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src/HOL/SMT.thy

author | boehmes |

Thu Aug 25 11:15:31 2011 +0200 (2011-08-25) | |

changeset 44488 | 587bf61a00a1 |

parent 44087 | 8e491cb8841c |

child 45392 | 828e08541cee |

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

improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization

1 (* Title: HOL/SMT.thy

2 Author: Sascha Boehme, TU Muenchen

3 *)

5 header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *}

7 theory SMT

8 imports Record

9 uses

10 "Tools/SMT/smt_utils.ML"

11 "Tools/SMT/smt_failure.ML"

12 "Tools/SMT/smt_config.ML"

13 ("Tools/SMT/smt_builtin.ML")

14 ("Tools/SMT/smt_datatypes.ML")

15 ("Tools/SMT/smt_normalize.ML")

16 ("Tools/SMT/smt_translate.ML")

17 ("Tools/SMT/smt_solver.ML")

18 ("Tools/SMT/smtlib_interface.ML")

19 ("Tools/SMT/z3_interface.ML")

20 ("Tools/SMT/z3_proof_parser.ML")

21 ("Tools/SMT/z3_proof_tools.ML")

22 ("Tools/SMT/z3_proof_literals.ML")

23 ("Tools/SMT/z3_proof_methods.ML")

24 ("Tools/SMT/z3_proof_reconstruction.ML")

25 ("Tools/SMT/z3_model.ML")

26 ("Tools/SMT/smt_setup_solvers.ML")

27 begin

31 subsection {* Triggers for quantifier instantiation *}

33 text {*

34 Some SMT solvers support patterns as a quantifier instantiation

35 heuristics. Patterns may either be positive terms (tagged by "pat")

36 triggering quantifier instantiations -- when the solver finds a

37 term matching a positive pattern, it instantiates the corresponding

38 quantifier accordingly -- or negative terms (tagged by "nopat")

39 inhibiting quantifier instantiations. A list of patterns

40 of the same kind is called a multipattern, and all patterns in a

41 multipattern are considered conjunctively for quantifier instantiation.

42 A list of multipatterns is called a trigger, and their multipatterns

43 act disjunctively during quantifier instantiation. Each multipattern

44 should mention at least all quantified variables of the preceding

45 quantifier block.

46 *}

48 datatype pattern = Pattern

50 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"

51 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"

53 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"

54 where "trigger _ P = P"

58 subsection {* Quantifier weights *}

60 text {*

61 Weight annotations to quantifiers influence the priority of quantifier

62 instantiations. They should be handled with care for solvers, which support

63 them, because incorrect choices of weights might render a problem unsolvable.

64 *}

66 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"

68 text {*

69 Weights must be non-negative. The value @{text 0} is equivalent to providing

70 no weight at all.

72 Weights should only be used at quantifiers and only inside triggers (if the

73 quantifier has triggers). Valid usages of weights are as follows:

75 \begin{itemize}

76 \item

77 @{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}

78 \item

79 @{term "\<forall>x. weight 3 (P x)"}

80 \end{itemize}

81 *}

85 subsection {* Higher-order encoding *}

87 text {*

88 Application is made explicit for constants occurring with varying

89 numbers of arguments. This is achieved by the introduction of the

90 following constant.

91 *}

93 definition fun_app where "fun_app f = f"

95 text {*

96 Some solvers support a theory of arrays which can be used to encode

97 higher-order functions. The following set of lemmas specifies the

98 properties of such (extensional) arrays.

99 *}

101 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other

102 fun_upd_upd fun_app_def

106 subsection {* First-order logic *}

108 text {*

109 Some SMT solvers only accept problems in first-order logic, i.e.,

110 where formulas and terms are syntactically separated. When

111 translating higher-order into first-order problems, all

112 uninterpreted constants (those not built-in in the target solver)

113 are treated as function symbols in the first-order sense. Their

114 occurrences as head symbols in atoms (i.e., as predicate symbols) are

115 turned into terms by logically equating such atoms with @{term True}.

116 For technical reasons, @{term True} and @{term False} occurring inside

117 terms are replaced by the following constants.

118 *}

120 definition term_true where "term_true = True"

121 definition term_false where "term_false = False"

125 subsection {* Integer division and modulo for Z3 *}

127 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where

128 "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"

130 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where

131 "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"

135 subsection {* Setup *}

137 use "Tools/SMT/smt_builtin.ML"

138 use "Tools/SMT/smt_datatypes.ML"

139 use "Tools/SMT/smt_normalize.ML"

140 use "Tools/SMT/smt_translate.ML"

141 use "Tools/SMT/smt_solver.ML"

142 use "Tools/SMT/smtlib_interface.ML"

143 use "Tools/SMT/z3_interface.ML"

144 use "Tools/SMT/z3_proof_parser.ML"

145 use "Tools/SMT/z3_proof_tools.ML"

146 use "Tools/SMT/z3_proof_literals.ML"

147 use "Tools/SMT/z3_proof_methods.ML"

148 use "Tools/SMT/z3_proof_reconstruction.ML"

149 use "Tools/SMT/z3_model.ML"

150 use "Tools/SMT/smt_setup_solvers.ML"

152 setup {*

153 SMT_Config.setup #>

154 SMT_Normalize.setup #>

155 SMT_Solver.setup #>

156 SMTLIB_Interface.setup #>

157 Z3_Interface.setup #>

158 Z3_Proof_Reconstruction.setup #>

159 SMT_Setup_Solvers.setup

160 *}

164 subsection {* Configuration *}

166 text {*

167 The current configuration can be printed by the command

168 @{text smt_status}, which shows the values of most options.

