2 \chapter{The Classical Reasoner}\label{chap:classical}
3 \index{classical reasoner|(}
4 \newcommand\ainfer[2]{\begin{array}{r@{\,}l}#2\\ \hline#1\end{array}}
6 Although Isabelle is generic, many users will be working in some
7 extension of classical first-order logic. Isabelle's set theory~{\tt
8 ZF} is built upon theory~\texttt{FOL}, while {\HOL}
9 conceptually contains first-order logic as a fragment.
10 Theorem-proving in predicate logic is undecidable, but many
11 researchers have developed strategies to assist in this task.
13 Isabelle's classical reasoner is an \ML{} functor that accepts certain
14 information about a logic and delivers a suite of automatic tactics. Each
15 tactic takes a collection of rules and executes a simple, non-clausal proof
16 procedure. They are slow and simplistic compared with resolution theorem
17 provers, but they can save considerable time and effort. They can prove
18 theorems such as Pelletier's~\cite{pelletier86} problems~40 and~41 in
20 \[ (\exists y. \forall x. J(y,x) \bimp \neg J(x,x))
21 \imp \neg (\forall x. \exists y. \forall z. J(z,y) \bimp \neg J(z,x)) \]
22 \[ (\forall z. \exists y. \forall x. F(x,y) \bimp F(x,z) \conj \neg F(x,x))
23 \imp \neg (\exists z. \forall x. F(x,z))
26 The tactics are generic. They are not restricted to first-order logic, and
27 have been heavily used in the development of Isabelle's set theory. Few
28 interactive proof assistants provide this much automation. The tactics can
29 be traced, and their components can be called directly; in this manner,
30 any proof can be viewed interactively.
32 The simplest way to apply the classical reasoner (to subgoal~$i$) is to type
36 This command quickly proves most simple formulas of the predicate calculus or
37 set theory. To attempt to prove \emph{all} subgoals using a combination of
38 rewriting and classical reasoning, try
42 To do all obvious logical steps, even if they do not prove the
43 subgoal, type one of the following:
45 by (Clarify_tac \(i\)); \emph{\textrm{applies to one subgoal}}
46 by Safe_tac; \emph{\textrm{applies to all subgoals}}
48 You need to know how the classical reasoner works in order to use it
49 effectively. There are many tactics to choose from, including {\tt
50 Fast_tac} and \texttt{Best_tac}.
52 We shall first discuss the underlying principles, then present the
53 classical reasoner. Finally, we shall see how to instantiate it for
54 new logics. The logics \FOL, \ZF, {\HOL} and {\HOLCF} have it already
58 \section{The sequent calculus}
59 \index{sequent calculus}
60 Isabelle supports natural deduction, which is easy to use for interactive
61 proof. But natural deduction does not easily lend itself to automation,
62 and has a bias towards intuitionism. For certain proofs in classical
63 logic, it can not be called natural. The {\bf sequent calculus}, a
64 generalization of natural deduction, is easier to automate.
66 A {\bf sequent} has the form $\Gamma\turn\Delta$, where $\Gamma$
67 and~$\Delta$ are sets of formulae.%
68 \footnote{For first-order logic, sequents can equivalently be made from
69 lists or multisets of formulae.} The sequent
70 \[ P@1,\ldots,P@m\turn Q@1,\ldots,Q@n \]
71 is {\bf valid} if $P@1\conj\ldots\conj P@m$ implies $Q@1\disj\ldots\disj
72 Q@n$. Thus $P@1,\ldots,P@m$ represent assumptions, each of which is true,
73 while $Q@1,\ldots,Q@n$ represent alternative goals. A sequent is {\bf
74 basic} if its left and right sides have a common formula, as in $P,Q\turn
75 Q,R$; basic sequents are trivially valid.
77 Sequent rules are classified as {\bf right} or {\bf left}, indicating which
78 side of the $\turn$~symbol they operate on. Rules that operate on the
79 right side are analogous to natural deduction's introduction rules, and
80 left rules are analogous to elimination rules.
81 Recall the natural deduction rules for
83 \iflabelundefined{fol-fig}{from {\it Introduction to Isabelle}}%
84 {Fig.\ts\ref{fol-fig}}.
