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doc-src/IsarRef/pure.tex

author | nipkow |

Wed, 04 Aug 2004 11:25:08 +0200 | |

changeset 15106 | e8cef6993701 |

parent 14955 | 08ee855c1d94 |

child 15686 | 406a98ee8027 |

permissions | -rw-r--r-- |

aded comment

\chapter{Basic language elements}\label{ch:pure-syntax} Subsequently, we introduce the main part of Pure theory and proof commands, together with fundamental proof methods and attributes. Chapter~\ref{ch:gen-tools} describes further Isar elements provided by generic tools and packages (such as the Simplifier) that are either part of Pure Isabelle or pre-installed in most object logics. Chapter~\ref{ch:logics} refers to object-logic specific elements (mainly for HOL and ZF). \medskip Isar commands may be either \emph{proper} document constructors, or \emph{improper commands}. Some proof methods and attributes introduced later are classified as improper as well. Improper Isar language elements, which are subsequently marked by ``$^*$'', are often helpful when developing proof documents, while their use is discouraged for the final human-readable outcome. Typical examples are diagnostic commands that print terms or theorems according to the current context; other commands emulate old-style tactical theorem proving. \section{Theory commands} \subsection{Defining theories}\label{sec:begin-thy} \indexisarcmd{header}\indexisarcmd{theory}\indexisarcmd{context}\indexisarcmd{end} \begin{matharray}{rcl} \isarcmd{header} & : & \isarkeep{toplevel} \\ \isarcmd{theory} & : & \isartrans{toplevel}{theory} \\ \isarcmd{context}^* & : & \isartrans{toplevel}{theory} \\ \isarcmd{end} & : & \isartrans{theory}{toplevel} \\ \end{matharray} Isabelle/Isar ``new-style'' theories are either defined via theory files or interactively. Both theory-level specifications and proofs are handled uniformly --- occasionally definitional mechanisms even require some explicit proof as well. In contrast, ``old-style'' Isabelle theories support batch processing only, with the proof scripts collected in separate ML files. The first ``real'' command of any theory has to be $\THEORY$, which starts a new theory based on the merge of existing ones. Just preceding $\THEORY$, there may be an optional $\isarkeyword{header}$ declaration, which is relevant to document preparation only; it acts very much like a special pre-theory markup command (cf.\ \S\ref{sec:markup-thy} and \S\ref{sec:markup-thy}). The $\END$ command concludes a theory development; it has to be the very last command of any theory file loaded in batch-mode. The theory context may be also changed interactively by $\CONTEXT$ without creating a new theory. \begin{rail} 'header' text ; 'theory' name '=' (name + '+') filespecs? ':' ; 'context' name ; filespecs: 'files' ((name | parname) +); \end{rail} \begin{descr} \item [$\isarkeyword{header}~text$] provides plain text markup just preceding the formal beginning of a theory. In actual document preparation the corresponding {\LaTeX} macro \verb,\isamarkupheader, may be redefined to produce chapter or section headings. See also \S\ref{sec:markup-thy} and \S\ref{sec:markup-prf} for further markup commands. \item [$\THEORY~A = B@1 + \cdots + B@n\colon$] starts a new theory $A$ based on the merge of existing theories $B@1, \dots, B@n$. Due to inclusion of several ancestors, the overall theory structure emerging in an Isabelle session forms a directed acyclic graph (DAG). Isabelle's theory loader ensures that the sources contributing to the development graph are always up-to-date. Changed files are automatically reloaded when processing theory headers interactively; batch-mode explicitly distinguishes \verb,update_thy, from \verb,use_thy,, see also \cite{isabelle-ref}. The optional $\isarkeyword{files}$ specification declares additional dependencies on ML files. Files will be loaded immediately, unless the name is put in parentheses, which merely documents the dependency to be resolved later in the text (typically via explicit $\isarcmd{use}$ in the body text, see \S\ref{sec:ML}). In reminiscence of the old-style theory system of Isabelle, \texttt{$A$.thy} may be also accompanied by an additional file \texttt{$A$.ML} consisting of ML code that is executed in the context of the \emph{finished} theory $A$. That file should not be included in the $\isarkeyword{files}$ dependency declaration, though. \item [$\CONTEXT~B$] enters an existing theory context, basically in read-only mode, so only a limited set of commands may be performed without destroying the theory. Just as for $\THEORY$, the theory loader ensures that $B$ is loaded and up-to-date. This command is occasionally useful for quick interactive experiments; normally one should always commence a new context via $\THEORY$. \item [$\END$] concludes the current theory definition or context switch. Note that this command cannot be undone, but the whole theory definition has to be retracted. \end{descr} \subsection{Markup commands}\label{sec:markup-thy} \indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection} \indexisarcmd{subsubsection}\indexisarcmd{text}\indexisarcmd{text-raw} \begin{matharray}{rcl} \isarcmd{chapter} & : & \isartrans{theory}{theory} \\ \isarcmd{section} & : & \isartrans{theory}{theory} \\ \isarcmd{subsection} & : & \isartrans{theory}{theory} \\ \isarcmd{subsubsection} & : & \isartrans{theory}{theory} \\ \isarcmd{text} & : & \isartrans{theory}{theory} \\ \isarcmd{text_raw} & : & \isartrans{theory}{theory} \\ \end{matharray} Apart from formal comments (see \S\ref{sec:comments}), markup commands provide a structured way to insert text into the document generated from a theory (see \cite{isabelle-sys} for more information on Isabelle's document preparation tools). \railalias{textraw}{text\_raw} \railterm{textraw} \begin{rail} ('chapter' | 'section' | 'subsection' | 'subsubsection' | 'text' | textraw) text ; \end{rail} \begin{descr} \item [$\isarkeyword{chapter}$, $\isarkeyword{section}$, $\isarkeyword{subsection}$, and $\isarkeyword{subsubsection}$] mark chapter and section headings. \item [$\TEXT$] specifies paragraphs of plain text, including references to formal entities (see also \S\ref{sec:antiq} on ``antiquotations''). \item [$\isarkeyword{text_raw}$] inserts {\LaTeX} source into the output, without additional markup. Thus the full range of document manipulations becomes available. \end{descr} Any of these markup elements corresponds to a {\LaTeX} command with the name prefixed by \verb,\isamarkup,. For the sectioning commands this is a plain macro with a single argument, e.g.\ \verb,\isamarkupchapter{,\dots\verb,}, for $\isarkeyword{chapter}$. The $\isarkeyword{text}$ markup results in a {\LaTeX} environment \verb,\begin{isamarkuptext}, {\dots} \verb,\end{isamarkuptext},, while $\isarkeyword{text_raw}$ causes the text to be inserted directly into the {\LaTeX} source. \medskip Additional markup commands are available for proofs (see \S\ref{sec:markup-prf}). Also note that the $\isarkeyword{header}$ declaration (see \S\ref{sec:begin-thy}) admits to insert section markup just preceding the actual theory definition. \subsection{Type classes and sorts}\label{sec:classes} \indexisarcmd{classes}\indexisarcmd{classrel}\indexisarcmd{defaultsort} \begin{matharray}{rcll} \isarcmd{classes} & : & \isartrans{theory}{theory} \\ \isarcmd{classrel} & : & \isartrans{theory}{theory} & (axiomatic!) \\ \isarcmd{defaultsort} & : & \isartrans{theory}{theory} \\ \end{matharray} \begin{rail} 'classes' (classdecl +) ; 'classrel' (nameref ('<' | subseteq) nameref + 'and') ; 'defaultsort' sort ; \end{rail} \begin{descr} \item [$\isarkeyword{classes}~c \subseteq \vec c$] declares class $c$ to be a subclass of existing classes $\vec c$. Cyclic class structures are ruled out. \item [$\isarkeyword{classrel}~c@1 \subseteq c@2$] states subclass relations between existing classes $c@1$ and $c@2$. This is done axiomatically! The $\INSTANCE$ command (see \S\ref{sec:axclass}) provides a way to introduce proven class relations. \item [$\isarkeyword{defaultsort}~s$] makes sort $s$ the new default sort for any type variables given without sort constraints. Usually, the default sort would be only changed when defining a new object-logic. \end{descr} \subsection{Primitive types and type abbreviations}\label{sec:types-pure} \indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities} \begin{matharray}{rcll} \isarcmd{types} & : & \isartrans{theory}{theory} \\ \isarcmd{typedecl} & : & \isartrans{theory}{theory} \\ \isarcmd{nonterminals} & : & \isartrans{theory}{theory} \\ \isarcmd{arities} & : & \isartrans{theory}{theory} & (axiomatic!) \\ \end{matharray} \begin{rail} 'types' (typespec '=' type infix? +) ; 'typedecl' typespec infix? ; 'nonterminals' (name +) ; 'arities' (nameref '::' arity +) ; \end{rail} \begin{descr} \item [$\TYPES~(\vec\alpha)t = \tau$] introduces \emph{type synonym} $(\vec\alpha)t$ for existing type $\tau$. Unlike actual type definitions, as are available in Isabelle/HOL for example, type synonyms are just purely syntactic abbreviations without any logical significance. Internally, type synonyms are fully expanded. \item [$\isarkeyword{typedecl}~(\vec\alpha)t$] declares a new type constructor $t$, intended as an actual logical type. Note that the Isabelle/HOL object-logic overrides $\isarkeyword{typedecl}$ by its own version (\S\ref{sec:hol-typedef}). \item [$\isarkeyword{nonterminals}~\vec c$] declares $0$-ary type constructors $\vec c$ to act as purely syntactic types, i.e.\ nonterminal symbols of Isabelle's inner syntax of terms or types. \item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's order-sorted signature of types by new type constructor arities. This is done axiomatically! The $\INSTANCE$ command (see \S\ref{sec:axclass}) provides a way to introduce proven type arities. \end{descr} \subsection{Constants and simple definitions}\label{sec:consts} \indexisarcmd{consts}\indexisarcmd{defs}\indexisarcmd{constdefs}\indexoutertoken{constdecl} \begin{matharray}{rcl} \isarcmd{consts} & : & \isartrans{theory}{theory} \\ \isarcmd{defs} & : & \isartrans{theory}{theory} \\ \isarcmd{constdefs} & : & \isartrans{theory}{theory} \\ \end{matharray} \begin{rail} 'consts' ((name '::' type mixfix?) +) ; 'defs' ('(' 'overloaded' ')')? (axmdecl prop +) ; \end{rail} \begin{rail} 'constdefs' structs? (constdecl? constdef +) ; structs: '(' 'structure' (vars + 'and') ')' ; constdecl: (name '::' type) mixfix | (name '::' type) | name 'where' | mixfix ; constdef: thmdecl? prop ; \end{rail} \begin{descr} \item [$\CONSTS~c::\sigma$] declares constant $c$ to have any instance of type scheme $\sigma$. The optional mixfix annotations may attach concrete syntax to the constants declared. \item [$\DEFS~name: eqn$] introduces $eqn$ as a definitional axiom for some existing constant. See \cite[\S6]{isabelle-ref} for more details on the form of equations admitted as constant definitions. The $(overloaded)$ option declares definitions to be potentially overloaded. Unless this option is given, a warning message would be issued for any definitional equation with a more special type than that of the corresponding constant declaration. \item [$\CONSTDEFS$] provides a streamlined combination of constants declarations and definitions: type-inference takes care of the most general typing of the given specification (the optional type constraint may refer to type-inference dummies ``$_$'' as usual). The resulting type declaration needs to agree with that of the specification; overloading is \emph{not} supported here! The constant name may be omitted altogether, if neither type nor syntax declarations are given. The canonical name of the definitional axiom for constant $c$ will be $c_def$, unless specified otherwise. Also note that the given list of specifications is processed in a strictly sequential manner, with type-checking being performed independently. An optional initial context of $(structure)$ declarations admits use of indexed syntax, using the special symbol \verb,\<index>, (printed as ``\i''). The latter concept is particularly useful with locales (see also \S\ref{sec:locale}). \end{descr} \subsection{Syntax and translations}\label{sec:syn-trans} \indexisarcmd{syntax}\indexisarcmd{translations} \begin{matharray}{rcl} \isarcmd{syntax} & : & \isartrans{theory}{theory} \\ \isarcmd{translations} & : & \isartrans{theory}{theory} \\ \end{matharray} \railalias{rightleftharpoons}{\isasymrightleftharpoons} \railterm{rightleftharpoons} \railalias{rightharpoonup}{\isasymrightharpoonup} \railterm{rightharpoonup} \railalias{leftharpoondown}{\isasymleftharpoondown} \railterm{leftharpoondown} \begin{rail} 'syntax' ('(' ( name | 'output' | name 'output' ) ')')? (constdecl +) ; 'translations' (transpat ('==' | '=>' | '<=' | rightleftharpoons | rightharpoonup | leftharpoondown) transpat +) ; transpat: ('(' nameref ')')? string ; \end{rail} \begin{descr} \item [$\isarkeyword{syntax}~(mode)~decls$] is similar to $\CONSTS~decls$, except that the actual logical signature extension is omitted. Thus the context free grammar of Isabelle's inner syntax may be augmented in arbitrary ways, independently of the logic. The $mode$ argument refers to the print mode that the grammar rules belong; unless the $\isarkeyword{output}$ indicator is given, all productions are added both to the input and output grammar. \item [$\isarkeyword{translations}~rules$] specifies syntactic translation rules (i.e.\ macros): parse~/ print rules (\isasymrightleftharpoons), parse rules (\isasymrightharpoonup), or print rules (\isasymleftharpoondown). Translation patterns may be prefixed by the syntactic category to be used for parsing; the default is $logic$. \end{descr} \subsection{Axioms and theorems}\label{sec:axms-thms} \indexisarcmd{axioms}\indexisarcmd{lemmas}\indexisarcmd{theorems} \begin{matharray}{rcll} \isarcmd{axioms} & : & \isartrans{theory}{theory} & (axiomatic!) \\ \isarcmd{lemmas} & : & \isartrans{theory}{theory} \\ \isarcmd{theorems} & : & \isartrans{theory}{theory} \\ \end{matharray} \begin{rail} 'axioms' (axmdecl prop +) ; ('lemmas' | 'theorems') locale? (thmdef? thmrefs + 'and') ; \end{rail} \begin{descr} \item [$\isarkeyword{axioms}~a: \phi$] introduces arbitrary statements as axioms of the meta-logic. In fact, axioms are ``axiomatic theorems'', and may be referred later just as any other theorem. Axioms are usually only introduced when declaring new logical systems. Everyday work is typically done the hard way, with proper definitions and proven theorems. \item [$\isarkeyword{lemmas}~a = \vec b$] retrieves and stores existing facts in the theory context, or the specified locale (see also \S\ref{sec:locale}). Typical applications would also involve attributes, to declare Simplifier rules, for example. \item [$\isarkeyword{theorems}$] is essentially the same as $\isarkeyword{lemmas}$, but marks the result as a different kind of facts. \end{descr} \subsection{Name spaces} \indexisarcmd{global}\indexisarcmd{local}\indexisarcmd{hide} \begin{matharray}{rcl} \isarcmd{global} & : & \isartrans{theory}{theory} \\ \isarcmd{local} & : & \isartrans{theory}{theory} \\ \isarcmd{hide} & : & \isartrans{theory}{theory} \\ \end{matharray} \begin{rail} 'hide' name (nameref + ) ; \end{rail} Isabelle organizes any kind of name declarations (of types, constants, theorems etc.) by separate hierarchically structured name spaces. Normally the user does not have to control the behavior of name spaces by hand, yet the following commands provide some way to do so. \begin{descr} \item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current name declaration mode. Initially, theories start in $\isarkeyword{local}$ mode, causing all names to be automatically qualified by the theory name. Changing this to $\isarkeyword{global}$ causes all names to be declared without the theory prefix, until $\isarkeyword{local}$ is declared again. Note that global names are prone to get hidden accidently later, when qualified names of the same base name are introduced. \item [$\isarkeyword{hide}~space~names$] removes declarations from a given name space (which may be $class$, $type$, or $const$). Hidden objects remain valid within the logic, but are inaccessible from user input. In output, the special qualifier ``$\mathord?