doc-src/IsarRef/generic.tex
author wenzelm
Thu Apr 13 15:02:02 2000 +0200 (2000-04-13)
changeset 8704 f76f41f24c44
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Simplifier options;
     1 
     2 \chapter{Generic Tools and Packages}\label{ch:gen-tools}
     3 
     4 \section{Axiomatic Type Classes}\label{sec:axclass}
     5 
     6 \indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes}
     7 \begin{matharray}{rcl}
     8   \isarcmd{axclass} & : & \isartrans{theory}{theory} \\
     9   \isarcmd{instance} & : & \isartrans{theory}{proof(prove)} \\
    10   intro_classes & : & \isarmeth \\
    11 \end{matharray}
    12 
    13 Axiomatic type classes are provided by Isabelle/Pure as a \emph{definitional}
    14 interface to type classes (cf.~\S\ref{sec:classes}).  Thus any object logic
    15 may make use of this light-weight mechanism of abstract theories
    16 \cite{Wenzel:1997:TPHOL}.  There is also a tutorial on \emph{Using Axiomatic
    17   Type Classes in Isabelle} that is part of the standard Isabelle
    18 documentation.
    19 %FIXME cite
    20 
    21 \begin{rail}
    22   'axclass' classdecl (axmdecl prop comment? +)
    23   ;
    24   'instance' (nameref '<' nameref | nameref '::' simplearity) comment?
    25   ;
    26 \end{rail}
    27 
    28 \begin{descr}
    29 \item [$\isarkeyword{axclass}~c < \vec c~axms$] defines an axiomatic type
    30   class as the intersection of existing classes, with additional axioms
    31   holding.  Class axioms may not contain more than one type variable.  The
    32   class axioms (with implicit sort constraints added) are bound to the given
    33   names.  Furthermore a class introduction rule is generated, which is
    34   employed by method $intro_classes$ to support instantiation proofs of this
    35   class.
    36   
    37 \item [$\isarkeyword{instance}~c@1 < c@2$ and $\isarkeyword{instance}~t ::
    38   (\vec s)c$] setup a goal stating a class relation or type arity.  The proof
    39   would usually proceed by $intro_classes$, and then establish the
    40   characteristic theorems of the type classes involved.  After finishing the
    41   proof, the theory will be augmented by a type signature declaration
    42   corresponding to the resulting theorem.
    43 \item [$intro_classes$] repeatedly expands all class introduction rules of
    44   this theory.
    45 \end{descr}
    46 
    47 
    48 \section{Calculational proof}\label{sec:calculation}
    49 
    50 \indexisarcmd{also}\indexisarcmd{finally}
    51 \indexisarcmd{moreover}\indexisarcmd{ultimately}
    52 \indexisaratt{trans}
    53 \begin{matharray}{rcl}
    54   \isarcmd{also} & : & \isartrans{proof(state)}{proof(state)} \\
    55   \isarcmd{finally} & : & \isartrans{proof(state)}{proof(chain)} \\
    56   \isarcmd{moreover} & : & \isartrans{proof(state)}{proof(state)} \\
    57   \isarcmd{ultimately} & : & \isartrans{proof(state)}{proof(chain)} \\
    58   trans & : & \isaratt \\
    59 \end{matharray}
    60 
    61 Calculational proof is forward reasoning with implicit application of
    62 transitivity rules (such those of $=$, $\le$, $<$).  Isabelle/Isar maintains
    63 an auxiliary register $calculation$\indexisarthm{calculation} for accumulating
    64 results obtained by transitivity composed with the current result.  Command
    65 $\ALSO$ updates $calculation$ involving $this$, while $\FINALLY$ exhibits the
    66 final $calculation$ by forward chaining towards the next goal statement.  Both
    67 commands require valid current facts, i.e.\ may occur only after commands that
    68 produce theorems such as $\ASSUMENAME$, $\NOTENAME$, or some finished proof of
    69 $\HAVENAME$, $\SHOWNAME$ etc.  The $\MOREOVER$ and $\ULTIMATELY$ commands are
    70 similar to $\ALSO$ and $\FINALLY$, but only collect further results in
    71 $calculation$ without applying any rules yet.
