src/HOL/SMT.thy
author boehmes
Sun Jan 19 22:38:17 2014 +0100 (2014-01-19)
changeset 55049 327eafb594ba
parent 55007 0c07990363a3
child 56046 683148f3ae48
permissions -rw-r--r--
removed obsolete remote_cvc3 and remote_z3
     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 begin
    11 
    12 ML_file "Tools/SMT/smt_utils.ML"
    13 ML_file "Tools/SMT/smt_failure.ML"
    14 ML_file "Tools/SMT/smt_config.ML"
    15 
    16 
    17 subsection {* Triggers for quantifier instantiation *}
    18 
    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 *}
    33 
    34 datatype pattern = Pattern
    35 
    36 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    37 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    38 
    39 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    40 where "trigger _ P = P"
    41 
    42 
    43 
    44 subsection {* Quantifier weights *}
    45 
    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 *}
    51 
    52 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
    53 
    54 text {*
    55 Weights must be non-negative.  The value @{text 0} is equivalent to providing
    56 no weight at all.
    57 
    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:
    60 
    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 *}
    68 
    69 
    70 
    71 subsection {* Higher-order encoding *}
    72 
    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 *}
    78 
    79 definition fun_app where "fun_app f = f"
    80 
    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 *}
    86 
    87 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
    88   fun_upd_upd fun_app_def
    89 
    90 
    91 
    92 subsection {* First-order logic *}
    93 
    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 *}
   105 
   106 definition term_true where "term_true = True"
   107 definition term_false where "term_false = False"
   108 
   109 
   110 
   111 subsection {* Integer division and modulo for Z3 *}
   112 
   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)))"
   115 
   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))"
   118 
   119 
   120 
   121 subsection {* Setup *}
   122 
   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"
   137 
   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 *}
   146 
   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"
   152 
   153 
   154 subsection {* Configuration *}
   155 
   156 text {*
   157 The current configuration can be printed by the command
   158 @{text smt_status}, which shows the values of most options.
   159 *}
   160 
   161 
   162 
   163 subsection {* General configuration options *}
   164 
   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.
   169 
   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 setting Isabelle system option @{text z3_non_commercial} to @{text yes}.
   173 *}
   174 
   175 declare [[ smt_solver = z3 ]]
   176 
   177 text {*
   178 Since SMT solvers are potentially non-terminating, there is a timeout
   179 (given in seconds) to restrict their runtime.  A value greater than
   180 120 (seconds) is in most cases not advisable.
   181 *}
   182 
   183 declare [[ smt_timeout = 20 ]]
   184 
   185 text {*
   186 SMT solvers apply randomized heuristics.  In case a problem is not
   187 solvable by an SMT solver, changing the following option might help.
   188 *}
   189 
   190 declare [[ smt_random_seed = 1 ]]
   191 
   192 text {*
   193 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   194 solvers are fully trusted without additional checks.  The following
   195 option can cause the SMT solver to run in proof-producing mode, giving
   196 a checkable certificate.  This is currently only implemented for Z3.
   197 *}
   198 
   199 declare [[ smt_oracle = false ]]
   200 
   201 text {*
   202 Each SMT solver provides several commandline options to tweak its
   203 behaviour.  They can be passed to the solver by setting the following
   204 options.
   205 *}
   206 
   207 declare [[ cvc3_options = "" ]]
   208 declare [[ yices_options = "" ]]
   209 declare [[ z3_options = "" ]]
   210 
   211 text {*
   212 Enable the following option to use built-in support for datatypes and
   213 records.  Currently, this is only implemented for Z3 running in oracle
   214 mode.
   215 *}
   216 
   217 declare [[ smt_datatypes = false ]]
   218 
   219 text {*
   220 The SMT method provides an inference mechanism to detect simple triggers
   221 in quantified formulas, which might increase the number of problems
   222 solvable by SMT solvers (note: triggers guide quantifier instantiations
   223 in the SMT solver).  To turn it on, set the following option.
   224 *}
   225 
   226 declare [[ smt_infer_triggers = false ]]
   227 
   228 text {*
   229 The SMT method monomorphizes the given facts, that is, it tries to
   230 instantiate all schematic type variables with fixed types occurring
   231 in the problem.  This is a (possibly nonterminating) fixed-point
   232 construction whose cycles are limited by the following option.
   233 *}
   234 
   235 declare [[ monomorph_max_rounds = 5 ]]
   236 
   237 text {*
   238 In addition, the number of generated monomorphic instances is limited
   239 by the following option.
   240 *}
   241 
   242 declare [[ monomorph_max_new_instances = 500 ]]
   243 
   244 
   245 
   246 subsection {* Certificates *}
   247 
   248 text {*
   249 By setting the option @{text smt_certificates} to the name of a file,
   250 all following applications of an SMT solver a cached in that file.
   251 Any further application of the same SMT solver (using the very same
   252 configuration) re-uses the cached certificate instead of invoking the
   253 solver.  An empty string disables caching certificates.
   254 
   255 The filename should be given as an explicit path.  It is good
   256 practice to use the name of the current theory (with ending
   257 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   258 Certificate files should be used at most once in a certain theory context,
   259 to avoid race conditions with other concurrent accesses.
   260 *}
   261 
   262 declare [[ smt_certificates = "" ]]
   263 
   264 text {*
   265 The option @{text smt_read_only_certificates} controls whether only
   266 stored certificates are should be used or invocation of an SMT solver
   267 is allowed.  When set to @{text true}, no SMT solver will ever be
   268 invoked and only the existing certificates found in the configured
   269 cache are used;  when set to @{text false} and there is no cached
   270 certificate for some proposition, then the configured SMT solver is
   271 invoked.
