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doc-src/Ref/classical.tex

author | wenzelm |

Sun Oct 31 20:11:23 1999 +0100 (1999-10-31) | |

changeset 7990 | 0a604b2fc2b1 |

parent 6592 | c120262044b6 |

child 8136 | 8c65f3ca13f2 |

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

updated;

1 %% $Id$

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.

8 Isabelle's set theory~{\tt ZF} is built upon theory~\texttt{FOL},

9 while {\HOL} 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

19 seconds:

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))

24 \]

25 %

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

33 \begin{ttbox}

34 by (Blast_tac \(i\));

35 \end{ttbox}

36 This command quickly proves most simple formulas of the predicate calculus or

37 set theory. To attempt to prove subgoals using a combination of

38 rewriting and classical reasoning, try

39 \begin{ttbox}

40 auto(); \emph{\textrm{applies to all subgoals}}

41 force i; \emph{\textrm{applies to one subgoal}}

42 \end{ttbox}

43 To do all obvious logical steps, even if they do not prove the

44 subgoal, type one of the following:

45 \begin{ttbox}

46 by Safe_tac; \emph{\textrm{applies to all subgoals}}

47 by (Clarify_tac \(i\)); \emph{\textrm{applies to one subgoal}}

48 \end{ttbox}

51 You need to know how the classical reasoner works in order to use it

52 effectively. There are many tactics to choose from, including

53 {\tt Fast_tac} and \texttt{Best_tac}.

55 We shall first discuss the underlying principles, then present the

56 classical reasoner. Finally, we shall see how to instantiate it for new logics.

57 The logics \FOL, \ZF, {\HOL} and {\HOLCF} have it already installed.

60 \section{The sequent calculus}

61 \index{sequent calculus}

62 Isabelle supports natural deduction, which is easy to use for interactive

63 proof. But natural deduction does not easily lend itself to automation,

64 and has a bias towards intuitionism. For certain proofs in classical

65 logic, it can not be called natural. The {\bf sequent calculus}, a

66 generalization of natural deduction, is easier to automate.

68 A {\bf sequent} has the form $\Gamma\turn\Delta$, where $\Gamma$

69 and~$\Delta$ are sets of formulae.%

70 \footnote{For first-order logic, sequents can equivalently be made from

71 lists or multisets of formulae.} The sequent

72 \[ P@1,\ldots,P@m\turn Q@1,\ldots,Q@n \]

73 is {\bf valid} if $P@1\conj\ldots\conj P@m$ implies $Q@1\disj\ldots\disj

74 Q@n$. Thus $P@1,\ldots,P@m$ represent assumptions, each of which is true,

75 while $Q@1,\ldots,Q@n$ represent alternative goals. A sequent is {\bf

76 basic} if its left and right sides have a common formula, as in $P,Q\turn

77 Q,R$; basic sequents are trivially valid.

79 Sequent rules are classified as {\bf right} or {\bf left}, indicating which

80 side of the $\turn$~symbol they operate on. Rules that operate on the

81 right side are analogous to natural deduction's introduction rules, and

82 left rules are analogous to elimination rules.

83 Recall the natural deduction rules for

84 first-order logic,

85 \iflabelundefined{fol-fig}{from {\it Introduction to Isabelle}}%

86 {Fig.\ts\ref{fol-fig}}.

87 The sequent calculus analogue of~$({\imp}I)$ is the rule

88 $$

89 \ainfer{\Gamma &\turn \Delta, P\imp Q}{P,\Gamma &\turn \Delta,Q}

90 \eqno({\imp}R)

91 $$

92 This breaks down some implication on the right side of a sequent; $\Gamma$

93 and $\Delta$ stand for the sets of formulae that are unaffected by the

94 inference. The analogue of the pair~$({\disj}I1)$ and~$({\disj}I2)$ is the

95 single rule

96 $$

97 \ainfer{\Gamma &\turn \Delta, P\disj Q}{\Gamma &\turn \Delta,P,Q}

98 \eqno({\disj}R)

99 $$

100 This breaks down some disjunction on the right side, replacing it by both

101 disjuncts. Thus, the sequent calculus is a kind of multiple-conclusion logic.

103 To illustrate the use of multiple formulae on the right, let us prove

104 the classical theorem $(P\imp Q)\disj(Q\imp P)$. Working backwards, we

105 reduce this formula to a basic sequent:

106 \[ \infer[(\disj)R]{\turn(P\imp Q)\disj(Q\imp P)}

107 {\infer[(\imp)R]{\turn(P\imp Q), (Q\imp P)\;}

108 {\infer[(\imp)R]{P \turn Q, (Q\imp P)\qquad}

109 {P, Q \turn Q, P\qquad\qquad}}}

110 \]

111 This example is typical of the sequent calculus: start with the desired

112 theorem and apply rules backwards in a fairly arbitrary manner. This yields a

113 surprisingly effective proof procedure. Quantifiers add few complications,

114 since Isabelle handles parameters and schematic variables. See Chapter~10

115 of {\em ML for the Working Programmer}~\cite{paulson-ml2} for further

116 discussion.