    72 
    73 Also note that the automatic term abbreviation ``$\dots$'' has its canonical
    74 application with calculational proofs.  It refers to the argument\footnote{The
    75   argument of a curried infix expression is its right-hand side.} of the
    76 preceding statement.
    77 
    78 Isabelle/Isar calculations are implicitly subject to block structure in the
    79 sense that new threads of calculational reasoning are commenced for any new
    80 block (as opened by a local goal, for example).  This means that, apart from
    81 being able to nest calculations, there is no separate \emph{begin-calculation}
    82 command required.
    83 
    84 \medskip
    85 
    86 The Isar calculation proof commands may be defined as
    87 follows:\footnote{Internal bookkeeping such as proper handling of
    88   block-structure has been suppressed.}
    89 \begin{matharray}{rcl}
    90   \ALSO@0 & \equiv & \NOTE{calculation}{this} \\
    91   \ALSO@{n+1} & \equiv & \NOTE{calculation}{trans~[OF~calculation~this]} \\
    92   \FINALLY & \equiv & \ALSO~\FROM{calculation} \\
    93   \MOREOVER & \equiv & \NOTE{calculation}{calculation~this} \\
    94   \ULTIMATELY & \equiv & \MOREOVER~\FROM{calculation} \\
    95 \end{matharray}
    96 
    97 \begin{rail}
    98   ('also' | 'finally') transrules? comment?
    99   ;
   100   ('moreover' | 'ultimately') comment?
   101   ;
   102   'trans' (() | 'add' | 'del')
   103   ;
   104 
   105   transrules: '(' thmrefs ')' interest?
   106   ;
   107 \end{rail}
   108 
   109 \begin{descr}
   110 \item [$\ALSO~(\vec a)$] maintains the auxiliary $calculation$ register as
   111   follows.  The first occurrence of $\ALSO$ in some calculational thread
   112   initializes $calculation$ by $this$. Any subsequent $\ALSO$ on the same
   113   level of block-structure updates $calculation$ by some transitivity rule
   114   applied to $calculation$ and $this$ (in that order).  Transitivity rules are
   115   picked from the current context plus those given as explicit arguments (the
   116   latter have precedence).
   117   
   118 \item [$\FINALLY~(\vec a)$] maintaining $calculation$ in the same way as
   119   $\ALSO$, and concludes the current calculational thread.  The final result
   120   is exhibited as fact for forward chaining towards the next goal. Basically,
   121   $\FINALLY$ just abbreviates $\ALSO~\FROM{calculation}$.  Note that
   122   ``$\FINALLY~\SHOW{}{\Var{thesis}}~\DOT$'' and
   123   ``$\FINALLY~\HAVE{}{\phi}~\DOT$'' are typical idioms for concluding
   124   calculational proofs.
   125   
   126 \item [$\MOREOVER$ and $\ULTIMATELY$] are analogous to $\ALSO$ and $\FINALLY$,
   127   but collect results only, without applying rules.
   128   
   129 \item [$trans$] declares theorems as transitivity rules.
   130 \end{descr}
   131 
   132 
   133 \section{Named local contexts (cases)}\label{sec:cases}
   134 
   135 \indexisarcmd{case}\indexisarcmd{print-cases}
   136 \indexisaratt{case-names}\indexisaratt{params}
   137 \begin{matharray}{rcl}
   138   \isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\
   139   \isarcmd{print_cases}^* & : & \isarkeep{proof} \\
   140   case_names & : & \isaratt \\
   141   params & : & \isaratt \\
   142 \end{matharray}
   143 
   144 Basically, Isar proof contexts are built up explicitly using commands like
   145 $\FIXNAME$, $\ASSUMENAME$ etc.\ (see \S\ref{sec:proof-context}).  In typical
   146 verification tasks this can become hard to manage, though.  In particular, a
   147 large number of local contexts may emerge from case analysis or induction over
   148 inductive sets and types.
