\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

   ;

   '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 a subclass relation

   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' (constdecl +)

   ;

   'defs' ('(overloaded)')? (axmdecl prop +)

   ;

   'constdefs' (constdecl prop +)

   ;

   constdecl: name '::' type mixfix?

   ;

 \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 [$\CONSTDEFS~c::\sigma~eqn$] combines declarations and definitions of

   constants, using the canonical name $c_def$ for the definitional axiom.

 \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 | 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.

 \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}

 See \cite[\S8]{isabelle-ref} for more information on syntax transformations.

 \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}

 \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 \\

 \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)?

   ;

 \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.  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.

 \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} \\

 \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

   ;

   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.

 \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 (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 theorems from the

   theory context containing all of the constants occurring in the terms $\vec

   t$.  Note that giving the empty list yields \emph{all} theorems of the

   current theory.

 \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}

 \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} \\

 \end{matharray}

 \begin{descr}

 \item [$\isarkeyword{cd}~name$] 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.

 \end{descr}

 These system commands are scarcely used when working with the Proof~General

 interface, since loading of theories is done transparently.

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