\mathord?$'' is prefixed to the full internal name. Unqualified (global) names may not be hidden. \end{descr} \subsection{Incorporating ML code}\label{sec:ML} \indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{ML-command} \indexisarcmd{ML-setup}\indexisarcmd{setup} \indexisarcmd{method-setup} \begin{matharray}{rcl} \isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\ \isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\ \isarcmd{ML_command} & : & \isartrans{\cdot}{\cdot} \\ \isarcmd{ML_setup} & : & \isartrans{theory}{theory} \\ \isarcmd{setup} & : & \isartrans{theory}{theory} \\ \isarcmd{method_setup} & : & \isartrans{theory}{theory} \\ \end{matharray} \railalias{MLsetup}{ML\_setup} \railterm{MLsetup} \railalias{methodsetup}{method\_setup} \railterm{methodsetup} \railalias{MLcommand}{ML\_command} \railterm{MLcommand} \begin{rail} 'use' name ; ('ML' | MLcommand | MLsetup | 'setup') text ; methodsetup name '=' text text ; \end{rail} \begin{descr} \item [$\isarkeyword{use}~file$] reads and executes ML commands from $file$. The current theory context (if present) is passed down to the ML session, but may not be modified. Furthermore, the file name is checked with the $\isarkeyword{files}$ dependency declaration given in the theory header (see also \S\ref{sec:begin-thy}). \item [$\isarkeyword{ML}~text$ and $\isarkeyword{ML_command}~text$] execute ML commands from $text$. The theory context is passed in the same way as for $\isarkeyword{use}$, but may not be changed. Note that the output of $\isarkeyword{ML_command}$ is less verbose than plain $\isarkeyword{ML}$. \item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$. The theory context is passed down to the ML session, and fetched back afterwards. Thus $text$ may actually change the theory as a side effect. \item [$\isarkeyword{setup}~text$] changes the current theory context by applying $text$, which refers to an ML expression of type \texttt{(theory~->~theory)~list}. The $\isarkeyword{setup}$ command is the canonical way to initialize any object-logic specific tools and packages written in ML. \item [$\isarkeyword{method_setup}~name = text~description$] defines a proof method in the current theory. The given $text$ has to be an ML expression of type \texttt{Args.src -> Proof.context -> Proof.method}. Parsing concrete method syntax from \texttt{Args.src} input can be quite tedious in general. The following simple examples are for methods without any explicit arguments, or a list of theorems, respectively. {\footnotesize \begin{verbatim} Method.no_args (Method.METHOD (fn facts => foobar_tac)) Method.thms_args (fn thms => Method.METHOD (fn facts => foobar_tac)) Method.ctxt_args (fn ctxt => Method.METHOD (fn facts => foobar_tac)) Method.thms_ctxt_args (fn thms => fn ctxt => Method.METHOD (fn facts => foobar_tac)) \end{verbatim} } Note that mere tactic emulations may ignore the \texttt{facts} parameter above. Proper proof methods would do something appropriate with the list of current facts, though. Single-rule methods usually do strict forward-chaining (e.g.\ by using \texttt{Method.multi_resolves}), while automatic ones just insert the facts using \texttt{Method.insert_tac} before applying the main tactic. \end{descr} \subsection{Syntax translation functions} \indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation} \indexisarcmd{print-translation}\indexisarcmd{typed-print-translation} \indexisarcmd{print-ast-translation}\indexisarcmd{token-translation} \begin{matharray}{rcl} \isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\ \isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\ \isarcmd{print_translation} & : & \isartrans{theory}{theory} \\ \isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\ \isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\ \isarcmd{token_translation} & : & \isartrans{theory}{theory} \\ \end{matharray} \railalias{parseasttranslation}{parse\_ast\_translation} \railterm{parseasttranslation} \railalias{parsetranslation}{parse\_translation} \railterm{parsetranslation} \railalias{printtranslation}{print\_translation} \railterm{printtranslation} \railalias{typedprinttranslation}{typed\_print\_translation} \railterm{typedprinttranslation} \railalias{printasttranslation}{print\_ast\_translation} \railterm{printasttranslation} \railalias{tokentranslation}{token\_translation} \railterm{tokentranslation} \begin{rail} ( parseasttranslation | parsetranslation | printtranslation | typedprinttranslation | printasttranslation ) ('(' 'advanced' ')')? text; tokentranslation text \end{rail} Syntax translation functions written in ML admit almost arbitrary manipulations of Isabelle's inner syntax. Any of the above commands have a single \railqtok{text} argument that refers to an ML expression of appropriate type, which are as follows by default: \begin{ttbox} val parse_ast_translation : (string * (ast list -> ast)) list val parse_translation : (string * (term list -> term)) list val print_translation : (string * (term list -> term)) list val typed_print_translation : (string * (bool -> typ -> term list -> term)) list val print_ast_translation : (string * (ast list -> ast)) list val token_translation : (string * string * (string -> string * real)) list \end{ttbox} In case that the $(advanced)$ option is given, the corresponding translation functions may depend on the signature of the current theory context. This allows to implement advanced syntax mechanisms, as translations functions may refer to specific theory declarations and auxiliary data. See also \cite[\S8]{isabelle-ref} for more information on the general concept of syntax transformations in Isabelle. \begin{ttbox} val parse_ast_translation: (string * (Sign.sg -> ast list -> ast)) list val parse_translation: (string * (Sign.sg -> term list -> term)) list val print_translation: (string * (Sign.sg -> term list -> term)) list val typed_print_translation: (string * (Sign.sg -> bool -> typ -> term list -> term)) list val print_ast_translation: (string * (Sign.sg -> ast list -> ast)) list \end{ttbox} \subsection{Oracles} \indexisarcmd{oracle} \begin{matharray}{rcl} \isarcmd{oracle} & : & \isartrans{theory}{theory} \\ \end{matharray} Oracles provide an interface to external reasoning systems, without giving up control completely --- each theorem carries a derivation object recording any oracle invocation. See \cite[\S6]{isabelle-ref} for more information. \begin{rail} 'oracle' name '=' text ; \end{rail} \begin{descr} \item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML function $text$, which has to be of type \texttt{Sign.sg~*~Object.T~->~term}. \end{descr} \section{Proof commands} Proof commands perform transitions of Isar/VM machine configurations, which are block-structured, consisting of a stack of nodes with three main components: logical proof context, current facts, and open goals. Isar/VM transitions are \emph{typed} according to the following three different modes of operation: \begin{descr} \item [$proof(prove)$] means that a new goal has just been stated that is now to be \emph{proven}; the next command may refine it by some proof method, and enter a sub-proof to establish the actual result. \item [$proof(state)$] is like a nested theory mode: the context may be augmented by \emph{stating} additional assumptions, intermediate results etc. \item [$proof(chain)$] is intermediate between $proof(state)$ and $proof(prove)$: existing facts (i.e.