   149 
   150 \medskip
   151 
   152 The $\CASENAME$ command provides a shorthand to refer to certain parts of
   153 logical context symbolically.  Proof methods may provide an environment of
   154 named ``cases'' of the form $c\colon \vec x, \vec \phi$.  Then the effect of
   155 $\CASE{c}$ is exactly the same as $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$.
   156 
   157 It is important to note that $\CASENAME$ does \emph{not} provide any means to
   158 peek at the current goal state, which is treated as strictly non-observable in
   159 Isar!  Instead, the cases considered here usually emerge in a canonical way
   160 from certain pieces of specification that appear in the theory somewhere else
   161 (e.g.\ in an inductive definition, or recursive function).  See also
   162 \S\ref{sec:induct-method} for more details of how this works in HOL.
   163 
   164 \medskip
   165 
   166 Named cases may be exhibited in the current proof context only if both the
   167 proof method and the rules involved support this.  Case names and parameters
   168 of basic rules may be declared by hand as well, by using appropriate
   169 attributes.  Thus variant versions of rules that have been derived manually
   170 may be used in advanced case analysis later.
   171 
   172 \railalias{casenames}{case\_names}
   173 \railterm{casenames}
   174 
   175 \begin{rail}
   176   'case' nameref attributes?
   177   ;
   178   casenames (name + )
   179   ;
   180   'params' ((name * ) + 'and')
   181   ;
   182 \end{rail}
   183 %FIXME bug in rail
   184 
   185 \begin{descr}
   186 \item [$\CASE{c}$] invokes a named local context $c\colon \vec x, \vec \phi$,
   187   as provided by an appropriate proof method (such as $cases$ and $induct$ in
   188   Isabelle/HOL, see \S\ref{sec:induct-method}).  The command $\CASE{c}$
   189   abbreviates $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$.
   190 \item [$\isarkeyword{print_cases}$] prints all local contexts of the current
   191   state, using Isar proof language notation.  This is a diagnostic command;
   192   $undo$ does not apply.
   193 \item [$case_names~\vec c$] declares names for the local contexts of premises
   194   of some theorem; $\vec c$ refers to the \emph{suffix} of the list premises.
   195 \item [$params~\vec p@1 \dots \vec p@n$] renames the innermost parameters of
   196   premises $1, \dots, n$ of some theorem.  An empty list of names may be given
   197   to skip positions, leaving the present parameters unchanged.
   198 \end{descr}
   199 
   200 
   201 \section{Generalized existence}
   202 
   203 \indexisarcmd{obtain}
   204 \begin{matharray}{rcl}
   205   \isarcmd{obtain} & : & \isartrans{proof(prove)}{proof(state)} \\
   206 \end{matharray}
   207 
   208 Generalized existence reasoning means that additional elements with certain
   209 properties are introduced, together with a soundness proof of that context
   210 change (the rest of the main goal is left unchanged).
   211 
   212 Syntactically, the $\OBTAINNAME$ language element is like an initial proof
   213 method to the present goal, followed by a proof of its additional claim,
   214 followed by the actual context commands (using the syntax of $\FIXNAME$ and
   215 $\ASSUMENAME$, see \S\ref{sec:proof-context}).
   216 
   217 \begin{rail}
   218   'obtain' (vars + 'and') comment? \\ 'where' (assm comment? + 'and')
   219   ;
   220 \end{rail}
   221 
   222 $\OBTAINNAME$ is defined as a derived Isar command as follows; here the
   223 preceding goal shall be $\psi$, with (optional) facts $\vec b$ indicated for
   224 forward chaining.