119 \section{Simulating sequents by natural deduction}

120 Isabelle can represent sequents directly, as in the object-logic~\texttt{LK}\@.

121 But natural deduction is easier to work with, and most object-logics employ

122 it. Fortunately, we can simulate the sequent $P@1,\ldots,P@m\turn

123 Q@1,\ldots,Q@n$ by the Isabelle formula

124 \[ \List{P@1;\ldots;P@m; \neg Q@2;\ldots; \neg Q@n}\Imp Q@1, \]

125 where the order of the assumptions and the choice of~$Q@1$ are arbitrary.

126 Elim-resolution plays a key role in simulating sequent proofs.

128 We can easily handle reasoning on the left.

129 As discussed in

130 \iflabelundefined{destruct}{{\it Introduction to Isabelle}}{\S\ref{destruct}},

131 elim-resolution with the rules $(\disj E)$, $(\bot E)$ and $(\exists E)$

132 achieves a similar effect as the corresponding sequent rules. For the

133 other connectives, we use sequent-style elimination rules instead of

134 destruction rules such as $({\conj}E1,2)$ and $(\forall E)$. But note that

135 the rule $(\neg L)$ has no effect under our representation of sequents!

136 $$

137 \ainfer{\neg P,\Gamma &\turn \Delta}{\Gamma &\turn \Delta,P}\eqno({\neg}L)

138 $$

139 What about reasoning on the right? Introduction rules can only affect the

140 formula in the conclusion, namely~$Q@1$. The other right-side formulae are

141 represented as negated assumptions, $\neg Q@2$, \ldots,~$\neg Q@n$.

142 \index{assumptions!negated}

143 In order to operate on one of these, it must first be exchanged with~$Q@1$.

144 Elim-resolution with the {\bf swap} rule has this effect:

145 $$ \List{\neg P; \; \neg R\Imp P} \Imp R \eqno(swap) $$

146 To ensure that swaps occur only when necessary, each introduction rule is

147 converted into a swapped form: it is resolved with the second premise

148 of~$(swap)$. The swapped form of~$({\conj}I)$, which might be

149 called~$({\neg\conj}E)$, is

150 \[ \List{\neg(P\conj Q); \; \neg R\Imp P; \; \neg R\Imp Q} \Imp R. \]

151 Similarly, the swapped form of~$({\imp}I)$ is

152 \[ \List{\neg(P\imp Q); \; \List{\neg R;P}\Imp Q} \Imp R \]

153 Swapped introduction rules are applied using elim-resolution, which deletes

154 the negated formula. Our representation of sequents also requires the use

155 of ordinary introduction rules. If we had no regard for readability, we

156 could treat the right side more uniformly by representing sequents as

157 \[ \List{P@1;\ldots;P@m; \neg Q@1;\ldots; \neg Q@n}\Imp \bot. \]

160 \section{Extra rules for the sequent calculus}

161 As mentioned, destruction rules such as $({\conj}E1,2)$ and $(\forall E)$

162 must be replaced by sequent-style elimination rules. In addition, we need

163 rules to embody the classical equivalence between $P\imp Q$ and $\neg P\disj

164 Q$. The introduction rules~$({\disj}I1,2)$ are replaced by a rule that

165 simulates $({\disj}R)$:

166 \[ (\neg Q\Imp P) \Imp P\disj Q \]

167 The destruction rule $({\imp}E)$ is replaced by

168 \[ \List{P\imp Q;\; \neg P\Imp R;\; Q\Imp R} \Imp R. \]

169 Quantifier replication also requires special rules. In classical logic,

170 $\exists x{.}P$ is equivalent to $\neg\forall x{.}\neg P$; the rules

171 $(\exists R)$ and $(\forall L)$ are dual:

172 \[ \ainfer{\Gamma &\turn \Delta, \exists x{.}P}

173 {\Gamma &\turn \Delta, \exists x{.}P, P[t/x]} \; (\exists R)

174 \qquad

175 \ainfer{\forall x{.}P, \Gamma &\turn \Delta}

176 {P[t/x], \forall x{.}P, \Gamma &\turn \Delta} \; (\forall L)

177 \]

178 Thus both kinds of quantifier may be replicated. Theorems requiring

179 multiple uses of a universal formula are easy to invent; consider

180 \[ (\forall x.P(x)\imp P(f(x))) \conj P(a) \imp P(f^n(a)), \]

181 for any~$n>1$. Natural examples of the multiple use of an existential

182 formula are rare; a standard one is $\exists x.\forall y. P(x)\imp P(y)$.

184 Forgoing quantifier replication loses completeness, but gains decidability,

185 since the search space becomes finite. Many useful theorems can be proved

186 without replication, and the search generally delivers its verdict in a

187 reasonable time. To adopt this approach, represent the sequent rules

188 $(\exists R)$, $(\exists L)$ and $(\forall R)$ by $(\exists I)$, $(\exists

189 E)$ and $(\forall I)$, respectively, and put $(\forall E)$ into elimination

190 form:

191 $$ \List{\forall x{.}P(x); P(t)\Imp Q} \Imp Q \eqno(\forall E@2) $$

192 Elim-resolution with this rule will delete the universal formula after a

193 single use. To replicate universal quantifiers, replace the rule by

194 $$

195 \List{\forall x{.}P(x);\; \List{P(t); \forall x{.}P(x)}\Imp Q} \Imp Q.

