doc-src/IsarRef/pure.tex
 author wenzelm Tue Sep 12 17:34:50 2000 +0200 (2000-09-12) changeset 9936 f080397656d8 parent 9727 5e18de753e0f child 10160 bb8f9412fec6 permissions -rw-r--r--
renamed "delrule" to "rule del";
     1

     2 \chapter{Basic Isar Language Elements}\label{ch:pure-syntax}

     3

     4 Subsequently, we introduce the main part of Pure Isar theory and proof

     5 commands, together with fundamental proof methods and attributes.

     6 Chapter~\ref{ch:gen-tools} describes further Isar elements provided by generic

     7 tools and packages (such as the Simplifier) that are either part of Pure

     8 Isabelle or pre-installed by most object logics.  Chapter~\ref{ch:hol-tools}

     9 refers to actual object-logic specific elements of Isabelle/HOL.

    10

    11 \medskip

    12

    13 Isar commands may be either \emph{proper} document constructors, or

    14 \emph{improper commands}.  Some proof methods and attributes introduced later

    15 are classified as improper as well.  Improper Isar language elements, which

    16 are subsequently marked by $^*$, are often helpful when developing proof

    17 documents, while their use is discouraged for the final outcome.  Typical

    18 examples are diagnostic commands that print terms or theorems according to the

    19 current context; other commands even emulate old-style tactical theorem

    20 proving.

    21

    22

    23 \section{Theory commands}

    24

    25 \subsection{Defining theories}\label{sec:begin-thy}

    26

    27 \indexisarcmd{header}\indexisarcmd{theory}\indexisarcmd{end}\indexisarcmd{context}

    28 \begin{matharray}{rcl}

    29   \isarcmd{header} & : & \isarkeep{toplevel} \\

    30   \isarcmd{theory} & : & \isartrans{toplevel}{theory} \\

    31   \isarcmd{context}^* & : & \isartrans{toplevel}{theory} \\

    32   \isarcmd{end} & : & \isartrans{theory}{toplevel} \\

    33 \end{matharray}

    34

    35 Isabelle/Isar new-style'' theories are either defined via theory files or

    36 interactively.  Both theory-level specifications and proofs are handled

    37 uniformly --- occasionally definitional mechanisms even require some explicit

    38 proof as well.  In contrast, old-style'' Isabelle theories support batch

    39 processing only, with the proof scripts collected in separate ML files.

    40

    41 The first actual command of any theory has to be $\THEORY$, starting a new

    42 theory based on the merge of existing ones.  Just preceding $\THEORY$, there

    43 may be an optional $\isarkeyword{header}$ declaration, which is relevant to

    44 document preparation only; it acts very much like a special pre-theory markup

    45 command (cf.\ \S\ref{sec:markup-thy} and \S\ref{sec:markup-thy}).  The theory

    46 context may be also changed by $\CONTEXT$ without creating a new theory.  In

    47 both cases, $\END$ concludes the theory development; it has to be the very

    48 last command of any theory file.

    49

    50 \begin{rail}

    51   'header' text

    52   ;

    53   'theory' name '=' (name + '+') filespecs? ':'

    54   ;

    55   'context' name

    56   ;

    57   'end'

    58   ;;

    59

    60   filespecs: 'files' ((name | parname) +);

    61 \end{rail}

    62

    63 \begin{descr}

    64 \item [$\isarkeyword{header}~text$] provides plain text markup just preceding

    65   the formal beginning of a theory.  In actual document preparation the

    66   corresponding {\LaTeX} macro \verb,\isamarkupheader, may be redefined to

    67   produce chapter or section headings.  See also \S\ref{sec:markup-thy} and

    68   \S\ref{sec:markup-prf} for further markup commands.

    69

    70 \item [$\THEORY~A = B@1 + \cdots + B@n\colon$] commences a new theory $A$

    71   based on existing ones $B@1 + \cdots + B@n$.  Isabelle's theory loader

    72   system ensures that any of the base theories are properly loaded (and fully

    73   up-to-date when $\THEORY$ is executed interactively).  The optional

    74   $\isarkeyword{files}$ specification declares additional dependencies on ML

    75   files.  Unless put in parentheses, any file will be loaded immediately via

    76   $\isarcmd{use}$ (see also \S\ref{sec:ML}).  The optional ML file

    77   \texttt{$A$.ML} that may be associated with any theory should \emph{not} be

    78   included in $\isarkeyword{files}$, though.

    79

    80 \item [$\CONTEXT~B$] enters an existing theory context, basically in read-only

    81   mode, so only a limited set of commands may be performed without destroying

    82   the theory.  Just as for $\THEORY$, the theory loader ensures that $B$ is

    83   loaded and up-to-date.

    84

    85 \item [$\END$] concludes the current theory definition or context switch.

    86 Note that this command cannot be undone, but the whole theory definition has

    87 to be retracted.

    88 \end{descr}

    89

    90

    91 \subsection{Theory markup commands}\label{sec:markup-thy}

    92

    93 \indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection}

    94 \indexisarcmd{subsubsection}\indexisarcmd{text}\indexisarcmd{text-raw}

    95 \begin{matharray}{rcl}

    96   \isarcmd{chapter} & : & \isartrans{theory}{theory} \\

    97   \isarcmd{section} & : & \isartrans{theory}{theory} \\

    98   \isarcmd{subsection} & : & \isartrans{theory}{theory} \\

    99   \isarcmd{subsubsection} & : & \isartrans{theory}{theory} \\

   100   \isarcmd{text} & : & \isartrans{theory}{theory} \\

   101   \isarcmd{text_raw} & : & \isartrans{theory}{theory} \\

   102 \end{matharray}

   103

   104 Apart from formal comments (see \S\ref{sec:comments}), markup commands provide

   105 a structured way to insert text into the document generated from a theory (see

   106 \cite{isabelle-sys} for more information on Isabelle's document preparation

   107 tools).

   108

   109 \railalias{textraw}{text\_raw}

   110 \railterm{textraw}

   111

   112 \begin{rail}

   113   ('chapter' | 'section' | 'subsection' | 'subsubsection' | 'text' | textraw) text

   114   ;

   115 \end{rail}

   116

   117 \begin{descr}

   118 \item [$\isarkeyword{chapter}$, $\isarkeyword{section}$,

   119   $\isarkeyword{subsection}$, and $\isarkeyword{subsubsection}$] mark chapter

   120   and section headings.

   121 \item [$\TEXT$] specifies paragraphs of plain text, including references to

   122   formal entities.\footnote{The latter feature is not yet supported.

   123     Nevertheless, any source text of the form

   124     \texttt{\at\ttlbrace$\dots$\ttrbrace}'' should be considered as reserved

   125     for future use.}

   126 \item [$\isarkeyword{text_raw}$] inserts {\LaTeX} source into the output,

   127   without additional markup.  Thus the full range of document manipulations

   128   becomes available.  A typical application would be to emit

   129   \verb,\begin{comment}, and \verb,\end{comment}, commands to exclude certain

   130   parts from the final document.\footnote{This requires the \texttt{comment}

   131     package to be included in {\LaTeX}, of course.}

   132 \end{descr}

   133

   134 Any of these markup elements corresponds to a {\LaTeX} command with the name

   135 prefixed by \verb,\isamarkup,.  For the sectioning commands this is a plain

   136 macro with a single argument, e.g.\ \verb,\isamarkupchapter{,\dots\verb,}, for

   137 $\isarkeyword{chapter}$.  The $\isarkeyword{text}$ markup results in a

   138 {\LaTeX} environment \verb,\begin{isamarkuptext}, {\dots}

   139   \verb,\end{isamarkuptext},, while $\isarkeyword{text_raw}$ causes the text

   140 to be inserted directly into the {\LaTeX} source.

