src/HOL/SMT.thy
 author blanchet Mon Sep 29 14:32:30 2014 +0200 (2014-09-29) changeset 58481 62bc7c79212b parent 58441 c1b489999de9 child 58598 d9892c88cb56 permissions -rw-r--r--
```     1 (*  Title:      HOL/SMT.thy
```
```     2     Author:     Sascha Boehme, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2 *}
```
```     6
```
```     7 theory SMT
```
```     8 imports Divides
```
```     9 keywords "smt_status" :: diag
```
```    10 begin
```
```    11
```
```    12 subsection {* A skolemization tactic and proof method *}
```
```    13
```
```    14 lemma choices:
```
```    15   "\<And>Q. \<forall>x. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x. Q x (f x) (fa x)"
```
```    16   "\<And>Q. \<forall>x. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x. Q x (f x) (fa x) (fb x)"
```
```    17   "\<And>Q. \<forall>x. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x. Q x (f x) (fa x) (fb x) (fc x)"
```
```    18   by metis+
```
```    19
```
```    20 lemma bchoices:
```
```    21   "\<And>Q. \<forall>x \<in> S. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x \<in> S. Q x (f x) (fa x)"
```
```    22   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x \<in> S. Q x (f x) (fa x) (fb x)"
```
```    23   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x)"
```
```    24   by metis+
```
```    25
```
```    26 ML {*
```
```    27 fun moura_tac ctxt =
```
```    28   Atomize_Elim.atomize_elim_tac ctxt THEN'
```
```    29   SELECT_GOAL (Clasimp.auto_tac (ctxt addSIs @{thms choice choices bchoice bchoices}) THEN
```
```    30     ALLGOALS (blast_tac ctxt));
```
```    31 *}
```
```    32
```
```    33 method_setup moura = {*
```
```    34  Scan.succeed (SIMPLE_METHOD' o moura_tac)
```
```    35 *} "solve skolemization goals, especially those arising from Z3 proofs"
```
```    36
```
```    37 hide_fact (open) choices bchoices
```
```    38
```
```    39
```
```    40 subsection {* Triggers for quantifier instantiation *}
```
```    41
```
```    42 text {*
```
```    43 Some SMT solvers support patterns as a quantifier instantiation
```
```    44 heuristics. Patterns may either be positive terms (tagged by "pat")
```
```    45 triggering quantifier instantiations -- when the solver finds a
```
```    46 term matching a positive pattern, it instantiates the corresponding
```
```    47 quantifier accordingly -- or negative terms (tagged by "nopat")
```
```    48 inhibiting quantifier instantiations. A list of patterns
```
```    49 of the same kind is called a multipattern, and all patterns in a
```
```    50 multipattern are considered conjunctively for quantifier instantiation.
```
```    51 A list of multipatterns is called a trigger, and their multipatterns
```
```    52 act disjunctively during quantifier instantiation. Each multipattern
```
```    53 should mention at least all quantified variables of the preceding
```
```    54 quantifier block.
```
```    55 *}
```
```    56
```
```    57 typedecl 'a symb_list
```
```    58
```
```    59 consts
```
```    60   Symb_Nil :: "'a symb_list"
```
```    61   Symb_Cons :: "'a \<Rightarrow> 'a symb_list \<Rightarrow> 'a symb_list"
```
```    62
```
```    63 typedecl pattern
```
```    64
```
```    65 consts
```
```    66   pat :: "'a \<Rightarrow> pattern"
```
```    67   nopat :: "'a \<Rightarrow> pattern"
```
```    68
```
```    69 definition trigger :: "pattern symb_list symb_list \<Rightarrow> bool \<Rightarrow> bool" where
```
```    70   "trigger _ P = P"
```
```    71
```
```    72
```
```    73 subsection {* Higher-order encoding *}
```
```    74
```
```    75 text {*
```
```    76 Application is made explicit for constants occurring with varying
```
```    77 numbers of arguments. This is achieved by the introduction of the
```
```    78 following constant.