196 \eqno(\forall E@3)

197 $$

198 To replicate existential quantifiers, replace $(\exists I)$ by

199 \[ \List{\neg(\exists x{.}P(x)) \Imp P(t)} \Imp \exists x{.}P(x). \]

200 All introduction rules mentioned above are also useful in swapped form.

202 Replication makes the search space infinite; we must apply the rules with

203 care. The classical reasoner distinguishes between safe and unsafe

204 rules, applying the latter only when there is no alternative. Depth-first

205 search may well go down a blind alley; best-first search is better behaved

206 in an infinite search space. However, quantifier replication is too

207 expensive to prove any but the simplest theorems.

210 \section{Classical rule sets}

211 \index{classical sets}

212 Each automatic tactic takes a {\bf classical set} --- a collection of

213 rules, classified as introduction or elimination and as {\bf safe} or {\bf

214 unsafe}. In general, safe rules can be attempted blindly, while unsafe

215 rules must be used with care. A safe rule must never reduce a provable

216 goal to an unprovable set of subgoals.

218 The rule~$({\disj}I1)$ is unsafe because it reduces $P\disj Q$ to~$P$. Any

219 rule is unsafe whose premises contain new unknowns. The elimination

220 rule~$(\forall E@2)$ is unsafe, since it is applied via elim-resolution,

221 which discards the assumption $\forall x{.}P(x)$ and replaces it by the

222 weaker assumption~$P(\Var{t})$. The rule $({\exists}I)$ is unsafe for

223 similar reasons. The rule~$(\forall E@3)$ is unsafe in a different sense:

224 since it keeps the assumption $\forall x{.}P(x)$, it is prone to looping.

225 In classical first-order logic, all rules are safe except those mentioned

226 above.

228 The safe/unsafe distinction is vague, and may be regarded merely as a way

229 of giving some rules priority over others. One could argue that

230 $({\disj}E)$ is unsafe, because repeated application of it could generate

231 exponentially many subgoals. Induction rules are unsafe because inductive

232 proofs are difficult to set up automatically. Any inference is unsafe that

233 instantiates an unknown in the proof state --- thus \ttindex{match_tac}

234 must be used, rather than \ttindex{resolve_tac}. Even proof by assumption

235 is unsafe if it instantiates unknowns shared with other subgoals --- thus

236 \ttindex{eq_assume_tac} must be used, rather than \ttindex{assume_tac}.

238 \subsection{Adding rules to classical sets}

239 Classical rule sets belong to the abstract type \mltydx{claset}, which

240 supports the following operations (provided the classical reasoner is

241 installed!):

242 \begin{ttbox}

243 empty_cs : claset

244 print_cs : claset -> unit

245 rep_cs : claset -> {safeEs: thm list, safeIs: thm list,

246 hazEs: thm list, hazIs: thm list,

247 swrappers: (string * wrapper) list,

248 uwrappers: (string * wrapper) list,

249 safe0_netpair: netpair, safep_netpair: netpair,

250 haz_netpair: netpair, dup_netpair: netpair}

251 addSIs : claset * thm list -> claset \hfill{\bf infix 4}

252 addSEs : claset * thm list -> claset \hfill{\bf infix 4}

253 addSDs : claset * thm list -> claset \hfill{\bf infix 4}

254 addIs : claset * thm list -> claset \hfill{\bf infix 4}

255 addEs : claset * thm list -> claset \hfill{\bf infix 4}

256 addDs : claset * thm list -> claset \hfill{\bf infix 4}

257 delrules : claset * thm list -> claset \hfill{\bf infix 4}

258 \end{ttbox}

259 The add operations ignore any rule already present in the claset with the same

260 classification (such as Safe Introduction). They print a warning if the rule

261 has already been added with some other classification, but add the rule

262 anyway. Calling \texttt{delrules} deletes all occurrences of a rule from the

263 claset, but see the warning below concerning destruction rules.

264 \begin{ttdescription}

265 \item[\ttindexbold{empty_cs}] is the empty classical set.

267 \item[\ttindexbold{print_cs} $cs$] displays the printable contents of~$cs$,

268 which is the rules. All other parts are non-printable.

270 \item[\ttindexbold{rep_cs} $cs$] decomposes $cs$ as a record of its internal

271 components, namely the safe introduction and elimination rules, the unsafe

272 introduction and elimination rules, the lists of safe and unsafe wrappers

273 (see \ref{sec:modifying-search}), and the internalized forms of the rules.

275 \item[$cs$ addSIs $rules$] \indexbold{*addSIs}

276 adds safe introduction~$rules$ to~$cs$.

278 \item[$cs$ addSEs $rules$] \indexbold{*addSEs}

279 adds safe elimination~$rules$ to~$cs$.

281 \item[$cs$ addSDs $rules$] \indexbold{*addSDs}

282 adds safe destruction~$rules$ to~$cs$.

284 \item[$cs$ addIs $rules$] \indexbold{*addIs}

285 adds unsafe introduction~$rules$ to~$cs$.

287 \item[$cs$ addEs $rules$] \indexbold{*addEs}

288 adds unsafe elimination~$rules$ to~$cs$.

290 \item[$cs$ addDs $rules$] \indexbold{*addDs}

291 adds unsafe destruction~$rules$ to~$cs$.

