src/HOL/SMT.thy
author blanchet
Tue Aug 09 09:05:21 2011 +0200 (2011-08-09)
changeset 44087 8e491cb8841c
parent 43929 61d432e51aff
child 44488 587bf61a00a1
permissions -rw-r--r--
load lambda-lifting structure earlier, so it can be used in Metis
     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 uses
    10   "Tools/SMT/smt_utils.ML"
    11   "Tools/SMT/smt_failure.ML"
    12   "Tools/SMT/smt_config.ML"
    13   ("Tools/SMT/smt_builtin.ML")
    14   ("Tools/SMT/smt_datatypes.ML")
    15   ("Tools/SMT/smt_normalize.ML")
    16   ("Tools/SMT/smt_translate.ML")
    17   ("Tools/SMT/smt_solver.ML")
    18   ("Tools/SMT/smtlib_interface.ML")
    19   ("Tools/SMT/z3_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/smt_setup_solvers.ML")
    27 begin
    28 
    29 
    30 
    31 subsection {* Triggers for quantifier instantiation *}
    32 
    33 text {*
    34 Some SMT solvers support patterns as a quantifier instantiation
    35 heuristics.  Patterns may either be positive terms (tagged by "pat")
    36 triggering quantifier instantiations -- when the solver finds a
    37 term matching a positive pattern, it instantiates the corresponding
    38 quantifier accordingly -- or negative terms (tagged by "nopat")
    39 inhibiting quantifier instantiations.  A list of patterns
    40 of the same kind is called a multipattern, and all patterns in a
    41 multipattern are considered conjunctively for quantifier instantiation.
    42 A list of multipatterns is called a trigger, and their multipatterns
    43 act disjunctively during quantifier instantiation.  Each multipattern
    44 should mention at least all quantified variables of the preceding
    45 quantifier block.
    46 *}
    47 
    48 datatype pattern = Pattern
    49 
    50 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    51 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    52 
    53 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    54 where "trigger _ P = P"
    55 
    56 
    57 
    58 subsection {* Quantifier weights *}
    59 
    60 text {*
    61 Weight annotations to quantifiers influence the priority of quantifier
    62 instantiations.  They should be handled with care for solvers, which support
    63 them, because incorrect choices of weights might render a problem unsolvable.
    64 *}
    65 
    66 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
    67 
    68 text {*
    69 Weights must be non-negative.  The value @{text 0} is equivalent to providing
    70 no weight at all.
    71 
    72 Weights should only be used at quantifiers and only inside triggers (if the
    73 quantifier has triggers).  Valid usages of weights are as follows:
    74 
    75 \begin{itemize}
    76 \item
    77 @{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}
    78 \item
    79 @{term "\<forall>x. weight 3 (P x)"}
    80 \end{itemize}
    81 *}
    82 
    83 
    84 
    85 subsection {* Higher-order encoding *}
    86 
    87 text {*
    88 Application is made explicit for constants occurring with varying
    89 numbers of arguments.  This is achieved by the introduction of the
    90 following constant.
    91 *}
    92 
    93 definition fun_app where "fun_app f = f"
    94 
    95 text {*
    96 Some solvers support a theory of arrays which can be used to encode
    97 higher-order functions.  The following set of lemmas specifies the
    98 properties of such (extensional) arrays.
    99 *}
   100 
   101 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
   102   fun_upd_upd fun_app_def
   103 
   104 
   105 
   106 subsection {* First-order logic *}
   107 
   108 text {*
   109 Some SMT solvers only accept problems in first-order logic, i.e.,
   110 where formulas and terms are syntactically separated. When
   111 translating higher-order into first-order problems, all
   112 uninterpreted constants (those not built-in in the target solver)
   113 are treated as function symbols in the first-order sense.  Their
   114 occurrences as head symbols in atoms (i.e., as predicate symbols) are
   115 turned into terms by logically equating such atoms with @{term True}.
   116 For technical reasons, @{term True} and @{term False} occurring inside
   117 terms are replaced by the following constants.
