src/HOL/Library/Old_SMT.thy
author wenzelm
Fri Oct 07 10:46:34 2016 +0200 (2016-10-07)
changeset 64078 0b22328a353c
parent 61585 a9599d3d7610
permissions -rw-r--r--
more official legacy status;
     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 \<open>0\<close> 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       (legacy_feature "Proof method \"old_smt\" will be discontinued soon -- use \"smt\" instead";
   146        METHOD (fn facts => HEADGOAL (Old_SMT_Solver.smt_tac ctxt (thms @ facts)))))
   147 \<close> "apply an SMT solver to the current goal"
   148 
   149 
   150 subsection \<open>Configuration\<close>
   151 
   152 text \<open>
   153 The current configuration can be printed by the command
   154 \<open>old_smt_status\<close>, which shows the values of most options.
   155 \<close>
   156 
   157 
   158 
   159 subsection \<open>General configuration options\<close>
   160 
   161 text \<open>
   162 The option \<open>old_smt_solver\<close> can be used to change the target SMT
   163 solver.  The possible values can be obtained from the \<open>old_smt_status\<close>
   164 command.
   165 
   166 Due to licensing restrictions, Yices and Z3 are not installed/enabled
   167 by default.  Z3 is free for non-commercial applications and can be enabled
   168 by setting the \<open>OLD_Z3_NON_COMMERCIAL\<close> environment variable to
   169 \<open>yes\<close>.
   170 \<close>
   171 
   172 declare [[ old_smt_solver = z3 ]]
   173 
   174 text \<open>
   175 Since SMT solvers are potentially non-terminating, there is a timeout
   176 (given in seconds) to restrict their runtime.  A value greater than
   177 120 (seconds) is in most cases not advisable.
   178 \<close>
   179 
   180 declare [[ old_smt_timeout = 20 ]]
   181 
   182 text \<open>
   183 SMT solvers apply randomized heuristics.  In case a problem is not
   184 solvable by an SMT solver, changing the following option might help.
   185 \<close>
   186 
   187 declare [[ old_smt_random_seed = 1 ]]
   188 
   189 text \<open>
   190 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   191 solvers are fully trusted without additional checks.  The following
   192 option can cause the SMT solver to run in proof-producing mode, giving
   193 a checkable certificate.  This is currently only implemented for Z3.
   194 \<close>
   195 
   196 declare [[ old_smt_oracle = false ]]
   197 
   198 text \<open>
   199 Each SMT solver provides several commandline options to tweak its
   200 behaviour.  They can be passed to the solver by setting the following
   201 options.
   202 \<close>
   203 
   204 declare [[ old_cvc3_options = "" ]]
   205 declare [[ old_yices_options = "" ]]
   206 declare [[ old_z3_options = "" ]]
   207 
   208 text \<open>
   209 Enable the following option to use built-in support for datatypes and
   210 records.  Currently, this is only implemented for Z3 running in oracle
   211 mode.
   212 \<close>
   213 
   214 declare [[ old_smt_datatypes = false ]]
   215 
   216 text \<open>
   217 The SMT method provides an inference mechanism to detect simple triggers
   218 in quantified formulas, which might increase the number of problems
   219 solvable by SMT solvers (note: triggers guide quantifier instantiations
   220 in the SMT solver).  To turn it on, set the following option.
   221 \<close>
   222 
   223 declare [[ old_smt_infer_triggers = false ]]
   224 
   225 text \<open>
   226 The SMT method monomorphizes the given facts, that is, it tries to
   227 instantiate all schematic type variables with fixed types occurring
   228 in the problem.  This is a (possibly nonterminating) fixed-point
   229 construction whose cycles are limited by the following option.
   230 \<close>
   231 
   232 declare [[ monomorph_max_rounds = 5 ]]
   233 
   234 text \<open>
   235 In addition, the number of generated monomorphic instances is limited
   236 by the following option.
   237 \<close>
   238 
   239 declare [[ monomorph_max_new_instances = 500 ]]
   240 
   241 
   242 
   243 subsection \<open>Certificates\<close>
   244 
   245 text \<open>
   246 By setting the option \<open>old_smt_certificates\<close> to the name of a file,
   247 all following applications of an SMT solver a cached in that file.
   248 Any further application of the same SMT solver (using the very same
   249 configuration) re-uses the cached certificate instead of invoking the
   250 solver.  An empty string disables caching certificates.
   251 
   252 The filename should be given as an explicit path.  It is good
   253 practice to use the name of the current theory (with ending
   254 \<open>.certs\<close> instead of \<open>.thy\<close>) as the certificates file.
   255 Certificate files should be used at most once in a certain theory context,
   256 to avoid race conditions with other concurrent accesses.
   257 \<close>
   258 
   259 declare [[ old_smt_certificates = "" ]]
   260 
   261 text \<open>
   262 The option \<open>old_smt_read_only_certificates\<close> controls whether only
   263 stored certificates are should be used or invocation of an SMT solver
   264 is allowed.  When set to \<open>true\<close>, no SMT solver will ever be
   265 invoked and only the existing certificates found in the configured
   266 cache are used;  when set to \<open>false\<close> and there is no cached
   267 certificate for some proposition, then the configured SMT solver is
   268 invoked.
   269 \<close>
   270 
   271 declare [[ old_smt_read_only_certificates = false ]]
   272 
   273 
   274 
   275 subsection \<open>Tracing\<close>
   276 
   277 text \<open>
   278 The SMT method, when applied, traces important information.  To
   279 make it entirely silent, set the following option to \<open>false\<close>.
