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doc-src/Ref/classical.tex

author | wenzelm |

Sun Jul 23 12:01:05 2000 +0200 (2000-07-23) | |

changeset 9408 | d3d56e1d2ec1 |

parent 8926 | 0c7f90147f5d |

child 9439 | a95343122ad0 |

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

classical atts now intro! / intro / intro?;

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 Function variables may only be applied to parameters of the subgoal.

455 (This restriction arises because the prover does not use higher-order

456 unification.) If other function variables are present then the prover will

457 fail with the message {\small\tt Function Var's argument not a bound variable}.

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

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

460 fast_tac} and the other tactics described below.

461 \end{itemize}

462 %

463 \begin{ttbox}

464 blast_tac : claset -> int -> tactic

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

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

467 \end{ttbox}

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

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

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

471 \begin{ttdescription}

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

473 subgoal~$i$, increasing the search bound using iterative

474 deepening~\cite{korf85}.

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

477 to prove subgoal~$i$ using a search bound of $lim$. Sometimes a slow

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

479 successful search bound to this tactic instead.

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

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

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

484 has been closed, extended or split.

485 \end{ttdescription}

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

489 \begin{ttbox}

490 type clasimpset = claset * simpset;

491 auto_tac : clasimpset -> tactic

492 force_tac : clasimpset -> int -> tactic

493 auto : unit -> unit

494 force : int -> unit

495 \end{ttbox}

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

497 simplification and classical reasoning.

498 \begin{ttdescription}

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

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

501 leaving the ones it cannot prove.

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

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

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

505 performing a rather exhaustive search.

506 \end{ttdescription}

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

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

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

510 For interactive use,

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

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

515 \subsection{Semi-automatic tactics}

516 \begin{ttbox}

517 clarify_tac : claset -> int -> tactic

518 clarify_step_tac : claset -> int -> tactic

519 clarsimp_tac : clasimpset -> int -> tactic

520 \end{ttbox}

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

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

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

524 instantiating quantifiers yourself.

525 \begin{ttdescription}

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

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

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

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

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

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

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

533 applied.

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

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

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

537 \end{ttdescription}

540 \subsection{Other classical tactics}

541 \begin{ttbox}

542 fast_tac : claset -> int -> tactic

543 best_tac : claset -> int -> tactic

544 slow_tac : claset -> int -> tactic

545 slow_best_tac : claset -> int -> tactic

546 \end{ttbox}

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

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

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

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

551 search.%

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

553 the step of proving a subgoal by assumption.}

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

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

557 that sets up the classical reasoner.

558 \begin{ttdescription}

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

560 depth-first search to prove subgoal~$i$.

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

563 best-first search to prove subgoal~$i$.

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

566 depth-first search to prove subgoal~$i$.

568 \item[\ttindexbold{slow_best_tac} $cs$ $i$] applies \texttt{slow_step_tac} with

569 best-first search to prove subgoal~$i$.

570 \end{ttdescription}

573 \subsection{Depth-limited automatic tactics}

574 \begin{ttbox}

575 depth_tac : claset -> int -> int -> tactic

576 deepen_tac : claset -> int -> int -> tactic

577 \end{ttbox}

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

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

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

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

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

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

585 \begin{ttdescription}

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

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

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

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

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

592 \end{ttdescription}

595 \subsection{Single-step tactics}

596 \begin{ttbox}

597 safe_step_tac : claset -> int -> tactic

598 safe_tac : claset -> tactic

599 inst_step_tac : claset -> int -> tactic

600 step_tac : claset -> int -> tactic

601 slow_step_tac : claset -> int -> tactic

602 \end{ttbox}

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

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

605 \begin{ttdescription}

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

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

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

609 unknowns), or substitution.

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

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

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

615 but allows unknowns to be instantiated.

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

618 procedure. The unsafe wrapper tacticals are applied to a tactic that tries

619 \texttt{safe_tac}, \texttt{inst_step_tac}, or applies an unsafe rule

620 from~$cs$.