   225 \begin{matharray}{l}
   226   \OBTAIN{\vec x}{a}{\vec \phi}~~\langle proof\rangle \equiv {} \\[0.5ex]
   227   \quad \PROOF{succeed} \\
   228   \qquad \DEF{}{thesis \equiv \psi} \\
   229   \qquad \PRESUME{that}{\All{\vec x} \vec\phi \Imp thesis} \\
   230   \qquad \FROM{\vec b}~\SHOW{}{thesis}~~\langle proof\rangle \\
   231   \quad \NEXT \\
   232   \qquad \FIX{\vec x}~\ASSUME{a}{\vec\phi} \\
   233 \end{matharray}
   234 
   235 Typically, the soundness proof is relatively straight-forward, often just by
   236 canonical automated tools such as $\BY{simp}$ (see \S\ref{sec:simp}) or
   237 $\BY{blast}$ (see \S\ref{sec:classical-auto}).  Note that the ``$that$''
   238 presumption above is usually declared as simplification and (unsafe)
   239 introduction rule, depending on the object-logic's policy,
   240 though.\footnote{HOL and HOLCF do this already.}
   241 
   242 The original goal statement is wrapped into a local definition in order to
   243 avoid any automated tools descending into it.  Usually, any statement would
   244 admit the intended reduction anyway; only in very rare cases $thesis_def$ has
   245 to be expanded to complete the soundness proof.
   246 
   247 \medskip
   248 
   249 In a sense, $\OBTAINNAME$ represents at the level of Isar proofs what would be
   250 meta-logical existential quantifiers and conjunctions.  This concept has a
   251 broad range of useful applications, ranging from plain elimination (or even
   252 introduction) of object-level existentials and conjunctions, to elimination
   253 over results of symbolic evaluation of recursive definitions, for example.
   254 
   255 
   256 \section{Miscellaneous methods and attributes}
   257 
   258 \indexisarmeth{unfold}\indexisarmeth{fold}
   259 \indexisarmeth{erule}\indexisarmeth{drule}\indexisarmeth{frule}
   260 \indexisarmeth{fail}\indexisarmeth{succeed}
   261 \begin{matharray}{rcl}
   262   unfold & : & \isarmeth \\
   263   fold & : & \isarmeth \\[0.5ex]
   264   erule^* & : & \isarmeth \\
   265   drule^* & : & \isarmeth \\
   266   frule^* & : & \isarmeth \\[0.5ex]
   267   succeed & : & \isarmeth \\
   268   fail & : & \isarmeth \\
   269 \end{matharray}
   270 
   271 \begin{rail}
   272   ('fold' | 'unfold' | 'erule' | 'drule' | 'frule') thmrefs
   273   ;
   274 \end{rail}
   275 
   276 \begin{descr}
   277 \item [$unfold~\vec a$ and $fold~\vec a$] expand and fold back again the given
   278   meta-level definitions throughout all goals; any facts provided are inserted
   279   into the goal and subject to rewriting as well.
   280 \item [$erule~\vec a$, $drule~\vec a$, and $frule~\vec a$] are similar to the
   281   basic $rule$ method (see \S\ref{sec:pure-meth-att}), but apply rules by
   282   elim-resolution, destruct-resolution, and forward-resolution, respectively
   283   \cite{isabelle-ref}.  These are improper method, mainly for experimentation
   284   and emulating tactic scripts.
   285   
   286   Different modes of basic rule application are usually expressed in Isar at
   287   the proof language level, rather than via implicit proof state
   288   manipulations.  For example, a proper single-step elimination would be done
   289   using the basic $rule$ method, with forward chaining of current facts.
   290 \item [$succeed$] yields a single (unchanged) result; it is the identity of
   291   the ``\texttt{,}'' method combinator (cf.\ \S\ref{sec:syn-meth}).
   292 \item [$fail$] yields an empty result sequence; it is the identity of the
   293   ``\texttt{|}'' method combinator (cf.\ \S\ref{sec:syn-meth}).