```
```    79 *}
```
```    80
```
```    81 definition fun_app :: "'a \<Rightarrow> 'a" where "fun_app f = f"
```
```    82
```
```    83 text {*
```
```    84 Some solvers support a theory of arrays which can be used to encode
```
```    85 higher-order functions. The following set of lemmas specifies the
```
```    86 properties of such (extensional) arrays.
```
```    87 *}
```
```    88
```
```    89 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def
```
```    90
```
```    91
```
```    92 subsection {* Normalization *}
```
```    93
```
```    94 lemma case_bool_if[abs_def]: "case_bool x y P = (if P then x else y)"
```
```    95   by simp
```
```    96
```
```    97 lemmas Ex1_def_raw = Ex1_def[abs_def]
```
```    98 lemmas Ball_def_raw = Ball_def[abs_def]
```
```    99 lemmas Bex_def_raw = Bex_def[abs_def]
```
```   100 lemmas abs_if_raw = abs_if[abs_def]
```
```   101 lemmas min_def_raw = min_def[abs_def]
```
```   102 lemmas max_def_raw = max_def[abs_def]
```
```   103
```
```   104
```
```   105 subsection {* Integer division and modulo for Z3 *}
```
```   106
```
```   107 text {*
```
```   108 The following Z3-inspired definitions are overspecified for the case where @{text "l = 0"}. This
```
```   109 Schönheitsfehler is corrected in the @{text div_as_z3div} and @{text mod_as_z3mod} theorems.
```
```   110 *}
```
```   111
```
```   112 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
```
```   113   "z3div k l = (if l \<ge> 0 then k div l else - (k div - l))"
```
```   114
```
```   115 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
```
```   116   "z3mod k l = k mod (if l \<ge> 0 then l else - l)"
```
```   117
```
```   118 lemma div_as_z3div:
```
```   119   "\<forall>k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"
```
```   120   by (simp add: z3div_def)
```
```   121
```
```   122 lemma mod_as_z3mod:
```
```   123   "\<forall>k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"
```
```   124   by (simp add: z3mod_def)
```
```   125
```
```   126
```
```   127 subsection {* Setup *}
```
```   128
```
```   129 ML_file "Tools/SMT/smt_util.ML"
```
```   130 ML_file "Tools/SMT/smt_failure.ML"
```
```   131 ML_file "Tools/SMT/smt_config.ML"
```
```   132 ML_file "Tools/SMT/smt_builtin.ML"
```
```   133 ML_file "Tools/SMT/smt_datatypes.ML"
```
```   134 ML_file "Tools/SMT/smt_normalize.ML"
```
```   135 ML_file "Tools/SMT/smt_translate.ML"
```
```   136 ML_file "Tools/SMT/smtlib.ML"
```
```   137 ML_file "Tools/SMT/smtlib_interface.ML"
```
```   138 ML_file "Tools/SMT/smtlib_proof.ML"
```
```   139 ML_file "Tools/SMT/smtlib_isar.ML"
```
```   140 ML_file "Tools/SMT/z3_proof.ML"
```
```   141 ML_file "Tools/SMT/z3_isar.ML"
```
```   142 ML_file "Tools/SMT/smt_solver.ML"
```
```   143 ML_file "Tools/SMT/cvc4_interface.ML"
```
```   144 ML_file "Tools/SMT/verit_proof.ML"
```
```   145 ML_file "Tools/SMT/verit_isar.ML"
```
```   146 ML_file "Tools/SMT/verit_proof_parse.ML"
```
```   147 ML_file "Tools/SMT/z3_interface.ML"
```
```   148 ML_file "Tools/SMT/z3_replay_util.ML"
```
```   149 ML_file "Tools/SMT/z3_replay_literals.ML"
```
```   150 ML_file "Tools/SMT/z3_replay_rules.ML"
```
```   151 ML_file "Tools/SMT/z3_replay_methods.ML"
```
```   152 ML_file "Tools/SMT/z3_replay.ML"
```
```   153 ML_file "Tools/SMT/smt_systems.ML"
```
```   154
```
```   155 method_setup smt = {*
```
```   156   Scan.optional Attrib.thms [] >>
```
```   157     (fn thms => fn ctxt =>
```
```   158       METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))
```
```   159 *} "apply an SMT solver to the current goal"
```
```   160
```
```   161
```
```   162 subsection {* Configuration *}
```
```   163
```
```   164 text {*
```
```   165 The current configuration can be printed by the command
```
```   166 @{text smt_status}, which shows the values of most options.