293 \item[$cs$ delrules $rules$] \indexbold{*delrules}

294 deletes~$rules$ from~$cs$. It prints a warning for those rules that are not

295 in~$cs$.

296 \end{ttdescription}

298 \begin{warn}

299 If you added $rule$ using \texttt{addSDs} or \texttt{addDs}, then you must delete

300 it as follows:

301 \begin{ttbox}

302 \(cs\) delrules [make_elim \(rule\)]

303 \end{ttbox}

304 \par\noindent

305 This is necessary because the operators \texttt{addSDs} and \texttt{addDs} convert

306 the destruction rules to elimination rules by applying \ttindex{make_elim},

307 and then insert them using \texttt{addSEs} and \texttt{addEs}, respectively.

308 \end{warn}

310 Introduction rules are those that can be applied using ordinary resolution.

311 The classical set automatically generates their swapped forms, which will

312 be applied using elim-resolution. Elimination rules are applied using

313 elim-resolution. In a classical set, rules are sorted by the number of new

314 subgoals they will yield; rules that generate the fewest subgoals will be

315 tried first (see \S\ref{biresolve_tac}).

317 For elimination and destruction rules there are variants of the add operations

318 adding a rule in a way such that it is applied only if also its second premise

319 can be unified with an assumption of the current proof state:

320 \indexbold{*addSE2}\indexbold{*addSD2}\indexbold{*addE2}\indexbold{*addD2}

321 \begin{ttbox}

322 addSE2 : claset * (string * thm) -> claset \hfill{\bf infix 4}

323 addSD2 : claset * (string * thm) -> claset \hfill{\bf infix 4}

324 addE2 : claset * (string * thm) -> claset \hfill{\bf infix 4}

325 addD2 : claset * (string * thm) -> claset \hfill{\bf infix 4}

326 \end{ttbox}

327 \begin{warn}

328 A rule to be added in this special way must be given a name, which is used

329 to delete it again -- when desired -- using \texttt{delSWrappers} or

330 \texttt{delWrappers}, respectively. This is because these add operations

331 are implemented as wrappers (see \ref{sec:modifying-search} below).

332 \end{warn}

335 \subsection{Modifying the search step}

336 \label{sec:modifying-search}

337 For a given classical set, the proof strategy is simple. Perform as many safe

338 inferences as possible; or else, apply certain safe rules, allowing

339 instantiation of unknowns; or else, apply an unsafe rule. The tactics also

340 eliminate assumptions of the form $x=t$ by substitution if they have been set

341 up to do so (see \texttt{hyp_subst_tacs} in~\S\ref{sec:classical-setup} below).

342 They may perform a form of Modus Ponens: if there are assumptions $P\imp Q$

343 and~$P$, then replace $P\imp Q$ by~$Q$.

345 The classical reasoning tactics --- except \texttt{blast_tac}! --- allow

346 you to modify this basic proof strategy by applying two lists of arbitrary

347 {\bf wrapper tacticals} to it.

348 The first wrapper list, which is considered to contain safe wrappers only,

349 affects \ttindex{safe_step_tac} and all the tactics that call it.

350 The second one, which may contain unsafe wrappers, affects the unsafe parts

351 of \ttindex{step_tac}, \ttindex{slow_step_tac}, and the tactics that call them.

352 A wrapper transforms each step of the search, for example

353 by attempting other tactics before or after the original step tactic.

354 All members of a wrapper list are applied in turn to the respective step tactic.

356 Initially the two wrapper lists are empty, which means no modification of the

357 step tactics. Safe and unsafe wrappers are added to a claset

358 with the functions given below, supplying them with wrapper names.

359 These names may be used to selectively delete wrappers.

361 \begin{ttbox}

362 type wrapper = (int -> tactic) -> (int -> tactic);

364 addSWrapper : claset * (string * wrapper ) -> claset \hfill{\bf infix 4}

365 addSbefore : claset * (string * (int -> tactic)) -> claset \hfill{\bf infix 4}

366 addSaltern : claset * (string * (int -> tactic)) -> claset \hfill{\bf infix 4}

367 delSWrapper : claset * string -> claset \hfill{\bf infix 4}

369 addWrapper : claset * (string * wrapper ) -> claset \hfill{\bf infix 4}

370 addbefore : claset * (string * (int -> tactic)) -> claset \hfill{\bf infix 4}

371 addaltern : claset * (string * (int -> tactic)) -> claset \hfill{\bf infix 4}

372 delWrapper : claset * string -> claset \hfill{\bf infix 4}

374 addSss : claset * simpset -> claset \hfill{\bf infix 4}

375 addss : claset * simpset -> claset \hfill{\bf infix 4}

376 \end{ttbox}

377 %

379 \begin{ttdescription}

380 \item[$cs$ addSWrapper $(name,wrapper)$] \indexbold{*addSWrapper}

381 adds a new wrapper, which should yield a safe tactic,

382 to modify the existing safe step tactic.

384 \item[$cs$ addSbefore $(name,tac)$] \indexbold{*addSbefore}

385 adds the given tactic as a safe wrapper, such that it is tried

386 {\em before} each safe step of the search.

388 \item[$cs$ addSaltern $(name,tac)$] \indexbold{*addSaltern}

389 adds the given tactic as a safe wrapper, such that it is tried

390 when a safe step of the search would fail.