   280 \<close>
   281 
   282 declare [[ old_smt_verbose = true ]]
   283 
   284 text \<open>
   285 For tracing the generated problem file given to the SMT solver as
   286 well as the returned result of the solver, the option
   287 \<open>old_smt_trace\<close> should be set to \<open>true\<close>.
   288 \<close>
   289 
   290 declare [[ old_smt_trace = false ]]
   291 
   292 text \<open>
   293 From the set of assumptions given to the SMT solver, those assumptions
   294 used in the proof are traced when the following option is set to
   295 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   296 (see options \<open>old_smt_solver\<close> and \<open>old_smt_oracle\<close> above).
   297 \<close>
   298 
   299 declare [[ old_smt_trace_used_facts = false ]]
   300 
   301 
   302 
   303 subsection \<open>Schematic rules for Z3 proof reconstruction\<close>
   304 
   305 text \<open>
   306 Several prof rules of Z3 are not very well documented.  There are two
   307 lemma groups which can turn failing Z3 proof reconstruction attempts
   308 into succeeding ones: the facts in \<open>z3_rule\<close> are tried prior to
   309 any implemented reconstruction procedure for all uncertain Z3 proof
   310 rules;  the facts in \<open>z3_simp\<close> are only fed to invocations of
   311 the simplifier when reconstructing theory-specific proof steps.
   312 \<close>
   313 
   314 lemmas [old_z3_rule] =
   315   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   316   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   317   if_True if_False not_not
   318 
   319 lemma [old_z3_rule]:
   320   "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
   321   "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
   322   "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
   323   "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
   324   "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
   325   "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
   326   "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
   327   "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
   328   by auto
   329 
   330 lemma [old_z3_rule]:
   331   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   332   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   333   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   334   "(True \<longrightarrow> P) = P"
   335   "(P \<longrightarrow> True) = True"
   336   "(False \<longrightarrow> P) = True"
   337   "(P \<longrightarrow> P) = True"
   338   by auto
   339 
   340 lemma [old_z3_rule]:
   341   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   342   by auto
   343 
   344 lemma [old_z3_rule]:
   345   "(\<not>True) = False"
   346   "(\<not>False) = True"
   347   "(x = x) = True"
   348   "(P = True) = P"
   349   "(True = P) = P"
   350   "(P = False) = (\<not>P)"
   351   "(False = P) = (\<not>P)"
   352   "((\<not>P) = P) = False"
   353   "(P = (\<not>P)) = False"
   354   "((\<not>P) = (\<not>Q)) = (P = Q)"
   355   "\<not>(P = (\<not>Q)) = (P = Q)"
   356   "\<not>((\<not>P) = Q) = (P = Q)"
   357   "(P \<noteq> Q) = (Q = (\<not>P))"
   358   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   359   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   360   by auto
   361 
   362 lemma [old_z3_rule]:
   363   "(if P then P else \<not>P) = True"
   364   "(if \<not>P then \<not>P else P) = True"
   365   "(if P then True else False) = P"
   366   "(if P then False else True) = (\<not>P)"
   367   "(if P then Q else True) = ((\<not>P) \<or> Q)"
   368   "(if P then Q else True) = (Q \<or> (\<not>P))"
   369   "(if P then Q else \<not>Q) = (P = Q)"
   370   "(if P then Q else \<not>Q) = (Q = P)"
   371   "(if P then \<not>Q else Q) = (P = (\<not>Q))"
   372   "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
   373   "(if \<not>P then x else y) = (if P then y else x)"
   374   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
   375   "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
   376   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
   377   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
   378   "(if P then x else if P then y else z) = (if P then x else z)"
   379   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
   380   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
   381   "(if P then x = y else x = z) = (x = (if P then y else z))"
   382   "(if P then x = y else y = z) = (y = (if P then x else z))"
   383   "(if P then x = y else z = y) = (y = (if P then x else z))"
   384   by auto
   385 
   386 lemma [old_z3_rule]:
   387   "0 + (x::int) = x"
   388   "x + 0 = x"
   389   "x + x = 2 * x"
   390   "0 * x = 0"
   391   "1 * x = x"
   392   "x + y = y + x"
   393   by auto
   394 
   395 lemma [old_z3_rule]:  (* for def-axiom *)
   396   "P = Q \<or> P \<or> Q"
   397   "P = Q \<or> \<not>P \<or> \<not>Q"
   398   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   399   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   400   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   401   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   402   "P \<noteq> Q \<or> P \<or> \<not>Q"
   403   "P \<noteq> Q \<or> \<not>P \<or> Q"
   404   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   405   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   406   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   407   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   408   "P \<or> \<not>Q \<or> P \<noteq> Q"
   409   "\<not>P \<or> Q \<or> P \<noteq> Q"
   410   "P \<or> y = (if P then x else y)"
   411   "P \<or> (if P then x else y) = y"
   412   "\<not>P \<or> x = (if P then x else y)"
   413   "\<not>P \<or>  (if P then x else y) = x"
   414   "P \<or> R \<or> \<not>(if P then Q else R)"
   415   "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"
   416   "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"
   417   "\<not>(if P then Q else R) \<or> P \<or> R"
   418   "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"
   419   "(if P then Q else R) \<or> P \<or> \<not>R"
   420   "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"
   421   "(if P then Q else \<not>R) \<or> P \<or> R"
   422   by auto
   423 
   424 ML_file "Old_SMT/old_smt_real.ML"
   425 ML_file "Old_SMT/old_smt_word.ML"
   426 
   427 hide_type (open) pattern
   428 hide_const fun_app term_true term_false z3div z3mod
   429 hide_const (open) trigger pat nopat weight
   430 
   431 end