   141

   142 \medskip

   143

   144 Additional markup commands are available for proofs (see

   145 \S\ref{sec:markup-prf}).  Also note that the $\isarkeyword{header}$

   146 declaration (see \S\ref{sec:begin-thy}) admits to insert section markup just

   147 preceding the actual theory definition.

   148

   149

   150 \subsection{Type classes and sorts}\label{sec:classes}

   151

   152 \indexisarcmd{classes}\indexisarcmd{classrel}\indexisarcmd{defaultsort}

   153 \begin{matharray}{rcl}

   154   \isarcmd{classes} & : & \isartrans{theory}{theory} \\

   155   \isarcmd{classrel} & : & \isartrans{theory}{theory} \\

   156   \isarcmd{defaultsort} & : & \isartrans{theory}{theory} \\

   157 \end{matharray}

   158

   159 \begin{rail}

   160   'classes' (classdecl comment? +)

   161   ;

   162   'classrel' nameref '<' nameref comment?

   163   ;

   164   'defaultsort' sort comment?

   165   ;

   166 \end{rail}

   167

   168 \begin{descr}

   169 \item [$\isarkeyword{classes}~c<\vec c$] declares class $c$ to be a subclass

   170   of existing classes $\vec c$.  Cyclic class structures are ruled out.

   171 \item [$\isarkeyword{classrel}~c@1<c@2$] states a subclass relation between

   172   existing classes $c@1$ and $c@2$.  This is done axiomatically!  The

   173   $\isarkeyword{instance}$ command (see \S\ref{sec:axclass}) provides a way to

   174   introduce proven class relations.

   175 \item [$\isarkeyword{defaultsort}~s$] makes sort $s$ the new default sort for

   176   any type variables given without sort constraints.  Usually, the default

   177   sort would be only changed when defining new object-logics.

   178 \end{descr}

   179

   180

   181 \subsection{Primitive types and type abbreviations}\label{sec:types-pure}

   182

   183 \indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities}

   184 \begin{matharray}{rcl}

   185   \isarcmd{types} & : & \isartrans{theory}{theory} \\

   186   \isarcmd{typedecl} & : & \isartrans{theory}{theory} \\

   187   \isarcmd{nonterminals} & : & \isartrans{theory}{theory} \\

   188   \isarcmd{arities} & : & \isartrans{theory}{theory} \\

   189 \end{matharray}

   190

   191 \begin{rail}

   192   'types' (typespec '=' type infix? comment? +)

   193   ;

   194   'typedecl' typespec infix? comment?

   195   ;

   196   'nonterminals' (name +) comment?

   197   ;

   198   'arities' (nameref '::' arity comment? +)

   199   ;

   200 \end{rail}

   201

   202 \begin{descr}

   203 \item [$\TYPES~(\vec\alpha)t = \tau$] introduces \emph{type synonym}

   204   $(\vec\alpha)t$ for existing type $\tau$.  Unlike actual type definitions,

   205   as are available in Isabelle/HOL for example, type synonyms are just purely

   206   syntactic abbreviations without any logical significance.  Internally, type

   207   synonyms are fully expanded.

   208 \item [$\isarkeyword{typedecl}~(\vec\alpha)t$] declares a new type constructor

   209   $t$, intended as an actual logical type.  Note that object-logics such as

   210   Isabelle/HOL override $\isarkeyword{typedecl}$ by their own version.

   211 \item [$\isarkeyword{nonterminals}~\vec c$] declares $0$-ary type constructors

   212   $\vec c$ to act as purely syntactic types, i.e.\ nonterminal symbols of

   213   Isabelle's inner syntax of terms or types.

   214 \item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's order-sorted

   215   signature of types by new type constructor arities.  This is done

   216   axiomatically!  The $\isarkeyword{instance}$ command (see

   217   \S\ref{sec:axclass}) provides a way to introduce proven type arities.

   218 \end{descr}

   219

   220

   221 \subsection{Constants and simple definitions}\label{sec:consts}

   222

   223 \indexisarcmd{consts}\indexisarcmd{defs}\indexisarcmd{constdefs}\indexoutertoken{constdecl}

   224 \begin{matharray}{rcl}

   225   \isarcmd{consts} & : & \isartrans{theory}{theory} \\

   226   \isarcmd{defs} & : & \isartrans{theory}{theory} \\

   227   \isarcmd{constdefs} & : & \isartrans{theory}{theory} \\

   228 \end{matharray}

   229

   230 \begin{rail}

   231   'consts' (constdecl +)

   232   ;

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

   234   ;

   235   'constdefs' (constdecl prop comment? +)

   236   ;

   237

   238   constdecl: name '::' type mixfix? comment?

   239   ;

   240 \end{rail}

   241

   242 \begin{descr}

   243 \item [$\CONSTS~c::\sigma$] declares constant $c$ to have any instance of type

   244   scheme $\sigma$.  The optional mixfix annotations may attach concrete syntax

   245   to the constants declared.

   246

   247 \item [$\DEFS~name: eqn$] introduces $eqn$ as a definitional axiom for some

   248   existing constant.  See \cite[\S6]{isabelle-ref} for more details on the

   249   form of equations admitted as constant definitions.

   250

   251   The $overloaded$ option declares definitions to be potentially overloaded.

   252   Unless this option is given, a warning message would be issued for any

   253   definitional equation with a more special type than that of the

   254   corresponding constant declaration.

   255

   256 \item [$\isarkeyword{constdefs}~c::\sigma~eqn$] combines declarations and

   257   definitions of constants, using the canonical name $c_def$ for the

   258   definitional axiom.

   259 \end{descr}

   260

   261

   262 \subsection{Syntax and translations}\label{sec:syn-trans}

   263

   264 \indexisarcmd{syntax}\indexisarcmd{translations}

   265 \begin{matharray}{rcl}

   266   \isarcmd{syntax} & : & \isartrans{theory}{theory} \\

   267   \isarcmd{translations} & : & \isartrans{theory}{theory} \\

   268 \end{matharray}

   269

   270 \begin{rail}

   271   'syntax' ('(' ( name | 'output' | name 'output' ) ')')? (constdecl +)

   272   ;

   273   'translations' (transpat ('==' | '=>' | '<=') transpat comment? +)

   274   ;

   275   transpat: ('(' nameref ')')? string

   276   ;

   277 \end{rail}

   278

   279 \begin{descr}

   280 \item [$\isarkeyword{syntax}~(mode)~decls$] is similar to $\CONSTS~decls$,

   281   except that the actual logical signature extension is omitted.  Thus the

   282   context free grammar of Isabelle's inner syntax may be augmented in

   283   arbitrary ways, independently of the logic.  The $mode$ argument refers to

   284   the print mode that the grammar rules belong; unless the \texttt{output}

   285   flag is given, all productions are added both to the input and output

   286   grammar.

   287 \item [$\isarkeyword{translations}~rules$] specifies syntactic translation

   288   rules (i.e.\ \emph{macros}): parse~/ print rules (\texttt{==}), parse rules

   289   (\texttt{=>}), or print rules (\texttt{<=}).  Translation patterns may be

   290   prefixed by the syntactic category to be used for parsing; the default is

   291   \texttt{logic}.

   292 \end{descr}

   293

   294

   295 \subsection{Axioms and theorems}\label{sec:axms-thms}

   296

   297 \indexisarcmd{axioms}\indexisarcmd{theorems}\indexisarcmd{lemmas}

   298 \begin{matharray}{rcl}

   299   \isarcmd{axioms} & : & \isartrans{theory}{theory} \\

   300   \isarcmd{theorems} & : & \isartrans{theory}{theory} \\

   301   \isarcmd{lemmas} & : & \isartrans{theory}{theory} \\

   302 \end{matharray}

   303

   304 \begin{rail}

   305   'axioms' (axmdecl prop comment? +)

   306   ;

   307   ('theorems' | 'lemmas') (thmdef? thmrefs comment? + 'and')

   308   ;

   309 \end{rail}

   310

   311 \begin{descr}

   312 \item [$\isarkeyword{axioms}~a: \phi$] introduces arbitrary statements as

   313   axioms of the meta-logic.  In fact, axioms are axiomatic theorems'', and

   314   may be referred later just as any other theorem.