   272 *}
   273 
   274 declare [[ smt_read_only_certificates = false ]]
   275 
   276 
   277 
   278 subsection {* Tracing *}
   279 
   280 text {*
   281 The SMT method, when applied, traces important information.  To
   282 make it entirely silent, set the following option to @{text false}.
   283 *}
   284 
   285 declare [[ smt_verbose = true ]]
   286 
   287 text {*
   288 For tracing the generated problem file given to the SMT solver as
   289 well as the returned result of the solver, the option
   290 @{text smt_trace} should be set to @{text true}.
   291 *}
   292 
   293 declare [[ smt_trace = false ]]
   294 
   295 text {*
   296 From the set of assumptions given to the SMT solver, those assumptions
   297 used in the proof are traced when the following option is set to
   298 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   299 (see options @{text smt_solver} and @{text smt_oracle} above).
   300 *}
   301 
   302 declare [[ smt_trace_used_facts = false ]]
   303 
   304 
   305 
   306 subsection {* Schematic rules for Z3 proof reconstruction *}
   307 
   308 text {*
   309 Several prof rules of Z3 are not very well documented.  There are two
   310 lemma groups which can turn failing Z3 proof reconstruction attempts
   311 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   312 any implemented reconstruction procedure for all uncertain Z3 proof
   313 rules;  the facts in @{text z3_simp} are only fed to invocations of
   314 the simplifier when reconstructing theory-specific proof steps.
   315 *}
   316 
   317 lemmas [z3_rule] =
   318   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   319   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   320   if_True if_False not_not
   321 
   322 lemma [z3_rule]:
   323   "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
   324   "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
   325   "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
   326   "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
   327   "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
   328   "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
   329   "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
   330   "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
   331   by auto
   332 
   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   "(True \<longrightarrow> P) = P"
   338   "(P \<longrightarrow> True) = True"
   339   "(False \<longrightarrow> P) = True"
   340   "(P \<longrightarrow> P) = True"
   341   by auto
   342 
   343 lemma [z3_rule]:
   344   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   345   by auto
   346 
   347 lemma [z3_rule]:
   348   "(\<not>True) = False"
   349   "(\<not>False) = True"
   350   "(x = x) = True"
   351   "(P = True) = P"
   352   "(True = P) = P"
   353   "(P = False) = (\<not>P)"
   354   "(False = P) = (\<not>P)"
   355   "((\<not>P) = P) = False"
   356   "(P = (\<not>P)) = False"
   357   "((\<not>P) = (\<not>Q)) = (P = Q)"
   358   "\<not>(P = (\<not>Q)) = (P = Q)"
   359   "\<not>((\<not>P) = Q) = (P = Q)"
   360   "(P \<noteq> Q) = (Q = (\<not>P))"
   361   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   362   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   363   by auto
   364 
   365 lemma [z3_rule]:
   366   "(if P then P else \<not>P) = True"
   367   "(if \<not>P then \<not>P else P) = True"
   368   "(if P then True else False) = P"
   369   "(if P then False else True) = (\<not>P)"
   370   "(if P then Q else True) = ((\<not>P) \<or> Q)"
   371   "(if P then Q else True) = (Q \<or> (\<not>P))"
   372   "(if P then Q else \<not>Q) = (P = Q)"
   373   "(if P then Q else \<not>Q) = (Q = P)"
   374   "(if P then \<not>Q else Q) = (P = (\<not>Q))"
   375   "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
   376   "(if \<not>P then x else y) = (if P then y else x)"
   377   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
   378   "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
   379   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
   380   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
   381   "(if P then x else if P then y else z) = (if P then x else z)"
   382   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
   383   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
   384   "(if P then x = y else x = z) = (x = (if P then y else z))"
   385   "(if P then x = y else y = z) = (y = (if P then x else z))"
   386   "(if P then x = y else z = y) = (y = (if P then x else z))"
   387   by auto
   388 
   389 lemma [z3_rule]:
   390   "0 + (x::int) = x"
   391   "x + 0 = x"
   392   "x + x = 2 * x"
   393   "0 * x = 0"
   394   "1 * x = x"
   395   "x + y = y + x"
   396   by auto
   397 
   398 lemma [z3_rule]:  (* for def-axiom *)
   399   "P = Q \<or> P \<or> Q"
   400   "P = Q \<or> \<not>P \<or> \<not>Q"
   401   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   402   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   403   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   404   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   405   "P \<noteq> Q \<or> P \<or> \<not>Q"
   406   "P \<noteq> Q \<or> \<not>P \<or> Q"
   407   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   408   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   409   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   410   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   411   "P \<or> \<not>Q \<or> P \<noteq> Q"
   412   "\<not>P \<or> Q \<or> P \<noteq> Q"
   413   "P \<or> y = (if P then x else y)"
   414   "P \<or> (if P then x else y) = y"
   415   "\<not>P \<or> x = (if P then x else y)"
   416   "\<not>P \<or>  (if P then x else y) = x"
   417   "P \<or> R \<or> \<not>(if P then Q else R)"
   418   "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"
   419   "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"
   420   "\<not>(if P then Q else R) \<or> P \<or> R"
   421   "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"
   422   "(if P then Q else R) \<or> P \<or> \<not>R"
   423   "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"
   424   "(if P then Q else \<not>R) \<or> P \<or> R"
   425   by auto
   426 
   427 
   428 
   429 hide_type (open) pattern
   430 hide_const Pattern fun_app term_true term_false z3div z3mod
   431 hide_const (open) trigger pat nopat weight
   432 
   433 end