src/HOL/SMT.thy
 author wenzelm Sat Aug 16 18:31:47 2014 +0200 (2014-08-16) changeset 57957 e6ee35b8f4b5 parent 57231 dca8d06ecbba permissions -rw-r--r--
updated to named_theorems;
modernized setup;
tuned;
     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 typedecl pattern

    35

    36 consts

    37   pat :: "'a \<Rightarrow> pattern"

    38   nopat :: "'a \<Rightarrow> pattern"

    39

    40 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool" where "trigger _ P = P"

    41

    42

    43 subsection {* Quantifier weights *}

    44

    45 text {*

    46 Weight annotations to quantifiers influence the priority of quantifier

    47 instantiations.  They should be handled with care for solvers, which support

    48 them, because incorrect choices of weights might render a problem unsolvable.

    49 *}

    50

    51 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"

    52

    53 text {*

    54 Weights must be non-negative.  The value @{text 0} is equivalent to providing

    55 no weight at all.

    56

    57 Weights should only be used at quantifiers and only inside triggers (if the

    58 quantifier has triggers).  Valid usages of weights are as follows:

    59

    60 \begin{itemize}

    61 \item

    62 @{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}

    63 \item

    64 @{term "\<forall>x. weight 3 (P x)"}

    65 \end{itemize}

    66 *}

    67

    68

    69 subsection {* Higher-order encoding *}

    70

    71 text {*

    72 Application is made explicit for constants occurring with varying

    73 numbers of arguments.  This is achieved by the introduction of the

    74 following constant.

    75 *}

    76

    77 definition fun_app where "fun_app f = f"

    78

    79 text {*

    80 Some solvers support a theory of arrays which can be used to encode

    81 higher-order functions.  The following set of lemmas specifies the

    82 properties of such (extensional) arrays.

    83 *}

    84

    85 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other

    86   fun_upd_upd fun_app_def

    87

    88

    89 subsection {* First-order logic *}

    90

    91 text {*

    92 Some SMT solvers only accept problems in first-order logic, i.e.,

    93 where formulas and terms are syntactically separated. When

    94 translating higher-order into first-order problems, all

    95 uninterpreted constants (those not built-in in the target solver)

    96 are treated as function symbols in the first-order sense.  Their

    97 occurrences as head symbols in atoms (i.e., as predicate symbols) are

    98 turned into terms by logically equating such atoms with @{term True}.

    99 For technical reasons, @{term True} and @{term False} occurring inside

   100 terms are replaced by the following constants.

   101 *}

   102

   103 definition term_true where "term_true = True"

   104 definition term_false where "term_false = False"

   105

   106

   107 subsection {* Integer division and modulo for Z3 *}

   108

   109 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where

   110   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"

   111

   112 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where

   113   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"

   114

   115

   116 subsection {* Setup *}

   117

   118 ML_file "Tools/SMT/smt_builtin.ML"

   119 ML_file "Tools/SMT/smt_datatypes.ML"

   120 ML_file "Tools/SMT/smt_normalize.ML"

   121 ML_file "Tools/SMT/smt_translate.ML"

   122 ML_file "Tools/SMT/smt_solver.ML"

   123 ML_file "Tools/SMT/smtlib_interface.ML"

   124 ML_file "Tools/SMT/z3_interface.ML"

   125 ML_file "Tools/SMT/z3_proof_parser.ML"

   126 ML_file "Tools/SMT/z3_proof_tools.ML"

   127 ML_file "Tools/SMT/z3_proof_literals.ML"

   128 ML_file "Tools/SMT/z3_proof_methods.ML"

   129 named_theorems z3_simp "simplification rules for Z3 proof reconstruction"

   130 ML_file "Tools/SMT/z3_proof_reconstruction.ML"

   131 ML_file "Tools/SMT/z3_model.ML"

   132 ML_file "Tools/SMT/smt_setup_solvers.ML"

   133

   134 setup {*

   135   SMT_Config.setup #>

   136   SMT_Normalize.setup #>

   137   SMTLIB_Interface.setup #>

   138   Z3_Interface.setup #>

   139   SMT_Setup_Solvers.setup

   140 *}

   141

   142 method_setup smt = {*

   143   Scan.optional Attrib.thms [] >>

   144     (fn thms => fn ctxt =>

   145       METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))

   146 *} "apply an SMT solver to the current goal"

   147

   148

   149 subsection {* Configuration *}

   150

   151 text {*

   152 The current configuration can be printed by the command

   153 @{text smt_status}, which shows the values of most options.

   154 *}

   155

   156

   157

   158 subsection {* General configuration options *}

   159

   160 text {*

   161 The option @{text smt_solver} can be used to change the target SMT

   162 solver.  The possible values can be obtained from the @{text smt_status}

   163 command.

   164

   165 Due to licensing restrictions, Yices and Z3 are not installed/enabled

   166 by default.  Z3 is free for non-commercial applications and can be enabled

   167 by setting Isabelle system option @{text z3_non_commercial} to @{text yes}.

