diff1 [simp]: "x \ B \ insert x A - B = A - B" by blast lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A" by blast lemma insert_Diff: "a \ A \ insert a (A - {a}) = A" by blast lemma Diff_insert_absorb: "x \ A \ (insert x A) - {x} = A" by auto lemma Diff_disjoint [simp]: "A \ (B - A) = {}" by blast lemma Diff_partition: "A \ B \ A \ (B - A) = B" by blast lemma double_diff: "A \ B \ B \ C \ B - (C - A) = A" by blast lemma Un_Diff_cancel [simp]: "A \ (B - A) = A \ B" by blast lemma Un_Diff_cancel2 [simp]: "(B - A) \ A = B \ A" by blast lemma Diff_Un: "A - (B \ C) = (A - B) \ (A - C)" by blast lemma Diff_Int: "A - (B \ C) = (A - B) \ (A - C)" by blast lemma Diff_Diff_Int: "A - (A - B) = A \ B" by blast lemma Un_Diff: "(A \ B) - C = (A - C) \ (B - C)" by blast lemma Int_Diff: "(A \ B) - C = A \ (B - C)" by blast lemma Diff_Int_distrib: "C \ (A - B) = (C \ A) - (C \ B)" by blast lemma Diff_Int_distrib2: "(A - B) \ C = (A \ C) - (B \ C)" by blast lemma Diff_Compl [simp]: "A - (- B) = A \ B" by auto lemma Compl_Diff_eq [simp]: "- (A - B) = - A \ B" by blast lemma subset_Compl_singleton [simp]: "A \ - {b} \ b \ A" by blast text \\<^medskip> Quantification over type \<^typ>\bool\.\ lemma bool_induct: "P True \ P False \ P x" by (cases x) auto lemma all_bool_eq: "(\b. P b) \ P True \ P False" by (auto intro: bool_induct) lemma bool_contrapos: "P x \ \ P False \ P True" by (cases x) auto lemma ex_bool_eq: "(\b. P b) \ P True \ P False" by (auto intro: bool_contrapos) lemma UNIV_bool: "UNIV = {False, True}" by (auto intro: bool_induct) text \\<^medskip> \Pow\\ lemma Pow_empty [simp]: "Pow {} = {{}}" by (auto simp add: Pow_def) lemma Pow_singleton_iff [simp]: "Pow X = {Y} \ X = {} \ Y = {}" by blast (* somewhat slow *) lemma Pow_insert: "Pow (insert a A) = Pow A \ (insert a ` Pow A)" by (blast intro: image_eqI [where ?x = "u - {a}" for u]) lemma Pow_Compl: "Pow (- A) = {- B | B. A \ Pow B}" by (blast intro: exI [where ?x = "- u" for u]) lemma Pow_UNIV [simp]: "Pow UNIV = UNIV" by blast lemma Un_Pow_subset: "Pow A \ Pow B \ Pow (A \ B)" by blast lemma Pow_Int_eq [simp]: "Pow (A \ B) = Pow A \ Pow B" by blast text \\<^medskip> Miscellany.\ lemma set_eq_subset: "A = B \ A \ B \ B \ A" by blast lemma subset_iff: "A \ B \ (\t. t \ A \ t \ B)" by blast lemma subset_iff_psubset_eq: "A \ B \ A \ B \ A = B" unfolding less_le by blast lemma all_not_in_conv [simp]: "(\x. x \ A) \ A = {}" by blast lemma ex_in_conv: "(\x. x \ A) \ A \ {}" by blast lemma ball_simps [simp, no_atp]: "\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)" "\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))" "\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))" "\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)" "\P. (\x\{}. P x) \ True" "\P. (\x\UNIV. P x) \ (\x. P x)" "\a B P. (\x\insert a B. P x) \ (P a \ (\x\B. P x))" "\P Q. (\x\Collect Q. P x) \ (\x. Q x \ P x)" "\A P f. (\x\f`A. P x) \ (\x\A. P (f x))" "\A P. (\ (\x\A. P x)) \ (\x\A. \ P x)" by auto lemma bex_simps [simp, no_atp]: "\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)" "\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))" "\P. (\x\{}. P x) \ False" "\P. (\x\UNIV. P x) \ (\x. P x)" "\a B P. (\x\insert a B. P x) \ (P a \ (\x\B. P x))" "\P Q. (\x\Collect Q. P x) \ (\x. Q x \ P x)" "\A P f. (\x\f`A. P x) \ (\x\A. P (f x))" "\A P. (\(\x\A. P x)) \ (\x\A. \ P x)" by auto lemma ex_image_cong_iff [simp, no_atp]: "(\x. x\f`A) \ A \ {}" "(\x. x\f`A \ P x) \ (\x\A. P (f x))" by auto subsubsection \Monotonicity of various operations\ lemma image_mono: "A \ B \ f ` A \ f ` B" by blast lemma Pow_mono: "A \ B \ Pow A \ Pow B" by blast lemma insert_mono: "C \ D \ insert a C \ insert a D" by blast lemma Un_mono: "A \ C \ B \ D \ A \ B \ C \ D" by (fact sup_mono) lemma Int_mono: "A \ C \ B \ D \ A \ B \ C \ D" by (fact inf_mono) lemma Diff_mono: "A \ C \ D \ B \ A - B \ C - D" by blast lemma Compl_anti_mono: "A \ B \ - B \ - A" by (fact compl_mono) text \\<^medskip> Monotonicity of implications.