   118 *}
   119 
   120 definition term_true where "term_true = True"
   121 definition term_false where "term_false = False"
   122 
   123 
   124 
   125 subsection {* Integer division and modulo for Z3 *}
   126 
   127 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
   128   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
   129 
   130 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
   131   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
   132 
   133 
   134 
   135 subsection {* Setup *}
   136 
   137 use "Tools/SMT/smt_builtin.ML"
   138 use "Tools/SMT/smt_datatypes.ML"
   139 use "Tools/SMT/smt_normalize.ML"
   140 use "Tools/SMT/smt_translate.ML"
   141 use "Tools/SMT/smt_solver.ML"
   142 use "Tools/SMT/smtlib_interface.ML"
   143 use "Tools/SMT/z3_interface.ML"
   144 use "Tools/SMT/z3_proof_parser.ML"
   145 use "Tools/SMT/z3_proof_tools.ML"
   146 use "Tools/SMT/z3_proof_literals.ML"
   147 use "Tools/SMT/z3_proof_methods.ML"
   148 use "Tools/SMT/z3_proof_reconstruction.ML"
   149 use "Tools/SMT/z3_model.ML"
   150 use "Tools/SMT/smt_setup_solvers.ML"
   151 
   152 setup {*
   153   SMT_Config.setup #>
   154   SMT_Normalize.setup #>
   155   SMT_Solver.setup #>
   156   SMTLIB_Interface.setup #>
   157   Z3_Interface.setup #>
   158   Z3_Proof_Reconstruction.setup #>
   159   SMT_Setup_Solvers.setup
   160 *}
   161 
   162 
   163 
   164 subsection {* Configuration *}
   165 
   166 text {*
   167 The current configuration can be printed by the command
   168 @{text smt_status}, which shows the values of most options.
   169 *}
   170 
   171 
   172 
   173 subsection {* General configuration options *}
   174 
   175 text {*
   176 The option @{text smt_solver} can be used to change the target SMT
   177 solver.  The possible values can be obtained from the @{text smt_status}
   178 command.
   179 
   180 Due to licensing restrictions, Yices and Z3 are not installed/enabled
   181 by default.  Z3 is free for non-commercial applications and can be enabled
   182 by simply setting the environment variable @{text Z3_NON_COMMERCIAL} to
   183 @{text yes}.
   184 *}
   185 
   186 declare [[ smt_solver = z3 ]]
   187 
   188 text {*
   189 Since SMT solvers are potentially non-terminating, there is a timeout
   190 (given in seconds) to restrict their runtime.  A value greater than
   191 120 (seconds) is in most cases not advisable.
   192 *}
   193 
   194 declare [[ smt_timeout = 20 ]]
   195 
   196 text {*
   197 SMT solvers apply randomized heuristics.  In case a problem is not
   198 solvable by an SMT solver, changing the following option might help.
   199 *}
   200 
   201 declare [[ smt_random_seed = 1 ]]
   202 
   203 text {*
   204 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   205 solvers are fully trusted without additional checks.  The following
   206 option can cause the SMT solver to run in proof-producing mode, giving
   207 a checkable certificate.  This is currently only implemented for Z3.
   208 *}
   209 
   210 declare [[ smt_oracle = false ]]
   211 
   212 text {*
   213 Each SMT solver provides several commandline options to tweak its
   214 behaviour.  They can be passed to the solver by setting the following
   215 options.
   216 *}
   217 
   218 declare [[ cvc3_options = "", remote_cvc3_options = "" ]]
   219 declare [[ yices_options = "" ]]
   220 declare [[ z3_options = "", remote_z3_options = "" ]]
   221 
   222 text {*
   223 Enable the following option to use built-in support for datatypes and
   224 records.  Currently, this is only implemented for Z3 running in oracle
   225 mode.
   226 *}
   227 
   228 declare [[ smt_datatypes = false ]]
   229 
   230 text {*
   231 The SMT method provides an inference mechanism to detect simple triggers
   232 in quantified formulas, which might increase the number of problems
   233 solvable by SMT solvers (note: triggers guide quantifier instantiations
   234 in the SMT solver).  To turn it on, set the following option.
   235 *}
   236 
   237 declare [[ smt_infer_triggers = false ]]
   238 
   239 text {*
   240 The SMT method monomorphizes the given facts, that is, it tries to
   241 instantiate all schematic type variables with fixed types occurring
   242 in the problem.  This is a (possibly nonterminating) fixed-point
   243 construction whose cycles are limited by the following option.
   244 *}
   245 
   246 declare [[ monomorph_max_rounds = 5 ]]
   247 
   248 text {*
   249 In addition, the number of generated monomorphic instances is limited
   250 by the following option.