85 The sequent calculus analogue of~$({\imp}I)$ is the rule
87 \ainfer{\Gamma &\turn \Delta, P\imp Q}{P,\Gamma &\turn \Delta,Q}
90 This breaks down some implication on the right side of a sequent; $\Gamma$
91 and $\Delta$ stand for the sets of formulae that are unaffected by the
92 inference. The analogue of the pair~$({\disj}I1)$ and~$({\disj}I2)$ is the
95 \ainfer{\Gamma &\turn \Delta, P\disj Q}{\Gamma &\turn \Delta,P,Q}
98 This breaks down some disjunction on the right side, replacing it by both
99 disjuncts. Thus, the sequent calculus is a kind of multiple-conclusion logic.
101 To illustrate the use of multiple formulae on the right, let us prove
102 the classical theorem $(P\imp Q)\disj(Q\imp P)$. Working backwards, we
103 reduce this formula to a basic sequent:
104 \[ \infer[(\disj)R]{\turn(P\imp Q)\disj(Q\imp P)}
105 {\infer[(\imp)R]{\turn(P\imp Q), (Q\imp P)\;}
106 {\infer[(\imp)R]{P \turn Q, (Q\imp P)\qquad}
107 {P, Q \turn Q, P\qquad\qquad}}}
109 This example is typical of the sequent calculus: start with the desired
110 theorem and apply rules backwards in a fairly arbitrary manner. This yields a
111 surprisingly effective proof procedure. Quantifiers add few complications,
112 since Isabelle handles parameters and schematic variables. See Chapter~10
113 of {\em ML for the Working Programmer}~\cite{paulson91} for further
117 \section{Simulating sequents by natural deduction}
118 Isabelle can represent sequents directly, as in the object-logic~\texttt{LK}\@.
119 But natural deduction is easier to work with, and most object-logics employ
120 it. Fortunately, we can simulate the sequent $P@1,\ldots,P@m\turn
121 Q@1,\ldots,Q@n$ by the Isabelle formula
122 \[ \List{P@1;\ldots;P@m; \neg Q@2;\ldots; \neg Q@n}\Imp Q@1, \]
123 where the order of the assumptions and the choice of~$Q@1$ are arbitrary.
124 Elim-resolution plays a key role in simulating sequent proofs.
126 We can easily handle reasoning on the left.
128 \iflabelundefined{destruct}{{\it Introduction to Isabelle}}{\S\ref{destruct}},
129 elim-resolution with the rules $(\disj E)$, $(\bot E)$ and $(\exists E)$
130 achieves a similar effect as the corresponding sequent rules. For the
131 other connectives, we use sequent-style elimination rules instead of
132 destruction rules such as $({\conj}E1,2)$ and $(\forall E)$. But note that
133 the rule $(\neg L)$ has no effect under our representation of sequents!
135 \ainfer{\neg P,\Gamma &\turn \Delta}{\Gamma &\turn \Delta,P}\eqno({\neg}L)
137 What about reasoning on the right? Introduction rules can only affect the
138 formula in the conclusion, namely~$Q@1$. The other right-side formulae are
139 represented as negated assumptions, $\neg Q@2$, \ldots,~$\neg Q@n$.