   315

   316   Axioms are usually only introduced when declaring new logical systems.

   317   Everyday work is typically done the hard way, with proper definitions and

   318   actual proven theorems.

   319 \item [$\isarkeyword{theorems}~a = \vec b$] stores lists of existing theorems.

   320   Typical applications would also involve attributes, to declare Simplifier

   321   rules, for example.

   322 \item [$\isarkeyword{lemmas}$] is similar to $\isarkeyword{theorems}$, but

   323   tags the results as lemma''.

   324 \end{descr}

   325

   326

   327 \subsection{Name spaces}

   328

   329 \indexisarcmd{global}\indexisarcmd{local}\indexisarcmd{hide}

   330 \begin{matharray}{rcl}

   331   \isarcmd{global} & : & \isartrans{theory}{theory} \\

   332   \isarcmd{local} & : & \isartrans{theory}{theory} \\

   333   \isarcmd{hide} & : & \isartrans{theory}{theory} \\

   334 \end{matharray}

   335

   336 \begin{rail}

   337   'global' comment?

   338   ;

   339   'local' comment?

   340   ;

   341   'hide' name (nameref + ) comment?

   342   ;

   343 \end{rail}

   344

   345 Isabelle organizes any kind of name declarations (of types, constants,

   346 theorems etc.) by separate hierarchically structured name spaces.  Normally

   347 the user does not have to control the behavior of name spaces by hand, yet the

   348 following commands provide some way to do so.

   349

   350 \begin{descr}

   351 \item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current

   352   name declaration mode.  Initially, theories start in $\isarkeyword{local}$

   353   mode, causing all names to be automatically qualified by the theory name.

   354   Changing this to $\isarkeyword{global}$ causes all names to be declared

   355   without the theory prefix, until $\isarkeyword{local}$ is declared again.

   356

   357   Note that global names are prone to get hidden accidently later, when

   358   qualified names of the same base name are introduced.

   359

   360 \item [$\isarkeyword{hide}~space~names$] removes declarations from a given

   361   name space (which may be $class$, $type$, or $const$).  Hidden objects

   362   remain valid within the logic, but are inaccessible from user input.  In

   363   output, the special qualifier $\mathord?\mathord?$'' is prefixed to the

   364   full internal name.

   365

   366   Unqualified (global) names may not be hidden deliberately.

   367 \end{descr}

   368

   369

   370 \subsection{Incorporating ML code}\label{sec:ML}

   371

   372 \indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{ML-command}

   373 \indexisarcmd{ML-setup}\indexisarcmd{setup}

   374 \indexisarcmd{method-setup}

   375 \begin{matharray}{rcl}

   376   \isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\

   377   \isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\

   378   \isarcmd{ML_command} & : & \isartrans{\cdot}{\cdot} \\

   379   \isarcmd{ML_setup} & : & \isartrans{theory}{theory} \\

   380   \isarcmd{setup} & : & \isartrans{theory}{theory} \\

   381   \isarcmd{method_setup} & : & \isartrans{theory}{theory} \\

   382 \end{matharray}

   383

   384 \railalias{MLsetup}{ML\_setup}

   385 \railterm{MLsetup}

   386

   387 \railalias{methodsetup}{method\_setup}

   388 \railterm{methodsetup}

   389

   390 \railalias{MLcommand}{ML\_command}

   391 \railterm{MLcommand}

   392

   393 \begin{rail}

   394   'use' name comment?

   395   ;

   396   ('ML' | MLcommand | MLsetup | 'setup') text comment?

   397   ;

   398   methodsetup name '=' text text comment?

   399   ;

   400 \end{rail}

   401

   402 \begin{descr}

   403 \item [$\isarkeyword{use}~file$] reads and executes ML commands from $file$.

   404   The current theory context (if present) is passed down to the ML session,

   405   but may not be modified.  Furthermore, the file name is checked with the

   406   $\isarkeyword{files}$ dependency declaration given in the theory header (see

   407   also \S\ref{sec:begin-thy}).

   408

   409 \item [$\isarkeyword{ML}~text$ and $\isarkeyword{ML_command}~text$] execute ML

   410   commands from $text$.  The theory context is passed in the same way as for

   411   $\isarkeyword{use}$, but may not be changed.  Note that

   412   $\isarkeyword{ML_command}$ is less verbose than plain $\isarkeyword{ML}$.

   413

   414 \item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$.  The

   415   theory context is passed down to the ML session, and fetched back

   416   afterwards.  Thus $text$ may actually change the theory as a side effect.

   417

   418 \item [$\isarkeyword{setup}~text$] changes the current theory context by

   419   applying $text$, which refers to an ML expression of type

   420   \texttt{(theory~->~theory)~list}.  The $\isarkeyword{setup}$ command is the

   421   canonical way to initialize any object-logic specific tools and packages

   422   written in ML.

   423

   424 \item [$\isarkeyword{method_setup}~name = text~description$] defines a proof

   425   method in the current theory.  The given $text$ has to be an ML expression

   426   of type \texttt{Args.src -> Proof.context -> Proof.method}.  Parsing

   427   concrete method syntax from \texttt{Args.src} input can be quite tedious in

   428   general.  The following simple examples are for methods without any explicit

   429   arguments, or a list of theorems, respectively.

   430

   431 {\footnotesize

   432 \begin{verbatim}

   433  Method.no_args (Method.METHOD (fn facts => foobar_tac))

   434  Method.thms_args (fn thms => Method.METHOD (fn facts => foobar_tac))

   435 \end{verbatim}

   436 }

   437

   438 Note that mere tactic emulations may ignore the \texttt{facts} parameter

   439 above.  Proper proof methods would do something appropriate'' with the list

   440 of current facts, though.  Single-rule methods usually do strict

   441 forward-chaining (e.g.\ by using \texttt{Method.multi_resolves}), while

   442 automatic ones just insert the facts using \texttt{Method.insert_tac} before

   443 applying the main tactic.

   444 \end{descr}

   445

   446

   447 \subsection{Syntax translation functions}

   448

   449 \indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation}

   450 \indexisarcmd{print-translation}\indexisarcmd{typed-print-translation}

   451 \indexisarcmd{print-ast-translation}\indexisarcmd{token-translation}

   452 \begin{matharray}{rcl}

   453   \isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\

   454   \isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\

   455   \isarcmd{print_translation} & : & \isartrans{theory}{theory} \\

   456   \isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\

   457   \isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\

   458   \isarcmd{token_translation} & : & \isartrans{theory}{theory} \\

   459 \end{matharray}

   460

   461 \railalias{parseasttranslation}{parse\_ast\_translation}

   462 \railterm{parseasttranslation}

   463

   464 \railalias{parsetranslation}{parse\_translation}

   465 \railterm{parsetranslation}

   466

   467 \railalias{printtranslation}{print\_translation}

   468 \railterm{printtranslation}

   469

   470 \railalias{typedprinttranslation}{typed\_print\_translation}

   471 \railterm{typedprinttranslation}

   472

   473 \railalias{printasttranslation}{print\_ast\_translation}

   474 \railterm{printasttranslation}

   475

   476 \railalias{tokentranslation}{token\_translation}

   477 \railterm{tokentranslation}

   478

   479 \begin{rail}

   480   ( parseasttranslation | parsetranslation | printtranslation | typedprinttranslation |

   481   printasttranslation | tokentranslation ) text comment?