   251 *}
   252 
   253 declare [[ monomorph_max_new_instances = 500 ]]
   254 
   255 
   256 
   257 subsection {* Certificates *}
   258 
   259 text {*
   260 By setting the option @{text smt_certificates} to the name of a file,
   261 all following applications of an SMT solver a cached in that file.
   262 Any further application of the same SMT solver (using the very same
   263 configuration) re-uses the cached certificate instead of invoking the
   264 solver.  An empty string disables caching certificates.
   265 
   266 The filename should be given as an explicit path.  It is good
   267 practice to use the name of the current theory (with ending
   268 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   269 *}
   270 
   271 declare [[ smt_certificates = "" ]]
   272 
   273 text {*
   274 The option @{text smt_fixed} controls whether only stored
   275 certificates are should be used or invocation of an SMT solver is
   276 allowed.  When set to @{text true}, no SMT solver will ever be
   277 invoked and only the existing certificates found in the configured
   278 cache are used;  when set to @{text false} and there is no cached
   279 certificate for some proposition, then the configured SMT solver is
   280 invoked.
   281 *}
   282 
   283 declare [[ smt_fixed = false ]]
   284 
   285 
   286 
   287 subsection {* Tracing *}
   288 
   289 text {*
   290 The SMT method, when applied, traces important information.  To
   291 make it entirely silent, set the following option to @{text false}.
   292 *}
   293 
   294 declare [[ smt_verbose = true ]]
   295 
   296 text {*
   297 For tracing the generated problem file given to the SMT solver as
   298 well as the returned result of the solver, the option
   299 @{text smt_trace} should be set to @{text true}.
   300 *}
   301 
   302 declare [[ smt_trace = false ]]
   303 
   304 text {*
   305 From the set of assumptions given to the SMT solver, those assumptions
   306 used in the proof are traced when the following option is set to
   307 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   308 (see options @{text smt_solver} and @{text smt_oracle} above).
   309 *}
   310 
   311 declare [[ smt_trace_used_facts = false ]]
   312 
   313 
   314 
   315 subsection {* Schematic rules for Z3 proof reconstruction *}
   316 
   317 text {*
   318 Several prof rules of Z3 are not very well documented.  There are two
   319 lemma groups which can turn failing Z3 proof reconstruction attempts
   320 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   321 any implemented reconstruction procedure for all uncertain Z3 proof
   322 rules;  the facts in @{text z3_simp} are only fed to invocations of
   323 the simplifier when reconstructing theory-specific proof steps.
   324 *}
   325 
   326 lemmas [z3_rule] =
   327   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   328   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   329   if_True if_False not_not
   330 
   331 lemma [z3_rule]:
   332   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   333   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   334   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   335   by auto
   336 
   337 lemma [z3_rule]:
   338   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   339   by auto
   340 
   341 lemma [z3_rule]:
   342   "((\<not>P) = P) = False"
   343   "(P = (\<not>P)) = False"
   344   "(P \<noteq> Q) = (Q = (\<not>P))"
   345   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   346   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   347   by auto
   348 
   349 lemma [z3_rule]:
   350   "(if P then P else \<not>P) = True"
   351   "(if \<not>P then \<not>P else P) = True"
   352   "(if P then True else False) = P"
   353   "(if P then False else True) = (\<not>P)"
   354   "(if \<not>P then x else y) = (if P then y else x)"
   355   "f (if P then x else y) = (if P then f x else f y)"
   356   by auto
   357 
   358 lemma [z3_rule]:
   359   "P = Q \<or> P \<or> Q"
   360   "P = Q \<or> \<not>P \<or> \<not>Q"
   361   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   362   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   363   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   364   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   365   "P \<noteq> Q \<or> P \<or> \<not>Q"
   366   "P \<noteq> Q \<or> \<not>P \<or> Q"
   367   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   368   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   369   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   370   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   371   "P \<or> \<not>Q \<or> P \<noteq> Q"
   372   "\<not>P \<or> Q \<or> P \<noteq> Q"
   373   by auto
   374 
   375 lemma [z3_rule]:
   376   "0 + (x::int) = x"
   377   "x + 0 = x"
   378   "0 * x = 0"
   379   "1 * x = x"
   380   "x + y = y + x"
   381   by auto
   382 
   383 
   384 
   385 hide_type (open) pattern
   386 hide_const Pattern fun_app term_true term_false z3div z3mod
   387 hide_const (open) trigger pat nopat weight
   388 
   389 end