src/HOL/SMT.thy
author boehmes
Mon Nov 22 15:45:42 2010 +0100 (2010-11-22)
changeset 40662 798aad2229c0
parent 40424 7550b2cba1cb
child 40664 e023788a91a1
permissions -rw-r--r--
added prove reconstruction for injective functions;
added SMT_Utils to collect frequently used functions
     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_failure.ML"
    12   "Tools/SMT/smt_config.ML"
    13   "Tools/SMT/smt_utils.ML"
    14   "Tools/SMT/smt_monomorph.ML"
    15   ("Tools/SMT/smt_builtin.ML")
    16   ("Tools/SMT/smt_normalize.ML")
    17   ("Tools/SMT/smt_translate.ML")
    18   ("Tools/SMT/smt_solver.ML")
    19   ("Tools/SMT/smtlib_interface.ML")
    20   ("Tools/SMT/z3_proof_parser.ML")
    21   ("Tools/SMT/z3_proof_tools.ML")
    22   ("Tools/SMT/z3_proof_literals.ML")
    23   ("Tools/SMT/z3_proof_methods.ML")
    24   ("Tools/SMT/z3_proof_reconstruction.ML")
    25   ("Tools/SMT/z3_model.ML")
    26   ("Tools/SMT/z3_interface.ML")
    27   ("Tools/SMT/smt_setup_solvers.ML")
    28 begin
    29 
    30 
    31 
    32 subsection {* Triggers for quantifier instantiation *}
    33 
    34 text {*
    35 Some SMT solvers support triggers for quantifier instantiation.
    36 Each trigger consists of one ore more patterns.  A pattern may either
    37 be a list of positive subterms (each being tagged by "pat"), or a
    38 list of negative subterms (each being tagged by "nopat").
    39 
    40 When an SMT solver finds a term matching a positive pattern (a
    41 pattern with positive subterms only), it instantiates the
    42 corresponding quantifier accordingly.  Negative patterns inhibit
    43 quantifier instantiations.  Each pattern should mention all preceding
    44 bound variables.
    45 *}
    46 
    47 datatype pattern = Pattern
    48 
    49 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
    50 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
    51 
    52 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
    53 where "trigger _ P = P"
    54 
    55 
    56 
    57 subsection {* Distinctness *}
    58 
    59 text {*
    60 As an abbreviation for a quadratic number of inequalities, SMT solvers
    61 provide a built-in @{text distinct}.  To avoid confusion with the
    62 already defined (and more general) @{term List.distinct}, a separate
    63 constant is defined.
    64 *}
    65 
    66 definition distinct :: "'a list \<Rightarrow> bool"
    67 where "distinct xs = List.distinct xs"
    68 
    69 
    70 
    71 subsection {* Higher-order encoding *}
    72 
    73 text {*
    74 Application is made explicit for constants occurring with varying
    75 numbers of arguments.  This is achieved by the introduction of the
    76 following constant.
    77 *}
    78 
    79 definition fun_app where "fun_app f x = f x"
    80 
    81 text {*
    82 Some solvers support a theory of arrays which can be used to encode
    83 higher-order functions.  The following set of lemmas specifies the
    84 properties of such (extensional) arrays.
    85 *}
    86 
    87 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
    88   fun_upd_upd fun_app_def
    89 
    90 
    91 
    92 subsection {* First-order logic *}
    93 
    94 text {*
    95 Some SMT solvers require a strict separation between formulas and
    96 terms.  When translating higher-order into first-order problems,
    97 all uninterpreted constants (those not builtin in the target solver)
    98 are treated as function symbols in the first-order sense.  Their
    99 occurrences as head symbols in atoms (i.e., as predicate symbols) is
   100 turned into terms by equating such atoms with @{term True} using the
   101 following term-level equation symbol.
   102 *}
   103 
   104 definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)"
   105 
   106 
   107 
   108 subsection {* Integer division and modulo for Z3 *}
   109 
   110 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
   111   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
   112 
   113 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
   114   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
   115 
   116 lemma div_by_z3div: "k div l = (
   117      if k = 0 \<or> l = 0 then 0
   118      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
   119      else z3div (-k) (-l))"
   120   by (auto simp add: z3div_def)
   121 
   122 lemma mod_by_z3mod: "k mod l = (
   123      if l = 0 then k
   124      else if k = 0 then 0
   125      else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
   126      else - z3mod (-k) (-l))"
   127   by (auto simp add: z3mod_def)
   128 
   129 
   130 
   131 subsection {* Setup *}
   132 
   133 use "Tools/SMT/smt_builtin.ML"
   134 use "Tools/SMT/smt_normalize.ML"
   135 use "Tools/SMT/smt_translate.ML"
   136 use "Tools/SMT/smt_solver.ML"
   137 use "Tools/SMT/smtlib_interface.ML"
   138 use "Tools/SMT/z3_interface.ML"
   139 use "Tools/SMT/z3_proof_parser.ML"
   140 use "Tools/SMT/z3_proof_tools.ML"
   141 use "Tools/SMT/z3_proof_literals.ML"
   142 use "Tools/SMT/z3_proof_methods.ML"
   143 use "Tools/SMT/z3_proof_reconstruction.ML"
   144 use "Tools/SMT/z3_model.ML"
   145 use "Tools/SMT/smt_setup_solvers.ML"
   146 
   147 setup {*
   148   SMT_Config.setup #>
   149   SMT_Solver.setup #>
   150   Z3_Proof_Reconstruction.setup #>
   151   SMT_Setup_Solvers.setup
   152 *}
   153 
   154 
   155 
   156 subsection {* Configuration *}
   157 
   158 text {*
   159 The current configuration can be printed by the command
   160 @{text smt_status}, which shows the values of most options.
