src/HOL/SMT.thy
author blanchet
Tue Mar 27 16:59:13 2012 +0300 (2012-03-27)
changeset 47152 446cfc760ccf
parent 46950 d0181abdbdac
child 47701 157e6108a342
permissions -rw-r--r--
renamed "smt_fixed" to "smt_read_only_certificates"
     1 (*  Title:      HOL/SMT.thy
     2     Author:     Sascha Boehme, TU Muenchen
     3 *)
     4 
     5 header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *}
     6 
     7 theory SMT
     8 imports Record
     9 keywords "smt_status" :: diag
    10 uses
    11   "Tools/SMT/smt_utils.ML"
    12   "Tools/SMT/smt_failure.ML"
    13   "Tools/SMT/smt_config.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
    29 
    30 
    31 
    32 subsection {* Triggers for quantifier instantiation *}
    33 
    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 *}
    48 
    49 datatype pattern = Pattern
    50 
    51 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    52 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    53 
    54 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    55 where "trigger _ P = P"
    56 
    57 
    58 
    59 subsection {* Quantifier weights *}
    60 
    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 *}
    66 
    67 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
    68 
    69 text {*
    70 Weights must be non-negative.  The value @{text 0} is equivalent to providing
    71 no weight at all.
    72 
    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:
    75 
    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 *}
    83 
    84 
    85 
    86 subsection {* Higher-order encoding *}
    87 
    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 *}
    93 
    94 definition fun_app where "fun_app f = f"
    95 
    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 *}
   101 
   102 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
   103   fun_upd_upd fun_app_def
   104 
   105 
   106 
   107 subsection {* First-order logic *}
   108 
   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 *}
   120 
   121 definition term_true where "term_true = True"
   122 definition term_false where "term_false = False"
   123 
   124 
   125 
   126 subsection {* Integer division and modulo for Z3 *}
   127 
   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)))"
   130 
   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))"
   133 
   134 
   135 
   136 subsection {* Setup *}
   137 
   138 use "Tools/SMT/smt_builtin.ML"
   139 use "Tools/SMT/smt_datatypes.ML"
   140 use "Tools/SMT/smt_normalize.ML"
   141 use "Tools/SMT/smt_translate.ML"
   142 use "Tools/SMT/smt_solver.ML"
   143 use "Tools/SMT/smtlib_interface.ML"
   144 use "Tools/SMT/z3_interface.ML"
   145 use "Tools/SMT/z3_proof_parser.ML"
   146 use "Tools/SMT/z3_proof_tools.ML"
   147 use "Tools/SMT/z3_proof_literals.ML"
   148 use "Tools/SMT/z3_proof_methods.ML"
   149 use "Tools/SMT/z3_proof_reconstruction.ML"
   150 use "Tools/SMT/z3_model.ML"
   151 use "Tools/SMT/smt_setup_solvers.ML"
   152 
   153 setup {*
   154   SMT_Config.setup #>
   155   SMT_Normalize.setup #>
   156   SMT_Solver.setup #>
   157   SMTLIB_Interface.setup #>
   158   Z3_Interface.setup #>
   159   Z3_Proof_Reconstruction.setup #>
   160   SMT_Setup_Solvers.setup
   161 *}
   162 
   163 
   164 
   165 subsection {* Configuration *}
   166 
   167 text {*
   168 The current configuration can be printed by the command
   169 @{text smt_status}, which shows the values of most options.
   170 *}
   171 
   172 
   173 
   174 subsection {* General configuration options *}
   175 
   176 text {*
   177 The option @{text smt_solver} can be used to change the target SMT
   178 solver.  The possible values can be obtained from the @{text smt_status}
   179 command.
   180 
   181 Due to licensing restrictions, Yices and Z3 are not installed/enabled
   182 by default.  Z3 is free for non-commercial applications and can be enabled
   183 by simply setting the environment variable @{text Z3_NON_COMMERCIAL} to
   184 @{text yes}.
   185 *}
   186 
   187 declare [[ smt_solver = z3 ]]
   188 
   189 text {*
   190 Since SMT solvers are potentially non-terminating, there is a timeout
   191 (given in seconds) to restrict their runtime.  A value greater than
   192 120 (seconds) is in most cases not advisable.
   193 *}
   194 
   195 declare [[ smt_timeout = 20 ]]
   196 
   197 text {*
   198 SMT solvers apply randomized heuristics.  In case a problem is not
   199 solvable by an SMT solver, changing the following option might help.
   200 *}
   201 
   202 declare [[ smt_random_seed = 1 ]]
   203 
   204 text {*
   205 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   206 solvers are fully trusted without additional checks.  The following
   207 option can cause the SMT solver to run in proof-producing mode, giving
   208 a checkable certificate.  This is currently only implemented for Z3.
   209 *}
   210 
   211 declare [[ smt_oracle = false ]]
   212 
   213 text {*
   214 Each SMT solver provides several commandline options to tweak its
   215 behaviour.  They can be passed to the solver by setting the following
   216 options.
   217 *}
   218 
   219 declare [[ cvc3_options = "", remote_cvc3_options = "" ]]
   220 declare [[ yices_options = "" ]]
   221 declare [[ z3_options = "", remote_z3_options = "" ]]
   222 
   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 *}
   228 
   229 declare [[ smt_datatypes = false ]]
   230 
   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 *}
   237 
   238 declare [[ smt_infer_triggers = false ]]
   239 
   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 *}
   246 
   247 declare [[ monomorph_max_rounds = 5 ]]
   248 
   249 text {*
   250 In addition, the number of generated monomorphic instances is limited
   251 by the following option.