\ the contents of the special ``$this$'' register) have been just picked up in order to be used when refining the goal claimed next. \end{descr} The proof mode indicator may be read as a verb telling the writer what kind of operation may be performed next. The corresponding typings of proof commands restricts the shape of well-formed proof texts to particular command sequences. So dynamic arrangements of commands eventually turn out as static texts of a certain structure. Appendix~\ref{ap:refcard} gives a simplified grammar of the overall (extensible) language emerging that way. \subsection{Markup commands}\label{sec:markup-prf} \indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsubsect} \indexisarcmd{txt}\indexisarcmd{txt-raw} \begin{matharray}{rcl} \isarcmd{sect} & : & \isartrans{proof}{proof} \\ \isarcmd{subsect} & : & \isartrans{proof}{proof} \\ \isarcmd{subsubsect} & : & \isartrans{proof}{proof} \\ \isarcmd{txt} & : & \isartrans{proof}{proof} \\ \isarcmd{txt_raw} & : & \isartrans{proof}{proof} \\ \end{matharray} These markup commands for proof mode closely correspond to the ones of theory mode (see \S\ref{sec:markup-thy}). \railalias{txtraw}{txt\_raw} \railterm{txtraw} \begin{rail} ('sect' | 'subsect' | 'subsubsect' | 'txt' | txtraw) text ; \end{rail} \subsection{Context elements}\label{sec:proof-context} \indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def} \begin{matharray}{rcl} \isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\ \end{matharray} The logical proof context consists of fixed variables and assumptions. The former closely correspond to Skolem constants, or meta-level universal quantification as provided by the Isabelle/Pure logical framework. Introducing some \emph{arbitrary, but fixed} variable via ``$\FIX x$'' results in a local value that may be used in the subsequent proof as any other variable or constant. Furthermore, any result $\edrv \phi[x]$ exported from the context will be universally closed wrt.\ $x$ at the outermost level: $\edrv \All x \phi$ (this is expressed using Isabelle's meta-variables). Similarly, introducing some assumption $\chi$ has two effects. On the one hand, a local theorem is created that may be used as a fact in subsequent proof steps. On the other hand, any result $\chi \drv \phi$ exported from the context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$. Thus, solving an enclosing goal using such a result would basically introduce a new subgoal stemming from the assumption. How this situation is handled depends on the actual version of assumption command used: while $\ASSUMENAME$ insists on solving the subgoal by unification with some premise of the goal, $\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the user. Local definitions, introduced by ``$\DEF{}{x \equiv t}$'', are achieved by combining ``$\FIX x$'' with another version of assumption that causes any hypothetical equation $x \equiv t$ to be eliminated by the reflexivity rule. Thus, exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$. \railalias{equiv}{\isasymequiv} \railterm{equiv} \begin{rail} 'fix' (vars + 'and') ; ('assume' | 'presume') (props + 'and') ; 'def' thmdecl? \\ name ('==' | equiv) term termpat? ; \end{rail} \begin{descr} \item [$\FIX{\vec x}$] introduces local \emph{arbitrary, but fixed} variables $\vec x$. \item [$\ASSUME{a}{\vec\phi}$ and $\PRESUME{a}{\vec\phi}$] introduce local theorems $\vec\phi$ by assumption. Subsequent results applied to an enclosing goal (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be able to unify with existing premises in the goal, while $\PRESUMENAME$ leaves $\vec\phi$ as new subgoals. Several lists of assumptions may be given (separated by $\isarkeyword{and}$); the resulting list of current facts consists of all of these concatenated. \item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition. In results exported from the context, $x$ is replaced by $t$. Basically, ``$\DEF{}{x \equiv t}$'' abbreviates ``$\FIX{x}~\ASSUME{}{x \equiv t}$'', with the resulting hypothetical equation solved by reflexivity. The default name for the definitional equation is $x_def$. \end{descr} The special name $prems$\indexisarthm{prems} refers to all assumptions of the current context as a list of theorems. \subsection{Facts and forward chaining} \indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with} \indexisarcmd{using} \begin{matharray}{rcl} \isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\ \isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\ \isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\ \isarcmd{using} & : & \isartrans{proof(prove)}{proof(prove)} \\ \end{matharray} New facts are established either by assumption or proof of local statements. Any fact will usually be involved in further proofs, either as explicit arguments of proof methods, or when forward chaining towards the next goal via $\THEN$ (and variants); $\FROMNAME$ and $\WITHNAME$ are composite forms involving $\NOTENAME$. The $\USINGNAME$ elements augments the collection of used facts \emph{after} a goal has been stated. Note that the special theorem name $this$\indexisarthm{this} refers to the most recently established facts, but only \emph{before} issuing a follow-up claim. \begin{rail} 'note' (thmdef? thmrefs + 'and') ; ('from' | 'with' | 'using') (thmrefs + 'and') ; \end{rail} \begin{descr} \item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result as $a$. Note that attributes may be involved as well, both on the left and right hand sides. \item [$\THEN$] indicates forward chaining by the current facts in order to establish the goal to be claimed next. The initial proof method invoked to refine that will be offered the facts to do ``anything appropriate'' (see also \S\ref{sec:proof-steps}). For example, method $rule$ (see \S\ref{sec:pure-meth-att}) would typically do an elimination rather than an introduction. Automatic methods usually insert the facts into the goal state before operation. This provides a simple scheme to control relevance of facts in automated proof search. \item [$\FROM{\vec b}$] abbreviates ``$\NOTE{}{\vec b}~\THEN$''; thus $\THEN$ is equivalent to ``$\FROM{this}$''. \item [$\WITH{\vec b}$] abbreviates ``$\FROM{\vec b~\AND~this}$''; thus the forward chaining is from earlier facts together with the current ones. \item [$\USING{\vec b}$] augments the facts being currently indicated for use by a subsequent refinement step (such as $\APPLYNAME$ or $\PROOFNAME$). \end{descr} Forward chaining with an empty list of theorems is the same as not chaining at all. Thus ``$\FROM{nothing}$'' has no effect apart from entering $prove(chain)$ mode, since $nothing$\indexisarthm{nothing} is bound to the empty list of theorems. Basic proof methods (such as $rule$) expect multiple facts to be given in their proper order, corresponding to a prefix of the premises of the rule involved. Note that positions may be easily skipped using something like $\FROM{\Text{\texttt{_}}~a~b}$, for example. This involves the trivial rule $\PROP\psi \Imp \PROP\psi$, which happens to be bound in Isabelle/Pure as ``\texttt{_}'' (underscore).