src/HOL/SMT.thy
author boehmes
Tue Oct 26 11:45:12 2010 +0200 (2010-10-26)
changeset 40162 7f58a9a843c2
parent 39483 9f0e5684f04b
child 40274 6486c610a549
permissions -rw-r--r--
joined setup of SMT solvers in one place; turned Z3-specific options into SMT options (renamed configuration options from z3_* to smt_*); more detailed SMT exception; improved SMT filter interface
     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   "Tools/Datatype/datatype_selectors.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/smt_setup_solvers.ML")
    23 begin
    24 
    25 
    26 
    27 subsection {* Triggers for quantifier instantiation *}
    28 
    29 text {*
    30 Some SMT solvers support triggers for quantifier instantiation.
    31 Each trigger consists of one ore more patterns.  A pattern may either
    32 be a list of positive subterms (each being tagged by "pat"), or a
    33 list of negative subterms (each being tagged by "nopat").
    34 
    35 When an SMT solver finds a term matching a positive pattern (a
    36 pattern with positive subterms only), it instantiates the
    37 corresponding quantifier accordingly.  Negative patterns inhibit
    38 quantifier instantiations.  Each pattern should mention all preceding
    39 bound variables.
    40 *}
    41 
    42 datatype pattern = Pattern
    43 
    44 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    45 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    46 
    47 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    48 where "trigger _ P = P"
    49 
    50 
    51 
    52 subsection {* Higher-order encoding *}
    53 
    54 text {*
    55 Application is made explicit for constants occurring with varying
    56 numbers of arguments.  This is achieved by the introduction of the
    57 following constant.
    58 *}
    59 
    60 definition fun_app where "fun_app f x = f x"
    61 
    62 text {*
    63 Some solvers support a theory of arrays which can be used to encode
    64 higher-order functions.  The following set of lemmas specifies the
    65 properties of such (extensional) arrays.
    66 *}
    67 
    68 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
    69   fun_upd_upd fun_app_def
    70 
    71 
    72 
    73 subsection {* First-order logic *}
    74 
    75 text {*
    76 Some SMT solvers require a strict separation between formulas and
    77 terms.  When translating higher-order into first-order problems,
    78 all uninterpreted constants (those not builtin in the target solver)
    79 are treated as function symbols in the first-order sense.  Their
    80 occurrences as head symbols in atoms (i.e., as predicate symbols) is
    81 turned into terms by equating such atoms with @{term True} using the
    82 following term-level equation symbol.
    83 *}
    84 
    85 definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)"
    86 
    87 
    88 
    89 subsection {* Integer division and modulo for Z3 *}
    90 
    91 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
    92   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
    93 
    94 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
    95   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
    96 
    97 lemma div_by_z3div: "k div l = (
    98      if k = 0 \<or> l = 0 then 0
    99      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
   100      else z3div (-k) (-l))"
   101   by (auto simp add: z3div_def)
   102 
   103 lemma mod_by_z3mod: "k mod l = (
   104      if l = 0 then k
   105      else if k = 0 then 0
   106      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
   107      else - z3mod (-k) (-l))"
   108   by (auto simp add: z3mod_def)
   109 
   110 
   111 
   112 subsection {* Setup *}
   113 
   114 use "Tools/SMT/smt_monomorph.ML"
   115 use "Tools/SMT/smt_normalize.ML"
   116 use "Tools/SMT/smt_translate.ML"
   117 use "Tools/SMT/smt_solver.ML"
   118 use "Tools/SMT/smtlib_interface.ML"
   119 use "Tools/SMT/z3_interface.ML"
   120 use "Tools/SMT/z3_proof_parser.ML"
   121 use "Tools/SMT/z3_proof_tools.ML"
   122 use "Tools/SMT/z3_proof_literals.ML"
   123 use "Tools/SMT/z3_proof_reconstruction.ML"
   124 use "Tools/SMT/z3_model.ML"
   125 use "Tools/SMT/smt_setup_solvers.ML"
   126 
   127 setup {*
   128   SMT_Solver.setup #>
   129   Z3_Proof_Reconstruction.setup #>
   130   SMT_Setup_Solvers.setup
   131 *}
   132 
   133 
   134 
   135 subsection {* Configuration *}
   136 
   137 text {*
   138 The current configuration can be printed by the command
   139 @{text smt_status}, which shows the values of most options.
   140 *}
   141 
   142 
   143 
   144 subsection {* General configuration options *}
   145 
   146 text {*
   147 The option @{text smt_solver} can be used to change the target SMT
   148 solver.  The possible values are @{text cvc3}, @{text yices}, and
   149 @{text z3}.  It is advisable to locally install the selected solver,
   150 although this is not necessary for @{text cvc3} and @{text z3}, which
   151 can also be used over an Internet-based service.
   152 
   153 When using local SMT solvers, the path to their binaries should be
   154 declared by setting the following environment variables:
   155 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
   156 *}
   157 
   158 declare [[ smt_solver = z3 ]]
   159 
   160 text {*
   161 Since SMT solvers are potentially non-terminating, there is a timeout
   162 (given in seconds) to restrict their runtime.  A value greater than
   163 120 (seconds) is in most cases not advisable.