```
```   167 *}
```
```   168
```
```   169
```
```   170 subsection {* General configuration options *}
```
```   171
```
```   172 text {*
```
```   173 The option @{text smt_solver} can be used to change the target SMT
```
```   174 solver. The possible values can be obtained from the @{text smt_status}
```
```   175 command.
```
```   176
```
```   177 Due to licensing restrictions, Z3 is not enabled by default. Z3 is free
```
```   178 for non-commercial applications and can be enabled by setting Isabelle
```
```   179 system option @{text z3_non_commercial} to @{text yes}.
```
```   180 *}
```
```   181
```
```   182 declare [[smt_solver = z3]]
```
```   183
```
```   184 text {*
```
```   185 Since SMT solvers are potentially nonterminating, there is a timeout
```
```   186 (given in seconds) to restrict their runtime.
```
```   187 *}
```
```   188
```
```   189 declare [[smt_timeout = 20]]
```
```   190
```
```   191 text {*
```
```   192 SMT solvers apply randomized heuristics. In case a problem is not
```
```   193 solvable by an SMT solver, changing the following option might help.
```
```   194 *}
```
```   195
```
```   196 declare [[smt_random_seed = 1]]
```
```   197
```
```   198 text {*
```
```   199 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
```
```   200 solvers are fully trusted without additional checks. The following
```
```   201 option can cause the SMT solver to run in proof-producing mode, giving
```
```   202 a checkable certificate. This is currently only implemented for Z3.
```
```   203 *}
```
```   204
```
```   205 declare [[smt_oracle = false]]
```
```   206
```
```   207 text {*
```
```   208 Each SMT solver provides several commandline options to tweak its
```
```   209 behaviour. They can be passed to the solver by setting the following
```
```   210 options.
```
```   211 *}
```
```   212
```
```   213 declare [[cvc3_options = ""]]
```
```   214 declare [[cvc4_options = "--full-saturate-quant --quant-cf"]]
```
```   215 declare [[veriT_options = ""]]
```
```   216 declare [[z3_options = ""]]
```
```   217
```
```   218 text {*
```
```   219 The SMT method provides an inference mechanism to detect simple triggers
```
```   220 in quantified formulas, which might increase the number of problems
```
```   221 solvable by SMT solvers (note: triggers guide quantifier instantiations
```
```   222 in the SMT solver). To turn it on, set the following option.
```
```   223 *}
```
```   224
```
```   225 declare [[smt_infer_triggers = false]]
```
```   226
```
```   227 text {*
```
```   228 Enable the following option to use built-in support for datatypes,
```
```   229 codatatypes, and records in CVC4. Currently, this is implemented only
```
```   230 in oracle mode.
```
```   231 *}
```
```   232
```
```   233 declare [[cvc4_extensions = false]]
```
```   234
```
```   235 text {*
```
```   236 Enable the following option to use built-in support for div/mod, datatypes,
```
```   237 and records in Z3. Currently, this is implemented only in oracle mode.
```
```   238 *}
```
```   239
```
```   240 declare [[z3_extensions = false]]
```
```   241
```
```   242
```
```   243 subsection {* Certificates *}
```
```   244
```
```   245 text {*
```
```   246 By setting the option @{text smt_certificates} to the name of a file,
```
```   247 all following applications of an SMT solver a cached in that file.
```
```   248 Any further application of the same SMT solver (using the very same
```
```   249 configuration) re-uses the cached certificate instead of invoking the
```
```   250 solver. An empty string disables caching certificates.
```
```   251
```
```   252 The filename should be given as an explicit path. It is good
```
```   253 practice to use the name of the current theory (with ending
```
```   254 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
```
```   255 Certificate files should be used at most once in a certain theory context,
```
```   256 to avoid race conditions with other concurrent accesses.