   482 \end{rail}

   483

   484 Syntax translation functions written in ML admit almost arbitrary

   485 manipulations of Isabelle's inner syntax.  Any of the above commands have a

   486 single \railqtoken{text} argument that refers to an ML expression of

   487 appropriate type.

   488

   489 \begin{ttbox}

   490 val parse_ast_translation   : (string * (ast list -> ast)) list

   491 val parse_translation       : (string * (term list -> term)) list

   492 val print_translation       : (string * (term list -> term)) list

   493 val typed_print_translation :

   494   (string * (bool -> typ -> term list -> term)) list

   495 val print_ast_translation   : (string * (ast list -> ast)) list

   496 val token_translation       :

   497   (string * string * (string -> string * real)) list

   498 \end{ttbox}

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

   500

   501

   502 \subsection{Oracles}

   503

   504 \indexisarcmd{oracle}

   505 \begin{matharray}{rcl}

   506   \isarcmd{oracle} & : & \isartrans{theory}{theory} \\

   507 \end{matharray}

   508

   509 Oracles provide an interface to external reasoning systems, without giving up

   510 control completely --- each theorem carries a derivation object recording any

   511 oracle invocation.  See \cite[\S6]{isabelle-ref} for more information.

   512

   513 \begin{rail}

   514   'oracle' name '=' text comment?

   515   ;

   516 \end{rail}

   517

   518 \begin{descr}

   519 \item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML

   520   function $text$, which has to be of type

   521   \texttt{Sign.sg~*~Object.T~->~term}.

   522 \end{descr}

   523

   524

   525 \section{Proof commands}

   526

   527 Proof commands perform transitions of Isar/VM machine configurations, which

   528 are block-structured, consisting of a stack of nodes with three main

   529 components: logical proof context, current facts, and open goals.  Isar/VM

   530 transitions are \emph{typed} according to the following three different modes

   531 of operation:

   532 \begin{descr}

   533 \item [$proof(prove)$] means that a new goal has just been stated that is now

   534   to be \emph{proven}; the next command may refine it by some proof method,

   535   and enter a sub-proof to establish the actual result.

   536 \item [$proof(state)$] is like an internal theory mode: the context may be

   537   augmented by \emph{stating} additional assumptions, intermediate results

   538   etc.

   539 \item [$proof(chain)$] is intermediate between $proof(state)$ and

   540   $proof(prove)$: existing facts (i.e.\ the contents of the special $this$''

   541   register) have been just picked up in order to be used when refining the

   542   goal claimed next.

   543 \end{descr}

   544

   545

   546 \subsection{Proof markup commands}\label{sec:markup-prf}

   547

   548 \indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsubsect}

   549 \indexisarcmd{txt}\indexisarcmd{txt-raw}

   550 \begin{matharray}{rcl}

   551   \isarcmd{sect} & : & \isartrans{proof}{proof} \\

   552   \isarcmd{subsect} & : & \isartrans{proof}{proof} \\

   553   \isarcmd{subsubsect} & : & \isartrans{proof}{proof} \\

   554   \isarcmd{txt} & : & \isartrans{proof}{proof} \\

   555   \isarcmd{txt_raw} & : & \isartrans{proof}{proof} \\

   556 \end{matharray}

   557

   558 These markup commands for proof mode closely correspond to the ones of theory

   559 mode (see \S\ref{sec:markup-thy}).

   560

   561 \railalias{txtraw}{txt\_raw}

   562 \railterm{txtraw}

   563

   564 \begin{rail}

   565   ('sect' | 'subsect' | 'subsubsect' | 'txt' | txtraw) text

   566   ;

   567 \end{rail}

   568

   569

   570 \subsection{Proof context}\label{sec:proof-context}

   571

   572 \indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def}

   573 \begin{matharray}{rcl}

   574   \isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\

   575   \isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\

   576   \isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\

   577   \isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\

   578 \end{matharray}

   579

   580 The logical proof context consists of fixed variables and assumptions.  The

   581 former closely correspond to Skolem constants, or meta-level universal

   582 quantification as provided by the Isabelle/Pure logical framework.

   583 Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in

   584 a local value that may be used in the subsequent proof as any other variable

   585 or constant.  Furthermore, any result $\edrv \phi[x]$ exported from the

   586 context will be universally closed wrt.\ $x$ at the outermost level: $\edrv   587 \All x \phi$ (this is expressed using Isabelle's meta-variables).

   588

   589 Similarly, introducing some assumption $\chi$ has two effects.  On the one

   590 hand, a local theorem is created that may be used as a fact in subsequent

   591 proof steps.  On the other hand, any result $\chi \drv \phi$ exported from the

   592 context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$.

   593 Thus, solving an enclosing goal using such a result would basically introduce

   594 a new subgoal stemming from the assumption.  How this situation is handled

   595 depends on the actual version of assumption command used: while $\ASSUMENAME$

   596 insists on solving the subgoal by unification with some premise of the goal,

   597 $\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the

   598 user.

   599

   600 Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by

   601 combining $\FIX x$ with another version of assumption that causes any

   602 hypothetical equation $x \equiv t$ to be eliminated by the reflexivity rule.

   603 Thus, exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$.

   604

   605 \begin{rail}

   606   'fix' (vars + 'and') comment?

   607   ;

   608   ('assume' | 'presume') (assm comment? + 'and')

   609   ;

   610   'def' thmdecl? \\ name '==' term termpat? comment?

   611   ;

   612

   613   var: name ('::' type)?

   614   ;

   615   vars: (name+) ('::' type)?

   616   ;

   617   assm: thmdecl? (prop proppat? +)

   618   ;

   619 \end{rail}

   620

   621 \begin{descr}

   622 \item [$\FIX{\vec x}$] introduces local \emph{arbitrary, but fixed} variables

   623   $\vec x$.

   624 \item [$\ASSUME{a}{\vec\phi}$ and $\PRESUME{a}{\vec\phi}$] introduce local

   625   theorems $\vec\phi$ by assumption.  Subsequent results applied to an

   626   enclosing goal (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$

   627   expects to be able to unify with existing premises in the goal, while

   628   $\PRESUMENAME$ leaves $\vec\phi$ as new subgoals.

   629

   630   Several lists of assumptions may be given (separated by

   631   $\isarkeyword{and}$); the resulting list of current facts consists of all of

   632   these concatenated.

   633 \item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition.

   634   In results exported from the context, $x$ is replaced by $t$.  Basically,

   635   $\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\ASSUME{}{x \equiv t}$, with the

   636   resulting hypothetical equation solved by reflexivity.

   637

   638   The default name for the definitional equation is $x_def$.

   639 \end{descr}

   640

   641 The special name $prems$\indexisarthm{prems} refers to all assumptions of the

   642 current context as a list of theorems.

   643

   644

   645 \subsection{Facts and forward chaining}

   646

   647 \indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with}

   648 \begin{matharray}{rcl}

   649   \isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\

   650   \isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\

   651   \isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\

   652   \isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\

   653 \end{matharray}

   654

   655 New facts are established either by assumption or proof of local statements.

   656 Any fact will usually be involved in further proofs, either as explicit

   657 arguments of proof methods, or when forward chaining towards the next goal via

   658 $\THEN$ (and variants).  Note that the special theorem name

   659 $this$\indexisarthm{this} refers to the most recently established facts.

   660 \begin{rail}

   661   'note' (thmdef? thmrefs comment? + 'and')

   662   ;

   663   'then' comment?

   664   ;

   665   ('from' | 'with') (thmrefs comment? + 'and')

   666   ;

   667 \end{rail}

   668

   669 \begin{descr}

   670 \item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result

   671   as $a$.  Note that attributes may be involved as well, both on the left and

   672   right hand sides.