   164 *}
   165 
   166 declare [[ smt_timeout = 20 ]]
   167 
   168 text {*
   169 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
   170 solvers are fully trusted without additional checks.  The following
   171 option can cause the SMT solver to run in proof-producing mode, giving
   172 a checkable certificate.  This is currently only implemented for Z3.
   173 *}
   174 
   175 declare [[ smt_oracle = false ]]
   176 
   177 text {*
   178 Each SMT solver provides several commandline options to tweak its
   179 behaviour.  They can be passed to the solver by setting the following
   180 options.
   181 *}
   182 
   183 declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]]
   184 
   185 text {*
   186 Enable the following option to use built-in support for datatypes and
   187 records.  Currently, this is only implemented for Z3 running in oracle
   188 mode.
   189 *}
   190 
   191 declare [[ smt_datatypes = false ]]
   192 
   193 
   194 
   195 subsection {* Certificates *}
   196 
   197 text {*
   198 By setting the option @{text smt_certificates} to the name of a file,
   199 all following applications of an SMT solver a cached in that file.
   200 Any further application of the same SMT solver (using the very same
   201 configuration) re-uses the cached certificate instead of invoking the
   202 solver.  An empty string disables caching certificates.
   203 
   204 The filename should be given as an explicit path.  It is good
   205 practice to use the name of the current theory (with ending
   206 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   207 *}
   208 
   209 declare [[ smt_certificates = "" ]]
   210 
   211 text {*
   212 The option @{text smt_fixed} controls whether only stored
   213 certificates are should be used or invocation of an SMT solver is
   214 allowed.  When set to @{text true}, no SMT solver will ever be
   215 invoked and only the existing certificates found in the configured
   216 cache are used;  when set to @{text false} and there is no cached
   217 certificate for some proposition, then the configured SMT solver is
   218 invoked.
   219 *}
   220 
   221 declare [[ smt_fixed = false ]]
   222 
   223 
   224 
   225 subsection {* Tracing *}
   226 
   227 text {*
   228 For tracing the generated problem file given to the SMT solver as
   229 well as the returned result of the solver, the option
   230 @{text smt_trace} should be set to @{text true}.
   231 *}
   232 
   233 declare [[ smt_trace = false ]]
   234 
   235 text {*
   236 From the set of assumptions given to the SMT solver, those assumptions
   237 used in the proof are traced when the following option is set to
   238 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   239 (see options @{text smt_solver} and @{text smt_oracle} above).
   240 *}
   241 
   242 declare [[ smt_trace_used_facts = false ]]
   243 
   244 
   245 
   246 subsection {* Schematic rules for Z3 proof reconstruction *}
   247 
   248 text {*
   249 Several prof rules of Z3 are not very well documented.  There are two
   250 lemma groups which can turn failing Z3 proof reconstruction attempts
   251 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   252 any implemented reconstruction procedure for all uncertain Z3 proof
   253 rules;  the facts in @{text z3_simp} are only fed to invocations of
   254 the simplifier when reconstructing theory-specific proof steps.
   255 *}
   256 
   257 lemmas [z3_rule] =
   258   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   259   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   260   if_True if_False not_not
   261 
   262 lemma [z3_rule]:
   263   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   264   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   265   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   266   by auto
   267 
   268 lemma [z3_rule]:
   269   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   270   by auto
   271 
   272 lemma [z3_rule]:
   273   "((\<not>P) = P) = False"
   274   "(P = (\<not>P)) = False"
   275   "(P \<noteq> Q) = (Q = (\<not>P))"
   276   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   277   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   278   by auto
   279 
   280 lemma [z3_rule]:
   281   "(if P then P else \<not>P) = True"
   282   "(if \<not>P then \<not>P else P) = True"
   283   "(if P then True else False) = P"
   284   "(if P then False else True) = (\<not>P)"
   285   "(if \<not>P then x else y) = (if P then y else x)"
   286   by auto
   287 
   288 lemma [z3_rule]:
   289   "P = Q \<or> P \<or> Q"
   290   "P = Q \<or> \<not>P \<or> \<not>Q"
   291   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   292   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   293   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   294   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   295   "P \<noteq> Q \<or> P \<or> \<not>Q"
   296   "P \<noteq> Q \<or> \<not>P \<or> Q"
   297   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   298   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   299   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   300   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   301   "P \<or> \<not>Q \<or> P \<noteq> Q"
   302   "\<not>P \<or> Q \<or> P \<noteq> Q"
   303   by auto
   304 
   305 lemma [z3_rule]:
   306   "0 + (x::int) = x"
   307   "x + 0 = x"
   308   "0 * x = 0"
   309   "1 * x = x"
   310   "x + y = y + x"
   311   by auto
   312 
   313 
   314 
   315 hide_type (open) pattern
   316 hide_const Pattern term_eq
   317 hide_const (open) trigger pat nopat fun_app z3div z3mod
   318 
   319 
   320 
   321 subsection {* Selectors for datatypes *}
   322 
   323 setup {* Datatype_Selectors.setup *}
   324 
   325 declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
   326 declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
   327 
   328 end