```
```   257 *}
```
```   258
```
```   259 declare [[smt_certificates = ""]]
```
```   260
```
```   261 text {*
```
```   262 The option @{text smt_read_only_certificates} controls whether only
```
```   263 stored certificates are should be used or invocation of an SMT solver
```
```   264 is allowed. When set to @{text true}, no SMT solver will ever be
```
```   265 invoked and only the existing certificates found in the configured
```
```   266 cache are used;  when set to @{text false} and there is no cached
```
```   267 certificate for some proposition, then the configured SMT solver is
```
```   268 invoked.
```
```   269 *}
```
```   270
```
```   271 declare [[smt_read_only_certificates = false]]
```
```   272
```
```   273
```
```   274 subsection {* Tracing *}
```
```   275
```
```   276 text {*
```
```   277 The SMT method, when applied, traces important information. To
```
```   278 make it entirely silent, set the following option to @{text false}.
```
```   279 *}
```
```   280
```
```   281 declare [[smt_verbose = true]]
```
```   282
```
```   283 text {*
```
```   284 For tracing the generated problem file given to the SMT solver as
```
```   285 well as the returned result of the solver, the option
```
```   286 @{text smt_trace} should be set to @{text true}.
```
```   287 *}
```
```   288
```
```   289 declare [[smt_trace = false]]
```
```   290
```
```   291
```
```   292 subsection {* Schematic rules for Z3 proof reconstruction *}
```
```   293
```
```   294 text {*
```
```   295 Several prof rules of Z3 are not very well documented. There are two
```
```   296 lemma groups which can turn failing Z3 proof reconstruction attempts
```
```   297 into succeeding ones: the facts in @{text z3_rule} are tried prior to
```
```   298 any implemented reconstruction procedure for all uncertain Z3 proof
```
```   299 rules;  the facts in @{text z3_simp} are only fed to invocations of
```
```   300 the simplifier when reconstructing theory-specific proof steps.
```
```   301 *}
```
```   302
```
```   303 lemmas [z3_rule] =
```
```   304   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
```
```   305   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
```
```   306   if_True if_False not_not
```
```   307
```
```   308 lemma [z3_rule]:
```
```   309   "(P \<and> Q) = (\<not> (\<not> P \<or> \<not> Q))"
```
```   310   "(P \<and> Q) = (\<not> (\<not> Q \<or> \<not> P))"
```
```   311   "(\<not> P \<and> Q) = (\<not> (P \<or> \<not> Q))"
```
```   312   "(\<not> P \<and> Q) = (\<not> (\<not> Q \<or> P))"
```
```   313   "(P \<and> \<not> Q) = (\<not> (\<not> P \<or> Q))"
```
```   314   "(P \<and> \<not> Q) = (\<not> (Q \<or> \<not> P))"
```
```   315   "(\<not> P \<and> \<not> Q) = (\<not> (P \<or> Q))"
```
```   316   "(\<not> P \<and> \<not> Q) = (\<not> (Q \<or> P))"
```
```   317   by auto
```
```   318
```
```   319 lemma [z3_rule]:
```
```   320   "(P \<longrightarrow> Q) = (Q \<or> \<not> P)"
```
```   321   "(\<not> P \<longrightarrow> Q) = (P \<or> Q)"
```
```   322   "(\<not> P \<longrightarrow> Q) = (Q \<or> P)"
```
```   323   "(True \<longrightarrow> P) = P"
```
```   324   "(P \<longrightarrow> True) = True"
```
```   325   "(False \<longrightarrow> P) = True"
```
```   326   "(P \<longrightarrow> P) = True"
```
```   327   by auto
```
```   328
```
```   329 lemma [z3_rule]:
```
```   330   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not> P)))"
```
```   331   by auto
```
```   332
```
```   333 lemma [z3_rule]:
```
```   334   "(\<not> True) = False"
```
```   335   "(\<not> False) = True"
```
```   336   "(x = x) = True"
```
```   337   "(P = True) = P"
```
```   338   "(True = P) = P"
```
