src/HOL/SMT.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 41174 10eb369f8c01
child 41280 a7de9d36f4f2
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
     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 equating such atoms with @{term True}.
   117 Whenever the boolean type occurs in first-order terms, it is replaced
   118 by the following type.
   119 *}
   120 
   121 typedecl term_bool
   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 lemma div_by_z3div:
   134   "\<forall>k l. k div l = (
   135     if k = 0 \<or> l = 0 then 0
   136     else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
   137     else z3div (-k) (-l))"
   138   by (auto simp add: z3div_def trigger_def)
   139 
   140 lemma mod_by_z3mod:
   141   "\<forall>k l. k mod l = (
   142     if l = 0 then k
   143     else if k = 0 then 0
   144     else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
   145     else - z3mod (-k) (-l))"
   146   by (auto simp add: z3mod_def trigger_def)
   147 
   148 
   149 
   150 subsection {* Setup *}
   151 
   152 use "Tools/SMT/smt_monomorph.ML"
   153 use "Tools/SMT/smt_builtin.ML"
   154 use "Tools/SMT/smt_normalize.ML"
   155 use "Tools/SMT/smt_translate.ML"
   156 use "Tools/SMT/smt_solver.ML"
   157 use "Tools/SMT/smtlib_interface.ML"
   158 use "Tools/SMT/z3_interface.ML"
   159 use "Tools/SMT/z3_proof_parser.ML"
   160 use "Tools/SMT/z3_proof_tools.ML"
   161 use "Tools/SMT/z3_proof_literals.ML"
   162 use "Tools/SMT/z3_proof_methods.ML"
   163 use "Tools/SMT/z3_proof_reconstruction.ML"
   164 use "Tools/SMT/z3_model.ML"
   165 use "Tools/SMT/smt_setup_solvers.ML"
   166 
   167 setup {*
   168   SMT_Config.setup #>
   169   SMT_Normalize.setup #>
   170   SMT_Solver.setup #>
   171   SMTLIB_Interface.setup #>
   172   Z3_Interface.setup #>
   173   Z3_Proof_Reconstruction.setup #>
   174   SMT_Setup_Solvers.setup
   175 *}
   176 
   177 
   178 
   179 subsection {* Configuration *}
   180 
   181 text {*
   182 The current configuration can be printed by the command
   183 @{text smt_status}, which shows the values of most options.
   184 *}
   185 
   186 
   187 
   188 subsection {* General configuration options *}
   189 
   190 text {*
   191 The option @{text smt_solver} can be used to change the target SMT
   192 solver.  The possible values are @{text cvc3}, @{text yices}, and
   193 @{text z3}.  It is advisable to locally install the selected solver,
   194 although this is not necessary for @{text cvc3} and @{text z3}, which
   195 can also be used over an Internet-based service.
   196 
   197 When using local SMT solvers, the path to their binaries should be
   198 declared by setting the following environment variables:
   199 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
   200 *}
   201 
   202 declare [[ smt_solver = z3 ]]
   203 
   204 text {*
   205 Since SMT solvers are potentially non-terminating, there is a timeout
   206 (given in seconds) to restrict their runtime.  A value greater than
   207 120 (seconds) is in most cases not advisable.
   208 *}
   209 
   210 declare [[ smt_timeout = 20 ]]
   211 
   212 text {*
   213 SMT solvers apply randomized heuristics.  In case a problem is not
   214 solvable by an SMT solver, changing the following option might help.
   215 *}
   216 
   217 declare [[ smt_random_seed = 1 ]]
   218 
   219 text {*
   220 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   221 solvers are fully trusted without additional checks.  The following
   222 option can cause the SMT solver to run in proof-producing mode, giving
   223 a checkable certificate.  This is currently only implemented for Z3.
   224 *}
   225 
   226 declare [[ smt_oracle = false ]]
   227 
   228 text {*
   229 Each SMT solver provides several commandline options to tweak its
   230 behaviour.  They can be passed to the solver by setting the following
   231 options.
   232 *}
   233 
   234 declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]]
   235 
   236 text {*
   237 Enable the following option to use built-in support for datatypes and
   238 records.  Currently, this is only implemented for Z3 running in oracle
   239 mode.
   240 *}
   241 
   242 declare [[ smt_datatypes = false ]]
   243 
   244 text {*
   245 The SMT method provides an inference mechanism to detect simple triggers
   246 in quantified formulas, which might increase the number of problems
   247 solvable by SMT solvers (note: triggers guide quantifier instantiations
   248 in the SMT solver).  To turn it on, set the following option.
   249 *}
   250 
   251 declare [[ smt_infer_triggers = false ]]
   252 
   253 text {*
   254 The SMT method monomorphizes the given facts, that is, it tries to
   255 instantiate all schematic type variables with fixed types occurring
   256 in the problem.  This is a (possibly nonterminating) fixed-point
   257 construction whose cycles are limited by the following option.