622 \item[\ttindexbold{slow_step_tac}]

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

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

625 The resulting search space is larger.

626 \end{ttdescription}

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

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

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

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

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

635 section, including its warnings.

637 The tactics

638 \begin{ttbox}

639 Blast_tac : int -> tactic

640 Auto_tac : tactic

641 Force_tac : int -> tactic

642 Fast_tac : int -> tactic

643 Best_tac : int -> tactic

644 Deepen_tac : int -> int -> tactic

645 Clarify_tac : int -> tactic

646 Clarify_step_tac : int -> tactic

647 Clarsimp_tac : int -> tactic

648 Safe_tac : tactic

649 Safe_step_tac : int -> tactic

650 Step_tac : int -> tactic

651 \end{ttbox}

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

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

654 \indexbold{*Deepen_tac}

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

656 \indexbold{*Safe_tac}\indexbold{*Safe_step_tac}

657 \indexbold{*Step_tac}

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

659 \begin{ttbox}

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

661 \end{ttbox}

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

663 The functions

664 \begin{ttbox}

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

666 \end{ttbox}

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

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

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

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

671 \begin{ttbox}

672 Delrules : thm list -> unit

673 \end{ttbox}

674 deletes rules from the current claset.

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

677 \begin{ttbox}

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

679 \end{ttbox}

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

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

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

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

684 ``$intro?$'', ``$elim?$'', and ``$dest?$''. Note that these extra rules do

685 not have any effect on classic Isabelle tactics.

688 \subsection{Accessing the current claset}

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

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

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

693 for a description.

694 \begin{ttbox}

695 claset : unit -> claset

696 claset_ref : unit -> claset ref

697 claset_of : theory -> claset

698 claset_ref_of : theory -> claset ref

699 print_claset : theory -> unit

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

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

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

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

704 \end{ttbox}

707 \subsection{Other useful tactics}

708 \index{tactics!for contradiction}

709 \index{tactics!for Modus Ponens}

710 \begin{ttbox}

711 contr_tac : int -> tactic

712 mp_tac : int -> tactic

713 eq_mp_tac : int -> tactic

714 swap_res_tac : thm list -> int -> tactic

715 \end{ttbox}

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

717 \begin{ttdescription}

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

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

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

721 tactic can produce multiple outcomes, enumerating all possible

722 contradictions.

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

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

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

727 $P\imp Q$ by~$Q$. It may instantiate unknowns. It fails if it can do

728 nothing.

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

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

732 is safe.

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

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

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

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

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

739 \end{ttdescription}

741 \subsection{Creating swapped rules}

742 \begin{ttbox}

743 swapify : thm list -> thm list

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

745 \end{ttbox}

746 \begin{ttdescription}

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

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

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

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

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

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

754 elim-resolution).

755 \end{ttdescription}

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

759 \index{classical reasoner!setting up}

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

761 have the classical reasoner already set up.

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

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

764 argument signature \texttt{CLASSICAL_DATA}:

765 \begin{ttbox}

766 signature CLASSICAL_DATA =

767 sig

768 val mp : thm

769 val not_elim : thm

770 val swap : thm

771 val sizef : thm -> int

772 val hyp_subst_tacs : (int -> tactic) list

773 end;

774 \end{ttbox}

775 Thus, the functor requires the following items:

776 \begin{ttdescription}

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

778 $\List{\Var{P}\imp\Var{Q};\; \Var{P}} \Imp \Var{Q}$.

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

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

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

784 $\List{\neg \Var{P}; \; \neg \Var{R}\Imp \Var{P}} \Imp \Var{R}$.

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

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

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

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

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

791 count the subgoals.

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

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

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

796 tactics are assumed to be safe!

797 \end{ttdescription}

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

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

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

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

803 \index{classical reasoner|)}

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

806 \label{sec:clasimp-setup}

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

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

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

811 argment, which can be contructed on the fly:

812 \begin{ttbox}

813 structure Clasimp = ClasimpFun

814 (structure Simplifier = Simplifier

815 and Classical = Classical

816 and Blast = Blast);

817 \end{ttbox}

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