140 \index{assumptions!negated}
141 In order to operate on one of these, it must first be exchanged with~$Q@1$.
142 Elim-resolution with the {\bf swap} rule has this effect:
143 $$ \List{\neg P; \; \neg R\Imp P} \Imp R \eqno(swap) $$
144 To ensure that swaps occur only when necessary, each introduction rule is
145 converted into a swapped form: it is resolved with the second premise
146 of~$(swap)$. The swapped form of~$({\conj}I)$, which might be
147 called~$({\neg\conj}E)$, is
148 \[ \List{\neg(P\conj Q); \; \neg R\Imp P; \; \neg R\Imp Q} \Imp R. \]
149 Similarly, the swapped form of~$({\imp}I)$ is
150 \[ \List{\neg(P\imp Q); \; \List{\neg R;P}\Imp Q} \Imp R \]
151 Swapped introduction rules are applied using elim-resolution, which deletes
152 the negated formula. Our representation of sequents also requires the use
153 of ordinary introduction rules. If we had no regard for readability, we
154 could treat the right side more uniformly by representing sequents as
155 \[ \List{P@1;\ldots;P@m; \neg Q@1;\ldots; \neg Q@n}\Imp \bot. \]
158 \section{Extra rules for the sequent calculus}
159 As mentioned, destruction rules such as $({\conj}E1,2)$ and $(\forall E)$
160 must be replaced by sequent-style elimination rules. In addition, we need
161 rules to embody the classical equivalence between $P\imp Q$ and $\neg P\disj
162 Q$. The introduction rules~$({\disj}I1,2)$ are replaced by a rule that
163 simulates $({\disj}R)$:
164 \[ (\neg Q\Imp P) \Imp P\disj Q \]
165 The destruction rule $({\imp}E)$ is replaced by
166 \[ \List{P\imp Q;\; \neg P\Imp R;\; Q\Imp R} \Imp R. \]
167 Quantifier replication also requires special rules. In classical logic,
168 $\exists x{.}P$ is equivalent to $\neg\forall x{.}\neg P$; the rules
169 $(\exists R)$ and $(\forall L)$ are dual:
170 \[ \ainfer{\Gamma &\turn \Delta, \exists x{.}P}
171 {\Gamma &\turn \Delta, \exists x{.}P, P[t/x]} \; (\exists R)
173 \ainfer{\forall x{.}P, \Gamma &\turn \Delta}
174 {P[t/x], \forall x{.}P, \Gamma &\turn \Delta} \; (\forall L)
176 Thus both kinds of quantifier may be replicated. Theorems requiring
177 multiple uses of a universal formula are easy to invent; consider
178 \[ (\forall x.P(x)\imp P(f(x))) \conj P(a) \imp P(f^n(a)), \]
179 for any~$n>1$. Natural examples of the multiple use of an existential
180 formula are rare; a standard one is $\exists x.\forall y. P(x)\imp P(y)$.
182 Forgoing quantifier replication loses completeness, but gains decidability,
183 since the search space becomes finite. Many useful theorems can be proved
184 without replication, and the search generally delivers its verdict in a
185 reasonable time. To adopt this approach, represent the sequent rules
186 $(\exists R)$, $(\exists L)$ and $(\forall R)$ by $(\exists I)$, $(\exists
187 E)$ and $(\forall I)$, respectively, and put $(\forall E)$ into elimination
189 $$ \List{\forall x{.}P(x); P(t)\Imp Q} \Imp Q \eqno(\forall E@2) $$
190 Elim-resolution with this rule will delete the universal formula after a
191 single use. To replicate universal quantifiers, replace the rule by
193 \List{\forall x{.}P(x);\; \List{P(t); \forall x{.}P(x)}\Imp Q} \Imp Q.
196 To replicate existential quantifiers, replace $(\exists I)$ by
197 \[ \List{\neg(\exists x{.}P(x)) \Imp P(t)} \Imp \exists x{.}P(x). \]
198 All introduction rules mentioned above are also useful in swapped form.
200 Replication makes the search space infinite; we must apply the rules with
201 care. The classical reasoner distinguishes between safe and unsafe
202 rules, applying the latter only when there is no alternative. Depth-first
203 search may well go down a blind alley; best-first search is better behaved
204 in an infinite search space. However, quantifier replication is too
205 expensive to prove any but the simplest theorems.
208 \section{Classical rule sets}
209 \index{classical sets}
210 Each automatic tactic takes a {\bf classical set} --- a collection of
211 rules, classified as introduction or elimination and as {\bf safe} or {\bf
212 unsafe}. In general, safe rules can be attempted blindly, while unsafe
213 rules must be used with care. A safe rule must never reduce a provable
214 goal to an unprovable set of subgoals.
216 The rule~$({\disj}I1)$ is unsafe because it reduces $P\disj Q$ to~$P$. Any
217 rule is unsafe whose premises contain new unknowns. The elimination
218 rule~$(\forall E@2)$ is unsafe, since it is applied via elim-resolution,
219 which discards the assumption $\forall x{.}P(x)$ and replaces it by the
220 weaker assumption~$P(\Var{t})$. The rule $({\exists}I)$ is unsafe for
221 similar reasons. The rule~$(\forall E@3)$ is unsafe in a different sense:
222 since it keeps the assumption $\forall x{.}P(x)$, it is prone to looping.