392 \item[$cs$ delSWrapper $name$] \indexbold{*delSWrapper}

393 deletes the safe wrapper with the given name.

395 \item[$cs$ addWrapper $(name,wrapper)$] \indexbold{*addWrapper}

396 adds a new wrapper to modify the existing (unsafe) step tactic.

398 \item[$cs$ addbefore $(name,tac)$] \indexbold{*addbefore}

399 adds the given tactic as an unsafe wrapper, such that it its result is

400 concatenated {\em before} the result of each unsafe step.

402 \item[$cs$ addaltern $(name,tac)$] \indexbold{*addaltern}

403 adds the given tactic as an unsafe wrapper, such that it its result is

404 concatenated {\em after} the result of each unsafe step.

406 \item[$cs$ delWrapper $name$] \indexbold{*delWrapper}

407 deletes the unsafe wrapper with the given name.

409 \item[$cs$ addSss $ss$] \indexbold{*addss}

410 adds the simpset~$ss$ to the classical set. The assumptions and goal will be

411 simplified, in a rather safe way, after each safe step of the search.

413 \item[$cs$ addss $ss$] \indexbold{*addss}

414 adds the simpset~$ss$ to the classical set. The assumptions and goal will be

415 simplified, before the each unsafe step of the search.

417 \end{ttdescription}

419 \index{simplification!from classical reasoner}

420 Strictly speaking, the operators \texttt{addss} and \texttt{addSss}

421 are not part of the classical reasoner.

422 , which are used as primitives

423 for the automatic tactics described in \S\ref{sec:automatic-tactics}, are

424 implemented as wrapper tacticals.

425 they

426 \begin{warn}

427 Being defined as wrappers, these operators are inappropriate for adding more

428 than one simpset at a time: the simpset added last overwrites any earlier ones.

429 When a simpset combined with a claset is to be augmented, this should done

430 {\em before} combining it with the claset.

431 \end{warn}

434 \section{The classical tactics}

435 \index{classical reasoner!tactics} If installed, the classical module provides

436 powerful theorem-proving tactics. Most of them have capitalized analogues

437 that use the default claset; see \S\ref{sec:current-claset}.

440 \subsection{The tableau prover}

441 The tactic \texttt{blast_tac} searches for a proof using a fast tableau prover,

442 coded directly in \ML. It then reconstructs the proof using Isabelle

443 tactics. It is faster and more powerful than the other classical

444 reasoning tactics, but has major limitations too.

445 \begin{itemize}

446 \item It does not use the wrapper tacticals described above, such as

447 \ttindex{addss}.

448 \item It ignores types, which can cause problems in \HOL. If it applies a rule

449 whose types are inappropriate, then proof reconstruction will fail.

450 \item It does not perform higher-order unification, as needed by the rule {\tt

451 rangeI} in {\HOL} and \texttt{RepFunI} in {\ZF}. There are often

452 alternatives to such rules, for example {\tt

453 range_eqI} and \texttt{RepFun_eqI}.

454 \item The message {\small\tt Function Var's argument not a bound variable\ }

455 relates to the lack of higher-order unification. Function variables

456 may only be applied to parameters of the subgoal.

457 \item Its proof strategy is more general than \texttt{fast_tac}'s but can be

458 slower. If \texttt{blast_tac} fails or seems to be running forever, try {\tt

459 fast_tac} and the other tactics described below.

460 \end{itemize}

461 %

462 \begin{ttbox}

463 blast_tac : claset -> int -> tactic

464 Blast.depth_tac : claset -> int -> int -> tactic

465 Blast.trace : bool ref \hfill{\bf initially false}

466 \end{ttbox}

467 The two tactics differ on how they bound the number of unsafe steps used in a

468 proof. While \texttt{blast_tac} starts with a bound of zero and increases it

469 successively to~20, \texttt{Blast.depth_tac} applies a user-supplied search bound.

470 \begin{ttdescription}

471 \item[\ttindexbold{blast_tac} $cs$ $i$] tries to prove

472 subgoal~$i$ using iterative deepening to increase the search bound.

474 \item[\ttindexbold{Blast.depth_tac} $cs$ $lim$ $i$] tries

475 to prove subgoal~$i$ using a search bound of $lim$. Often a slow

476 proof using \texttt{blast_tac} can be made much faster by supplying the

477 successful search bound to this tactic instead.

479 \item[set \ttindexbold{Blast.trace};] \index{tracing!of classical prover}

480 causes the tableau prover to print a trace of its search. At each step it

481 displays the formula currently being examined and reports whether the branch

482 has been closed, extended or split.

483 \end{ttdescription}

486 \subsection{Automatic tactics}\label{sec:automatic-tactics}

487 \begin{ttbox}

488 type clasimpset = claset * simpset;

489 auto_tac : clasimpset -> tactic

490 force_tac : clasimpset -> int -> tactic

491 auto : unit -> unit

492 force : int -> unit

493 \end{ttbox}

494 The automatic tactics attempt to prove goals using a combination of

495 simplification and classical reasoning.

496 \begin{ttdescription}

497 \item[\ttindexbold{auto_tac $(cs,ss)$}] is intended for situations where

498 there are a lot of mostly trivial subgoals; it proves all the easy ones,

499 leaving the ones it cannot prove.