   168 *}

   169

   170 declare [[ smt_solver = z3 ]]

   171

   172 text {*

   173 Since SMT solvers are potentially non-terminating, there is a timeout

   174 (given in seconds) to restrict their runtime.  A value greater than

   175 120 (seconds) is in most cases not advisable.

   176 *}

   177

   178 declare [[ smt_timeout = 20 ]]

   179

   180 text {*

   181 SMT solvers apply randomized heuristics.  In case a problem is not

   182 solvable by an SMT solver, changing the following option might help.

   183 *}

   184

   185 declare [[ smt_random_seed = 1 ]]

   186

   187 text {*

   188 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT

   189 solvers are fully trusted without additional checks.  The following

   190 option can cause the SMT solver to run in proof-producing mode, giving

   191 a checkable certificate.  This is currently only implemented for Z3.

   192 *}

   193

   194 declare [[ smt_oracle = false ]]

   195

   196 text {*

   197 Each SMT solver provides several commandline options to tweak its

   198 behaviour.  They can be passed to the solver by setting the following

   199 options.

   200 *}

   201

   202 declare [[ cvc3_options = "" ]]

   203 declare [[ yices_options = "" ]]

   204 declare [[ z3_options = "" ]]

   205

   206 text {*

   207 Enable the following option to use built-in support for datatypes and

   208 records.  Currently, this is only implemented for Z3 running in oracle

   209 mode.

   210 *}

   211

   212 declare [[ smt_datatypes = false ]]

   213

   214 text {*

   215 The SMT method provides an inference mechanism to detect simple triggers

   216 in quantified formulas, which might increase the number of problems

   217 solvable by SMT solvers (note: triggers guide quantifier instantiations

   218 in the SMT solver).  To turn it on, set the following option.

   219 *}

   220

   221 declare [[ smt_infer_triggers = false ]]

   222

   223 text {*

   224 The SMT method monomorphizes the given facts, that is, it tries to

   225 instantiate all schematic type variables with fixed types occurring

   226 in the problem.  This is a (possibly nonterminating) fixed-point

   227 construction whose cycles are limited by the following option.

   228 *}

   229

   230 declare [[ monomorph_max_rounds = 5 ]]

   231

   232 text {*

   233 In addition, the number of generated monomorphic instances is limited

   234 by the following option.

   235 *}

   236

   237 declare [[ monomorph_max_new_instances = 500 ]]

   238

   239

   240

   241 subsection {* Certificates *}

   242

   243 text {*

   244 By setting the option @{text smt_certificates} to the name of a file,

   245 all following applications of an SMT solver a cached in that file.

   246 Any further application of the same SMT solver (using the very same

   247 configuration) re-uses the cached certificate instead of invoking the

   248 solver.  An empty string disables caching certificates.

   249

   250 The filename should be given as an explicit path.  It is good

   251 practice to use the name of the current theory (with ending

   252 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.

   253 Certificate files should be used at most once in a certain theory context,

   254 to avoid race conditions with other concurrent accesses.

   255 *}

   256

   257 declare [[ smt_certificates = "" ]]

   258

   259 text {*

   260 The option @{text smt_read_only_certificates} controls whether only

   261 stored certificates are should be used or invocation of an SMT solver

   262 is allowed.  When set to @{text true}, no SMT solver will ever be

   263 invoked and only the existing certificates found in the configured

   264 cache are used;  when set to @{text false} and there is no cached

   265 certificate for some proposition, then the configured SMT solver is

   266 invoked.

   267 *}

   268

   269 declare [[ smt_read_only_certificates = false ]]

   270

   271

   272

   273 subsection {* Tracing *}

   274

   275 text {*

   276 The SMT method, when applied, traces important information.  To

   277 make it entirely silent, set the following option to @{text false}.

   278 *}

   279

   280 declare [[ smt_verbose = true ]]

   281

   282 text {*

   283 For tracing the generated problem file given to the SMT solver as

   284 well as the returned result of the solver, the option

   285 @{text smt_trace} should be set to @{text true}.

   286 *}

   287

   288 declare [[ smt_trace = false ]]

   289

   290 text {*

   291 From the set of assumptions given to the SMT solver, those assumptions

   292 used in the proof are traced when the following option is set to

   293 @{term true}.  This only works for Z3 when it runs in non-oracle mode

   294 (see options @{text smt_solver} and @{text smt_oracle} above).

   295 *}

   296

   297 declare [[ smt_trace_used_facts = false ]]

   298

   299

   300

   301 subsection {* Schematic rules for Z3 proof reconstruction *}

   302

   303 text {*

   304 Several prof rules of Z3 are not very well documented.  There are two

   305 lemma groups which can turn failing Z3 proof reconstruction attempts

   306 into succeeding ones: the facts in @{text z3_rule} are tried prior to

   307 any implemented reconstruction procedure for all uncertain Z3 proof

   308 rules;  the facts in @{text z3_simp} are only fed to invocations of

   309 the simplifier when reconstructing theory-specific proof steps.