```   339   "(P = False) = (\<not> P)"
```
```   340   "(False = P) = (\<not> P)"
```
```   341   "((\<not> P) = P) = False"
```
```   342   "(P = (\<not> P)) = False"
```
```   343   "((\<not> P) = (\<not> Q)) = (P = Q)"
```
```   344   "\<not> (P = (\<not> Q)) = (P = Q)"
```
```   345   "\<not> ((\<not> P) = Q) = (P = Q)"
```
```   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
```
```   350
```
```   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 P then Q else True) = ((\<not> P) \<or> Q)"
```
```   357   "(if P then Q else True) = (Q \<or> (\<not> P))"
```
```   358   "(if P then Q else \<not> Q) = (P = Q)"
```
```   359   "(if P then Q else \<not> Q) = (Q = P)"
```
```   360   "(if P then \<not> Q else Q) = (P = (\<not> Q))"
```
```   361   "(if P then \<not> Q else Q) = ((\<not> Q) = P)"
```
```   362   "(if \<not> P then x else y) = (if P then y else x)"
```
```   363   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not> Q) then y else x)"
```
```   364   "(if P then (if Q then x else y) else x) = (if (\<not> Q) \<and> P then y else x)"
```
```   365   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
```
```   366   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
```
```   367   "(if P then x else if P then y else z) = (if P then x else z)"
```
```   368   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
```
```   369   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
```
```   370   "(if P then x = y else x = z) = (x = (if P then y else z))"
```
```   371   "(if P then x = y else y = z) = (y = (if P then x else z))"
```
```   372   "(if P then x = y else z = y) = (y = (if P then x else z))"
```
```   373   by auto
```
```   374
```
```   375 lemma [z3_rule]:
```
```   376   "0 + (x::int) = x"
```
```   377   "x + 0 = x"
```
```   378   "x + x = 2 * x"
```
```   379   "0 * x = 0"
```
```   380   "1 * x = x"
```
```   381   "x + y = y + x"
```
```   382   by (auto simp add: mult_2)
```
```   383
```
```   384 lemma [z3_rule]:  (* for def-axiom *)
```
```   385   "P = Q \<or> P \<or> Q"
```
```   386   "P = Q \<or> \<not> P \<or> \<not> Q"
```
```   387   "(\<not> P) = Q \<or> \<not> P \<or> Q"
```
```   388   "(\<not> P) = Q \<or> P \<or> \<not> Q"
```
```   389   "P = (\<not> Q) \<or> \<not> P \<or> Q"
```
```   390   "P = (\<not> Q) \<or> P \<or> \<not> Q"
```
```   391   "P \<noteq> Q \<or> P \<or> \<not> Q"
```
```   392   "P \<noteq> Q \<or> \<not> P \<or> Q"
```
```   393   "P \<noteq> (\<not> Q) \<or> P \<or> Q"
```
```   394   "(\<not> P) \<noteq> Q \<or> P \<or> Q"
```
```   395   "P \<or> Q \<or> P \<noteq> (\<not> Q)"
```
```   396   "P \<or> Q \<or> (\<not> P) \<noteq> Q"
```
```   397   "P \<or> \<not> Q \<or> P \<noteq> Q"
```
```   398   "\<not> P \<or> Q \<or> P \<noteq> Q"
```
```   399   "P \<or> y = (if P then x else y)"
```
```   400   "P \<or> (if P then x else y) = y"
```
```   401   "\<not> P \<or> x = (if P then x else y)"
```
```   402   "\<not> P \<or> (if P then x else y) = x"
```
```   403   "P \<or> R \<or> \<not> (if P then Q else R)"
```
```   404   "\<not> P \<or> Q \<or> \<not> (if P then Q else R)"
```
```   405   "\<not> (if P then Q else R) \<or> \<not> P \<or> Q"
```
```   406   "\<not> (if P then Q else R) \<or> P \<or> R"
```
```   407   "(if P then Q else R) \<or> \<not> P \<or> \<not> Q"
```
```   408   "(if P then Q else R) \<or> P \<or> \<not> R"
```
```   409   "(if P then \<not> Q else R) \<or> \<not> P \<or> Q"
```
```   410   "(if P then Q else \<not> R) \<or> P \<or> R"
```
```   411   by auto
```
```   412
```
```   413 hide_type (open) symb_list pattern
```
```   414 hide_const (open) Symb_Nil Symb_Cons trigger pat nopat fun_app z3div z3mod
```
```   415
```
```   416 end
```