500 (Unfortunately, attempting to prove the hard ones may take a long time.)

501 \item[\ttindexbold{force_tac} $(cs,ss)$ $i$] is intended to prove subgoal~$i$

502 completely. It tries to apply all fancy tactics it knows about,

503 performing a rather exhaustive search.

504 \end{ttdescription}

505 They must be supplied both a simpset and a claset; therefore

506 they are most easily called as \texttt{Auto_tac} and \texttt{Force_tac}, which

507 use the default claset and simpset (see \S\ref{sec:current-claset} below).

508 For interactive use,

509 the shorthand \texttt{auto();} abbreviates \texttt{by Auto_tac;}

510 while \texttt{force 1;} abbreviates \texttt{by (Force_tac 1);}

513 \subsection{Semi-automatic tactics}

514 \begin{ttbox}

515 clarify_tac : claset -> int -> tactic

516 clarify_step_tac : claset -> int -> tactic

517 clarsimp_tac : clasimpset -> int -> tactic

518 \end{ttbox}

519 Use these when the automatic tactics fail. They perform all the obvious

520 logical inferences that do not split the subgoal. The result is a

521 simpler subgoal that can be tackled by other means, such as by

522 instantiating quantifiers yourself.

523 \begin{ttdescription}

524 \item[\ttindexbold{clarify_tac} $cs$ $i$] performs a series of safe steps on

525 subgoal~$i$ by repeatedly calling \texttt{clarify_step_tac}.

526 \item[\ttindexbold{clarify_step_tac} $cs$ $i$] performs a safe step on

527 subgoal~$i$. No splitting step is applied; for example, the subgoal $A\conj

528 B$ is left as a conjunction. Proof by assumption, Modus Ponens, etc., may be

529 performed provided they do not instantiate unknowns. Assumptions of the

530 form $x=t$ may be eliminated. The user-supplied safe wrapper tactical is

531 applied.

532 \item[\ttindexbold{clarsimp_tac} $cs$ $i$] acts like \texttt{clarify_tac}, but

533 also does simplification with the given simpset. note that if the simpset

534 includes a splitter for the premises, the subgoal may still be split.

535 \end{ttdescription}

538 \subsection{Other classical tactics}

539 \begin{ttbox}

540 fast_tac : claset -> int -> tactic

541 best_tac : claset -> int -> tactic

542 slow_tac : claset -> int -> tactic

543 slow_best_tac : claset -> int -> tactic

544 \end{ttbox}

545 These tactics attempt to prove a subgoal using sequent-style reasoning.

546 Unlike \texttt{blast_tac}, they construct proofs directly in Isabelle. Their

547 effect is restricted (by \texttt{SELECT_GOAL}) to one subgoal; they either prove

548 this subgoal or fail. The \texttt{slow_} versions conduct a broader

549 search.%

550 \footnote{They may, when backtracking from a failed proof attempt, undo even

551 the step of proving a subgoal by assumption.}

553 The best-first tactics are guided by a heuristic function: typically, the

554 total size of the proof state. This function is supplied in the functor call

555 that sets up the classical reasoner.

556 \begin{ttdescription}

557 \item[\ttindexbold{fast_tac} $cs$ $i$] applies \texttt{step_tac} using

558 depth-first search, to prove subgoal~$i$.

560 \item[\ttindexbold{best_tac} $cs$ $i$] applies \texttt{step_tac} using

561 best-first search, to prove subgoal~$i$.

563 \item[\ttindexbold{slow_tac} $cs$ $i$] applies \texttt{slow_step_tac} using

564 depth-first search, to prove subgoal~$i$.

566 \item[\ttindexbold{slow_best_tac} $cs$ $i$] applies \texttt{slow_step_tac} using

567 best-first search, to prove subgoal~$i$.

568 \end{ttdescription}

571 \subsection{Depth-limited automatic tactics}

572 \begin{ttbox}

573 depth_tac : claset -> int -> int -> tactic

574 deepen_tac : claset -> int -> int -> tactic

575 \end{ttbox}

576 These work by exhaustive search up to a specified depth. Unsafe rules are

577 modified to preserve the formula they act on, so that it be used repeatedly.

578 They can prove more goals than \texttt{fast_tac} can but are much

579 slower, for example if the assumptions have many universal quantifiers.

581 The depth limits the number of unsafe steps. If you can estimate the minimum

582 number of unsafe steps needed, supply this value as~$m$ to save time.

583 \begin{ttdescription}

584 \item[\ttindexbold{depth_tac} $cs$ $m$ $i$]

585 tries to prove subgoal~$i$ by exhaustive search up to depth~$m$.

587 \item[\ttindexbold{deepen_tac} $cs$ $m$ $i$]

588 tries to prove subgoal~$i$ by iterative deepening. It calls \texttt{depth_tac}

589 repeatedly with increasing depths, starting with~$m$.

590 \end{ttdescription}

593 \subsection{Single-step tactics}

594 \begin{ttbox}

595 safe_step_tac : claset -> int -> tactic

596 safe_tac : claset -> tactic

597 inst_step_tac : claset -> int -> tactic

598 step_tac : claset -> int -> tactic

599 slow_step_tac : claset -> int -> tactic

600 \end{ttbox}

601 The automatic proof procedures call these tactics. By calling them

602 yourself, you can execute these procedures one step at a time.

