src/HOL/Library/Old_SMT.thy
 author wenzelm Wed Jun 17 11:03:05 2015 +0200 (2015-06-17) changeset 60500 903bb1495239 parent 59966 c01cea2ba71e child 61585 a9599d3d7610 permissions -rw-r--r--
isabelle update_cartouches;
     1 (*  Title:      HOL/Library/Old_SMT.thy

     2     Author:     Sascha Boehme, TU Muenchen

     3 *)

     4

     5 section \<open>Old Version of Bindings to Satisfiability Modulo Theories (SMT) solvers\<close>

     6

     7 theory Old_SMT

     8 imports "../Real" "../Word/Word"

     9 keywords "old_smt_status" :: diag

    10 begin

    11

    12 ML_file "Old_SMT/old_smt_utils.ML"

    13 ML_file "Old_SMT/old_smt_failure.ML"

    14 ML_file "Old_SMT/old_smt_config.ML"

    15

    16

    17 subsection \<open>Triggers for quantifier instantiation\<close>

    18

    19 text \<open>

    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 \<close>

    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 \<open>Quantifier weights\<close>

    44

    45 text \<open>

    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 \<close>

    50

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

    52

    53 text \<open>

    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 \<close>

    67

    68

    69 subsection \<open>Higher-order encoding\<close>

    70

    71 text \<open>

    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 \<close>

    76

    77 definition fun_app where "fun_app f = f"

    78

    79 text \<open>

    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 \<close>

    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 \<open>First-order logic\<close>

    90

    91 text \<open>

    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 \<close>

   102

   103 definition term_true where "term_true = True"

   104 definition term_false where "term_false = False"

   105

   106

   107 subsection \<open>Integer division and modulo for Z3\<close>

   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 \<open>Setup\<close>

   117

   118 ML_file "Old_SMT/old_smt_builtin.ML"

   119 ML_file "Old_SMT/old_smt_datatypes.ML"

   120 ML_file "Old_SMT/old_smt_normalize.ML"

   121 ML_file "Old_SMT/old_smt_translate.ML"

   122 ML_file "Old_SMT/old_smt_solver.ML"

   123 ML_file "Old_SMT/old_smtlib_interface.ML"

   124 ML_file "Old_SMT/old_z3_interface.ML"

   125 ML_file "Old_SMT/old_z3_proof_parser.ML"

   126 ML_file "Old_SMT/old_z3_proof_tools.ML"

   127 ML_file "Old_SMT/old_z3_proof_literals.ML"

   128 ML_file "Old_SMT/old_z3_proof_methods.ML"

   129 named_theorems old_z3_simp "simplification rules for Z3 proof reconstruction"

   130 ML_file "Old_SMT/old_z3_proof_reconstruction.ML"

   131 ML_file "Old_SMT/old_z3_model.ML"

   132 ML_file "Old_SMT/old_smt_setup_solvers.ML"

   133

   134 setup \<open>

   135   Old_SMT_Config.setup #>

   136   Old_SMT_Normalize.setup #>

   137   Old_SMTLIB_Interface.setup #>

   138   Old_Z3_Interface.setup #>

   139   Old_SMT_Setup_Solvers.setup

   140 \<close>

   141

   142 method_setup old_smt = \<open>

   143   Scan.optional Attrib.thms [] >>

   144     (fn thms => fn ctxt =>

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

   146 \<close> "apply an SMT solver to the current goal"

   147

   148

   149 subsection \<open>Configuration\<close>

   150

   151 text \<open>

   152 The current configuration can be printed by the command

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

   154 \<close>

   155

   156

   157

   158 subsection \<open>General configuration options\<close>

   159

   160 text \<open>

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

   162 solver.  The possible values can be obtained from the @{text old_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 the @{text OLD_Z3_NON_COMMERCIAL} environment variable to

   168 @{text yes}.

   169 \<close>

   170

   171 declare [[ old_smt_solver = z3 ]]

   172

   173 text \<open>

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

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

   176 120 (seconds) is in most cases not advisable.

   177 \<close>

   178

   179 declare [[ old_smt_timeout = 20 ]]

   180

   181 text \<open>

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

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

   184 \<close>

   185

   186 declare [[ old_smt_random_seed = 1 ]]

   187

   188 text \<open>

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

   190 solvers are fully trusted without additional checks.  The following

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

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

   193 \<close>

   194

   195 declare [[ old_smt_oracle = false ]]

   196

   197 text \<open>

   198 Each SMT solver provides several commandline options to tweak its

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

   200 options.