   673 \item [$\THEN$] indicates forward chaining by the current facts in order to

   674   establish the goal to be claimed next.  The initial proof method invoked to

   675   refine that will be offered the facts to do anything appropriate'' (cf.\

   676   also \S\ref{sec:proof-steps}).  For example, method $rule$ (see

   677   \S\ref{sec:pure-meth-att}) would typically do an elimination rather than an

   678   introduction.  Automatic methods usually insert the facts into the goal

   679   state before operation.  This provides a simple scheme to control relevance

   680   of facts in automated proof search.

   681 \item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is

   682   equivalent to $\FROM{this}$.

   683 \item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward

   684   chaining is from earlier facts together with the current ones.

   685 \end{descr}

   686

   687 Basic proof methods (such as $rule$, see \S\ref{sec:pure-meth-att}) expect

   688 multiple facts to be given in their proper order, corresponding to a prefix of

   689 the premises of the rule involved.  Note that positions may be easily skipped

   690 using something like $\FROM{\Text{\texttt{_}}~a~b}$, for example.  This

   691 involves the trivial rule $\PROP\psi \Imp \PROP\psi$, which happens to be

   692 bound in Isabelle/Pure as \texttt{_}''

   693 (underscore).\indexisarthm{_@\texttt{_}}

   694

   695 Forward chaining with an empty list of theorems is the same as not chaining.

   696 Thus $\FROM{nothing}$ has no effect apart from entering $prove(chain)$ mode,

   697 since $nothing$\indexisarthm{nothing} is bound to the empty list of facts.

   698

   699

   700 \subsection{Goal statements}

   701

   702 \indexisarcmd{theorem}\indexisarcmd{lemma}

   703 \indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus}

   704 \begin{matharray}{rcl}

   705   \isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\

   706   \isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\

   707   \isarcmd{have} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\

   708   \isarcmd{show} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\

   709   \isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\

   710   \isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\

   711 \end{matharray}

   712

   713 Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$

   714 and $\LEMMANAME$.  New local goals may be claimed within proof mode as well.

   715 Four variants are available, indicating whether the result is meant to solve

   716 some pending goal or whether forward chaining is indicated.

   717

   718 \begin{rail}

   719   ('theorem' | 'lemma') goal

   720   ;

   721   ('have' | 'show' | 'hence' | 'thus') goal

   722   ;

   723

   724   goal: thmdecl? prop proppat? comment?

   725   ;

   726 \end{rail}

   727

   728 \begin{descr}

   729 \item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal,

   730   eventually resulting in some theorem $\turn \phi$ to be put back into the

   731   theory.

   732 \item [$\LEMMA{a}{\phi}$] is similar to $\THEOREMNAME$, but tags the result as

   733   lemma''.

   734 \item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a

   735   theorem with the current assumption context as hypotheses.

   736 \item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some

   737   pending goal with the result \emph{exported} into the corresponding context

   738   (cf.\ \S\ref{sec:proof-context}).

   739 \item [$\HENCENAME$] abbreviates $\THEN~\HAVENAME$, i.e.\ claims a local goal

   740   to be proven by forward chaining the current facts.  Note that $\HENCENAME$

   741   is also equivalent to $\FROM{this}~\HAVENAME$.

   742 \item [$\THUSNAME$] abbreviates $\THEN~\SHOWNAME$.  Note that $\THUSNAME$ is

   743   also equivalent to $\FROM{this}~\SHOWNAME$.

   744 \end{descr}

   745

   746 Note that any goal statement causes some term abbreviations (such as

   747 $\Var{thesis}$, $\dots$) to be bound automatically, see also

   748 \S\ref{sec:term-abbrev}.  Furthermore, the local context of a (non-atomic)

   749 goal is provided via the case name $antecedent$\indexisarcase{antecedent}, see

   750 also \S\ref{sec:cases}.

   751

   752

   753 \subsection{Initial and terminal proof steps}\label{sec:proof-steps}

   754

   755 \indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by}

   756 \indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry}

   757 \begin{matharray}{rcl}

   758   \isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\

   759   \isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\

   760   \isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

   761   \isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

   762   \isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

   763   \isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

   764 \end{matharray}

   765

   766 Arbitrary goal refinement via tactics is considered harmful.  Properly, the

   767 Isar framework admits proof methods to be invoked in two places only.

   768 \begin{enumerate}

   769 \item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated

   770   goal to a number of sub-goals that are to be solved later.  Facts are passed

   771   to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode.

   772

   773 \item A \emph{terminal} conclusion step $\QED{m@2}$ is intended to solve

   774   remaining goals.  No facts are passed to $m@2$.

   775 \end{enumerate}

   776

   777 The only other proper way to affect pending goals is by $\SHOWNAME$, which

   778 involves an explicit statement of what is to be solved.

   779

   780 \medskip

   781

   782 Also note that initial proof methods should either solve the goal completely,

   783 or constitute some well-understood reduction to new sub-goals.  Arbitrary

   784 automatic proof tools that are prone leave a large number of badly structured

   785 sub-goals are no help in continuing the proof document in any intelligible

   786 way.

   787

   788 \medskip

   789

   790 Unless given explicitly by the user, the default initial method is $rule$'',

   791 which applies a single standard elimination or introduction rule according to

   792 the topmost symbol involved.  There is no separate default terminal method.

   793 Any remaining goals are always solved by assumption in the very last step.

   794

   795 \begin{rail}

   796   'proof' interest? meth? comment?

   797   ;

   798   'qed' meth? comment?

   799   ;

   800   'by' meth meth? comment?

   801   ;

   802   ('.' | '..' | 'sorry') comment?

   803   ;

   804

   805   meth: method interest?

   806   ;

   807 \end{rail}

   808

   809 \begin{descr}

   810 \item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for

   811   forward chaining are passed if so indicated by $proof(chain)$ mode.

   812 \item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and

   813   concludes the sub-proof by assumption.  If the goal had been $\SHOWNAME$ (or

   814   $\THUSNAME$), some pending sub-goal is solved as well by the rule resulting

   815   from the result \emph{exported} into the enclosing goal context.  Thus

   816   $\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting

   817   rule does not fit to any pending goal\footnote{This includes any additional

   818     strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing

   819   context.  Debugging such a situation might involve temporarily changing

   820   $\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing

   821   some occurrences of $\ASSUMENAME$ by $\PRESUMENAME$.

   822 \item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it

   823   abbreviates $\PROOF{m@1}~\QED{m@2}$, with backtracking across both methods,

   824   though.  Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done

   825   by expanding its definition; in many cases $\PROOF{m@1}$ is already

   826   sufficient to see what is going wrong.

   827 \item [$\DDOT$''] is a \emph{default proof}\index{proof!default}; it

   828   abbreviates $\BY{rule}$.

   829 \item [$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it

   830   abbreviates $\BY{this}$.

   831 \item [$\SORRY$] is a \emph{fake proof}\index{proof!fake}; provided that the

   832   \texttt{quick_and_dirty} flag is enabled, $\SORRY$ pretends to solve the

   833   goal without further ado.  Of course, the result would be a fake theorem

   834   only, involving some oracle in its internal derivation object (this is

   835   indicated as $[!]$'' in the printed result).  The main application of

   836   $\SORRY$ is to support experimentation and top-down proof development.

   837 \end{descr}

   838

   839

   840 \subsection{Fundamental methods and attributes}\label{sec:pure-meth-att}

   841

   842 The following proof methods and attributes refer to basic logical operations

   843 of Isar.  Further methods and attributes are provided by several generic and

   844 object-logic specific tools and packages (see chapters \ref{ch:gen-tools} and

   845 \ref{ch:hol-tools}).