   252 *}
   253 
   254 declare [[ monomorph_max_new_instances = 500 ]]
   255 
   256 
   257 
   258 subsection {* Certificates *}
   259 
   260 text {*
   261 By setting the option @{text smt_certificates} to the name of a file,
   262 all following applications of an SMT solver a cached in that file.
   263 Any further application of the same SMT solver (using the very same
   264 configuration) re-uses the cached certificate instead of invoking the
   265 solver.  An empty string disables caching certificates.
   266 
   267 The filename should be given as an explicit path.  It is good
   268 practice to use the name of the current theory (with ending
   269 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   270 *}
   271 
   272 declare [[ smt_certificates = "" ]]
   273 
   274 text {*
   275 The option @{text smt_read_only_certificates} controls whether only
   276 stored certificates are should be used or invocation of an SMT solver
   277 is allowed.  When set to @{text true}, no SMT solver will ever be
   278 invoked and only the existing certificates found in the configured
   279 cache are used;  when set to @{text false} and there is no cached
   280 certificate for some proposition, then the configured SMT solver is
   281 invoked.
   282 *}
   283 
   284 declare [[ smt_read_only_certificates = false ]]
   285 
   286 
   287 
   288 subsection {* Tracing *}
   289 
   290 text {*
   291 The SMT method, when applied, traces important information.  To
   292 make it entirely silent, set the following option to @{text false}.
   293 *}
   294 
   295 declare [[ smt_verbose = true ]]
   296 
   297 text {*
   298 For tracing the generated problem file given to the SMT solver as
   299 well as the returned result of the solver, the option
   300 @{text smt_trace} should be set to @{text true}.
   301 *}
   302 
   303 declare [[ smt_trace = false ]]
   304 
   305 text {*
   306 From the set of assumptions given to the SMT solver, those assumptions
   307 used in the proof are traced when the following option is set to
   308 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   309 (see options @{text smt_solver} and @{text smt_oracle} above).
   310 *}
   311 
   312 declare [[ smt_trace_used_facts = false ]]
   313 
   314 
   315 
   316 subsection {* Schematic rules for Z3 proof reconstruction *}
   317 
   318 text {*
   319 Several prof rules of Z3 are not very well documented.  There are two
   320 lemma groups which can turn failing Z3 proof reconstruction attempts
   321 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   322 any implemented reconstruction procedure for all uncertain Z3 proof
   323 rules;  the facts in @{text z3_simp} are only fed to invocations of
   324 the simplifier when reconstructing theory-specific proof steps.
   325 *}
   326 
   327 lemmas [z3_rule] =
   328   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   329   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   330   if_True if_False not_not
   331 
   332 lemma [z3_rule]:
   333   "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
   334   "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
   335   "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
   336   "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
   337   "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
   338   "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
   339   "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
   340   "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
   341   by auto
   342 
   343 lemma [z3_rule]:
   344   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   345   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   346   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   347   "(True \<longrightarrow> P) = P"
   348   "(P \<longrightarrow> True) = True"
   349   "(False \<longrightarrow> P) = True"
   350   "(P \<longrightarrow> P) = True"
   351   by auto
   352 
   353 lemma [z3_rule]:
   354   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   355   by auto
   356 
   357 lemma [z3_rule]:
   358   "(\<not>True) = False"
   359   "(\<not>False) = True"
   360   "(x = x) = True"
   361   "(P = True) = P"
   362   "(True = P) = P"
   363   "(P = False) = (\<not>P)"
   364   "(False = P) = (\<not>P)"
   365   "((\<not>P) = P) = False"
   366   "(P = (\<not>P)) = False"
   367   "((\<not>P) = (\<not>Q)) = (P = Q)"
   368   "\<not>(P = (\<not>Q)) = (P = Q)"
   369   "\<not>((\<not>P) = Q) = (P = Q)"
   370   "(P \<noteq> Q) = (Q = (\<not>P))"
   371   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   372   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   373   by auto
   374 
   375 lemma [z3_rule]:
   376   "(if P then P else \<not>P) = True"
   377   "(if \<not>P then \<not>P else P) = True"
   378   "(if P then True else False) = P"
   379   "(if P then False else True) = (\<not>P)"
   380   "(if P then Q else True) = ((\<not>P) \<or> Q)"
   381   "(if P then Q else True) = (Q \<or> (\<not>P))"
   382   "(if P then Q else \<not>Q) = (P = Q)"
   383   "(if P then Q else \<not>Q) = (Q = P)"
   384   "(if P then \<not>Q else Q) = (P = (\<not>Q))"
   385   "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
   386   "(if \<not>P then x else y) = (if P then y else x)"
   387   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
   388   "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
   389   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
   390   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
   391   "(if P then x else if P then y else z) = (if P then x else z)"
   392   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
   393   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
   394   "(if P then x = y else x = z) = (x = (if P then y else z))"
   395   "(if P then x = y else y = z) = (y = (if P then x else z))"
   396   "(if P then x = y else z = y) = (y = (if P then x else z))"
   397   by auto
   398 
   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
   407 
   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
   436 
   437 
   438 
   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
   442 
   443 end