223 In classical first-order logic, all rules are safe except those mentioned
226 The safe/unsafe distinction is vague, and may be regarded merely as a way
227 of giving some rules priority over others. One could argue that
228 $({\disj}E)$ is unsafe, because repeated application of it could generate
229 exponentially many subgoals. Induction rules are unsafe because inductive
230 proofs are difficult to set up automatically. Any inference is unsafe that
231 instantiates an unknown in the proof state --- thus \ttindex{match_tac}
232 must be used, rather than \ttindex{resolve_tac}. Even proof by assumption
233 is unsafe if it instantiates unknowns shared with other subgoals --- thus
234 \ttindex{eq_assume_tac} must be used, rather than \ttindex{assume_tac}.
236 \subsection{Adding rules to classical sets}
237 Classical rule sets belong to the abstract type \mltydx{claset}, which
238 supports the following operations (provided the classical reasoner is
242 print_cs : claset -> unit
243 addSIs : claset * thm list -> claset \hfill{\bf infix 4}
244 addSEs : claset * thm list -> claset \hfill{\bf infix 4}
245 addSDs : claset * thm list -> claset \hfill{\bf infix 4}
246 addIs : claset * thm list -> claset \hfill{\bf infix 4}
247 addEs : claset * thm list -> claset \hfill{\bf infix 4}
248 addDs : claset * thm list -> claset \hfill{\bf infix 4}
249 delrules : claset * thm list -> claset \hfill{\bf infix 4}
251 The add operations ignore any rule already present in the claset with the same
252 classification (such as Safe Introduction). They print a warning if the rule
253 has already been added with some other classification, but add the rule
254 anyway. Calling \texttt{delrules} deletes all occurrences of a rule from the
255 claset, but see the warning below concerning destruction rules.
256 \begin{ttdescription}
257 \item[\ttindexbold{empty_cs}] is the empty classical set.
259 \item[\ttindexbold{print_cs} $cs$] prints the rules of~$cs$.
261 \item[$cs$ addSIs $rules$] \indexbold{*addSIs}
262 adds safe introduction~$rules$ to~$cs$.
264 \item[$cs$ addSEs $rules$] \indexbold{*addSEs}
265 adds safe elimination~$rules$ to~$cs$.
267 \item[$cs$ addSDs $rules$] \indexbold{*addSDs}
268 adds safe destruction~$rules$ to~$cs$.
270 \item[$cs$ addIs $rules$] \indexbold{*addIs}
271 adds unsafe introduction~$rules$ to~$cs$.
273 \item[$cs$ addEs $rules$] \indexbold{*addEs}
274 adds unsafe elimination~$rules$ to~$cs$.
276 \item[$cs$ addDs $rules$] \indexbold{*addDs}
277 adds unsafe destruction~$rules$ to~$cs$.
279 \item[$cs$ delrules $rules$] \indexbold{*delrules}
280 deletes~$rules$ from~$cs$. It prints a warning for those rules that are not
285 If you added $rule$ using \texttt{addSDs} or \texttt{addDs}, then you must delete
288 \(cs\) delrules [make_elim \(rule\)]
291 This is necessary because the operators \texttt{addSDs} and \texttt{addDs} convert
292 the destruction rules to elimination rules by applying \ttindex{make_elim},
293 and then insert them using \texttt{addSEs} and \texttt{addEs}, respectively.
296 Introduction rules are those that can be applied using ordinary resolution.
297 The classical set automatically generates their swapped forms, which will
298 be applied using elim-resolution. Elimination rules are applied using
299 elim-resolution. In a classical set, rules are sorted by the number of new
300 subgoals they will yield; rules that generate the fewest subgoals will be
301 tried first (see \S\ref{biresolve_tac}).
304 \subsection{Modifying the search step}
305 For a given classical set, the proof strategy is simple. Perform as many safe
306 inferences as possible; or else, apply certain safe rules, allowing
307 instantiation of unknowns; or else, apply an unsafe rule. The tactics also
308 eliminate assumptions of the form $x=t$ by substitution if they have been set
309 up to do so (see \texttt{hyp_subst_tacs} in~\S\ref{sec:classical-setup} below).
310 They may perform a form of Modus Ponens: if there are assumptions $P\imp Q$
311 and~$P$, then replace $P\imp Q$ by~$Q$.
313 The classical reasoning tactics --- except \texttt{blast_tac}! --- allow
314 you to modify this basic proof strategy by applying two arbitrary {\bf
315 wrapper tacticals} to it. This affects each step of the search.