   846

   847 \indexisarmeth{assumption}\indexisarmeth{this}\indexisarmeth{rule}\indexisarmeth{$-$}

   848 \indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest}\indexisaratt{rule}

   849 \indexisaratt{OF}\indexisaratt{of}

   850 \begin{matharray}{rcl}

   851   assumption & : & \isarmeth \\

   852   this & : & \isarmeth \\

   853   rule & : & \isarmeth \\

   854   - & : & \isarmeth \\

   855   OF & : & \isaratt \\

   856   of & : & \isaratt \\

   857   intro & : & \isaratt \\

   858   elim & : & \isaratt \\

   859   dest & : & \isaratt \\

   860   rule & : & \isaratt \\

   861 \end{matharray}

   862

   863 \begin{rail}

   864   'rule' thmrefs?

   865   ;

   866   'OF' thmrefs

   867   ;

   868   'of' insts ('concl' ':' insts)?

   869   ;

   870   'rule' 'del'

   871   ;

   872 \end{rail}

   873

   874 \begin{descr}

   875 \item [$assumption$] solves some goal by a single assumption step.  Any facts

   876   given (${} \le 1$) are guaranteed to participate in the refinement.  Recall

   877   that $\QEDNAME$ (see \S\ref{sec:proof-steps}) already concludes any

   878   remaining sub-goals by assumption.

   879 \item [$this$] applies all of the current facts directly as rules.  Recall

   880   that $\DOT$'' (dot) abbreviates $\BY{this}$.

   881 \item [$rule~\vec a$] applies some rule given as argument in backward manner;

   882   facts are used to reduce the rule before applying it to the goal.  Thus

   883   $rule$ without facts is plain \emph{introduction}, while with facts it

   884   becomes \emph{elimination}.

   885

   886   When no arguments are given, the $rule$ method tries to pick appropriate

   887   rules automatically, as declared in the current context using the $intro$,

   888   $elim$, $dest$ attributes (see below).  This is the default behavior of

   889   $\PROOFNAME$ and $\DDOT$'' (double-dot) steps (see

   890   \S\ref{sec:proof-steps}).

   891 \item [$-$''] does nothing but insert the forward chaining facts as premises

   892   into the goal.  Note that command $\PROOFNAME$ without any method actually

   893   performs a single reduction step using the $rule$ method; thus a plain

   894   \emph{do-nothing} proof step would be $\PROOF{-}$ rather than $\PROOFNAME$

   895   alone.

   896 \item [$OF~\vec a$] applies some theorem to given rules $\vec a$ (in

   897   parallel).  This corresponds to the \texttt{MRS} operator in ML

   898   \cite[\S5]{isabelle-ref}, but note the reversed order.  Positions may be

   899   skipped by including $\_$'' (underscore) as argument.

   900 \item [$of~\vec t$] performs positional instantiation.  The terms $\vec t$ are

   901   substituted for any schematic variables occurring in a theorem from left to

   902   right; \texttt{_}'' (underscore) indicates to skip a position.  Arguments

   903   following a $concl\colon$'' specification refer to positions of the

   904   conclusion of a rule.

   905 \item [$intro$, $elim$, and $dest$] declare introduction, elimination, and

   906   destruct rules, respectively.  Note that the classical reasoner (see

   907   \S\ref{sec:classical-basic}) introduces different versions of these

   908   attributes, and the $rule$ method, too.  In object-logics with classical

   909   reasoning enabled, the latter version should be used all the time to avoid

   910   confusion!

   911 \item [$rule~del$] undeclares introduction, elimination, or destruct rules.

   912 \end{descr}

   913

   914

   915 \subsection{Term abbreviations}\label{sec:term-abbrev}

   916

   917 \indexisarcmd{let}

   918 \begin{matharray}{rcl}

   919   \isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\

   920   \isarkeyword{is} & : & syntax \\

   921 \end{matharray}

   922

   923 Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements,

   924 or by annotating assumptions or goal statements with a list of patterns

   925 $\ISS{p@1\;\dots}{p@n}$.  In both cases, higher-order matching is invoked to

   926 bind extra-logical term variables, which may be either named schematic

   927 variables of the form $\Var{x}$, or nameless dummies \texttt{_}''

   928 (underscore).\indexisarvar{_@\texttt{_}} Note that in the $\LETNAME$ form the

   929 patterns occur on the left-hand side, while the $\ISNAME$ patterns are in

   930 postfix position.

   931

   932 Polymorphism of term bindings is handled in Hindley-Milner style, as in ML.

   933 Type variables referring to local assumptions or open goal statements are

   934 \emph{fixed}, while those of finished results or bound by $\LETNAME$ may occur

   935 in \emph{arbitrary} instances later.  Even though actual polymorphism should

   936 be rarely used in practice, this mechanism is essential to achieve proper

   937 incremental type-inference, as the user proceeds to build up the Isar proof

   938 text.

   939

   940 \medskip

   941

   942 Term abbreviations are quite different from actual local definitions as

   943 introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}).  The latter are

   944 visible within the logic as actual equations, while abbreviations disappear

   945 during the input process just after type checking.  Also note that $\DEFNAME$

   946 does not support polymorphism.

   947

   948 \begin{rail}

   949   'let' ((term + 'and') '=' term comment? + 'and')

   950   ;

   951 \end{rail}

   952

   953 The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or

   954 \railnonterm{proppat} (see \S\ref{sec:term-pats}).

   955

   956 \begin{descr}

   957 \item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$

   958   by simultaneous higher-order matching against terms $\vec t$.

   959 \item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the

   960   preceding statement.  Also note that $\ISNAME$ is not a separate command,

   961   but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.).

   962 \end{descr}

   963

   964 A few \emph{automatic} term abbreviations\index{term abbreviations} for goals

   965 and facts are available as well.  For any open goal,

   966 $\Var{thesis_prop}$\indexisarvar{thesis-prop} refers to the full proposition

   967 (which may be a rule), $\Var{thesis_concl}$\indexisarvar{thesis-concl} to its

   968 (atomic) conclusion, and $\Var{thesis}$\indexisarvar{thesis} to its

   969 object-level statement.  The latter two abstract over any meta-level

   970 parameters.

   971

   972 Fact statements resulting from assumptions or finished goals are bound as

   973 $\Var{this_prop}$\indexisarvar{this-prop},

   974 $\Var{this_concl}$\indexisarvar{this-concl}, and

   975 $\Var{this}$\indexisarvar{this}, similar to $\Var{thesis}$ above.  In case

   976 $\Var{this}$ refers to an object-logic statement that is an application

   977 $f(t)$, then $t$ is bound to the special text variable

   978 $\dots$''\indexisarvar{\dots} (three dots).  The canonical application of

   979 the latter are calculational proofs (see \S\ref{sec:calculation}).

   980

   981

   982 \subsection{Block structure}

   983

   984 \indexisarcmd{next}\indexisarcmd{\{}\indexisarcmd{\}}

   985 \begin{matharray}{rcl}

   986   \NEXT & : & \isartrans{proof(state)}{proof(state)} \\

   987   \BG & : & \isartrans{proof(state)}{proof(state)} \\

   988   \EN & : & \isartrans{proof(state)}{proof(state)} \\

   989 \end{matharray}

   990

   991 \railalias{lbrace}{\ttlbrace}

   992 \railterm{lbrace}

   993

   994 \railalias{rbrace}{\ttrbrace}

   995 \railterm{rbrace}

   996

   997 \begin{rail}

   998   'next' comment?

   999   ;

  1000   lbrace comment?

  1001   ;

  1002   rbrace comment?

  1003   ;

  1004 \end{rail}

  1005

  1006 While Isar is inherently block-structured, opening and closing blocks is

  1007 mostly handled rather casually, with little explicit user-intervention.  Any

  1008 local goal statement automatically opens \emph{two} blocks, which are closed

  1009 again when concluding the sub-proof (by $\QEDNAME$ etc.).  Sections of

  1010 different context within a sub-proof may be switched via $\NEXT$, which is

  1011 just a single block-close followed by block-open again.  Thus the effect of

  1012 $\NEXT$ to reset the local proof context. There is no goal focus involved

  1013 here!

