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src/HOL/SMT.thy

author | boehmes |

Sun Dec 19 18:54:29 2010 +0100 (2010-12-19) | |

changeset 41281 | 679118e35378 |

parent 41280 | a7de9d36f4f2 |

child 41328 | 6792a5c92a58 |

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

removed odd decoration of built-in symbols as Vars (instead provide built-in desctructor functions along with their inverse functions);

removed odd retyping during folify (instead, keep all terms well-typed)

removed odd retyping during folify (instead, keep all terms well-typed)

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 List

9 uses

10 "Tools/Datatype/datatype_selectors.ML"

11 "Tools/SMT/smt_utils.ML"

12 "Tools/SMT/smt_failure.ML"

13 "Tools/SMT/smt_config.ML"

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

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

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

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

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

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

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

28 begin

32 subsection {* Triggers for quantifier instantiation *}

34 text {*

35 Some SMT solvers support patterns as a quantifier instantiation

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

37 triggering quantifier instantiations -- when the solver finds a

38 term matching a positive pattern, it instantiates the corresponding

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

40 inhibiting quantifier instantiations. A list of patterns

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

42 multipattern are considered conjunctively for quantifier instantiation.

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

44 act disjunctively during quantifier instantiation. Each multipattern

45 should mention at least all quantified variables of the preceding

46 quantifier block.

47 *}

49 datatype pattern = Pattern

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

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

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

55 where "trigger _ P = P"

59 subsection {* Quantifier weights *}

61 text {*

62 Weight annotations to quantifiers influence the priority of quantifier

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

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

65 *}

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

69 text {*

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

71 no weight at all.

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

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

76 \begin{itemize}

77 \item

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

79 \item

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

81 \end{itemize}

82 *}

86 subsection {* Higher-order encoding *}

88 text {*

89 Application is made explicit for constants occurring with varying

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

91 following constant.

92 *}

94 definition fun_app where "fun_app f = f"

96 text {*

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

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

99 properties of such (extensional) arrays.

100 *}

102 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other

103 fun_upd_upd fun_app_def

107 subsection {* First-order logic *}

109 text {*

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

111 where formulas and terms are syntactically separated. When

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

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

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

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

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

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

118 terms are replaced by the following constants.

119 *}

121 definition term_true where "term_true = True"

122 definition term_false where "term_false = False"

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

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

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

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

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

137 subsection {* Setup *}

139 use "Tools/SMT/smt_monomorph.ML"

140 use "Tools/SMT/smt_builtin.ML"

141 use "Tools/SMT/smt_normalize.ML"

142 use "Tools/SMT/smt_translate.ML"

143 use "Tools/SMT/smt_solver.ML"

144 use "Tools/SMT/smtlib_interface.ML"

145 use "Tools/SMT/z3_interface.ML"

146 use "Tools/SMT/z3_proof_parser.ML"

147 use "Tools/SMT/z3_proof_tools.ML"

148 use "Tools/SMT/z3_proof_literals.ML"

149 use "Tools/SMT/z3_proof_methods.ML"

150 use "Tools/SMT/z3_proof_reconstruction.ML"

151 use "Tools/SMT/z3_model.ML"

152 use "Tools/SMT/smt_setup_solvers.ML"

154 setup {*

155 SMT_Config.setup #>

156 SMT_Normalize.setup #>

157 SMT_Solver.setup #>

158 SMTLIB_Interface.setup #>

159 Z3_Interface.setup #>

160 Z3_Proof_Reconstruction.setup #>

161 SMT_Setup_Solvers.setup

162 *}

166 subsection {* Configuration *}

168 text {*

169 The current configuration can be printed by the command

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

171 *}

175 subsection {* General configuration options *}

177 text {*

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

179 solver. The possible values are @{text cvc3}, @{text yices}, and

180 @{text z3}. It is advisable to locally install the selected solver,

181 although this is not necessary for @{text cvc3} and @{text z3}, which

182 can also be used over an Internet-based service.

184 When using local SMT solvers, the path to their binaries should be

185 declared by setting the following environment variables:

186 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.

187 *}

189 declare [[ smt_solver = z3 ]]

191 text {*

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

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

194 120 (seconds) is in most cases not advisable.

195 *}

197 declare [[ smt_timeout = 20 ]]

199 text {*

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

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

202 *}

204 declare [[ smt_random_seed = 1 ]]

206 text {*

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

208 solvers are fully trusted without additional checks. The following

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

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

211 *}

213 declare [[ smt_oracle = false ]]

215 text {*

216 Each SMT solver provides several commandline options to tweak its

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

218 options.

219 *}

221 declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]]

223 text {*

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

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

226 mode.

227 *}

229 declare [[ smt_datatypes = false ]]

231 text {*

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

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

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

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

236 *}

238 declare [[ smt_infer_triggers = false ]]

240 text {*

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

242 instantiate all schematic type variables with fixed types occurring

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

244 construction whose cycles are limited by the following option.

245 *}

247 declare [[ smt_monomorph_limit = 10 ]]

251 subsection {* Certificates *}

253 text {*

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

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

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

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

258 solver. An empty string disables caching certificates.

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

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

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

263 *}

265 declare [[ smt_certificates = "" ]]

267 text {*

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

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

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

271 invoked and only the existing certificates found in the configured

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

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

274 invoked.

275 *}

277 declare [[ smt_fixed = false ]]

281 subsection {* Tracing *}

283 text {*

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

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

286 *}

288 declare [[ smt_verbose = true ]]

290 text {*

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

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

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

294 *}

296 declare [[ smt_trace = false ]]

298 text {*

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

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

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

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

303 *}

305 declare [[ smt_trace_used_facts = false ]]

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

311 text {*

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

313 lemma groups which can turn failing Z3 proof reconstruction attempts

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

315 any implemented reconstruction procedure for all uncertain Z3 proof

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

317 the simplifier when reconstructing theory-specific proof steps.

318 *}

320 lemmas [z3_rule] =

321 refl eq_commute conj_commute disj_commute simp_thms nnf_simps

322 ring_distribs field_simps times_divide_eq_right times_divide_eq_left

323 if_True if_False not_not

325 lemma [z3_rule]:

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

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

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

329 by auto

331 lemma [z3_rule]:

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

333 by auto

335 lemma [z3_rule]:

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

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

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

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

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

341 by auto

343 lemma [z3_rule]:

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

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

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

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

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

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

350 by auto

352 lemma [z3_rule]:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

367 by auto

369 lemma [z3_rule]:

370 "0 + (x::int) = x"

371 "x + 0 = x"

372 "0 * x = 0"

373 "1 * x = x"

374 "x + y = y + x"

375 by auto

379 hide_type (open) pattern

380 hide_const Pattern fun_app term_true term_false z3div z3mod

381 hide_const (open) trigger pat nopat weight

385 subsection {* Selectors for datatypes *}

387 setup {* Datatype_Selectors.setup *}

389 declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]

390 declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]

392 end