\indexisarthm{_@\texttt{_}} Automated methods (such as $simp$ or $auto$) just insert any given facts before their usual operation. Depending on the kind of procedure involved, the order of facts is less significant here. \subsection{Goal statements}\label{sec:goals} \indexisarcmd{lemma}\indexisarcmd{theorem}\indexisarcmd{corollary} \indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus} \begin{matharray}{rcl} \isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\ \isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\ \isarcmd{corollary} & : & \isartrans{theory}{proof(prove)} \\ \isarcmd{have} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\ \isarcmd{show} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\ \isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\ \isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\ \end{matharray} From a theory context, proof mode is entered by an initial goal command such as $\LEMMANAME$, $\THEOREMNAME$, or $\COROLLARYNAME$. Within a proof, new claims may be introduced locally as well; four variants are available here to indicate whether forward chaining of facts should be performed initially (via $\THEN$), and whether the final result is meant to solve some pending goal. Goals may consist of multiple statements, resulting in a list of facts eventually. A pending multi-goal is internally represented as a meta-level conjunction (printed as \verb,&&,), which is usually split into the corresponding number of sub-goals prior to an initial method application, via $\PROOFNAME$ (\S\ref{sec:proof-steps}) or $\APPLYNAME$ (\S\ref{sec:tactic-commands}). The $induct$ method covered in \S\ref{sec:cases-induct-meth} acts on multiple claims simultaneously. Claims at the theory level may be either in short or long form. A short goal merely consists of several simultaneous propositions (often just one). A long goal includes an explicit context specification for the subsequent conclusions, involving local parameters; here the role of each part of the statement is explicitly marked by separate keywords (see also \S\ref{sec:locale}). \begin{rail} ('lemma' | 'theorem' | 'corollary') locale? (goal | longgoal) ; ('have' | 'show' | 'hence' | 'thus') goal ; goal: (props + 'and') ; longgoal: thmdecl? (contextelem *) 'shows' goal ; \end{rail} \begin{descr} \item [$\LEMMA{a}{\vec\phi}$] enters proof mode with $\vec\phi$ as main goal, eventually resulting in some fact $\turn \vec\phi$ to be put back into the theory context, or into the specified locale (cf.\ \S\ref{sec:locale}). An additional \railnonterm{context} specification may build up an initial proof context for the subsequent claim; this includes local definitions and syntax as well, see the definition of $contextelem$ in \S\ref{sec:locale}. \item [$\THEOREM{a}{\vec\phi}$ and $\COROLLARY{a}{\vec\phi}$] are essentially the same as $\LEMMA{a}{\vec\phi}$, but the facts are internally marked as being of a different kind. This discrimination acts like a formal comment. \item [$\HAVE{a}{\vec\phi}$] claims a local goal, eventually resulting in a fact within the current logical context. This operation is completely independent of any pending sub-goals of an enclosing goal statements, so $\HAVENAME$ may be freely used for experimental exploration of potential results within a proof body. \item [$\SHOW{a}{\vec\phi}$] is like $\HAVE{a}{\vec\phi}$ plus a second stage to refine some pending sub-goal for each one of the finished result, after having been exported into the corresponding context (at the head of the sub-proof of this $\SHOWNAME$ command). To accommodate interactive debugging, resulting rules are printed before being applied internally. Even more, interactive execution of $\SHOWNAME$ predicts potential failure and displays the resulting error as a warning beforehand. Watch out for the following message: \begin{ttbox} Problem! Local statement will fail to solve any pending goal \end{ttbox} \item [$\HENCENAME$] abbreviates ``$\THEN~\HAVENAME$'', i.e.\ claims a local goal to be proven by forward chaining the current facts. Note that $\HENCENAME$ is also equivalent to ``$\FROM{this}~\HAVENAME$''. \item [$\THUSNAME$] abbreviates ``$\THEN~\SHOWNAME$''. Note that $\THUSNAME$ is also equivalent to ``$\FROM{this}~\SHOWNAME$''. \end{descr} Any goal statement causes some term abbreviations (such as $\Var{thesis}$) to be bound automatically, see also \S\ref{sec:term-abbrev}. Furthermore, the local context of a (non-atomic) goal is provided via the $rule_context$\indexisarcase{rule-context} case. \medskip \begin{warn} Isabelle/Isar suffers theory-level goal statements to contain \emph{unbound schematic variables}, although this does not conform to the aim of human-readable proof documents! The main problem with schematic goals is that the actual outcome is usually hard to predict, depending on the behavior of the proof methods applied during the course of reasoning. Note that most semi-automated methods heavily depend on several kinds of implicit rule declarations within the current theory context. As this would also result in non-compositional checking of sub-proofs, \emph{local goals} are not allowed to be schematic at all. Nevertheless, schematic goals do have their use in Prolog-style interactive synthesis of proven results, usually by stepwise refinement via emulation of traditional Isabelle tactic scripts (see also \S\ref{sec:tactic-commands}). In any case, users should know what they are doing. \end{warn} \subsection{Initial and terminal proof steps}\label{sec:proof-steps} \indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by} \indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry} \begin{matharray}{rcl} \isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\ \isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\ \isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ \isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ \isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ \isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ \end{matharray} Arbitrary goal refinement via tactics is considered harmful. Properly, the Isar framework admits proof methods to be invoked in two places only. \begin{enumerate} \item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated goal to a number of sub-goals that are to be solved later. Facts are passed to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode. \item A \emph{terminal} conclusion step $\QED{m@2}$ is intended to solve remaining goals. No facts are passed to $m@2$. \end{enumerate} The only other (proper) way to affect pending goals in a proof body is by $\SHOWNAME$, which involves an explicit statement of what is to be solved eventually. Thus we avoid the fundamental problem of unstructured tactic scripts that consist of numerous consecutive goal transformations, with invisible effects. \medskip As a general rule of thumb for good proof style, initial proof methods should either solve the goal completely, or constitute some well-understood reduction to new sub-goals. Arbitrary automatic proof tools that are prone leave a large number of badly structured sub-goals are no help in continuing the proof document in an intelligible manner. Unless given explicitly by the user, the default initial method is ``$rule$'', which applies a single standard elimination or introduction rule according to the topmost symbol involved. There is no separate default terminal method. Any remaining goals are always solved by assumption in the very last step. \begin{rail} 'proof' method? ; 'qed' method? ; 'by' method method? ; ('.' | '..' | 'sorry') ; \end{rail} \begin{descr} \item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for forward chaining are passed if so indicated by $proof(chain)$ mode. \item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and concludes the sub-proof by assumption. If the goal had been $\SHOWNAME$ (or $\THUSNAME$), some pending sub-goal is solved as well by the rule resulting from the result \emph{exported} into the enclosing goal context. Thus $\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting rule does not fit to any pending goal\footnote{This includes any additional ``strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing context. Debugging such a situation might involve temporarily changing $\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing occurrences of $\ASSUMENAME$ by $\PRESUMENAME$. \item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it abbreviates $\PROOF{m@1}~\QED{m@2}$, but with backtracking across both methods. Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done by expanding its definition; in many cases $\PROOF{m@1}$ (or even $\APPLY{m@1}$) is already sufficient to see the problem. \item [``$\DDOT$''] is a \emph{default proof}\index{proof!default}; it abbreviates $\BY{rule}$. \item [``$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it abbreviates $\BY{this}$. \item [$\SORRY$] is a \emph{fake proof}\index{proof!fake} pretending to solve the pending claim without further ado. This only works in interactive development, or if the \texttt{quick_and_dirty} flag is enabled. Facts emerging from fake proofs are not the real thing. Internally, each theorem container is tainted by an oracle invocation, which is indicated as ``$[!]$'' in the printed result. The most important application of $\SORRY$ is to support experimentation and top-down proof development. \end{descr} \subsection{Fundamental methods and attributes}\label{sec:pure-meth-att} The following proof methods and attributes refer to basic logical operations of Isar. Further methods and attributes are provided by several generic and object-logic specific tools and packages (see chapters \ref{ch:gen-tools} and \ref{ch:logics}). \indexisarmeth{$-$}\indexisarmeth{assumption} \indexisarmeth{this}\indexisarmeth{rule}\indexisarmeth{rules} \indexisarattof{Pure}{intro}\indexisarattof{Pure}{elim} \indexisarattof{Pure}{dest}\indexisarattof{Pure}{rule} \indexisaratt{OF}\indexisaratt{of}\indexisaratt{where} \begin{matharray}{rcl} - & : & \isarmeth \\ assumption & : & \isarmeth \\ this & : & \isarmeth \\ rule & : & \isarmeth \\ rules & : & \isarmeth \\[0.5ex] intro & : & \isaratt \\ elim & : & \isaratt \\ dest & : & \isaratt \\ rule & : & \isaratt \\[0.5ex] OF & : & \isaratt \\ of & : & \isaratt \\ where & : & \isaratt \\ \end{matharray} \begin{rail} 'rule' thmrefs? ; 'rules' ('!' ?) (rulemod *) ; rulemod: ('intro' | 'elim' | 'dest') ((('!' | () | '?') nat?) | 'del') ':' thmrefs ; ('intro' | 'elim' | 'dest') ('!' | () | '?') nat? ; 'rule' 'del' ; 'OF' thmrefs ; 'of' insts ('concl' ':' insts)? ; 'where' (name '=' term * 'and') ; \end{rail} \begin{descr} \item [``$-$''] does nothing but insert the forward chaining facts as premises into the goal. Note that command $\PROOFNAME$ without any method actually performs a single reduction step using the $rule$ method; thus a plain \emph{do-nothing} proof step would be ``$\PROOF{-}$'' rather than $\PROOFNAME$ alone. \item [$assumption$] solves some goal by a single assumption step. All given facts are guaranteed to participate in the refinement; this means there may be only $0$ or $1$ in the first place. Recall that $\QEDNAME$ (see \S\ref{sec:proof-steps}) already concludes any remaining sub-goals by assumption, so structured proofs usually need not quote the $assumption$ method at all. \item [$this$] applies all of the current facts directly as rules. Recall that ``$\DOT$'' (dot) abbreviates ``$\BY{this}$''. \item [$rule~\vec a$] applies some rule given as argument in backward manner; facts are used to reduce the rule before applying it to the goal. Thus $rule$ without facts is plain introduction, while with facts it becomes elimination. When no arguments are given, the $rule$ method tries to pick appropriate rules automatically, as declared in the current context using the $intro$, $elim$, $dest$ attributes (see below). This is the default behavior of $\PROOFNAME$ and ``$\DDOT$'' (double-dot) steps (see \S\ref{sec:proof-steps}). \item [$rules$] performs intuitionistic proof search, depending on specifically declared rules from the context, or given as explicit arguments. Chained facts are inserted into the goal before commencing proof search; ``$rules!$'' means to include the current $prems$ as well. Rules need to be classified as $intro$, $elim$, or $dest$; here the ``$!$'' indicator refers to ``safe'' rules, which may be applied aggressively (without considering back-tracking later). Rules declared with ``$?$'' are ignored in proof search (the single-step $rule$ method still observes these). An explicit weight annotation may be given as well; otherwise the number of rule premises will be taken into account here. \item [$intro$, $elim$, and $dest$] declare introduction, elimination, and destruct rules, to be used with the $rule$ and $rules$ methods. Note that the latter will ignore rules declared with ``$?$'', while ``$!$'' are used most aggressively. The classical reasoner (see \S\ref{sec:classical}) introduces its own variants of these attributes; use qualified names to access the present versions of Isabelle/Pure, i.e.\ $Pure{\dtt}intro$ or $CPure{\dtt}intro$. \item [$rule~del$] undeclares introduction, elimination, or destruct rules. \item [$OF~\vec a$] applies some theorem to given rules $\vec a$ (in parallel). This corresponds to the \texttt{MRS} operator in ML \cite[\S5]{isabelle-ref}, but note the reversed order. Positions may be effectively skipped by including ``$\_$'' (underscore) as argument. \item [$of~\vec t$] performs positional instantiation of term variables. The terms $\vec t$ are substituted for any schematic variables occurring in a theorem from left to right; ``\texttt{_}'' (underscore) indicates to skip a position. Arguments following a ``$concl\colon$'' specification refer to positions of the conclusion of a rule. \item [$where~\vec x = \vec t$] performs named instantiation of schematic type and term variables occurring in a theorem. Schematic variables have to be specified on the left-hand side (e.g.\ $?x1\!.\!3$). The question mark may be omitted if the variable name is neither a keyword nor contains a dot. Types are instantiated before terms, and instantiations have to be written in that order. Because type instantiations are inferred from term instantiations, explicit type instantiations are seldom necessary. \end{descr} \subsection{Term abbreviations}\label{sec:term-abbrev} \indexisarcmd{let} \begin{matharray}{rcl} \isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\ \isarkeyword{is} & : & syntax \\ \end{matharray} Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements, or by annotating assumptions or goal statements with a list of patterns ``$\ISS{p@1\;\dots}{p@n}$''. In both cases, higher-order matching is invoked to bind extra-logical term variables, which may be either named schematic variables of the form $\Var{x}$, or nameless dummies ``\texttt{_}'' (underscore).\indexisarvar{_@\texttt{_}} Note that in the $\LETNAME$ form the patterns occur on the left-hand side, while the $\ISNAME$ patterns are in postfix position. Polymorphism of term bindings is handled in Hindley-Milner style, similar to ML. Type variables referring to local assumptions or open goal statements are \emph{fixed}, while those of finished results or bound by $\LETNAME$ may occur in \emph{arbitrary} instances later. Even though actual polymorphism should be rarely used in practice, this mechanism is essential to achieve proper incremental type-inference, as the user proceeds to build up the Isar proof text from left to right. \medskip Term abbreviations are quite different from local definitions as introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}). The latter are visible within the logic as actual equations, while abbreviations disappear during the input process just after type checking. Also note that $\DEFNAME$ does not support polymorphism. \begin{rail} 'let' ((term + 'and') '=' term + 'and') ; \end{rail} The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or \railnonterm{proppat} (see \S\ref{sec:term-decls}). \begin{descr} \item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$ by simultaneous higher-order matching against terms $\vec t$. \item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the preceding statement. Also note that $\ISNAME$ is not a separate command, but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.). \end{descr} Some \emph{automatic} term abbreviations\index{term abbreviations} for goals and facts are available as well. For any open goal, $\Var{thesis}$\indexisarvar{thesis} refers to its object-level statement, abstracted over any meta-level parameters (if present). Likewise, $\Var{this}$\indexisarvar{this} is bound for fact statements resulting from assumptions