  1014

  1015 For slightly more advanced applications, there are explicit block parentheses

  1016 as well.  These typically achieve a stronger forward style of reasoning.

  1017

  1018 \begin{descr}

  1019 \item [$\NEXT$] switches to a fresh block within a sub-proof, resetting the

  1020   local context to the initial one.

  1021 \item [$\BG$ and $\EN$] explicitly open and close blocks.  Any current facts

  1022   pass through $\BG$'' unchanged, while $\EN$'' causes any result to be

  1023   \emph{exported} into the enclosing context.  Thus fixed variables are

  1024   generalized, assumptions discharged, and local definitions unfolded (cf.\

  1025   \S\ref{sec:proof-context}).  There is no difference of $\ASSUMENAME$ and

  1026   $\PRESUMENAME$ in this mode of forward reasoning --- in contrast to plain

  1027   backward reasoning with the result exported at $\SHOWNAME$ time.

  1028 \end{descr}

  1029

  1030

  1031 \subsection{Emulating tactic scripts}\label{sec:tactic-commands}

  1032

  1033 The Isar provides separate commands to accommodate tactic-style proof scripts

  1034 within the same system.  While being outside the orthodox Isar proof language,

  1035 these might come in handy for interactive exploration and debugging, or even

  1036 actual tactical proof within new-style theories (to benefit from document

  1037 preparation, for example).  See also \S\ref{sec:tactics} for actual tactics,

  1038 that have been encapsulated as proof methods.  Proper proof methods may be

  1039 used in scripts, too.

  1040

  1041 \indexisarcmd{apply}\indexisarcmd{apply-end}\indexisarcmd{done}

  1042 \indexisarcmd{defer}\indexisarcmd{prefer}\indexisarcmd{back}

  1043 \indexisarcmd{declare}

  1044 \begin{matharray}{rcl}

  1045   \isarcmd{apply}^* & : & \isartrans{proof(prove)}{proof(prove)} \\

  1046   \isarcmd{apply_end}^* & : & \isartrans{proof(state)}{proof(state)} \\

  1047   \isarcmd{done}^* & : & \isartrans{proof(prove)}{proof(state)} \\

  1048   \isarcmd{defer}^* & : & \isartrans{proof}{proof} \\

  1049   \isarcmd{prefer}^* & : & \isartrans{proof}{proof} \\

  1050   \isarcmd{back}^* & : & \isartrans{proof}{proof} \\

  1051   \isarcmd{declare}^* & : & \isartrans{theory}{theory} \\

  1052 \end{matharray}

  1053

  1054 \railalias{applyend}{apply\_end}

  1055 \railterm{applyend}

  1056

  1057 \begin{rail}

  1058   ( 'apply' | applyend ) method comment?

  1059   ;

  1060   'done' comment?

  1061   ;

  1062   'defer' nat? comment?

  1063   ;

  1064   'prefer' nat comment?

  1065   ;

  1066   'back' comment?

  1067   ;

  1068   'declare' thmrefs comment?

  1069   ;

  1070 \end{rail}

  1071

  1072 \begin{descr}

  1073 \item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in initial

  1074   position, but unlike $\PROOFNAME$ it retains $proof(prove)$'' mode.  Thus

  1075   consecutive method applications may be given just as in tactic scripts.

  1076

  1077   Facts are passed to $m$ as indicated by the goal's forward-chain mode, and

  1078   are \emph{consumed} afterwards.  Thus any further $\isarkeyword{apply}$

  1079   command would always work in a purely backward manner.

  1080

  1081 \item [$\isarkeyword{apply_end}~(m)$] applies proof method $m$ as if in

  1082   terminal position.  Basically, this simulates a multi-step tactic script for

  1083   $\QEDNAME$, but may be given anywhere within the proof body.

  1084

  1085   No facts are passed to $m$.  Furthermore, the static context is that of the

  1086   enclosing goal (as for actual $\QEDNAME$).  Thus the proof method may not

  1087   refer to any assumptions introduced in the current body, for example.

  1088

  1089 \item [$\isarkeyword{done}$] completes a proof script, provided that the

  1090   current goal state is already solved completely.  Note that actual

  1091   structured proof commands (e.g.\ $\DOT$'' or $\SORRY$) may be used to

  1092   conclude proof scripts as well.

  1093

  1094 \item [$\isarkeyword{defer}~n$ and $\isarkeyword{prefer}~n$] shuffle the list

  1095   of pending goals: $defer$ puts off goal $n$ to the end of the list ($n = 1$

  1096   by default), while $prefer$ brings goal $n$ to the top.

  1097

  1098 \item [$\isarkeyword{back}$] does back-tracking over the result sequence of

  1099   the latest proof command.\footnote{Unlike the ML function \texttt{back}

  1100     \cite{isabelle-ref}, the Isar command does not search upwards for further

  1101     branch points.} Basically, any proof command may return multiple results.

  1102

  1103 \item [$\isarkeyword{declare}~thms$] declares theorems to the current theory

  1104   context.  No theorem binding is involved here, unlike

  1105   $\isarkeyword{theorems}$ or $\isarkeyword{lemmas}$ (cf.\

  1106   \S\ref{sec:axms-thms}).  So $\isarkeyword{declare}$ only has the effect of

  1107   applying attributes as included in the theorem specification.

  1108 \end{descr}

  1109

  1110 Any proper Isar proof method may be used with tactic script commands such as

  1111 $\isarkeyword{apply}$.  A few additional emulations of actual tactics are

  1112 provided as well; these would be never used in actual structured proofs, of

  1113 course.

  1114

  1115

  1116 \subsection{Meta-linguistic features}

  1117

  1118 \indexisarcmd{oops}

  1119 \begin{matharray}{rcl}

  1120   \isarcmd{oops} & : & \isartrans{proof}{theory} \\

  1121 \end{matharray}

  1122

  1123 The $\OOPS$ command discontinues the current proof attempt, while considering

  1124 the partial proof text as properly processed.  This is conceptually quite

  1125 different from faking'' actual proofs via $\SORRY$ (see

  1126 \S\ref{sec:proof-steps}): $\OOPS$ does not observe the proof structure at all,

  1127 but goes back right to the theory level.  Furthermore, $\OOPS$ does not

  1128 produce any result theorem --- there is no claim to be able to complete the

  1129 proof anyhow.

  1130

  1131 A typical application of $\OOPS$ is to explain Isar proofs \emph{within} the

  1132 system itself, in conjunction with the document preparation tools of Isabelle

  1133 described in \cite{isabelle-sys}.  Thus partial or even wrong proof attempts

  1134 can be discussed in a logically sound manner.  Note that the Isabelle {\LaTeX}

  1135 macros can be easily adapted to print something like $\dots$'' instead of an

  1136 $\OOPS$'' keyword.

  1137

  1138 \medskip The $\OOPS$ command is undoable, unlike $\isarkeyword{kill}$ (see

  1139 \S\ref{sec:history}).  The effect is to get back to the theory \emph{before}

  1140 the opening of the proof.

  1141

  1142

  1143 \section{Other commands}

  1144

  1145 \subsection{Diagnostics}

  1146

  1147 \indexisarcmd{pr}\indexisarcmd{thm}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{typ}

  1148 \begin{matharray}{rcl}

  1149   \isarcmd{pr}^* & : & \isarkeep{\cdot} \\

  1150   \isarcmd{thm}^* & : & \isarkeep{theory~|~proof} \\

  1151   \isarcmd{term}^* & : & \isarkeep{theory~|~proof} \\

  1152   \isarcmd{prop}^* & : & \isarkeep{theory~|~proof} \\

  1153   \isarcmd{typ}^* & : & \isarkeep{theory~|~proof} \\

  1154 \end{matharray}

  1155

  1156 These diagnostic commands assist interactive development.  Note that $undo$

  1157 does not apply here, the theory or proof configuration is not changed.