169 *}

173 subsection {* General configuration options *}

175 text {*

176 The option @{text smt_solver} can be used to change the target SMT

177 solver. The possible values can be obtained from the @{text smt_status}

178 command.

180 Due to licensing restrictions, Yices and Z3 are not installed/enabled

181 by default. Z3 is free for non-commercial applications and can be enabled

182 by simply setting the environment variable @{text Z3_NON_COMMERCIAL} to

183 @{text yes}.

184 *}

186 declare [[ smt_solver = z3 ]]

188 text {*

189 Since SMT solvers are potentially non-terminating, there is a timeout

190 (given in seconds) to restrict their runtime. A value greater than

191 120 (seconds) is in most cases not advisable.

192 *}

194 declare [[ smt_timeout = 20 ]]

196 text {*

197 SMT solvers apply randomized heuristics. In case a problem is not

198 solvable by an SMT solver, changing the following option might help.

199 *}

201 declare [[ smt_random_seed = 1 ]]

203 text {*

204 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT

205 solvers are fully trusted without additional checks. The following

206 option can cause the SMT solver to run in proof-producing mode, giving

207 a checkable certificate. This is currently only implemented for Z3.

208 *}

210 declare [[ smt_oracle = false ]]

212 text {*

213 Each SMT solver provides several commandline options to tweak its

214 behaviour. They can be passed to the solver by setting the following

215 options.

216 *}

218 declare [[ cvc3_options = "", remote_cvc3_options = "" ]]

219 declare [[ yices_options = "" ]]

220 declare [[ z3_options = "", remote_z3_options = "" ]]

222 text {*

223 Enable the following option to use built-in support for datatypes and

224 records. Currently, this is only implemented for Z3 running in oracle

225 mode.

226 *}

228 declare [[ smt_datatypes = false ]]

230 text {*

231 The SMT method provides an inference mechanism to detect simple triggers

232 in quantified formulas, which might increase the number of problems

233 solvable by SMT solvers (note: triggers guide quantifier instantiations

234 in the SMT solver). To turn it on, set the following option.

235 *}

237 declare [[ smt_infer_triggers = false ]]

239 text {*

240 The SMT method monomorphizes the given facts, that is, it tries to

241 instantiate all schematic type variables with fixed types occurring

242 in the problem. This is a (possibly nonterminating) fixed-point

243 construction whose cycles are limited by the following option.

244 *}

246 declare [[ monomorph_max_rounds = 5 ]]

248 text {*

249 In addition, the number of generated monomorphic instances is limited

250 by the following option.

251 *}

253 declare [[ monomorph_max_new_instances = 500 ]]

257 subsection {* Certificates *}

259 text {*

260 By setting the option @{text smt_certificates} to the name of a file,

261 all following applications of an SMT solver a cached in that file.

262 Any further application of the same SMT solver (using the very same

263 configuration) re-uses the cached certificate instead of invoking the

264 solver. An empty string disables caching certificates.

266 The filename should be given as an explicit path. It is good

267 practice to use the name of the current theory (with ending

268 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.

269 *}

271 declare [[ smt_certificates = "" ]]

273 text {*

274 The option @{text smt_fixed} controls whether only stored

275 certificates are should be used or invocation of an SMT solver is

276 allowed. When set to @{text true}, no SMT solver will ever be

277 invoked and only the existing certificates found in the configured

278 cache are used; when set to @{text false} and there is no cached

279 certificate for some proposition, then the configured SMT solver is

280 invoked.

281 *}

283 declare [[ smt_fixed = false ]]

287 subsection {* Tracing *}

289 text {*

290 The SMT method, when applied, traces important information. To

291 make it entirely silent, set the following option to @{text false}.

292 *}

294 declare [[ smt_verbose = true ]]

296 text {*

297 For tracing the generated problem file given to the SMT solver as

298 well as the returned result of the solver, the option

299 @{text smt_trace} should be set to @{text true}.

300 *}

302 declare [[ smt_trace = false ]]

304 text {*

305 From the set of assumptions given to the SMT solver, those assumptions

306 used in the proof are traced when the following option is set to

307 @{term true}. This only works for Z3 when it runs in non-oracle mode

308 (see options @{text smt_solver} and @{text smt_oracle} above).

309 *}

311 declare [[ smt_trace_used_facts = false ]]

315 subsection {* Schematic rules for Z3 proof reconstruction *}

317 text {*

318 Several prof rules of Z3 are not very well documented. There are two

319 lemma groups which can turn failing Z3 proof reconstruction attempts

320 into succeeding ones: the facts in @{text z3_rule} are tried prior to

321 any implemented reconstruction procedure for all uncertain Z3 proof

322 rules; the facts in @{text z3_simp} are only fed to invocations of

323 the simplifier when reconstructing theory-specific proof steps.

324 *}

326 lemmas [z3_rule] =

327 refl eq_commute conj_commute disj_commute simp_thms nnf_simps

328 ring_distribs field_simps times_divide_eq_right times_divide_eq_left

329 if_True if_False not_not

331 lemma [z3_rule]:

332 "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"

333 "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"

334 "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"

335 "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"

336 "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"

337 "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"

338 "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"

339 "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"

340 by auto

342 lemma [z3_rule]:

343 "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"

344 "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"

345 "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"

346 "(True \<longrightarrow> P) = P"

347 "(P \<longrightarrow> True) = True"

348 "(False \<longrightarrow> P) = True"

349 "(P \<longrightarrow> P) = True"

350 by auto

352 lemma [z3_rule]:

353 "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"

354 by auto

356 lemma [z3_rule]:

357 "(\<not>True) = False"

358 "(\<not>False) = True"

359 "(x = x) = True"

360 "(P = True) = P"

361 "(True = P) = P"

362 "(P = False) = (\<not>P)"

363 "(False = P) = (\<not>P)"

364 "((\<not>P) = P) = False"

365 "(P = (\<not>P)) = False"

366 "((\<not>P) = (\<not>Q)) = (P = Q)"

367 "\<not>(P = (\<not>Q)) = (P = Q)"

368 "\<not>((\<not>P) = Q) = (P = Q)"

369 "(P \<noteq> Q) = (Q = (\<not>P))"

370 "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"

371 "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"

372 by auto

374 lemma [z3_rule]:

375 "(if P then P else \<not>P) = True"

376 "(if \<not>P then \<not>P else P) = True"

377 "(if P then True else False) = P"

378 "(if P then False else True) = (\<not>P)"

379 "(if P then Q else True) = ((\<not>P) \<or> Q)"

380 "(if P then Q else True) = (Q \<or> (\<not>P))"

381 "(if P then Q else \<not>Q) = (P = Q)"

382 "(if P then Q else \<not>Q) = (Q = P)"

383 "(if P then \<not>Q else Q) = (P = (\<not>Q))"

384 "(if P then \<not>Q else Q) = ((\<not>Q) = P)"

385 "(if \<not>P then x else y) = (if P then y else x)"

386 "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"

387 "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"

388 "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"

389 "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"

390 "(if P then x else if P then y else z) = (if P then x else z)"

391 "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"

392 "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"

393 "(if P then x = y else x = z) = (x = (if P then y else z))"

394 "(if P then x = y else y = z) = (y = (if P then x else z))"

395 "(if P then x = y else z = y) = (y = (if P then x else z))"

396 "f (if P then x else y) = (if P then f x else f y)"

397 by auto

399 lemma [z3_rule]:

400 "0 + (x::int) = x"

401 "x + 0 = x"

402 "x + x = 2 * x"

403 "0 * x = 0"

404 "1 * x = x"

405 "x + y = y + x"

406 by auto

408 lemma [z3_rule]: (* for def-axiom *)

409 "P = Q \<or> P \<or> Q"

410 "P = Q \<or> \<not>P \<or> \<not>Q"

411 "(\<not>P) = Q \<or> \<not>P \<or> Q"

412 "(\<not>P) = Q \<or> P \<or> \<not>Q"

413 "P = (\<not>Q) \<or> \<not>P \<or> Q"

414 "P = (\<not>Q) \<or> P \<or> \<not>Q"

415 "P \<noteq> Q \<or> P \<or> \<not>Q"

416 "P \<noteq> Q \<or> \<not>P \<or> Q"

417 "P \<noteq> (\<not>Q) \<or> P \<or> Q"

418 "(\<not>P) \<noteq> Q \<or> P \<or> Q"

419 "P \<or> Q \<or> P \<noteq> (\<not>Q)"

420 "P \<or> Q \<or> (\<not>P) \<noteq> Q"

421 "P \<or> \<not>Q \<or> P \<noteq> Q"

422 "\<not>P \<or> Q \<or> P \<noteq> Q"

423 "P \<or> y = (if P then x else y)"

424 "P \<or> (if P then x else y) = y"

425 "\<not>P \<or> x = (if P then x else y)"

426 "\<not>P \<or> (if P then x else y) = x"

427 "P \<or> R \<or> \<not>(if P then Q else R)"

428 "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"

429 "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"

430 "\<not>(if P then Q else R) \<or> P \<or> R"

431 "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"

432 "(if P then Q else R) \<or> P \<or> \<not>R"

433 "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"

434 "(if P then Q else \<not>R) \<or> P \<or> R"

435 by auto

439 hide_type (open) pattern

440 hide_const Pattern fun_app term_true term_false z3div z3mod

441 hide_const (open) trigger pat nopat weight

443 end