   310 *}

   311

   312 lemmas [z3_rule] =

   313   refl eq_commute conj_commute disj_commute simp_thms nnf_simps

   314   ring_distribs field_simps times_divide_eq_right times_divide_eq_left

   315   if_True if_False not_not

   316

   317 lemma [z3_rule]:

   318   "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"

   319   "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"

   320   "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"

   321   "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"

   322   "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"

   323   "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"

   324   "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"

   325   "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"

   326   by auto

   327

   328 lemma [z3_rule]:

   329   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"

   330   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"

   331   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"

   332   "(True \<longrightarrow> P) = P"

   333   "(P \<longrightarrow> True) = True"

   334   "(False \<longrightarrow> P) = True"

   335   "(P \<longrightarrow> P) = True"

   336   by auto

   337

   338 lemma [z3_rule]:

   339   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"

   340   by auto

   341

   342 lemma [z3_rule]:

   343   "(\<not>True) = False"

   344   "(\<not>False) = True"

   345   "(x = x) = True"

   346   "(P = True) = P"

   347   "(True = P) = P"

   348   "(P = False) = (\<not>P)"

   349   "(False = P) = (\<not>P)"

   350   "((\<not>P) = P) = False"

   351   "(P = (\<not>P)) = False"

   352   "((\<not>P) = (\<not>Q)) = (P = Q)"

   353   "\<not>(P = (\<not>Q)) = (P = Q)"

   354   "\<not>((\<not>P) = Q) = (P = Q)"

   355   "(P \<noteq> Q) = (Q = (\<not>P))"

   356   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"

   357   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"

   358   by auto

   359

   360 lemma [z3_rule]:

   361   "(if P then P else \<not>P) = True"

   362   "(if \<not>P then \<not>P else P) = True"

   363   "(if P then True else False) = P"

   364   "(if P then False else True) = (\<not>P)"

   365   "(if P then Q else True) = ((\<not>P) \<or> Q)"

   366   "(if P then Q else True) = (Q \<or> (\<not>P))"

   367   "(if P then Q else \<not>Q) = (P = Q)"

   368   "(if P then Q else \<not>Q) = (Q = P)"

   369   "(if P then \<not>Q else Q) = (P = (\<not>Q))"

   370   "(if P then \<not>Q else Q) = ((\<not>Q) = P)"

   371   "(if \<not>P then x else y) = (if P then y else x)"

   372   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"

   373   "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"

   374   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"

   375   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"

   376   "(if P then x else if P then y else z) = (if P then x else z)"

   377   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"

   378   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"

   379   "(if P then x = y else x = z) = (x = (if P then y else z))"

   380   "(if P then x = y else y = z) = (y = (if P then x else z))"

   381   "(if P then x = y else z = y) = (y = (if P then x else z))"

   382   by auto

   383

   384 lemma [z3_rule]:

   385   "0 + (x::int) = x"

   386   "x + 0 = x"

   387   "x + x = 2 * x"

   388   "0 * x = 0"

   389   "1 * x = x"

   390   "x + y = y + x"

   391   by auto

   392

   393 lemma [z3_rule]:  (* for def-axiom *)

   394   "P = Q \<or> P \<or> Q"

   395   "P = Q \<or> \<not>P \<or> \<not>Q"

   396   "(\<not>P) = Q \<or> \<not>P \<or> Q"

   397   "(\<not>P) = Q \<or> P \<or> \<not>Q"

   398   "P = (\<not>Q) \<or> \<not>P \<or> Q"

   399   "P = (\<not>Q) \<or> P \<or> \<not>Q"

   400   "P \<noteq> Q \<or> P \<or> \<not>Q"

   401   "P \<noteq> Q \<or> \<not>P \<or> Q"

   402   "P \<noteq> (\<not>Q) \<or> P \<or> Q"

   403   "(\<not>P) \<noteq> Q \<or> P \<or> Q"

   404   "P \<or> Q \<or> P \<noteq> (\<not>Q)"

   405   "P \<or> Q \<or> (\<not>P) \<noteq> Q"

   406   "P \<or> \<not>Q \<or> P \<noteq> Q"

   407   "\<not>P \<or> Q \<or> P \<noteq> Q"

   408   "P \<or> y = (if P then x else y)"

   409   "P \<or> (if P then x else y) = y"

   410   "\<not>P \<or> x = (if P then x else y)"

   411   "\<not>P \<or>  (if P then x else y) = x"

   412   "P \<or> R \<or> \<not>(if P then Q else R)"

   413   "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"

   414   "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"

   415   "\<not>(if P then Q else R) \<or> P \<or> R"

   416   "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"

   417   "(if P then Q else R) \<or> P \<or> \<not>R"

   418   "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"

   419   "(if P then Q else \<not>R) \<or> P \<or> R"

   420   by auto

   421

   422 hide_type (open) pattern

   423 hide_const fun_app term_true term_false z3div z3mod

   424 hide_const (open) trigger pat nopat weight

   425

   426 end