   294 \end{descr}
   295 
   296 
   297 \indexisaratt{standard}
   298 \indexisaratt{elimify}
   299 
   300 \indexisaratt{RS}\indexisaratt{COMP}
   301 \indexisaratt{where}
   302 \indexisaratt{tag}\indexisaratt{untag}
   303 \indexisaratt{transfer}
   304 \indexisaratt{export}
   305 \indexisaratt{unfold}\indexisaratt{fold}
   306 \begin{matharray}{rcl}
   307   tag & : & \isaratt \\
   308   untag & : & \isaratt \\[0.5ex]
   309   RS & : & \isaratt \\
   310   COMP & : & \isaratt \\[0.5ex]
   311   where & : & \isaratt \\[0.5ex]
   312   unfold & : & \isaratt \\
   313   fold & : & \isaratt \\[0.5ex]
   314   standard & : & \isaratt \\
   315   elimify & : & \isaratt \\
   316   export^* & : & \isaratt \\
   317   transfer & : & \isaratt \\[0.5ex]
   318 \end{matharray}
   319 
   320 \begin{rail}
   321   'tag' (nameref+)
   322   ;
   323   'untag' name
   324   ;
   325   ('RS' | 'COMP') nat? thmref
   326   ;
   327   'where' (name '=' term * 'and')
   328   ;
   329   ('unfold' | 'fold') thmrefs
   330   ;
   331 \end{rail}
   332 
   333 \begin{descr}
   334 \item [$tag~name~args$ and $untag~name$] add and remove $tags$ of some
   335   theorem.  Tags may be any list of strings that serve as comment for some
   336   tools (e.g.\ $\LEMMANAME$ causes the tag ``$lemma$'' to be added to the
   337   result).  The first string is considered the tag name, the rest its
   338   arguments.  Note that untag removes any tags of the same name.
   339 \item [$RS~n~a$ and $COMP~n~a$] compose rules.  $RS$ resolves with the $n$-th
   340   premise of $a$; $COMP$ is a version of $RS$ that skips the automatic lifting
   341   process that is normally intended (cf.\ \texttt{RS} and \texttt{COMP} in
   342   \cite[\S5]{isabelle-ref}).
   343 \item [$where~\vec x = \vec t$] perform named instantiation of schematic
   344   variables occurring in a theorem.  Unlike instantiation tactics (such as
   345   \texttt{res_inst_tac}, see \cite{isabelle-ref}), actual schematic variables
   346   have to be specified (e.g.\ $\Var{x@3}$).
   347   
   348 \item [$unfold~\vec a$ and $fold~\vec a$] expand and fold back again the given
   349   meta-level definitions throughout a rule.
   350  
   351 \item [$standard$] puts a theorem into the standard form of object-rules, just
   352   as the ML function \texttt{standard} (see \cite[\S5]{isabelle-ref}).
   353   
   354 \item [$elimify$] turns an destruction rule into an elimination, just as the
   355   ML function \texttt{make\_elim} (see \cite{isabelle-ref}).
   356   
   357 \item [$export$] lifts a local result out of the current proof context,
   358   generalizing all fixed variables and discharging all assumptions.  Note that
   359   proper incremental export is already done as part of the basic Isar
   360   machinery.  This attribute is mainly for experimentation.
   361   
   362 \item [$transfer$] promotes a theorem to the current theory context, which has
   363   to enclose the former one.  This is done automatically whenever rules are
   364   joined by inference.