316 Usually they are the identity tacticals, but they could apply another
317 tactic before or after the step tactic. The first one, which is
318 considered to be safe, affects \ttindex{safe_step_tac} and all the
319 tactics that call it. The the second one, which may be unsafe, affects
320 \ttindex{step_tac}, \ttindex{slow_step_tac} and the tactics that call
324 addss : claset * simpset -> claset \hfill{\bf infix 4}
325 addSbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
326 addSaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
327 setSWrapper : claset * ((int -> tactic) ->
328 (int -> tactic)) -> claset \hfill{\bf infix 4}
329 compSWrapper : claset * ((int -> tactic) ->
330 (int -> tactic)) -> claset \hfill{\bf infix 4}
331 addbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
332 addaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
333 setWrapper : claset * ((int -> tactic) ->
334 (int -> tactic)) -> claset \hfill{\bf infix 4}
335 compWrapper : claset * ((int -> tactic) ->
336 (int -> tactic)) -> claset \hfill{\bf infix 4}
339 \index{simplification!from classical reasoner} The wrapper tacticals
340 underly the operator addss, which combines each search step by
341 simplification. Strictly speaking, \texttt{addss} is not part of the
342 classical reasoner. It should be defined (using \texttt{addSaltern
343 (CHANGED o (safe_asm_more_full_simp_tac ss)}) when the simplifier is
346 \begin{ttdescription}
347 \item[$cs$ addss $ss$] \indexbold{*addss}
348 adds the simpset~$ss$ to the classical set. The assumptions and goal will be
349 simplified, in a safe way, after the safe steps of the search.
351 \item[$cs$ addSbefore $tac$] \indexbold{*addSbefore}
352 changes the safe wrapper tactical to apply the given tactic {\em before}
353 each safe step of the search.
355 \item[$cs$ addSaltern $tac$] \indexbold{*addSaltern}
356 changes the safe wrapper tactical to apply the given tactic when a safe step
357 of the search would fail.
359 \item[$cs$ setSWrapper $tactical$] \indexbold{*setSWrapper}
360 specifies a new safe wrapper tactical.
362 \item[$cs$ compSWrapper $tactical$] \indexbold{*compSWrapper}
363 composes the $tactical$ with the existing safe wrapper tactical,
364 to combine their effects.
366 \item[$cs$ addbefore $tac$] \indexbold{*addbefore}
367 changes the (unsafe) wrapper tactical to apply the given tactic, which should
368 be safe, {\em before} each step of the search.
370 \item[$cs$ addaltern $tac$] \indexbold{*addaltern}
371 changes the (unsafe) wrapper tactical to apply the given tactic
372 {\em alternatively} after each step of the search.
374 \item[$cs$ setWrapper $tactical$] \indexbold{*setWrapper}
375 specifies a new (unsafe) wrapper tactical.
377 \item[$cs$ compWrapper $tactical$] \indexbold{*compWrapper}
378 composes the $tactical$ with the existing (unsafe) wrapper tactical,
379 to combine their effects.
383 \section{The classical tactics}
384 \index{classical reasoner!tactics} If installed, the classical module provides
385 powerful theorem-proving tactics. Most of them have capitalized analogues
386 that use the default claset; see \S\ref{sec:current-claset}.
388 \subsection{Semi-automatic tactics}
390 clarify_tac : claset -> int -> tactic
391 clarify_step_tac : claset -> int -> tactic
393 Use these when the automatic tactics fail. They perform all the obvious
394 logical inferences that do not split the subgoal. The result is a
395 simpler subgoal that can be tackled by other means, such as by
396 instantiating quantifiers yourself.
397 \begin{ttdescription}
398 \item[\ttindexbold{clarify_tac} $cs$ $i$] performs a series of safe steps on
399 subgoal~$i$, using \texttt{clarify_step_tac}.
401 \item[\ttindexbold{clarify_step_tac} $cs$ $i$] performs a safe step on
402 subgoal~$i$. No splitting step is applied; for example, the subgoal $A\conj
403 B$ is left as a conjunction. Proof by assumption, Modus Ponens, etc., may be
404 performed provided they do not instantiate unknowns. Assumptions of the
405 form $x=t$ may be eliminated. The user-supplied safe wrapper tactical is
410 \subsection{The tableau prover}
411 The tactic \texttt{blast_tac} searches for a proof using a fast tableau prover,
412 coded directly in \ML. It then reconstructs the proof using Isabelle
413 tactics. It is faster and more powerful than the other classical
414 reasoning tactics, but has major limitations too.
