src/HOL/SMT.thy
author boehmes
Thu May 27 16:29:33 2010 +0200 (2010-05-27)
changeset 37153 8feed34275ce
parent 37151 3e9e8dfb3c98
child 37157 86872cbae9e9
permissions -rw-r--r--
renamed constant "apply" to "fun_app" (which is closer to the related "fun_upd")
     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   "~~/src/Tools/cache_io.ML"
    11   ("Tools/SMT/smt_monomorph.ML")
    12   ("Tools/SMT/smt_normalize.ML")
    13   ("Tools/SMT/smt_translate.ML")
    14   ("Tools/SMT/smt_solver.ML")
    15   ("Tools/SMT/smtlib_interface.ML")
    16   ("Tools/SMT/z3_proof_parser.ML")
    17   ("Tools/SMT/z3_proof_tools.ML")
    18   ("Tools/SMT/z3_proof_literals.ML")
    19   ("Tools/SMT/z3_proof_reconstruction.ML")
    20   ("Tools/SMT/z3_model.ML")
    21   ("Tools/SMT/z3_interface.ML")
    22   ("Tools/SMT/z3_solver.ML")
    23   ("Tools/SMT/cvc3_solver.ML")
    24   ("Tools/SMT/yices_solver.ML")
    25 begin
    26 
    27 
    28 
    29 subsection {* Triggers for quantifier instantiation *}
    30 
    31 text {*
    32 Some SMT solvers support triggers for quantifier instantiation.
    33 Each trigger consists of one ore more patterns.  A pattern may either
    34 be a list of positive subterms (each being tagged by "pat"), or a
    35 list of negative subterms (each being tagged by "nopat").
    36 
    37 When an SMT solver finds a term matching a positive pattern (a
    38 pattern with positive subterms only), it instantiates the
    39 corresponding quantifier accordingly.  Negative patterns inhibit
    40 quantifier instantiations.  Each pattern should mention all preceding
    41 bound variables.
    42 *}
    43 
    44 datatype pattern = Pattern
    45 
    46 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    47 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    48 
    49 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    50 where "trigger _ P = P"
    51 
    52 
    53 
    54 subsection {* Higher-order encoding *}
    55 
    56 text {*
    57 Application is made explicit for constants occurring with varying
    58 numbers of arguments.  This is achieved by the introduction of the
    59 following constant.
    60 *}
    61 
    62 definition fun_app where "fun_app f x = f x"
    63 
    64 text {*
    65 Some solvers support a theory of arrays which can be used to encode
    66 higher-order functions.  The following set of lemmas specifies the
    67 properties of such (extensional) arrays.
    68 *}
    69 
    70 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
    71   fun_upd_upd
    72 
    73 
    74 
    75 subsection {* First-order logic *}
    76 
    77 text {*
    78 Some SMT solvers require a strict separation between formulas and
    79 terms.  When translating higher-order into first-order problems,
    80 all uninterpreted constants (those not builtin in the target solver)
    81 are treated as function symbols in the first-order sense.  Their
    82 occurrences as head symbols in atoms (i.e., as predicate symbols) is
    83 turned into terms by equating such atoms with @{term True} using the
    84 following term-level equation symbol.
    85 *}
    86 
    87 definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)"
    88 
    89 
    90 
    91 subsection {* Integer division and modulo for Z3 *}
    92 
    93 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
    94   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
    95 
    96 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
    97   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
    98 
    99 lemma div_by_z3div: "k div l = (
   100      if k = 0 \<or> l = 0 then 0
   101      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
   102      else z3div (-k) (-l))"
   103   by (auto simp add: z3div_def)
   104 
   105 lemma mod_by_z3mod: "k mod l = (
   106      if l = 0 then k
   107      else if k = 0 then 0
   108      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
   109      else - z3mod (-k) (-l))"
   110   by (auto simp add: z3mod_def)
   111 
   112 
   113 
   114 subsection {* Setup *}
   115 
   116 use "Tools/SMT/smt_monomorph.ML"
   117 use "Tools/SMT/smt_normalize.ML"
   118 use "Tools/SMT/smt_translate.ML"
   119 use "Tools/SMT/smt_solver.ML"
   120 use "Tools/SMT/smtlib_interface.ML"
   121 use "Tools/SMT/z3_interface.ML"
   122 use "Tools/SMT/z3_proof_parser.ML"
   123 use "Tools/SMT/z3_proof_tools.ML"
   124 use "Tools/SMT/z3_proof_literals.ML"
   125 use "Tools/SMT/z3_proof_reconstruction.ML"
   126 use "Tools/SMT/z3_model.ML"
   127 use "Tools/SMT/z3_solver.ML"
   128 use "Tools/SMT/cvc3_solver.ML"
   129 use "Tools/SMT/yices_solver.ML"
   130 
   131 setup {*
   132   SMT_Solver.setup #>
   133   Z3_Proof_Reconstruction.setup #>
   134   Z3_Solver.setup #>
   135   CVC3_Solver.setup #>
   136   Yices_Solver.setup
   137 *}
   138 
   139 
   140 
   141 subsection {* Configuration *}
   142 
   143 text {*
   144 The current configuration can be printed by the command
   145 @{text smt_status}, which shows the values of most options.
