src/HOL/SMT.thy
author boehmes
Sun Dec 19 18:54:29 2010 +0100 (2010-12-19)
changeset 41281 679118e35378
parent 41280 a7de9d36f4f2
child 41328 6792a5c92a58
permissions -rw-r--r--
removed odd decoration of built-in symbols as Vars (instead provide built-in desctructor functions along with their inverse functions);
removed odd retyping during folify (instead, keep all terms well-typed)
     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 List
     9 uses
    10   "Tools/Datatype/datatype_selectors.ML"
    11   "Tools/SMT/smt_utils.ML"
    12   "Tools/SMT/smt_failure.ML"
    13   "Tools/SMT/smt_config.ML"
    14   ("Tools/SMT/smt_monomorph.ML")
    15   ("Tools/SMT/smt_builtin.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_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/z3_interface.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 
   127 subsection {* Integer division and modulo for Z3 *}
   128 
   129 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
   130   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
   131 
   132 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
   133   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
   134 
   135 
   136 
   137 subsection {* Setup *}
   138 
   139 use "Tools/SMT/smt_monomorph.ML"
   140 use "Tools/SMT/smt_builtin.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 are @{text cvc3}, @{text yices}, and
   180 @{text z3}.  It is advisable to locally install the selected solver,
   181 although this is not necessary for @{text cvc3} and @{text z3}, which
   182 can also be used over an Internet-based service.
   183 
   184 When using local SMT solvers, the path to their binaries should be
   185 declared by setting the following environment variables:
   186 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
   187 *}
   188 
   189 declare [[ smt_solver = z3 ]]
   190 
   191 text {*
   192 Since SMT solvers are potentially non-terminating, there is a timeout
   193 (given in seconds) to restrict their runtime.  A value greater than
   194 120 (seconds) is in most cases not advisable.
   195 *}
   196 
   197 declare [[ smt_timeout = 20 ]]
   198 
   199 text {*
   200 SMT solvers apply randomized heuristics.  In case a problem is not
   201 solvable by an SMT solver, changing the following option might help.
   202 *}
   203 
   204 declare [[ smt_random_seed = 1 ]]
   205 
   206 text {*
   207 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   208 solvers are fully trusted without additional checks.  The following
   209 option can cause the SMT solver to run in proof-producing mode, giving
   210 a checkable certificate.  This is currently only implemented for Z3.
   211 *}
   212 
   213 declare [[ smt_oracle = false ]]
   214 
   215 text {*
   216 Each SMT solver provides several commandline options to tweak its
   217 behaviour.  They can be passed to the solver by setting the following
   218 options.
   219 *}
   220 
   221 declare [[ cvc3_options = "", yices_options = "", 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 [[ smt_monomorph_limit = 10 ]]
   248 
   249 
   250 
   251 subsection {* Certificates *}
   252 
   253 text {*
   254 By setting the option @{text smt_certificates} to the name of a file,
   255 all following applications of an SMT solver a cached in that file.
   256 Any further application of the same SMT solver (using the very same
   257 configuration) re-uses the cached certificate instead of invoking the
   258 solver.  An empty string disables caching certificates.
   259 
   260 The filename should be given as an explicit path.  It is good
   261 practice to use the name of the current theory (with ending
   262 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   263 *}
   264 
   265 declare [[ smt_certificates = "" ]]
   266 
   267 text {*
   268 The option @{text smt_fixed} controls whether only stored
   269 certificates are should be used or invocation of an SMT solver is
   270 allowed.  When set to @{text true}, no SMT solver will ever be
   271 invoked and only the existing certificates found in the configured
   272 cache are used;  when set to @{text false} and there is no cached
   273 certificate for some proposition, then the configured SMT solver is
   274 invoked.
   275 *}
   276 
   277 declare [[ smt_fixed = false ]]
   278 
   279 
   280 
   281 subsection {* Tracing *}
   282 
   283 text {*
   284 The SMT method, when applied, traces important information.  To
   285 make it entirely silent, set the following option to @{text false}.
   286 *}
   287 
   288 declare [[ smt_verbose = true ]]
   289 
   290 text {*
   291 For tracing the generated problem file given to the SMT solver as
   292 well as the returned result of the solver, the option
   293 @{text smt_trace} should be set to @{text true}.
   294 *}
   295 
   296 declare [[ smt_trace = false ]]
   297 
   298 text {*
   299 From the set of assumptions given to the SMT solver, those assumptions
   300 used in the proof are traced when the following option is set to
   301 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   302 (see options @{text smt_solver} and @{text smt_oracle} above).
   303 *}
   304 
   305 declare [[ smt_trace_used_facts = false ]]
   306 
   307 
   308 
   309 subsection {* Schematic rules for Z3 proof reconstruction *}
   310 
   311 text {*
   312 Several prof rules of Z3 are not very well documented.  There are two
   313 lemma groups which can turn failing Z3 proof reconstruction attempts
   314 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   315 any implemented reconstruction procedure for all uncertain Z3 proof
   316 rules;  the facts in @{text z3_simp} are only fed to invocations of
   317 the simplifier when reconstructing theory-specific proof steps.
   318 *}
   319 
   320 lemmas [z3_rule] =
   321   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   322   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   323   if_True if_False not_not
   324 
   325 lemma [z3_rule]:
   326   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   327   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   328   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   329   by auto
   330 
   331 lemma [z3_rule]:
   332   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   333   by auto
   334 
   335 lemma [z3_rule]:
   336   "((\<not>P) = P) = False"
   337   "(P = (\<not>P)) = False"
   338   "(P \<noteq> Q) = (Q = (\<not>P))"
   339   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   340   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   341   by auto
   342 
   343 lemma [z3_rule]:
   344   "(if P then P else \<not>P) = True"
   345   "(if \<not>P then \<not>P else P) = True"
   346   "(if P then True else False) = P"
   347   "(if P then False else True) = (\<not>P)"
   348   "(if \<not>P then x else y) = (if P then y else x)"
   349   "f (if P then x else y) = (if P then f x else f y)"
   350   by auto
   351 
   352 lemma [z3_rule]:
   353   "P = Q \<or> P \<or> Q"
   354   "P = Q \<or> \<not>P \<or> \<not>Q"
   355   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   356   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   357   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   358   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   359   "P \<noteq> Q \<or> P \<or> \<not>Q"
   360   "P \<noteq> Q \<or> \<not>P \<or> Q"
   361   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   362   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   363   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   364   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   365   "P \<or> \<not>Q \<or> P \<noteq> Q"
   366   "\<not>P \<or> Q \<or> P \<noteq> Q"
   367   by auto
   368 
   369 lemma [z3_rule]:
   370   "0 + (x::int) = x"
   371   "x + 0 = x"
   372   "0 * x = 0"
   373   "1 * x = x"
   374   "x + y = y + x"
   375   by auto
   376 
   377 
   378 
   379 hide_type (open) pattern
   380 hide_const Pattern fun_app term_true term_false z3div z3mod
   381 hide_const (open) trigger pat nopat weight
   382 
   383 
   384 
   385 subsection {* Selectors for datatypes *}
   386 
   387 setup {* Datatype_Selectors.setup *}
   388 
   389 declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
   390 declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
   391 
   392 end