   365 
   366 \end{descr}
   367 
   368 
   369 \section{The Simplifier}
   370 
   371 \subsection{Simplification methods}\label{sec:simp}
   372 
   373 \indexisarmeth{simp}\indexisarmeth{simp-all}
   374 \begin{matharray}{rcl}
   375   simp & : & \isarmeth \\
   376   simp_all & : & \isarmeth \\
   377 \end{matharray}
   378 
   379 \railalias{simpall}{simp\_all}
   380 \railterm{simpall}
   381 
   382 \railalias{noasm}{no\_asm}
   383 \railterm{noasm}
   384 
   385 \railalias{noasmsimp}{no\_asm\_simp}
   386 \railterm{noasmsimp}
   387 
   388 \railalias{noasmuse}{no\_asm\_use}
   389 \railterm{noasmuse}
   390 
   391 \begin{rail}
   392   ('simp' | simpall) ('!' ?) simpopt? (simpmod * )
   393   ;
   394 
   395   simpopt: (noasm | noasmsimp | noasmuse) ':'
   396   ;
   397   simpmod: ('add' | 'del' | 'only' | 'split' (() | 'add' | 'del') | 'other') ':' thmrefs
   398   ;
   399 \end{rail}
   400 
   401 \begin{descr}
   402 \item [$simp$] invokes Isabelle's simplifier, after declaring additional rules
   403   according to the arguments given.  Note that the \railtterm{only} modifier
   404   first removes all other rewrite rules, congruences, and looper tactics
   405   (including splits), and then behaves like \railtterm{add}.
   406   
   407   The \railtterm{split} modifiers add or delete rules for the Splitter (see
   408   also \cite{isabelle-ref}), the default is to add.  This works only if the
   409   Simplifier method has been properly setup to include the Splitter (all major
   410   object logics such HOL, HOLCF, FOL, ZF do this already).
   411   
   412   The \railtterm{other} modifier ignores its arguments.  Nevertheless,
   413   additional kinds of rules may be declared by including appropriate
   414   attributes in the specification.
   415 \item [$simp_all$] is similar to $simp$, but acts on all goals.
   416 \end{descr}
   417 
   418 By default, the Simplifier methods are based on \texttt{asm_full_simp_tac}
   419 internally \cite[\S10]{isabelle-ref}.  In structured proofs this is usually
   420 quite well behaved in practice: just the local premises of the actual goal are
   421 involved, additional facts may inserted via explicit forward-chaining (using
   422 $\THEN$, $\FROMNAME$ etc.).  The full context of assumptions is only included
   423 if the ``$!$'' (bang) argument is given, which should be used with some care,
   424 though.
   425 
   426 Additional Simplifier options may be specified to tune the behavior even
   427 further: $no_asm$ means assumptions are ignored completely (cf.\ 
   428 \texttt{simp_tac}), $no_asm_simp$ means assumptions are used in the
   429 simplification of the conclusion but are not themselves simplified (cf.\ 
   430 \texttt{asm_simp_tac}), and $no_asm_use$ means assumptions are simplified but
   431 are not used in the simplification of each other or the conclusion (cf.
   432 \texttt{full_simp_tac}).
   433 
   434 \medskip
   435 
   436 The Splitter package is usually configured to work as part of the Simplifier.
   437 There is no separate $split$ method available.  The effect of repeatedly
   438 applying \texttt{split_tac} can be simulated by
   439 $(simp~only\colon~split\colon~\vec a)$.
   440 
   441 
   442 \subsection{Declaring rules}
   443 
   444 \indexisarcmd{print-simpset}
   445 \indexisaratt{simp}\indexisaratt{split}\indexisaratt{cong}
   446 \begin{matharray}{rcl}
   447   print_simpset & : & \isarkeep{theory~|~proof} \\
   448   simp & : & \isaratt \\
   449   split & : & \isaratt \\
   450   cong & : & \isaratt \\
   451 \end{matharray}
   452 
   453 \begin{rail}
   454   ('simp' | 'split' | 'cong') (() | 'add' | 'del')
   455   ;
   456 \end{rail}
   457 
   458 \begin{descr}
   459 \item [$print_simpset$] prints the collection of rules declared to the
   460   Simplifier, which is also known as ``simpset'' internally
   461   \cite{isabelle-ref}.  This is a diagnostic command; $undo$ does not apply.
   462 \item [$simp$] declares simplification rules.
   463 \item [$split$] declares split rules.
   464 \item [$cong$] declares congruence rules.