\ lemma in_mono: "A \ B \ x \ A \ x \ B" by (rule impI) (erule subsetD) lemma conj_mono: "P1 \ Q1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)" by iprover lemma disj_mono: "P1 \ Q1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)" by iprover lemma imp_mono: "Q1 \ P1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)" by iprover lemma imp_refl: "P \ P" .. lemma not_mono: "Q \ P \ \ P \ \ Q" by iprover lemma ex_mono: "(\x. P x \ Q x) \ (\x. P x) \ (\x. Q x)" by iprover lemma all_mono: "(\x. P x \ Q x) \ (\x. P x) \ (\x. Q x)" by iprover lemma Collect_mono: "(\x. P x \ Q x) \ Collect P \ Collect Q" by blast lemma Int_Collect_mono: "A \ B \ (\x. x \ A \ P x \ Q x) \ A \ Collect P \ B \ Collect Q" by blast lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono lemma eq_to_mono: "a = b \ c = d \ b \ d \ a \ c" by iprover subsubsection \Inverse image of a function\ definition vimage :: "('a \ 'b) \ 'b set \ 'a set" (infixr "-`" 90) where "f -` B \ {x. f x \ B}" lemma vimage_eq [simp]: "a \ f -` B \ f a \ B" unfolding vimage_def by blast lemma vimage_singleton_eq: "a \ f -` {b} \ f a = b" by simp lemma vimageI [intro]: "f a = b \ b \ B \ a \ f -` B" unfolding vimage_def by blast lemma vimageI2: "f a \ A \ a \ f -` A" unfolding vimage_def by fast lemma vimageE [elim!]: "a \ f -` B \ (\x. f a = x \ x \ B \ P) \ P" unfolding vimage_def by blast lemma vimageD: "a \ f -` A \ f a \ A" unfolding vimage_def by fast lemma vimage_empty [simp]: "f -` {} = {}" by blast lemma vimage_Compl: "f -` (- A) = - (f -` A)" by blast lemma vimage_Un [simp]: "f -` (A \ B) = (f -` A) \ (f -` B)" by blast lemma vimage_Int [simp]: "f -` (A \ B) = (f -` A) \ (f -` B)" by fast lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}" by blast lemma vimage_Collect: "(\x. P (f x) = Q x) \ f -` (Collect P) = Collect Q" by blast lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \ (f -` B)" \ \NOT suitable for rewriting because of the recurrence of \{a}\.\ by blast lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)" by blast lemma vimage_UNIV [simp]: "f -` UNIV = UNIV" by blast lemma vimage_mono: "A \ B \ f -` A \ f -` B" \ \monotonicity\ by blast lemma vimage_image_eq: "f -` (f ` A) = {y. \x\A. f x = f y}" by (blast intro: sym) lemma image_vimage_subset: "f ` (f -` A) \ A" by blast lemma image_vimage_eq [simp]: "f ` (f -` A) = A \ range f" by blast lemma image_subset_iff_subset_vimage: "f ` A \ B \ A \ f -` B" by blast lemma subset_vimage_iff: "A \ f -` B \ (\x\A. f x \ B)" by auto lemma vimage_const [simp]: "((\x. c) -` A) = (if c \ A then UNIV else {})" by auto lemma vimage_if [simp]: "((\x. if x \ B then c else d) -` A) = (if c \ A then (if d \ A then UNIV else B) else if d \ A then - B else {})" by (auto simp add: vimage_def) lemma vimage_inter_cong: "(\ w. w \ S \ f w = g w) \ f -` y \ S = g -` y \ S" by auto lemma vimage_ident [simp]: "(\x. x) -` Y = Y" by blast subsubsection \Singleton sets\ definition is_singleton :: "'a set \ bool" where "is_singleton A \ (\x. A = {x})" lemma is_singletonI [simp, intro!]: "is_singleton {x}" unfolding is_singleton_def by simp lemma is_singletonI': "A \ {} \ (\x y. x \ A \ y \ A \ x = y) \ is_singleton A" unfolding is_singleton_def by blast lemma is_singletonE: "is_singleton A \ (\x. A = {x} \ P) \ P" unfolding is_singleton_def by blast subsubsection \Getting the contents of a singleton set\ definition the_elem :: "'a set \ 'a" where "the_elem X = (THE x. X = {x})" lemma the_elem_eq [simp]: "the_elem {x} = x" by (simp add: the_elem_def) lemma is_singleton_the_elem: "is_singleton A \ A = {the_elem A}" by (auto simp: is_singleton_def) lemma the_elem_image_unique: assumes "A \ {}" and *: "\y. y \ A \ f y = f x" shows "the_elem (f ` A) = f x" unfolding the_elem_def proof (rule the1_equality) from \A \ {}\ obtain y where "y \ A" by auto with * have "f x = f y" by simp with \y \ A\ have "f x \ f ` A" by blast with * show "f ` A = {f x}" by auto then show "\!x. f ` A = {x}" by auto qed subsubsection \Least value operator\ lemma Least_mono: "mono f \ \x\S. \y\S. x \ y \ (LEAST y. y \ f ` S) = f (LEAST x. x \ S)" for f :: "'a::order \ 'b::order" \ \Courtesy of Stephan Merz\ apply clarify apply (erule_tac P = "\x. x \ S" in LeastI2_order) apply fast apply (rule LeastI2_order) apply (auto elim: monoD intro!: order_antisym) done subsubsection \Monad operation\ definition bind :: "'a set \ ('a \ 'b set) \ 'b set" where "bind A f = {x. \B \ f`A. x \ B}" hide_const (open) bind lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\x. Set.bind (B x) C)" for A :: "'a set" by (auto simp: bind_def) lemma empty_bind [simp]: "Set.bind {} f = {}" by (simp add: bind_def) lemma nonempty_bind_const: "A \ {} \ Set.bind A (\_. B) = B" by (auto simp: bind_def) lemma bind_const: "Set.bind A (\_. B) = (if A = {} then {} else B)" by (auto simp: bind_def) lemma bind_singleton_conv_image: "Set.bind A (\x. {f x}) = f ` A" by (auto simp: bind_def) subsubsection \Operations for execution\ definition is_empty :: "'a set \ bool" where [code_abbrev]: "is_empty A \ A = {}" hide_const (open) is_empty definition remove :: "'a \ 'a set \ 'a set" where [code_abbrev]: "remove x A = A - {x}" hide_const (open) remove lemma member_remove [simp]: "x \ Set.remove y A \ x \ A \ x \ y" by (simp add: remove_def) definition filter :: "('a \ bool) \ 'a set \ 'a set" where [code_abbrev]: "filter P A = {a \ A. P a}" hide_const (open) filter lemma member_filter [simp]: "x \ Set.filter P A \ x \ A \ P x" by (simp add: filter_def) instantiation set :: (equal) equal begin definition "HOL.equal A B \ A \ B \ B \ A" instance by standard (auto simp add: equal_set_def) end text \Misc\ definition pairwise :: "('a \ 'a \ bool) \ 'a set \ bool" where "pairwise R S \ (\x \ S. \y \ S. x \ y \ R x y)" lemma pairwise_alt: "pairwise R S \ (\x\S. \y\S-{x}. R x y)" by (auto simp add: pairwise_def) lemma pairwise_trivial [simp]: "pairwise (\i j. j \ i) I" by (auto simp: pairwise_def) lemma pairwiseI [intro?]: "pairwise R S" if "\x y. x \ S \ y \ S \ x \ y \ R x y" using that by (simp add: pairwise_def) lemma pairwiseD: "R x y" and "R y x" if "pairwise R S" "x \ S" and "y \ S" and "x \ y" using that by (simp_all add: pairwise_def) lemma pairwise_empty [simp]: "pairwise P {}" by (simp add: pairwise_def) lemma pairwise_singleton [simp]: "pairwise P {A}" by (simp add: pairwise_def) lemma pairwise_insert: "pairwise r (insert x s) \ (\y. y \ s \ y \ x \ r x y \ r y x) \ pairwise r s" by (force simp: pairwise_def) lemma pairwise_subset: "pairwise P S \ T \ S \ pairwise P T" by (force simp: pairwise_def) lemma pairwise_mono: "\pairwise P A; \x y. P x y \ Q x y; B \ A\ \ pairwise Q B" by (fastforce simp: pairwise_def) lemma pairwise_imageI: "pairwise P (f ` A)" if "\x y. x \ A \ y \ A \ x \ y \ f x \ f y \ P (f x) (f y)" using that by (auto intro: pairwiseI) lemma pairwise_image: "pairwise r (f ` s) \ pairwise (\x y. (f x \ f y) \ r (f x) (f