416 \item It does not use the wrapper tacticals described above, such as
418 \item It ignores types, which can cause problems in \HOL. If it applies a rule
419 whose types are inappropriate, then proof reconstruction will fail.
420 \item It does not perform higher-order unification, as needed by the rule {\tt
421 rangeI} in {\HOL} and \texttt{RepFunI} in {\ZF}. There are often
422 alternatives to such rules, for example {\tt
423 range_eqI} and \texttt{RepFun_eqI}.
424 \item The message {\small\tt Function Var's argument not a bound variable\ }
425 relates to the lack of higher-order unification. Function variables
426 may only be applied to parameters of the subgoal.
427 \item Its proof strategy is more general than \texttt{fast_tac}'s but can be
428 slower. If \texttt{blast_tac} fails or seems to be running forever, try {\tt
429 fast_tac} and the other tactics described below.
433 blast_tac : claset -> int -> tactic
434 Blast.depth_tac : claset -> int -> int -> tactic
435 Blast.trace : bool ref \hfill{\bf initially false}
437 The two tactics differ on how they bound the number of unsafe steps used in a
438 proof. While \texttt{blast_tac} starts with a bound of zero and increases it
439 successively to~20, \texttt{Blast.depth_tac} applies a user-supplied search bound.
440 \begin{ttdescription}
441 \item[\ttindexbold{blast_tac} $cs$ $i$] tries to prove
442 subgoal~$i$ using iterative deepening to increase the search bound.
444 \item[\ttindexbold{Blast.depth_tac} $cs$ $lim$ $i$] tries
445 to prove subgoal~$i$ using a search bound of $lim$. Often a slow
446 proof using \texttt{blast_tac} can be made much faster by supplying the
447 successful search bound to this tactic instead.
449 \item[set \ttindexbold{Blast.trace};] \index{tracing!of classical prover}
450 causes the tableau prover to print a trace of its search. At each step it
451 displays the formula currently being examined and reports whether the branch
452 has been closed, extended or split.
456 \subsection{An automatic tactic}
458 auto_tac : claset * simpset -> tactic
461 The auto-tactic attempts to prove all subgoals using a combination of
462 simplification and classical reasoning. It is intended for situations where
463 there are a lot of mostly trivial subgoals; it proves all the easy ones,
464 leaving the ones it cannot prove. (Unfortunately, attempting to prove the
465 hard ones may take a long time.) It must be supplied both a simpset and a
466 claset; therefore it is most easily called as \texttt{Auto_tac}, which uses
467 the default claset and simpset (see \S\ref{sec:current-claset} below). For
468 interactive use, the shorthand \texttt{auto();} abbreviates
473 \subsection{Other classical tactics}
475 fast_tac : claset -> int -> tactic
476 best_tac : claset -> int -> tactic
477 slow_tac : claset -> int -> tactic
478 slow_best_tac : claset -> int -> tactic
480 These tactics attempt to prove a subgoal using sequent-style reasoning.
481 Unlike \texttt{blast_tac}, they construct proofs directly in Isabelle. Their
482 effect is restricted (by \texttt{SELECT_GOAL}) to one subgoal; they either prove
483 this subgoal or fail. The \texttt{slow_} versions conduct a broader
485 \footnote{They may, when backtracking from a failed proof attempt, undo even
486 the step of proving a subgoal by assumption.}
488 The best-first tactics are guided by a heuristic function: typically, the
489 total size of the proof state. This function is supplied in the functor call
490 that sets up the classical reasoner.
491 \begin{ttdescription}
492 \item[\ttindexbold{fast_tac} $cs$ $i$] applies \texttt{step_tac} using
493 depth-first search, to prove subgoal~$i$.
495 \item[\ttindexbold{best_tac} $cs$ $i$] applies \texttt{step_tac} using
496 best-first search, to prove subgoal~$i$.
498 \item[\ttindexbold{slow_tac} $cs$ $i$] applies \texttt{slow_step_tac} using
499 depth-first search, to prove subgoal~$i$.
501 \item[\ttindexbold{slow_best_tac} $cs$ $i$] applies \texttt{slow_step_tac} using
502 best-first search, to prove subgoal~$i$.