   465 \end{descr}
   466 
   467 
   468 \subsection{Forward simplification}
   469 
   470 \indexisaratt{simplify}\indexisaratt{asm-simplify}
   471 \indexisaratt{full-simplify}\indexisaratt{asm-full-simplify}
   472 \begin{matharray}{rcl}
   473   simplify & : & \isaratt \\
   474   asm_simplify & : & \isaratt \\
   475   full_simplify & : & \isaratt \\
   476   asm_full_simplify & : & \isaratt \\
   477 \end{matharray}
   478 
   479 These attributes provide forward rules for simplification, which should be
   480 used only very rarely.  There are no separate options for declaring
   481 simplification rules locally.
   482 
   483 See the ML functions of the same name in \cite[\S10]{isabelle-ref} for more
   484 information.
   485 
   486 
   487 \section{The Classical Reasoner}
   488 
   489 \subsection{Basic methods}\label{sec:classical-basic}
   490 
   491 \indexisarmeth{rule}\indexisarmeth{intro}
   492 \indexisarmeth{elim}\indexisarmeth{default}\indexisarmeth{contradiction}
   493 \begin{matharray}{rcl}
   494   rule & : & \isarmeth \\
   495   intro & : & \isarmeth \\
   496   elim & : & \isarmeth \\
   497   contradiction & : & \isarmeth \\
   498 \end{matharray}
   499 
   500 \begin{rail}
   501   ('rule' | 'intro' | 'elim') thmrefs?
   502   ;
   503 \end{rail}
   504 
   505 \begin{descr}
   506 \item [$rule$] as offered by the classical reasoner is a refinement over the
   507   primitive one (see \S\ref{sec:pure-meth-att}).  In case that no rules are
   508   provided as arguments, it automatically determines elimination and
   509   introduction rules from the context (see also \S\ref{sec:classical-mod}).
   510   This is made the default method for basic proof steps, such as $\PROOFNAME$
   511   and ``$\DDOT$'' (two dots), see also \S\ref{sec:proof-steps} and
   512   \S\ref{sec:pure-meth-att}.
   513   
   514 \item [$intro$ and $elim$] repeatedly refine some goal by intro- or
   515   elim-resolution, after having inserted any facts.  Omitting the arguments
   516   refers to any suitable rules declared in the context, otherwise only the
   517   explicitly given ones may be applied.  The latter form admits better control
   518   of what actually happens, thus it is very appropriate as an initial method
   519   for $\PROOFNAME$ that splits up certain connectives of the goal, before
   520   entering the actual sub-proof.
   521   
   522 \item [$contradiction$] solves some goal by contradiction, deriving any result
   523   from both $\neg A$ and $A$.  Facts, which are guaranteed to participate, may
   524   appear in either order.
   525 \end{descr}
   526 
   527 
   528 \subsection{Automated methods}\label{sec:classical-auto}
   529 
   530 \indexisarmeth{blast}
   531 \indexisarmeth{fast}\indexisarmeth{best}\indexisarmeth{slow}\indexisarmeth{slow-best}
   532 \begin{matharray}{rcl}
   533  blast & : & \isarmeth \\
   534  fast & : & \isarmeth \\
   535  best & : & \isarmeth \\
   536  slow & : & \isarmeth \\
   537  slow_best & : & \isarmeth \\
   538 \end{matharray}
   539 
   540 \railalias{slowbest}{slow\_best}
   541 \railterm{slowbest}
   542 
   543 \begin{rail}
   544   'blast' ('!' ?) nat? (clamod * )
   545   ;
   546   ('fast' | 'best' | 'slow' | slowbest) ('!' ?) (clamod * )
   547   ;
   548 
   549   clamod: (('intro' | 'elim' | 'dest') (() | '?' | '??') | 'del') ':' thmrefs
   550   ;
   551 \end{rail}
   552 
   553 \begin{descr}
   554 \item [$blast$] refers to the classical tableau prover (see \texttt{blast_tac}
   555   in \cite[\S11]{isabelle-ref}).  The optional argument specifies a
   556   user-supplied search bound (default 20).