or finished goals. In case $\Var{this}$ refers to an object-logic statement that is an application $f(t)$, then $t$ is bound to the special text variable ``$\dots$''\indexisarvar{\dots} (three dots). The canonical application of the latter are calculational proofs (see \S\ref{sec:calculation}). \subsection{Block structure} \indexisarcmd{next}\indexisarcmd{\{}\indexisarcmd{\}} \begin{matharray}{rcl} \NEXT & : & \isartrans{proof(state)}{proof(state)} \\ \BG & : & \isartrans{proof(state)}{proof(state)} \\ \EN & : & \isartrans{proof(state)}{proof(state)} \\ \end{matharray} While Isar is inherently block-structured, opening and closing blocks is mostly handled rather casually, with little explicit user-intervention. Any local goal statement automatically opens \emph{two} blocks, which are closed again when concluding the sub-proof (by $\QEDNAME$ etc.). Sections of different context within a sub-proof may be switched via $\NEXT$, which is just a single block-close followed by block-open again. The effect of $\NEXT$ is to reset the local proof context; there is no goal focus involved here! For slightly more advanced applications, there are explicit block parentheses as well. These typically achieve a stronger forward style of reasoning. \begin{descr} \item [$\NEXT$] switches to a fresh block within a sub-proof, resetting the local context to the initial one. \item [$\BG$ and $\EN$] explicitly open and close blocks. Any current facts pass through ``$\BG$'' unchanged, while ``$\EN$'' causes any result to be \emph{exported} into the enclosing context. Thus fixed variables are generalized, assumptions discharged, and local definitions unfolded (cf.\ \S\ref{sec:proof-context}). There is no difference of $\ASSUMENAME$ and $\PRESUMENAME$ in this mode of forward reasoning --- in contrast to plain backward reasoning with the result exported at $\SHOWNAME$ time. \end{descr} \subsection{Emulating tactic scripts}\label{sec:tactic-commands} The Isar provides separate commands to accommodate tactic-style proof scripts within the same system. While being outside the orthodox Isar proof language, these might come in handy for interactive exploration and debugging, or even actual tactical proof within new-style theories (to benefit from document preparation, for example). See also \S\ref{sec:tactics} for actual tactics, that have been encapsulated as proof methods. Proper proof methods may be used in scripts, too. \indexisarcmd{apply}\indexisarcmd{apply-end}\indexisarcmd{done} \indexisarcmd{defer}\indexisarcmd{prefer}\indexisarcmd{back} \indexisarcmd{declare} \begin{matharray}{rcl} \isarcmd{apply}^* & : & \isartrans{proof(prove)}{proof(prove)} \\ \isarcmd{apply_end}^* & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{done}^* & : & \isartrans{proof(prove)}{proof(state)} \\ \isarcmd{defer}^* & : & \isartrans{proof}{proof} \\ \isarcmd{prefer}^* & : & \isartrans{proof}{proof} \\ \isarcmd{back}^* & : & \isartrans{proof}{proof} \\ \isarcmd{declare}^* & : & \isartrans{theory}{theory} \\ \end{matharray} \railalias{applyend}{apply\_end} \railterm{applyend} \begin{rail} ( 'apply' | applyend ) method ; 'defer' nat? ; 'prefer' nat ; 'declare' locale? (thmrefs + 'and') ; \end{rail} \begin{descr} \item [$\APPLY{m}$] applies proof method $m$ in initial position, but unlike $\PROOFNAME$ it retains ``$proof(prove)$'' mode. Thus consecutive method applications may be given just as in tactic scripts. Facts are passed to $m$ as indicated by the goal's forward-chain mode, and are \emph{consumed} afterwards. Thus any further $\APPLYNAME$ command would always work in a purely backward manner. \item [$\isarkeyword{apply_end}~(m)$] applies proof method $m$ as if in terminal position. Basically, this simulates a multi-step tactic script for $\QEDNAME$, but may be given anywhere within the proof body. No facts are passed to $m$. Furthermore, the static context is that of the enclosing goal (as for actual $\QEDNAME$). Thus the proof method may not refer to any assumptions introduced in the current body, for example. \item [$\isarkeyword{done}$] completes a proof script, provided that the current goal state is solved completely. Note that actual structured proof commands (e.g.\ ``$\DOT$'' or $\SORRY$) may be used to conclude proof scripts as well. \item [$\isarkeyword{defer}~n$ and $\isarkeyword{prefer}~n$] shuffle the list of pending goals: $defer$ puts off goal $n$ to the end of the list ($n = 1$ by default), while $prefer$ brings goal $n$ to the top. \item [$\isarkeyword{back}$] does back-tracking over the result sequence of the latest proof command. Basically, any proof command may return multiple results. \item [$\isarkeyword{declare}~thms$] declares theorems to the current theory context (or the specified locale, see also \S\ref{sec:locale}). No theorem binding is involved here, unlike $\isarkeyword{theorems}$ or $\isarkeyword{lemmas}$ (cf.\ \S\ref{sec:axms-thms}), so $\isarkeyword{declare}$ only has the effect of applying attributes as included in the theorem specification. \end{descr} Any proper Isar proof method may be used with tactic script commands such as $\APPLYNAME$. A few additional emulations of actual tactics are provided as well; these would be never used in actual structured proofs, of course. \subsection{Meta-linguistic features} \indexisarcmd{oops} \begin{matharray}{rcl} \isarcmd{oops} & : & \isartrans{proof}{theory} \\ \end{matharray} The $\OOPS$ command discontinues the current proof attempt, while considering the partial proof text as properly processed. This is conceptually quite different from ``faking'' actual proofs via $\SORRY$ (see \S\ref{sec:proof-steps}): $\OOPS$ does not observe the proof structure at all, but goes back right to the theory level. Furthermore, $\OOPS$ does not produce any result theorem --- there is no intended claim to be able to complete the proof anyhow. A typical application of $\OOPS$ is to explain Isar proofs \emph{within} the system itself, in conjunction with the document preparation tools of Isabelle described in \cite{isabelle-sys}. Thus partial or even wrong proof attempts can be discussed in a logically sound manner. Note that the Isabelle {\LaTeX} macros can be easily adapted to print something like ``$\dots$'' instead of an ``$\OOPS$'' keyword. \medskip The $\OOPS$ command is undo-able, unlike $\isarkeyword{kill}$ (see \S\ref{sec:history}). The effect is to get back to the theory just before the opening of the proof. \section{Other commands} \subsection{Diagnostics} \indexisarcmd{pr}\indexisarcmd{thm}\indexisarcmd{term} \indexisarcmd{prop}\indexisarcmd{typ} \begin{matharray}{rcl} \isarcmd{pr}^* & : & \isarkeep{\cdot} \\ \isarcmd{thm}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{term}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{prop}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{typ}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{prf}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{full_prf}^* & : & \isarkeep{theory~|~proof} \\ \end{matharray} These diagnostic commands assist interactive development. Note that $undo$ does not apply here, the theory or proof configuration is not changed. \begin{rail} 'pr' modes? nat? (',' nat)? ; 'thm' modes? thmrefs ; 'term' modes? term ; 'prop' modes? prop ; 'typ' modes? type ; 'prf' modes? thmrefs? ; 'full\_prf' modes? thmrefs? ; modes: '(' (name + ) ')' ; \end{rail} \begin{descr} \item [$\isarkeyword{pr}~goals, prems$] prints the current proof state (if present), including the proof context, current facts and goals. The optional limit arguments affect the number of goals and premises to be displayed, which is initially 10 for both. Omitting limit values leaves the current setting unchanged. \item [$\isarkeyword{thm}~\vec a$] retrieves theorems from the current theory or proof context. Note that any attributes included in the theorem specifications are applied to a temporary context derived from the current theory or proof; the result is discarded, i.e.