src/HOL/SMT.thy
author boehmes
Thu Jan 06 17:51:56 2011 +0100 (2011-01-06)
changeset 41432 3214c39777ab
parent 41426 09615ed31f04
child 41459 f0db8f40d656
permissions -rw-r--r--
differentiate between local and remote SMT solvers (e.g., "z3" vs. "remote_z3");
turned individual SMT solvers into components;
made CVC3 the default SMT solver (Z3 is licensed as "non-commercial only");
tuned smt_filter interface
     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_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
    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_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"
   153 
   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 *}
   163 
   164 
   165 
   166 subsection {* Configuration *}
   167 
   168 text {*
   169 The current configuration can be printed by the command
   170 @{text smt_status}, which shows the values of most options.
   171 *}
   172 
   173 
   174 
   175 subsection {* General configuration options *}
   176 
   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.
   181 *}
   182 
   183 declare [[ smt_solver = cvc3 ]]
   184 
   185 text {*
   186 Since SMT solvers are potentially non-terminating, there is a timeout
   187 (given in seconds) to restrict their runtime.  A value greater than
   188 120 (seconds) is in most cases not advisable.
   189 *}
   190 
   191 declare [[ smt_timeout = 20 ]]
   192 
   193 text {*
   194 SMT solvers apply randomized heuristics.  In case a problem is not
   195 solvable by an SMT solver, changing the following option might help.
   196 *}
   197 
   198 declare [[ smt_random_seed = 1 ]]
   199 
   200 text {*
   201 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   202 solvers are fully trusted without additional checks.  The following
   203 option can cause the SMT solver to run in proof-producing mode, giving
   204 a checkable certificate.  This is currently only implemented for Z3.
   205 *}
   206 
   207 declare [[ smt_oracle = false ]]
   208 
   209 text {*
   210 Each SMT solver provides several commandline options to tweak its
   211 behaviour.  They can be passed to the solver by setting the following
   212 options.
   213 *}
   214 
   215 declare [[ cvc3_options = "", remote_cvc3_options = "" ]]
   216 declare [[ yices_options = "" ]]
   217 declare [[ z3_options = "", remote_z3_options = "" ]]
   218 
   219 text {*
   220 Enable the following option to use built-in support for datatypes and
   221 records.  Currently, this is only implemented for Z3 running in oracle
   222 mode.
   223 *}
   224 
   225 declare [[ smt_datatypes = false ]]
   226 
   227 text {*
   228 The SMT method provides an inference mechanism to detect simple triggers
   229 in quantified formulas, which might increase the number of problems
   230 solvable by SMT solvers (note: triggers guide quantifier instantiations
   231 in the SMT solver).  To turn it on, set the following option.
   232 *}
   233 
   234 declare [[ smt_infer_triggers = false ]]
   235 
   236 text {*
   237 The SMT method monomorphizes the given facts, that is, it tries to
   238 instantiate all schematic type variables with fixed types occurring
   239 in the problem.  This is a (possibly nonterminating) fixed-point
   240 construction whose cycles are limited by the following option.
   241 *}
   242 
   243 declare [[ smt_monomorph_limit = 10 ]]
   244 
   245 
   246 
   247 subsection {* Certificates *}
   248 
   249 text {*
   250 By setting the option @{text smt_certificates} to the name of a file,
   251 all following applications of an SMT solver a cached in that file.
   252 Any further application of the same SMT solver (using the very same
   253 configuration) re-uses the cached certificate instead of invoking the
   254 solver.  An empty string disables caching certificates.
   255 
   256 The filename should be given as an explicit path.  It is good
   257 practice to use the name of the current theory (with ending
   258 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   259 *}
   260 
   261 declare [[ smt_certificates = "" ]]
   262 
   263 text {*
   264 The option @{text smt_fixed} controls whether only stored
   265 certificates are should be used or invocation of an SMT solver is
   266 allowed.  When set to @{text true}, no SMT solver will ever be
   267 invoked and only the existing certificates found in the configured
   268 cache are used;  when set to @{text false} and there is no cached
   269 certificate for some proposition, then the configured SMT solver is
   270 invoked.
   271 *}
   272 
   273 declare [[ smt_fixed = false ]]
   274 
   275 
   276 
   277 subsection {* Tracing *}
   278 
   279 text {*
   280 The SMT method, when applied, traces important information.  To
   281 make it entirely silent, set the following option to @{text false}.
   282 *}
   283 
   284 declare [[ smt_verbose = true ]]
   285 
   286 text {*
   287 For tracing the generated problem file given to the SMT solver as
   288 well as the returned result of the solver, the option
   289 @{text smt_trace} should be set to @{text true}.
   290 *}
   291 
   292 declare [[ smt_trace = false ]]
   293 
   294 text {*
   295 From the set of assumptions given to the SMT solver, those assumptions
   296 used in the proof are traced when the following option is set to
   297 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   298 (see options @{text smt_solver} and @{text smt_oracle} above).
   299 *}
   300 
   301 declare [[ smt_trace_used_facts = false ]]
   302 
   303 
   304 
   305 subsection {* Schematic rules for Z3 proof reconstruction *}
   306 
   307 text {*
   308 Several prof rules of Z3 are not very well documented.  There are two
   309 lemma groups which can turn failing Z3 proof reconstruction attempts
   310 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   311 any implemented reconstruction procedure for all uncertain Z3 proof
   312 rules;  the facts in @{text z3_simp} are only fed to invocations of
   313 the simplifier when reconstructing theory-specific proof steps.
   314 *}
   315 
   316 lemmas [z3_rule] =
   317   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   318   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   319   if_True if_False not_not
   320 
   321 lemma [z3_rule]:
   322   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   323   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   324   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   325   by auto
   326 
   327 lemma [z3_rule]:
   328   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   329   by auto
   330 
   331 lemma [z3_rule]:
   332   "((\<not>P) = P) = False"
   333   "(P = (\<not>P)) = False"
   334   "(P \<noteq> Q) = (Q = (\<not>P))"
   335   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   336   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   337   by auto
   338 
   339 lemma [z3_rule]:
   340   "(if P then P else \<not>P) = True"
   341   "(if \<not>P then \<not>P else P) = True"
   342   "(if P then True else False) = P"
   343   "(if P then False else True) = (\<not>P)"
   344   "(if \<not>P then x else y) = (if P then y else x)"
   345   "f (if P then x else y) = (if P then f x else f y)"
   346   by auto
   347 
   348 lemma [z3_rule]:
   349   "P = Q \<or> P \<or> Q"
   350   "P = Q \<or> \<not>P \<or> \<not>Q"
   351   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   352   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   353   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   354   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   355   "P \<noteq> Q \<or> P \<or> \<not>Q"
   356   "P \<noteq> Q \<or> \<not>P \<or> Q"
   357   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   358   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   359   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   360   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   361   "P \<or> \<not>Q \<or> P \<noteq> Q"
   362   "\<not>P \<or> Q \<or> P \<noteq> Q"
   363   by auto
   364 
   365 lemma [z3_rule]:
   366   "0 + (x::int) = x"
   367   "x + 0 = x"
   368   "0 * x = 0"
   369   "1 * x = x"
   370   "x + y = y + x"
   371   by auto
   372 
   373 
   374 
   375 hide_type (open) pattern
   376 hide_const Pattern fun_app term_true term_false z3div z3mod
   377 hide_const (open) trigger pat nopat weight
   378 
   379 end