506 \subsection{Depth-limited automatic tactics}
508 depth_tac : claset -> int -> int -> tactic
509 deepen_tac : claset -> int -> int -> tactic
511 These work by exhaustive search up to a specified depth. Unsafe rules are
512 modified to preserve the formula they act on, so that it be used repeatedly.
513 They can prove more goals than \texttt{fast_tac} can but are much
514 slower, for example if the assumptions have many universal quantifiers.
516 The depth limits the number of unsafe steps. If you can estimate the minimum
517 number of unsafe steps needed, supply this value as~$m$ to save time.
518 \begin{ttdescription}
519 \item[\ttindexbold{depth_tac} $cs$ $m$ $i$]
520 tries to prove subgoal~$i$ by exhaustive search up to depth~$m$.
522 \item[\ttindexbold{deepen_tac} $cs$ $m$ $i$]
523 tries to prove subgoal~$i$ by iterative deepening. It calls \texttt{depth_tac}
524 repeatedly with increasing depths, starting with~$m$.
528 \subsection{Single-step tactics}
530 safe_step_tac : claset -> int -> tactic
531 safe_tac : claset -> tactic
532 inst_step_tac : claset -> int -> tactic
533 step_tac : claset -> int -> tactic
534 slow_step_tac : claset -> int -> tactic
536 The automatic proof procedures call these tactics. By calling them
537 yourself, you can execute these procedures one step at a time.
538 \begin{ttdescription}
539 \item[\ttindexbold{safe_step_tac} $cs$ $i$] performs a safe step on
540 subgoal~$i$. The safe wrapper tactical is applied to a tactic that may
541 include proof by assumption or Modus Ponens (taking care not to instantiate
542 unknowns), or substitution.
544 \item[\ttindexbold{safe_tac} $cs$] repeatedly performs safe steps on all
545 subgoals. It is deterministic, with at most one outcome.
547 \item[\ttindexbold{inst_step_tac} $cs$ $i$] is like \texttt{safe_step_tac},
548 but allows unknowns to be instantiated.
550 \item[\ttindexbold{step_tac} $cs$ $i$] is the basic step of the proof
551 procedure. The (unsafe) wrapper tactical is applied to a tactic that tries
552 \texttt{safe_tac}, \texttt{inst_step_tac}, or applies an unsafe rule from~$cs$.
554 \item[\ttindexbold{slow_step_tac}]
555 resembles \texttt{step_tac}, but allows backtracking between using safe
556 rules with instantiation (\texttt{inst_step_tac}) and using unsafe rules.
557 The resulting search space is larger.
560 \subsection{The current claset}\label{sec:current-claset}
561 Some logics (\FOL, {\HOL} and \ZF) support the concept of a current
562 claset\index{claset!current}. This is a default set of classical rules. The
563 underlying idea is quite similar to that of a current simpset described in
564 \S\ref{sec:simp-for-dummies}; please read that section, including its
565 warnings. Just like simpsets, clasets can be associated with theories. The
568 Blast_tac : int -> tactic
569 Auto_tac : unit -> tactic
570 Fast_tac : int -> tactic
571 Best_tac : int -> tactic
572 Deepen_tac : int -> int -> tactic
573 Clarify_tac : int -> tactic
574 Clarify_step_tac : int -> tactic
576 Safe_step_tac : int -> tactic
577 Step_tac : int -> tactic
579 \indexbold{*Blast_tac}\indexbold{*Auto_tac}
580 \indexbold{*Best_tac}\indexbold{*Fast_tac}%
581 \indexbold{*Deepen_tac}
582 \indexbold{*Clarify_tac}\indexbold{*Clarify_step_tac}
583 \indexbold{*Safe_tac}\indexbold{*Safe_step_tac}
584 \indexbold{*Step_tac}
585 make use of the current claset. For example, \texttt{Blast_tac} is defined as
587 fun Blast_tac i st = blast_tac (!claset) i st;
589 and gets the current claset, \ttindex{!claset}, only after it is applied to a
590 proof state. The functions
592 AddSIs, AddSEs, AddSDs, AddIs, AddEs, AddDs: thm list -> unit
594 \indexbold{*AddSIs} \indexbold{*AddSEs} \indexbold{*AddSDs}
595 \indexbold{*AddIs} \indexbold{*AddEs} \indexbold{*AddDs}
596 are used to add rules to the current claset. They work exactly like their
597 lower case counterparts, such as \texttt{addSIs}. Calling
599 Delrules : thm list -> unit
601 deletes rules from the current claset.
