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