  1158

  1159 \begin{rail}

  1160   'pr' modes? nat? (',' nat)?

  1161   ;

  1162   'thm' modes? thmrefs

  1163   ;

  1164   'term' modes? term

  1165   ;

  1166   'prop' modes? prop

  1167   ;

  1168   'typ' modes? type

  1169   ;

  1170

  1171   modes: '(' (name + ) ')'

  1172   ;

  1173 \end{rail}

  1174

  1175 \begin{descr}

  1176 \item [$\isarkeyword{pr}~goals, prems$] prints the current proof state (if

  1177   present), including the proof context, current facts and goals.  The

  1178   optional limit arguments affect the number of goals and premises to be

  1179   displayed, which is initially 10 for both.  Omitting limit values leaves the

  1180   current setting unchanged.

  1181 \item [$\isarkeyword{thm}~\vec a$] retrieves theorems from the current theory

  1182   or proof context.  Note that any attributes included in the theorem

  1183   specifications are applied to a temporary context derived from the current

  1184   theory or proof; the result is discarded, i.e.\ attributes involved in $\vec   1185 a$ do not have any permanent effect.

  1186 \item [$\isarkeyword{term}~t$ and $\isarkeyword{prop}~\phi$] read, type-check

  1187   and print terms or propositions according to the current theory or proof

  1188   context; the inferred type of $t$ is output as well.  Note that these

  1189   commands are also useful in inspecting the current environment of term

  1190   abbreviations.

  1191 \item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic

  1192   according to the current theory or proof context.

  1193 \end{descr}

  1194

  1195 All of the diagnostic commands above admit a list of $modes$ to be specified,

  1196 which is appended to the current print mode (see also \cite{isabelle-ref}).

  1197 Thus the output behavior may be modified according particular print mode

  1198 features.  For example, $\isarkeyword{pr}~(latex~xsymbols~symbols)$ would

  1199 print the current proof state with mathematical symbols and special characters

  1200 represented in {\LaTeX} source, according to the Isabelle style

  1201 \cite{isabelle-sys}.

  1202

  1203 Note that antiquotations (cf.\ \S\ref{sec:antiq}) provide a more systematic

  1204 way to include formal items into the printed text document.

  1205

  1206

  1207 \subsection{Inspecting the context}

  1208

  1209 \indexisarcmd{print-facts}\indexisarcmd{print-binds}

  1210 \indexisarcmd{print-commands}\indexisarcmd{print-syntax}

  1211 \indexisarcmd{print-methods}\indexisarcmd{print-attributes}

  1212 \begin{matharray}{rcl}

  1213   \isarcmd{print_commands}^* & : & \isarkeep{\cdot} \\

  1214   \isarcmd{print_syntax}^* & : & \isarkeep{theory~|~proof} \\

  1215   \isarcmd{print_methods}^* & : & \isarkeep{theory~|~proof} \\

  1216   \isarcmd{print_attributes}^* & : & \isarkeep{theory~|~proof} \\

  1217   \isarcmd{print_facts}^* & : & \isarkeep{proof} \\

  1218   \isarcmd{print_binds}^* & : & \isarkeep{proof} \\

  1219 \end{matharray}

  1220

  1221 These commands print parts of the theory and proof context.  Note that there

  1222 are some further ones available, such as for the set of rules declared for

  1223 simplifications.

  1224

  1225 \begin{descr}

  1226 \item [$\isarkeyword{print_commands}$] prints Isabelle's outer theory syntax,

  1227   including keywords and command.

  1228 \item [$\isarkeyword{print_syntax}$] prints the inner syntax of types and

  1229   terms, depending on the current context.  The output can be very verbose,

  1230   including grammar tables and syntax translation rules.  See \cite[\S7,

  1231   \S8]{isabelle-ref} for further information on Isabelle's inner syntax.

  1232 \item [$\isarkeyword{print_methods}$] all proof methods available in the

  1233   current theory context.

  1234 \item [$\isarkeyword{print_attributes}$] all attributes available in the

  1235   current theory context.

  1236 \item [$\isarkeyword{print_facts}$] prints any named facts of the current

  1237   context, including assumptions and local results.

  1238 \item [$\isarkeyword{print_binds}$] prints all term abbreviations present in

  1239   the context.

  1240 \end{descr}

  1241

  1242

  1243 \subsection{History commands}\label{sec:history}

  1244

  1245 \indexisarcmd{undo}\indexisarcmd{redo}\indexisarcmd{kill}

  1246 \begin{matharray}{rcl}

  1247   \isarcmd{undo}^{{*}{*}} & : & \isarkeep{\cdot} \\

  1248   \isarcmd{redo}^{{*}{*}} & : & \isarkeep{\cdot} \\

  1249   \isarcmd{kill}^{{*}{*}} & : & \isarkeep{\cdot} \\

  1250 \end{matharray}

  1251

  1252 The Isabelle/Isar top-level maintains a two-stage history, for theory and

  1253 proof state transformation.  Basically, any command can be undone using

  1254 $\isarkeyword{undo}$, excluding mere diagnostic elements.  Its effect may be

  1255 revoked via $\isarkeyword{redo}$, unless the corresponding the

  1256 $\isarkeyword{undo}$ step has crossed the beginning of a proof or theory.  The

  1257 $\isarkeyword{kill}$ command aborts the current history node altogether,

  1258 discontinuing a proof or even the whole theory.  This operation is \emph{not}

  1259 undoable.

  1260

  1261 \begin{warn}

  1262   History commands should never be used with user interfaces such as

  1263   Proof~General \cite{proofgeneral,Aspinall:TACAS:2000}, which takes care of

  1264   stepping forth and back itself.  Interfering by manual $\isarkeyword{undo}$,

  1265   $\isarkeyword{redo}$, or even $\isarkeyword{kill}$ commands would quickly

  1266   result in utter confusion.

  1267 \end{warn}

  1268

  1269

  1270 \subsection{System operations}

  1271

  1272 \indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only}

  1273 \indexisarcmd{update-thy}\indexisarcmd{update-thy-only}

  1274 \begin{matharray}{rcl}

  1275   \isarcmd{cd}^* & : & \isarkeep{\cdot} \\

  1276   \isarcmd{pwd}^* & : & \isarkeep{\cdot} \\

  1277   \isarcmd{use_thy}^* & : & \isarkeep{\cdot} \\

  1278   \isarcmd{use_thy_only}^* & : & \isarkeep{\cdot} \\

  1279   \isarcmd{update_thy}^* & : & \isarkeep{\cdot} \\

  1280   \isarcmd{update_thy_only}^* & : & \isarkeep{\cdot} \\

  1281 \end{matharray}

  1282

  1283 \begin{descr}

  1284 \item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle

  1285   process.

  1286 \item [$\isarkeyword{pwd}~$] prints the current working directory.

  1287 \item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$,

  1288   $\isarkeyword{update_thy}$, $\isarkeyword{update_thy_only}$] load some

  1289   theory given as $name$ argument.  These commands are basically the same as

  1290   the corresponding ML functions\footnote{The ML versions also change the

  1291     implicit theory context to that of the theory loaded.}  (see also

  1292   \cite[\S1,\S6]{isabelle-ref}).  Note that both the ML and Isar versions may

  1293   load new- and old-style theories alike.

  1294 \end{descr}

  1295

  1296 These system commands are scarcely used when working with the Proof~General

  1297 interface, since loading of theories is done fully transparently.

  1298

  1299

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