603 \subsection{Other useful tactics}
604 \index{tactics!for contradiction}
605 \index{tactics!for Modus Ponens}
607 contr_tac : int -> tactic
608 mp_tac : int -> tactic
609 eq_mp_tac : int -> tactic
610 swap_res_tac : thm list -> int -> tactic
612 These can be used in the body of a specialized search.
613 \begin{ttdescription}
614 \item[\ttindexbold{contr_tac} {\it i}]\index{assumptions!contradictory}
615 solves subgoal~$i$ by detecting a contradiction among two assumptions of
616 the form $P$ and~$\neg P$, or fail. It may instantiate unknowns. The
617 tactic can produce multiple outcomes, enumerating all possible
620 \item[\ttindexbold{mp_tac} {\it i}]
621 is like \texttt{contr_tac}, but also attempts to perform Modus Ponens in
622 subgoal~$i$. If there are assumptions $P\imp Q$ and~$P$, then it replaces
623 $P\imp Q$ by~$Q$. It may instantiate unknowns. It fails if it can do
626 \item[\ttindexbold{eq_mp_tac} {\it i}]
627 is like \texttt{mp_tac} {\it i}, but may not instantiate unknowns --- thus, it
630 \item[\ttindexbold{swap_res_tac} {\it thms} {\it i}] refines subgoal~$i$ of
631 the proof state using {\it thms}, which should be a list of introduction
632 rules. First, it attempts to prove the goal using \texttt{assume_tac} or
633 \texttt{contr_tac}. It then attempts to apply each rule in turn, attempting
634 resolution and also elim-resolution with the swapped form.
637 \subsection{Creating swapped rules}
639 swapify : thm list -> thm list
640 joinrules : thm list * thm list -> (bool * thm) list
642 \begin{ttdescription}
643 \item[\ttindexbold{swapify} {\it thms}] returns a list consisting of the
644 swapped versions of~{\it thms}, regarded as introduction rules.
646 \item[\ttindexbold{joinrules} ({\it intrs}, {\it elims})]
647 joins introduction rules, their swapped versions, and elimination rules for
648 use with \ttindex{biresolve_tac}. Each rule is paired with~\texttt{false}
649 (indicating ordinary resolution) or~\texttt{true} (indicating
654 \section{Setting up the classical reasoner}\label{sec:classical-setup}
655 \index{classical reasoner!setting up}
656 Isabelle's classical object-logics, including \texttt{FOL} and \texttt{HOL}, have
657 the classical reasoner already set up. When defining a new classical logic,
658 you should set up the reasoner yourself. It consists of the \ML{} functor
659 \ttindex{ClassicalFun}, which takes the argument
663 signature CLASSICAL_DATA =
668 val sizef : thm -> int
669 val hyp_subst_tacs : (int -> tactic) list
672 Thus, the functor requires the following items:
673 \begin{ttdescription}
674 \item[\tdxbold{mp}] should be the Modus Ponens rule
675 $\List{\Var{P}\imp\Var{Q};\; \Var{P}} \Imp \Var{Q}$.
677 \item[\tdxbold{not_elim}] should be the contradiction rule
678 $\List{\neg\Var{P};\; \Var{P}} \Imp \Var{R}$.
680 \item[\tdxbold{swap}] should be the swap rule
681 $\List{\neg \Var{P}; \; \neg \Var{R}\Imp \Var{P}} \Imp \Var{R}$.
683 \item[\ttindexbold{sizef}] is the heuristic function used for best-first
684 search. It should estimate the size of the remaining subgoals. A good
685 heuristic function is \ttindex{size_of_thm}, which measures the size of the
686 proof state. Another size function might ignore certain subgoals (say,
687 those concerned with type checking). A heuristic function might simply
690 \item[\ttindexbold{hyp_subst_tacs}] is a list of tactics for substitution in
691 the hypotheses, typically created by \ttindex{HypsubstFun} (see
692 Chapter~\ref{substitution}). This list can, of course, be empty. The
693 tactics are assumed to be safe!
695 The functor is not at all sensitive to the formalization of the
696 object-logic. It does not even examine the rules, but merely applies
697 them according to its fixed strategy. The functor resides in {\tt
698 Provers/classical.ML} in the Isabelle sources.
700 \index{classical reasoner|)}