   201 \<close>

   202

   203 declare [[ old_cvc3_options = "" ]]

   204 declare [[ old_yices_options = "" ]]

   205 declare [[ old_z3_options = "" ]]

   206

   207 text \<open>

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

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

   210 mode.

   211 \<close>

   212

   213 declare [[ old_smt_datatypes = false ]]

   214

   215 text \<open>

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

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

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

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

   220 \<close>

   221

   222 declare [[ old_smt_infer_triggers = false ]]

   223

   224 text \<open>

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

   226 instantiate all schematic type variables with fixed types occurring

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

   228 construction whose cycles are limited by the following option.

   229 \<close>

   230

   231 declare [[ monomorph_max_rounds = 5 ]]

   232

   233 text \<open>

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

   235 by the following option.

   236 \<close>

   237

   238 declare [[ monomorph_max_new_instances = 500 ]]

   239

   240

   241

   242 subsection \<open>Certificates\<close>

   243

   244 text \<open>

   245 By setting the option @{text old_smt_certificates} to the name of a file,

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

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

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

   249 solver.  An empty string disables caching certificates.

   250

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

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

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

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

   255 to avoid race conditions with other concurrent accesses.

   256 \<close>

   257

   258 declare [[ old_smt_certificates = "" ]]

   259

   260 text \<open>

   261 The option @{text old_smt_read_only_certificates} controls whether only

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

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

   264 invoked and only the existing certificates found in the configured

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

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

   267 invoked.

   268 \<close>

   269

   270 declare [[ old_smt_read_only_certificates = false ]]

   271

   272

   273

   274 subsection \<open>Tracing\<close>

   275

   276 text \<open>

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

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

   279 \<close>

   280

   281 declare [[ old_smt_verbose = true ]]

   282

   283 text \<open>

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

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

   286 @{text old_smt_trace} should be set to @{text true}.

   287 \<close>

   288

   289 declare [[ old_smt_trace = false ]]

   290

   291 text \<open>

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

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

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

   295 (see options @{text old_smt_solver} and @{text old_smt_oracle} above).

   296 \<close>

   297

   298 declare [[ old_smt_trace_used_facts = false ]]

   299

   300

   301

   302 subsection \<open>Schematic rules for Z3 proof reconstruction\<close>

   303

   304 text \<open>

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

   306 lemma groups which can turn failing Z3 proof reconstruction attempts

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

   308 any implemented reconstruction procedure for all uncertain Z3 proof

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

   310 the simplifier when reconstructing theory-specific proof steps.

   311 \<close>

   312

   313 lemmas [old_z3_rule] =

   314   refl eq_commute conj_commute disj_commute simp_thms nnf_simps

   315   ring_distribs field_simps times_divide_eq_right times_divide_eq_left

   316   if_True if_False not_not

   317

   318 lemma [old_z3_rule]:

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

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

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

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

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

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

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

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

   327   by auto

   328

   329 lemma [old_z3_rule]:

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

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

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

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

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

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

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

   337   by auto

   338

   339 lemma [old_z3_rule]:

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

   341   by auto

   342

   343 lemma [old_z3_rule]:

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

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

   346   "(x = x) = True"

   347   "(P = True) = P"

   348   "(True = P) = P"

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

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

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

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

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

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

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

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

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

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

   359   by auto

   360

   361 lemma [old_z3_rule]:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   383   by auto

   384

   385 lemma [old_z3_rule]:

   386   "0 + (x::int) = x"

   387   "x + 0 = x"

   388   "x + x = 2 * x"

   389   "0 * x = 0"

   390   "1 * x = x"

   391   "x + y = y + x"

   392   by auto

   393

   394 lemma [old_z3_rule]:  (* for def-axiom *)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   421   by auto

   422

   423 ML_file "Old_SMT/old_smt_real.ML"

   424 ML_file "Old_SMT/old_smt_word.ML"

   425

   426 hide_type (open) pattern

   427 hide_const fun_app term_true term_false z3div z3mod

   428 hide_const (open) trigger pat nopat weight

   429

   430 end