   557 \item [$fast$, $best$, $slow$, $slow_best$] refer to the generic classical
   558   reasoner (see \cite[\S11]{isabelle-ref}, tactic \texttt{fast_tac} etc).
   559 \end{descr}
   560 
   561 Any of above methods support additional modifiers of the context of classical
   562 rules.  Their semantics is analogous to the attributes given in
   563 \S\ref{sec:classical-mod}.  Facts provided by forward chaining are
   564 inserted\footnote{These methods usually cannot make proper use of actual rules
   565   inserted that way, though.} into the goal before doing the search.  The
   566 ``!''~argument causes the full context of assumptions to be included as well.
   567 This is slightly less hazardous than for the Simplifier (see
   568 \S\ref{sec:simp}).
   569 
   570 
   571 \subsection{Combined automated methods}
   572 
   573 \indexisarmeth{auto}\indexisarmeth{force}
   574 \begin{matharray}{rcl}
   575   force & : & \isarmeth \\
   576   auto & : & \isarmeth \\
   577 \end{matharray}
   578 
   579 \begin{rail}
   580   ('force' | 'auto') ('!' ?) (clasimpmod * )
   581   ;
   582 
   583   clasimpmod: ('simp' (() | 'add' | 'del' | 'only') | 'other' |
   584     ('split' (() | 'add' | 'del')) |
   585     (('intro' | 'elim' | 'dest') (() | '?' | '??') | 'del')) ':' thmrefs
   586 \end{rail}
   587 
   588 \begin{descr}
   589 \item [$force$ and $auto$] provide access to Isabelle's combined
   590   simplification and classical reasoning tactics.  See \texttt{force_tac} and
   591   \texttt{auto_tac} in \cite[\S11]{isabelle-ref} for more information.  The
   592   modifier arguments correspond to those given in \S\ref{sec:simp} and
   593   \S\ref{sec:classical-auto}.  Just note that the ones related to the
   594   Simplifier are prefixed by \railtterm{simp} here.
   595   
   596   Facts provided by forward chaining are inserted into the goal before doing
   597   the search.  The ``!''~argument causes the full context of assumptions to be
   598   included as well.
   599 \end{descr}
   600 
   601 
   602 \subsection{Declaring rules}\label{sec:classical-mod}
   603 
   604 \indexisarcmd{print-claset}
   605 \indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest}
   606 \indexisaratt{iff}\indexisaratt{delrule}
   607 \begin{matharray}{rcl}
   608   print_claset & : & \isarkeep{theory~|~proof} \\
   609   intro & : & \isaratt \\
   610   elim & : & \isaratt \\
   611   dest & : & \isaratt \\
   612   iff & : & \isaratt \\
   613   delrule & : & \isaratt \\
   614 \end{matharray}
   615 
   616 \begin{rail}
   617   ('intro' | 'elim' | 'dest') (() | '?' | '??')
   618   ;
   619   'iff' (() | 'add' | 'del')
   620 \end{rail}
   621 
   622 \begin{descr}
   623 \item [$print_claset$] prints the collection of rules declared to the
   624   Classical Reasoner, which is also known as ``simpset'' internally
   625   \cite{isabelle-ref}.  This is a diagnostic command; $undo$ does not apply.
   626 \item [$intro$, $elim$, and $dest$] declare introduction, elimination, and
   627   destruct rules, respectively.  By default, rules are considered as
   628   \emph{safe}, while a single ``?'' classifies as \emph{unsafe}, and ``??'' as
   629   \emph{extra} (i.e.\ not applied in the search-oriented automated methods,
   630   but only in single-step methods such as $rule$).
   631   
   632 \item [$iff$] declares equations both as rules for the Simplifier and
   633   Classical Reasoner.
   634 
   635 \item [$delrule$] deletes introduction or elimination rules from the context.
   636   Note that destruction rules would have to be turned into elimination rules
   637   first, e.g.\ by using the $elimify$ attribute.
   638 \end{descr}
   639 
   640 
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