y)) s" by (force simp: pairwise_def) definition disjnt :: "'a set \ 'a set \ bool" where "disjnt A B \ A \ B = {}" lemma disjnt_self_iff_empty [simp]: "disjnt S S \ S = {}" by (auto simp: disjnt_def) lemma disjnt_iff: "disjnt A B \ (\x. \ (x \ A \ x \ B))" by (force simp: disjnt_def) lemma disjnt_sym: "disjnt A B \ disjnt B A" using disjnt_iff by blast lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}" by (auto simp: disjnt_def) lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y \ a \ Y \ disjnt X Y" by (simp add: disjnt_def) lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) \ a \ Y \ disjnt Y X" by (simp add: disjnt_def) lemma disjnt_subset1 : "\disjnt X Y; Z \ X\ \ disjnt Z Y" by (auto simp: disjnt_def) lemma disjnt_subset2 : "\disjnt X Y; Z \ Y\ \ disjnt X Z" by (auto simp: disjnt_def) lemma disjnt_Un1 [simp]: "disjnt (A \ B) C \ disjnt A C \ disjnt B C" by (auto simp: disjnt_def) lemma disjnt_Un2 [simp]: "disjnt C (A \ B) \ disjnt C A \ disjnt C B" by (auto simp: disjnt_def) lemma disjoint_image_subset: "\pairwise disjnt \; \X. X \ \ \ f X \ X\ \ pairwise disjnt (f `\)" unfolding disjnt_def pairwise_def by fast lemma pairwise_disjnt_iff: "pairwise disjnt \ \ (\x. \\<^sub>\\<^sub>1 X. X \ \ \ x \ X)" by (auto simp: Uniq_def disjnt_iff pairwise_def) lemma disjnt_insert: \<^marker>\contributor \Lars Hupel\\ \disjnt (insert x M) N\ if \x \ N\ \disjnt M N\ using that by (simp add: disjnt_def) lemma Int_emptyI: "(\x. x \ A \ x \ B \ False) \ A \ B = {}" by blast lemma in_image_insert_iff: assumes "\C. C \ B \ x \ C" shows "A \ insert x ` B \ x \ A \ A - {x} \ B" (is "?P \ ?Q") proof assume ?P then show ?Q using assms by auto next assume ?Q then have "x \ A" and "A - {x} \ B" by simp_all from \A - {x} \ B\ have "insert x (A - {x}) \ insert x ` B" by (rule imageI) also from \x \ A\ have "insert x (A - {x}) = A" by auto finally show ?P . qed hide_const (open) member not_member lemmas equalityI = subset_antisym lemmas set_mp = subsetD lemmas set_rev_mp = rev_subsetD ML \ val Ball_def = @{thm Ball_def} val Bex_def = @{thm Bex_def} val CollectD = @{thm CollectD} val CollectE = @{thm CollectE} val CollectI = @{thm CollectI} val Collect_conj_eq = @{thm Collect_conj_eq} val Collect_mem_eq = @{thm Collect_mem_eq} val IntD1 = @{thm IntD1} val IntD2 = @{thm IntD2} val IntE = @{thm IntE} val IntI = @{thm IntI} val Int_Collect = @{thm Int_Collect} val UNIV_I = @{thm UNIV_I} val UNIV_witness = @{thm UNIV_witness} val UnE = @{thm UnE} val UnI1 = @{thm UnI1} val UnI2 = @{thm UnI2} val ballE = @{thm ballE} val ballI = @{thm ballI} val bexCI = @{thm bexCI} val bexE = @{thm bexE} val bexI = @{thm bexI} val bex_triv = @{thm bex_triv} val bspec = @{thm bspec} val contra_subsetD = @{thm contra_subsetD} val equalityCE = @{thm equalityCE} val equalityD1 = @{thm equalityD1} val equalityD2 = @{thm equalityD2} val equalityE = @{thm equalityE} val equalityI = @{thm equalityI} val imageE = @{thm imageE} val imageI = @{thm imageI} val image_Un = @{thm image_Un} val image_insert = @{thm image_insert} val insert_commute = @{thm insert_commute} val insert_iff = @{thm insert_iff} val mem_Collect_eq = @{thm mem_Collect_eq} val rangeE = @{thm rangeE} val rangeI = @{thm rangeI} val range_eqI = @{thm range_eqI} val subsetCE = @{thm subsetCE} val subsetD = @{thm subsetD} val subsetI = @{thm subsetI} val subset_refl = @{thm subset_refl} val subset_trans = @{thm subset_trans} val vimageD = @{thm vimageD} val vimageE = @{thm vimageE} val vimageI = @{thm vimageI} val vimageI2 = @{thm vimageI2} val vimage_Collect = @{thm vimage_Collect} val vimage_Int = @{thm vimage_Int} val vimage_Un = @{thm vimage_Un} \ end