   258 *}
   259 
   260 declare [[ smt_monomorph_limit = 10 ]]
   261 
   262 
   263 
   264 subsection {* Certificates *}
   265 
   266 text {*
   267 By setting the option @{text smt_certificates} to the name of a file,
   268 all following applications of an SMT solver a cached in that file.
   269 Any further application of the same SMT solver (using the very same
   270 configuration) re-uses the cached certificate instead of invoking the
   271 solver.  An empty string disables caching certificates.
   272 
   273 The filename should be given as an explicit path.  It is good
   274 practice to use the name of the current theory (with ending
   275 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   276 *}
   277 
   278 declare [[ smt_certificates = "" ]]
   279 
   280 text {*
   281 The option @{text smt_fixed} controls whether only stored
   282 certificates are should be used or invocation of an SMT solver is
   283 allowed.  When set to @{text true}, no SMT solver will ever be
   284 invoked and only the existing certificates found in the configured
   285 cache are used;  when set to @{text false} and there is no cached
   286 certificate for some proposition, then the configured SMT solver is
   287 invoked.
   288 *}
   289 
   290 declare [[ smt_fixed = false ]]
   291 
   292 
   293 
   294 subsection {* Tracing *}
   295 
   296 text {*
   297 The SMT method, when applied, traces important information.  To
   298 make it entirely silent, set the following option to @{text false}.
   299 *}
   300 
   301 declare [[ smt_verbose = true ]]
   302 
   303 text {*
   304 For tracing the generated problem file given to the SMT solver as
   305 well as the returned result of the solver, the option
   306 @{text smt_trace} should be set to @{text true}.
   307 *}
   308 
   309 declare [[ smt_trace = false ]]
   310 
   311 text {*
   312 From the set of assumptions given to the SMT solver, those assumptions
   313 used in the proof are traced when the following option is set to
   314 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   315 (see options @{text smt_solver} and @{text smt_oracle} above).
   316 *}
   317 
   318 declare [[ smt_trace_used_facts = false ]]
   319 
   320 
   321 
   322 subsection {* Schematic rules for Z3 proof reconstruction *}
   323 
   324 text {*
   325 Several prof rules of Z3 are not very well documented.  There are two
   326 lemma groups which can turn failing Z3 proof reconstruction attempts
   327 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   328 any implemented reconstruction procedure for all uncertain Z3 proof
   329 rules;  the facts in @{text z3_simp} are only fed to invocations of
   330 the simplifier when reconstructing theory-specific proof steps.
   331 *}
   332 
   333 lemmas [z3_rule] =
   334   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   335   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   336   if_True if_False not_not
   337 
   338 lemma [z3_rule]:
   339   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   340   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   341   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   342   by auto
   343 
   344 lemma [z3_rule]:
   345   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   346   by auto
   347 
   348 lemma [z3_rule]:
   349   "((\<not>P) = P) = False"
   350   "(P = (\<not>P)) = False"
   351   "(P \<noteq> Q) = (Q = (\<not>P))"
   352   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   353   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   354   by auto
   355 
   356 lemma [z3_rule]:
   357   "(if P then P else \<not>P) = True"
   358   "(if \<not>P then \<not>P else P) = True"
   359   "(if P then True else False) = P"
   360   "(if P then False else True) = (\<not>P)"
   361   "(if \<not>P then x else y) = (if P then y else x)"
   362   "f (if P then x else y) = (if P then f x else f y)"
   363   by auto
   364 
   365 lemma [z3_rule]:
   366   "P = Q \<or> P \<or> Q"
   367   "P = Q \<or> \<not>P \<or> \<not>Q"
   368   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   369   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   370   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   371   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   372   "P \<noteq> Q \<or> P \<or> \<not>Q"
   373   "P \<noteq> Q \<or> \<not>P \<or> Q"
   374   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   375   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   376   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   377   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   378   "P \<or> \<not>Q \<or> P \<noteq> Q"
   379   "\<not>P \<or> Q \<or> P \<noteq> Q"
   380   by auto
   381 
   382 lemma [z3_rule]:
   383   "0 + (x::int) = x"
   384   "x + 0 = x"
   385   "0 * x = 0"
   386   "1 * x = x"
   387   "x + y = y + x"
   388   by auto
   389 
   390 
   391 
   392 hide_type term_bool
   393 hide_type (open) pattern
   394 hide_const Pattern fun_app
   395 hide_const (open) trigger pat nopat weight z3div z3mod
   396 
   397 
   398 
   399 subsection {* Selectors for datatypes *}
   400 
   401 setup {* Datatype_Selectors.setup *}
   402 
   403 declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
   404 declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
   405 
   406 end