   146 *}
   147 
   148 
   149 
   150 subsection {* General configuration options *}
   151 
   152 text {*
   153 The option @{text smt_solver} can be used to change the target SMT
   154 solver.  The possible values are @{text cvc3}, @{text yices}, and
   155 @{text z3}.  It is advisable to locally install the selected solver,
   156 although this is not necessary for @{text cvc3} and @{text z3}, which
   157 can also be used over an Internet-based service.
   158 
   159 When using local SMT solvers, the path to their binaries should be
   160 declared by setting the following environment variables:
   161 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
   162 *}
   163 
   164 declare [[ smt_solver = z3 ]]
   165 
   166 text {*
   167 Since SMT solvers are potentially non-terminating, there is a timeout
   168 (given in seconds) to restrict their runtime.  A value greater than
   169 120 (seconds) is in most cases not advisable.
   170 *}
   171 
   172 declare [[ smt_timeout = 20 ]]
   173 
   174 
   175 
   176 subsection {* Certificates *}
   177 
   178 text {*
   179 By setting the option @{text smt_certificates} to the name of a file,
   180 all following applications of an SMT solver a cached in that file.
   181 Any further application of the same SMT solver (using the very same
   182 configuration) re-uses the cached certificate instead of invoking the
   183 solver.  An empty string disables caching certificates.
   184 
   185 The filename should be given as an explicit path.  It is good
   186 practice to use the name of the current theory (with ending
   187 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   188 *}
   189 
   190 declare [[ smt_certificates = "" ]]
   191 
   192 text {*
   193 The option @{text smt_fixed} controls whether only stored
   194 certificates are should be used or invocation of an SMT solver is
   195 allowed.  When set to @{text true}, no SMT solver will ever be
   196 invoked and only the existing certificates found in the configured
   197 cache are used;  when set to @{text false} and there is no cached
   198 certificate for some proposition, then the configured SMT solver is
   199 invoked.
   200 *}
   201 
   202 declare [[ smt_fixed = false ]]
   203 
   204 
   205 
   206 subsection {* Tracing *}
   207 
   208 text {*
   209 For tracing the generated problem file given to the SMT solver as
   210 well as the returned result of the solver, the option
   211 @{text smt_trace} should be set to @{text true}.
   212 *}
   213 
   214 declare [[ smt_trace = false ]]
   215 
   216 
   217 
   218 subsection {* Z3-specific options *}
   219 
   220 text {*
   221 Z3 is the only SMT solver whose proofs are checked (or reconstructed)
   222 in Isabelle (all other solvers are implemented as oracles).  Enabling
   223 or disabling proof reconstruction for Z3 is controlled by the option
   224 @{text z3_proofs}. 
   225 *}
   226 
   227 declare [[ z3_proofs = true ]]
   228 
   229 text {*
   230 From the set of assumptions given to Z3, those assumptions used in
   231 the proof are traced when the option @{text z3_trace_assms} is set to
   232 @{term true}.
   233 *}
   234 
   235 declare [[ z3_trace_assms = false ]]
   236 
   237 text {*
   238 Z3 provides several commandline options to tweak its behaviour.  They
   239 can be configured by writing them literally as value for the option
   240 @{text z3_options}.
   241 *}
   242 
   243 declare [[ z3_options = "" ]]
   244 
   245 
   246 
   247 subsection {* Schematic rules for Z3 proof reconstruction *}
   248 
   249 text {*
   250 Several prof rules of Z3 are not very well documented.  There are two
   251 lemma groups which can turn failing Z3 proof reconstruction attempts
   252 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   253 any implemented reconstruction procedure for all uncertain Z3 proof
   254 rules;  the facts in @{text z3_simp} are only fed to invocations of
   255 the simplifier when reconstructing theory-specific proof steps.
   256 *}
   257 
   258 lemmas [z3_rule] =
   259   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   260   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   261   if_True if_False not_not
   262 
   263 lemma [z3_rule]:
   264   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   265   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   266   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   267   by auto
   268 
   269 lemma [z3_rule]:
   270   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   271   by auto
   272 
   273 lemma [z3_rule]:
   274   "((\<not>P) = P) = False"
   275   "(P = (\<not>P)) = False"
   276   "(P \<noteq> Q) = (Q = (\<not>P))"
   277   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   278   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   279   by auto
   280 
   281 lemma [z3_rule]:
   282   "(if P then P else \<not>P) = True"
   283   "(if \<not>P then \<not>P else P) = True"
   284   "(if P then True else False) = P"
   285   "(if P then False else True) = (\<not>P)"
   286   "(if \<not>P then x else y) = (if P then y else x)"
   287   by auto
   288 
   289 lemma [z3_rule]:
   290   "P = Q \<or> P \<or> Q"
   291   "P = Q \<or> \<not>P \<or> \<not>Q"
   292   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   293   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   294   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   295   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   296   "P \<noteq> Q \<or> P \<or> \<not>Q"
   297   "P \<noteq> Q \<or> \<not>P \<or> Q"
   298   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   299   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   300   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   301   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   302   "P \<or> \<not>Q \<or> P \<noteq> Q"
   303   "\<not>P \<or> Q \<or> P \<noteq> Q"
   304   by auto
   305 
   306 lemma [z3_rule]:
   307   "0 + (x::int) = x"
   308   "x + 0 = x"
   309   "0 * x = 0"
   310   "1 * x = x"
   311   "x + y = y + x"
   312   by auto
   313 
   314 
   315 
   316 hide_type (open) pattern
   317 hide_const Pattern term_eq
   318 hide_const (open) trigger pat nopat fun_app z3div z3mod
   319 
   320 end