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src/HOL/SMT.thy

author | wenzelm |

Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) | |

changeset 53374 | a14d2a854c02 |

parent 50317 | 4d1590544b91 |

child 55007 | 0c07990363a3 |

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

tuned proofs -- clarified flow of facts wrt. calculation;

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 keywords "smt_status" :: diag

10 begin

12 ML_file "Tools/SMT/smt_utils.ML"

13 ML_file "Tools/SMT/smt_failure.ML"

14 ML_file "Tools/SMT/smt_config.ML"

17 subsection {* Triggers for quantifier instantiation *}

19 text {*

20 Some SMT solvers support patterns as a quantifier instantiation

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

22 triggering quantifier instantiations -- when the solver finds a

23 term matching a positive pattern, it instantiates the corresponding

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

25 inhibiting quantifier instantiations. A list of patterns

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

27 multipattern are considered conjunctively for quantifier instantiation.

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

29 act disjunctively during quantifier instantiation. Each multipattern

30 should mention at least all quantified variables of the preceding

31 quantifier block.

32 *}

34 datatype pattern = Pattern

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

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

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

40 where "trigger _ P = P"

44 subsection {* Quantifier weights *}

46 text {*

47 Weight annotations to quantifiers influence the priority of quantifier

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

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

50 *}

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

54 text {*

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

56 no weight at all.

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

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

61 \begin{itemize}

62 \item

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

64 \item

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

66 \end{itemize}

67 *}

71 subsection {* Higher-order encoding *}

73 text {*

74 Application is made explicit for constants occurring with varying

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

76 following constant.

77 *}

79 definition fun_app where "fun_app f = f"

81 text {*

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

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

84 properties of such (extensional) arrays.

85 *}

87 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other

88 fun_upd_upd fun_app_def

92 subsection {* First-order logic *}

94 text {*

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

96 where formulas and terms are syntactically separated. When

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

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

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

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

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

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

103 terms are replaced by the following constants.

104 *}

106 definition term_true where "term_true = True"

107 definition term_false where "term_false = False"

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

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

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

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

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

121 subsection {* Setup *}

123 ML_file "Tools/SMT/smt_builtin.ML"

124 ML_file "Tools/SMT/smt_datatypes.ML"

125 ML_file "Tools/SMT/smt_normalize.ML"

126 ML_file "Tools/SMT/smt_translate.ML"

127 ML_file "Tools/SMT/smt_solver.ML"

128 ML_file "Tools/SMT/smtlib_interface.ML"

129 ML_file "Tools/SMT/z3_interface.ML"

130 ML_file "Tools/SMT/z3_proof_parser.ML"

131 ML_file "Tools/SMT/z3_proof_tools.ML"

132 ML_file "Tools/SMT/z3_proof_literals.ML"

133 ML_file "Tools/SMT/z3_proof_methods.ML"

134 ML_file "Tools/SMT/z3_proof_reconstruction.ML"

135 ML_file "Tools/SMT/z3_model.ML"

136 ML_file "Tools/SMT/smt_setup_solvers.ML"

138 setup {*

139 SMT_Config.setup #>

140 SMT_Normalize.setup #>

141 SMTLIB_Interface.setup #>

142 Z3_Interface.setup #>

143 Z3_Proof_Reconstruction.setup #>

144 SMT_Setup_Solvers.setup

145 *}

147 method_setup smt = {*

148 Scan.optional Attrib.thms [] >>

149 (fn thms => fn ctxt =>

150 METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))

151 *} "apply an SMT solver to the current goal"

154 subsection {* Configuration *}

156 text {*

157 The current configuration can be printed by the command

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

159 *}

163 subsection {* General configuration options *}

165 text {*

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

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

168 command.

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

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

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

173 @{text yes}.

174 *}

176 declare [[ smt_solver = z3 ]]

178 text {*

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

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

181 120 (seconds) is in most cases not advisable.

182 *}

184 declare [[ smt_timeout = 20 ]]

186 text {*

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

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

189 *}

191 declare [[ smt_random_seed = 1 ]]

193 text {*

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

195 solvers are fully trusted without additional checks. The following

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

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

198 *}

200 declare [[ smt_oracle = false ]]

202 text {*

203 Each SMT solver provides several commandline options to tweak its

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

205 options.

206 *}

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

209 declare [[ yices_options = "" ]]

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

212 text {*

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

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

215 mode.

216 *}

218 declare [[ smt_datatypes = false ]]

220 text {*

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

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

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

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

225 *}

227 declare [[ smt_infer_triggers = false ]]

229 text {*

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

231 instantiate all schematic type variables with fixed types occurring

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

233 construction whose cycles are limited by the following option.

234 *}

236 declare [[ monomorph_max_rounds = 5 ]]

238 text {*

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

240 by the following option.

241 *}

243 declare [[ monomorph_max_new_instances = 500 ]]

247 subsection {* Certificates *}

249 text {*

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

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

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

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

254 solver. An empty string disables caching certificates.

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

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

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

259 Certificate files should be used at most once in a certain theory context,

260 to avoid race conditions with other concurrent accesses.

261 *}

263 declare [[ smt_certificates = "" ]]

265 text {*

266 The option @{text smt_read_only_certificates} controls whether only

267 stored certificates are should be used or invocation of an SMT solver

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

269 invoked and only the existing certificates found in the configured

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

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

272 invoked.

273 *}

275 declare [[ smt_read_only_certificates = false ]]

279 subsection {* Tracing *}

281 text {*

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

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

284 *}

286 declare [[ smt_verbose = true ]]

288 text {*

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

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

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

292 *}

294 declare [[ smt_trace = false ]]

296 text {*

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

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

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

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

301 *}

303 declare [[ smt_trace_used_facts = false ]]

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

309 text {*

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

311 lemma groups which can turn failing Z3 proof reconstruction attempts

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

313 any implemented reconstruction procedure for all uncertain Z3 proof

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

315 the simplifier when reconstructing theory-specific proof steps.

316 *}

318 lemmas [z3_rule] =

319 refl eq_commute conj_commute disj_commute simp_thms nnf_simps

320 ring_distribs field_simps times_divide_eq_right times_divide_eq_left

321 if_True if_False not_not

323 lemma [z3_rule]:

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

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

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

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

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

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

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

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

332 by auto

334 lemma [z3_rule]:

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

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

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

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

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

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

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

342 by auto

344 lemma [z3_rule]:

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

346 by auto

348 lemma [z3_rule]:

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

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

351 "(x = x) = True"

352 "(P = True) = P"

353 "(True = P) = P"

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

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

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

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

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

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

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

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

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

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

364 by auto

366 lemma [z3_rule]:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

388 by auto

390 lemma [z3_rule]:

391 "0 + (x::int) = x"

392 "x + 0 = x"

393 "x + x = 2 * x"

394 "0 * x = 0"

395 "1 * x = x"

396 "x + y = y + x"

397 by auto

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

426 by auto

430 hide_type (open) pattern

431 hide_const Pattern fun_app term_true term_false z3div z3mod

432 hide_const (open) trigger pat nopat weight

434 end