603 \begin{ttdescription}

604 \item[\ttindexbold{safe_step_tac} $cs$ $i$] performs a safe step on

605 subgoal~$i$. The safe wrapper tacticals are applied to a tactic that may

606 include proof by assumption or Modus Ponens (taking care not to instantiate

607 unknowns), or substitution.

609 \item[\ttindexbold{safe_tac} $cs$] repeatedly performs safe steps on all

610 subgoals. It is deterministic, with at most one outcome.

612 \item[\ttindexbold{inst_step_tac} $cs$ $i$] is like \texttt{safe_step_tac},

613 but allows unknowns to be instantiated.

615 \item[\ttindexbold{step_tac} $cs$ $i$] is the basic step of the proof

616 procedure. The (unsafe) wrapper tacticals are applied to a tactic that tries

617 \texttt{safe_tac}, \texttt{inst_step_tac}, or applies an unsafe rule from~$cs$.

619 \item[\ttindexbold{slow_step_tac}]

620 resembles \texttt{step_tac}, but allows backtracking between using safe

621 rules with instantiation (\texttt{inst_step_tac}) and using unsafe rules.

622 The resulting search space is larger.

623 \end{ttdescription}

626 \subsection{The current claset}\label{sec:current-claset}

628 Each theory is equipped with an implicit \emph{current claset}

629 \index{claset!current}. This is a default set of classical

630 rules. The underlying idea is quite similar to that of a current

631 simpset described in \S\ref{sec:simp-for-dummies}; please read that

632 section, including its warnings.

634 The tactics

635 \begin{ttbox}

636 Blast_tac : int -> tactic

637 Auto_tac : tactic

638 Force_tac : int -> tactic

639 Fast_tac : int -> tactic

640 Best_tac : int -> tactic

641 Deepen_tac : int -> int -> tactic

642 Clarify_tac : int -> tactic

643 Clarify_step_tac : int -> tactic

644 Clarsimp_tac : int -> tactic

645 Safe_tac : tactic

646 Safe_step_tac : int -> tactic

647 Step_tac : int -> tactic

648 \end{ttbox}

649 \indexbold{*Blast_tac}\indexbold{*Auto_tac}\indexbold{*Force_tac}

650 \indexbold{*Best_tac}\indexbold{*Fast_tac}%

651 \indexbold{*Deepen_tac}

652 \indexbold{*Clarify_tac}\indexbold{*Clarify_step_tac}\indexbold{*Clarsimp_tac}

653 \indexbold{*Safe_tac}\indexbold{*Safe_step_tac}

654 \indexbold{*Step_tac}

655 make use of the current claset. For example, \texttt{Blast_tac} is defined as

656 \begin{ttbox}

657 fun Blast_tac i st = blast_tac (claset()) i st;

658 \end{ttbox}

659 and gets the current claset, only after it is applied to a proof state.

660 The functions

661 \begin{ttbox}

662 AddSIs, AddSEs, AddSDs, AddIs, AddEs, AddDs: thm list -> unit

663 \end{ttbox}

664 \indexbold{*AddSIs} \indexbold{*AddSEs} \indexbold{*AddSDs}

665 \indexbold{*AddIs} \indexbold{*AddEs} \indexbold{*AddDs}

666 are used to add rules to the current claset. They work exactly like their

667 lower case counterparts, such as \texttt{addSIs}. Calling

668 \begin{ttbox}

669 Delrules : thm list -> unit

670 \end{ttbox}

671 deletes rules from the current claset.

673 \medskip A few further functions are available as uppercase versions only:

674 \begin{ttbox}

675 AddXIs, AddXEs, AddXDs: thm list -> unit

676 \end{ttbox}

677 \indexbold{*AddXIs} \indexbold{*AddXEs} \indexbold{*AddXDs} augment the

678 current claset by \emph{extra} introduction, elimination, or destruct rules.

679 These provide additional hints for the basic non-automated proof methods of

680 Isabelle/Isar \cite{isabelle-isar-ref}. The corresponding Isar attributes are

681 ``$intro!!$'', ``$elim!!$'', and ``$dest!!$''. Note that these extra rules do

682 not have any effect on classic Isabelle tactics.

685 \subsection{Accessing the current claset}

686 \label{sec:access-current-claset}

688 the functions to access the current claset are analogous to the functions

689 for the current simpset, so please see \ref{sec:access-current-simpset}

690 for a description.