\ attributes involved in $\vec a$ do not have any permanent effect. \item [$\isarkeyword{term}~t$ and $\isarkeyword{prop}~\phi$] read, type-check and print terms or propositions according to the current theory or proof context; the inferred type of $t$ is output as well. Note that these commands are also useful in inspecting the current environment of term abbreviations. \item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic according to the current theory or proof context. \item [$\isarkeyword{prf}$] displays the (compact) proof term of the current proof state (if present), or of the given theorems. Note that this requires proof terms to be switched on for the current object logic (see the ``Proof terms'' section of the Isabelle reference manual for information on how to do this). \item [$\isarkeyword{full_prf}$] is like $\isarkeyword{prf}$, but displays the full proof term, i.e.\ also displays information omitted in the compact proof term, which is denoted by ``$_$'' placeholders there. \end{descr} All of the diagnostic commands above admit a list of $modes$ to be specified, which is appended to the current print mode (see also \cite{isabelle-ref}). Thus the output behavior may be modified according particular print mode features. For example, $\isarkeyword{pr}~(latex~xsymbols~symbols)$ would print the current proof state with mathematical symbols and special characters represented in {\LaTeX} source, according to the Isabelle style \cite{isabelle-sys}. Note that antiquotations (cf.\ \S\ref{sec:antiq}) provide a more systematic way to include formal items into the printed text document. \subsection{Inspecting the context} \indexisarcmd{print-facts}\indexisarcmd{print-binds} \indexisarcmd{print-commands}\indexisarcmd{print-syntax} \indexisarcmd{print-methods}\indexisarcmd{print-attributes} \indexisarcmd{thms-containing}\indexisarcmd{thm-deps} \indexisarcmd{print-theorems} \begin{matharray}{rcl} \isarcmd{print_commands}^* & : & \isarkeep{\cdot} \\ \isarcmd{print_syntax}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{print_methods}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{print_attributes}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{print_theorems}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{thms_containing}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{thms_deps}^* & : & \isarkeep{theory~|~proof} \\ \isarcmd{print_facts}^* & : & \isarkeep{proof} \\ \isarcmd{print_binds}^* & : & \isarkeep{proof} \\ \end{matharray} \railalias{thmscontaining}{thms\_containing} \railterm{thmscontaining} \railalias{thmdeps}{thm\_deps} \railterm{thmdeps} \begin{rail} thmscontaining ('(' nat ')')? (term * ) ; thmdeps thmrefs ; \end{rail} These commands print certain parts of the theory and proof context. Note that there are some further ones available, such as for the set of rules declared for simplifications. \begin{descr} \item [$\isarkeyword{print_commands}$] prints Isabelle's outer theory syntax, including keywords and command. \item [$\isarkeyword{print_syntax}$] prints the inner syntax of types and terms, depending on the current context. The output can be very verbose, including grammar tables and syntax translation rules. See \cite[\S7, \S8]{isabelle-ref} for further information on Isabelle's inner syntax. \item [$\isarkeyword{print_methods}$] prints all proof methods available in the current theory context. \item [$\isarkeyword{print_attributes}$] prints all attributes available in the current theory context. \item [$\isarkeyword{print_theorems}$] prints theorems available in the current theory context. In interactive mode this actually refers to the theorems left by the last transaction; this allows to inspect the result of advanced definitional packages, such as $\isarkeyword{datatype}$. \item [$\isarkeyword{thms_containing}~\vec t$] retrieves facts from the theory or proof context containing all of the constants or variables occurring in terms $\vec t$ (which may contain occurrences of ``$\_$''). Note that giving the empty list yields \emph{all} currently known facts. An optional limit for the number printed facts may be given; the default is 40. \item [$\isarkeyword{thm_deps}~\vec a$] visualizes dependencies of facts, using Isabelle's graph browser tool (see also \cite{isabelle-sys}). \item [$\isarkeyword{print_facts}$] prints any named facts of the current context, including assumptions and local results. \item [$\isarkeyword{print_binds}$] prints all term abbreviations present in the context. \end{descr} \subsection{History commands}\label{sec:history} \indexisarcmd{undo}\indexisarcmd{redo}\indexisarcmd{kill} \begin{matharray}{rcl} \isarcmd{undo}^{{*}{*}} & : & \isarkeep{\cdot} \\ \isarcmd{redo}^{{*}{*}} & : & \isarkeep{\cdot} \\ \isarcmd{kill}^{{*}{*}} & : & \isarkeep{\cdot} \\ \end{matharray} The Isabelle/Isar top-level maintains a two-stage history, for theory and proof state transformation. Basically, any command can be undone using $\isarkeyword{undo}$, excluding mere diagnostic elements. Its effect may be revoked via $\isarkeyword{redo}$, unless the corresponding $\isarkeyword{undo}$ step has crossed the beginning of a proof or theory. The $\isarkeyword{kill}$ command aborts the current history node altogether, discontinuing a proof or even the whole theory. This operation is \emph{not} undo-able. \begin{warn} History commands should never be used with user interfaces such as Proof~General \cite{proofgeneral,Aspinall:TACAS:2000}, which takes care of stepping forth and back itself. Interfering by manual $\isarkeyword{undo}$, $\isarkeyword{redo}$, or even $\isarkeyword{kill}$ commands would quickly result in utter confusion. \end{warn} \subsection{System operations} \indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only} \indexisarcmd{update-thy}\indexisarcmd{update-thy-only}\indexisarcmd{display-drafts} \indexisarcmd{print-drafts} \begin{matharray}{rcl} \isarcmd{cd}^* & : & \isarkeep{\cdot} \\ \isarcmd{pwd}^* & : & \isarkeep{\cdot} \\ \isarcmd{use_thy}^* & : & \isarkeep{\cdot} \\ \isarcmd{use_thy_only}^* & : & \isarkeep{\cdot} \\ \isarcmd{update_thy}^* & : & \isarkeep{\cdot} \\ \isarcmd{update_thy_only}^* & : & \isarkeep{\cdot} \\ \isarcmd{display_drafts}^* & : & \isarkeep{\cdot} \\ \isarcmd{print_drafts}^* & : & \isarkeep{\cdot} \\ \end{matharray} \railalias{usethy}{use\_thy} \railterm{usethy} \railalias{usethyonly}{use\_thy\_only} \railterm{usethyonly} \railalias{updatethy}{update\_thy} \railterm{updatethy} \railalias{updatethyonly}{update\_thy\_only} \railterm{updatethyonly} \railalias{displaydrafts}{display\_drafts} \railterm{displaydrafts} \railalias{printdrafts}{print\_drafts} \railterm{printdrafts} \begin{rail} ('cd' | usethy | usethyonly | updatethy | updatethyonly) name ; (displaydrafts | printdrafts) (name +) ; \end{rail} \begin{descr} \item [$\isarkeyword{cd}~path$] changes the current directory of the Isabelle process. \item [$\isarkeyword{pwd}~$] prints the current working directory. \item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$, $\isarkeyword{update_thy}$, $\isarkeyword{update_thy_only}$] load some theory given as $name$ argument. These commands are basically the same as the corresponding ML functions\footnote{The ML versions also change the implicit theory context to that of the theory loaded.} (see also \cite[\S1,\S6]{isabelle-ref}). Note that both the ML and Isar versions may load new- and old-style theories alike. \item [$\isarkeyword{display_drafts}~paths$ and $\isarkeyword{print_drafts}~paths$] perform simple output of a given list of raw source files. Only those symbols that do not require additional {\LaTeX} packages are displayed properly, everything else is left verbatim. \end{descr} These system commands are scarcely used when working with the Proof~General interface, since loading of theories is done transparently. %%% Local Variables: %%% mode: latex %%% TeX-master: "isar-ref" %%% End: