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src/HOL/SMT.thy

author | boehmes |

Tue Jun 07 10:24:16 2011 +0200 (2011-06-07) | |

changeset 43230 | dabf6e311213 |

parent 41762 | 00060198de12 |

child 43927 | 3a87cb597832 |

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

clarified meaning of monomorphization configuration option by renaming it

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_monomorph.ML")

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

15 ("Tools/SMT/smt_datatypes.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_interface.ML")

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

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

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

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

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

26 ("Tools/SMT/z3_model.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"

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

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

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

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

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

136 subsection {* Setup *}

138 use "Tools/SMT/smt_monomorph.ML"

139 use "Tools/SMT/smt_builtin.ML"

140 use "Tools/SMT/smt_datatypes.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 can be obtained from the @{text smt_status}

180 command.

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

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

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

185 @{text yes}.

186 *}

188 declare [[ smt_solver = z3 ]]

190 text {*

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

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

193 120 (seconds) is in most cases not advisable.

194 *}

196 declare [[ smt_timeout = 20 ]]

198 text {*

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

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

201 *}

203 declare [[ smt_random_seed = 1 ]]

205 text {*

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

207 solvers are fully trusted without additional checks. The following

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

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

210 *}

212 declare [[ smt_oracle = false ]]

214 text {*

215 Each SMT solver provides several commandline options to tweak its

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

217 options.

218 *}

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

221 declare [[ yices_options = "" ]]

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

224 text {*

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

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

227 mode.

228 *}

230 declare [[ smt_datatypes = false ]]

232 text {*

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

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

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

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

237 *}

239 declare [[ smt_infer_triggers = false ]]

241 text {*

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

243 instantiate all schematic type variables with fixed types occurring

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

245 construction whose cycles are limited by the following option.

246 *}

248 declare [[ monomorph_max_rounds = 5 ]]

250 text {*

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

252 by the following option.

253 *}

255 declare [[ monomorph_max_new_instances = 500 ]]

259 subsection {* Certificates *}

261 text {*

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

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

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

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

266 solver. An empty string disables caching certificates.

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

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

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

271 *}

273 declare [[ smt_certificates = "" ]]

275 text {*

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

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

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

279 invoked and only the existing certificates found in the configured

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

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

282 invoked.

283 *}

285 declare [[ smt_fixed = false ]]

289 subsection {* Tracing *}

291 text {*

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

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

294 *}

296 declare [[ smt_verbose = true ]]

298 text {*

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

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

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

302 *}

304 declare [[ smt_trace = false ]]

306 text {*

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

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

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

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

311 *}

313 declare [[ smt_trace_used_facts = false ]]

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

319 text {*

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

321 lemma groups which can turn failing Z3 proof reconstruction attempts

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

323 any implemented reconstruction procedure for all uncertain Z3 proof

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

325 the simplifier when reconstructing theory-specific proof steps.

326 *}

328 lemmas [z3_rule] =

329 refl eq_commute conj_commute disj_commute simp_thms nnf_simps

330 ring_distribs field_simps times_divide_eq_right times_divide_eq_left

331 if_True if_False not_not

333 lemma [z3_rule]:

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

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

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

337 by auto

339 lemma [z3_rule]:

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

341 by auto

343 lemma [z3_rule]:

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

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

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

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

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

349 by auto

351 lemma [z3_rule]:

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

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

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

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

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

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

358 by auto

360 lemma [z3_rule]:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

375 by auto

377 lemma [z3_rule]:

378 "0 + (x::int) = x"

379 "x + 0 = x"

380 "0 * x = 0"

381 "1 * x = x"

382 "x + y = y + x"

383 by auto

387 hide_type (open) pattern

388 hide_const Pattern fun_app term_true term_false z3div z3mod

389 hide_const (open) trigger pat nopat weight

391 end