691 \begin{ttbox}

692 claset : unit -> claset

693 claset_ref : unit -> claset ref

694 claset_of : theory -> claset

695 claset_ref_of : theory -> claset ref

696 print_claset : theory -> unit

697 CLASET :(claset -> tactic) -> tactic

698 CLASET' :(claset -> 'a -> tactic) -> 'a -> tactic

699 CLASIMPSET :(clasimpset -> tactic) -> tactic

700 CLASIMPSET' :(clasimpset -> 'a -> tactic) -> 'a -> tactic

701 \end{ttbox}

704 \subsection{Other useful tactics}

705 \index{tactics!for contradiction}

706 \index{tactics!for Modus Ponens}

707 \begin{ttbox}

708 contr_tac : int -> tactic

709 mp_tac : int -> tactic

710 eq_mp_tac : int -> tactic

711 swap_res_tac : thm list -> int -> tactic

712 \end{ttbox}

713 These can be used in the body of a specialized search.

714 \begin{ttdescription}

715 \item[\ttindexbold{contr_tac} {\it i}]\index{assumptions!contradictory}

716 solves subgoal~$i$ by detecting a contradiction among two assumptions of

717 the form $P$ and~$\neg P$, or fail. It may instantiate unknowns. The

718 tactic can produce multiple outcomes, enumerating all possible

719 contradictions.

721 \item[\ttindexbold{mp_tac} {\it i}]

722 is like \texttt{contr_tac}, but also attempts to perform Modus Ponens in

723 subgoal~$i$. If there are assumptions $P\imp Q$ and~$P$, then it replaces

724 $P\imp Q$ by~$Q$. It may instantiate unknowns. It fails if it can do

725 nothing.

727 \item[\ttindexbold{eq_mp_tac} {\it i}]

728 is like \texttt{mp_tac} {\it i}, but may not instantiate unknowns --- thus, it

729 is safe.

731 \item[\ttindexbold{swap_res_tac} {\it thms} {\it i}] refines subgoal~$i$ of

732 the proof state using {\it thms}, which should be a list of introduction

733 rules. First, it attempts to prove the goal using \texttt{assume_tac} or

734 \texttt{contr_tac}. It then attempts to apply each rule in turn, attempting

735 resolution and also elim-resolution with the swapped form.

736 \end{ttdescription}

738 \subsection{Creating swapped rules}

739 \begin{ttbox}

740 swapify : thm list -> thm list

741 joinrules : thm list * thm list -> (bool * thm) list

742 \end{ttbox}

743 \begin{ttdescription}

744 \item[\ttindexbold{swapify} {\it thms}] returns a list consisting of the

745 swapped versions of~{\it thms}, regarded as introduction rules.

747 \item[\ttindexbold{joinrules} ({\it intrs}, {\it elims})]

748 joins introduction rules, their swapped versions, and elimination rules for

749 use with \ttindex{biresolve_tac}. Each rule is paired with~\texttt{false}

750 (indicating ordinary resolution) or~\texttt{true} (indicating

751 elim-resolution).

752 \end{ttdescription}

755 \section{Setting up the classical reasoner}\label{sec:classical-setup}

756 \index{classical reasoner!setting up}

757 Isabelle's classical object-logics, including \texttt{FOL} and \texttt{HOL},

758 have the classical reasoner already set up.

759 When defining a new classical logic, you should set up the reasoner yourself.

760 It consists of the \ML{} functor \ttindex{ClassicalFun}, which takes the

761 argument signature \texttt{CLASSICAL_DATA}:

762 \begin{ttbox}

763 signature CLASSICAL_DATA =

764 sig

765 val mp : thm

766 val not_elim : thm

767 val swap : thm

768 val sizef : thm -> int

769 val hyp_subst_tacs : (int -> tactic) list

770 end;

771 \end{ttbox}

772 Thus, the functor requires the following items:

773 \begin{ttdescription}

774 \item[\tdxbold{mp}] should be the Modus Ponens rule

775 $\List{\Var{P}\imp\Var{Q};\; \Var{P}} \Imp \Var{Q}$.

777 \item[\tdxbold{not_elim}] should be the contradiction rule

778 $\List{\neg\Var{P};\; \Var{P}} \Imp \Var{R}$.

780 \item[\tdxbold{swap}] should be the swap rule

781 $\List{\neg \Var{P}; \; \neg \Var{R}\Imp \Var{P}} \Imp \Var{R}$.

783 \item[\ttindexbold{sizef}] is the heuristic function used for best-first

784 search. It should estimate the size of the remaining subgoals. A good

785 heuristic function is \ttindex{size_of_thm}, which measures the size of the

786 proof state. Another size function might ignore certain subgoals (say,

787 those concerned with type-checking). A heuristic function might simply

788 count the subgoals.

790 \item[\ttindexbold{hyp_subst_tacs}] is a list of tactics for substitution in

791 the hypotheses, typically created by \ttindex{HypsubstFun} (see

792 Chapter~\ref{substitution}). This list can, of course, be empty. The

793 tactics are assumed to be safe!

794 \end{ttdescription}

795 The functor is not at all sensitive to the formalization of the

796 object-logic. It does not even examine the rules, but merely applies

797 them according to its fixed strategy. The functor resides in {\tt

798 Provers/classical.ML} in the Isabelle sources.

800 \index{classical reasoner|)}

802 \section{Setting up the combination with the simplifier}

803 \label{sec:clasimp-setup}

805 To combine the classical reasoner and the simplifier, we simply call the

806 \ML{} functor \ttindex{ClasimpFun} that assembles the parts as required.

807 It takes a structure (of signature \texttt{CLASIMP_DATA}) as

808 argment, which can be contructed on the fly:

809 \begin{ttbox}

810 structure Clasimp = ClasimpFun

811 (structure Simplifier = Simplifier

812 and Classical = Classical

813 and Blast = Blast);

814 \end{ttbox}

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