   161 *}
   162 
   163 
   164 
   165 subsection {* General configuration options *}
   166 
   167 text {*
   168 The option @{text smt_solver} can be used to change the target SMT
   169 solver.  The possible values are @{text cvc3}, @{text yices}, and
   170 @{text z3}.  It is advisable to locally install the selected solver,
   171 although this is not necessary for @{text cvc3} and @{text z3}, which
   172 can also be used over an Internet-based service.
   173 
   174 When using local SMT solvers, the path to their binaries should be
   175 declared by setting the following environment variables:
   176 @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
   177 *}
   178 
   179 declare [[ smt_solver = z3 ]]
   180 
   181 text {*
   182 Since SMT solvers are potentially non-terminating, there is a timeout
   183 (given in seconds) to restrict their runtime.  A value greater than
   184 120 (seconds) is in most cases not advisable.
   185 *}
   186 
   187 declare [[ smt_timeout = 20 ]]
   188 
   189 text {*
   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 *}
   195 
   196 declare [[ smt_oracle = false ]]
   197 
   198 text {*
   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 *}
   203 
   204 declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]]
   205 
   206 text {*
   207 Enable the following option to use built-in support for datatypes and
   208 records.  Currently, this is only implemented for Z3 running in oracle
   209 mode.
   210 *}
   211 
   212 declare [[ smt_datatypes = false ]]
   213 
   214 
   215 
   216 subsection {* Certificates *}
   217 
   218 text {*
   219 By setting the option @{text smt_certificates} to the name of a file,
   220 all following applications of an SMT solver a cached in that file.
   221 Any further application of the same SMT solver (using the very same
   222 configuration) re-uses the cached certificate instead of invoking the
   223 solver.  An empty string disables caching certificates.
   224 
   225 The filename should be given as an explicit path.  It is good
   226 practice to use the name of the current theory (with ending
   227 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
   228 *}
   229 
   230 declare [[ smt_certificates = "" ]]
   231 
   232 text {*
   233 The option @{text smt_fixed} controls whether only stored
   234 certificates are should be used or invocation of an SMT solver is
   235 allowed.  When set to @{text true}, no SMT solver will ever be
   236 invoked and only the existing certificates found in the configured
   237 cache are used;  when set to @{text false} and there is no cached
   238 certificate for some proposition, then the configured SMT solver is
   239 invoked.
   240 *}
   241 
   242 declare [[ smt_fixed = false ]]
   243 
   244 
   245 
   246 subsection {* Tracing *}
   247 
   248 text {*
   249 The SMT method, when applied, traces important information.  To
   250 make it entirely silent, set the following option to @{text false}.
   251 *}
   252 
   253 declare [[ smt_verbose = true ]]
   254 
   255 text {*
   256 For tracing the generated problem file given to the SMT solver as
   257 well as the returned result of the solver, the option
   258 @{text smt_trace} should be set to @{text true}.
   259 *}
   260 
   261 declare [[ smt_trace = false ]]
   262 
   263 text {*
   264 From the set of assumptions given to the SMT solver, those assumptions
   265 used in the proof are traced when the following option is set to
   266 @{term true}.  This only works for Z3 when it runs in non-oracle mode
   267 (see options @{text smt_solver} and @{text smt_oracle} above).
   268 *}
   269 
   270 declare [[ smt_trace_used_facts = false ]]
   271 
   272 
   273 
   274 subsection {* Schematic rules for Z3 proof reconstruction *}
   275 
   276 text {*
   277 Several prof rules of Z3 are not very well documented.  There are two
   278 lemma groups which can turn failing Z3 proof reconstruction attempts
   279 into succeeding ones: the facts in @{text z3_rule} are tried prior to
   280 any implemented reconstruction procedure for all uncertain Z3 proof
   281 rules;  the facts in @{text z3_simp} are only fed to invocations of
   282 the simplifier when reconstructing theory-specific proof steps.
   283 *}
   284 
   285 lemmas [z3_rule] =
   286   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   287   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   288   if_True if_False not_not
   289 
   290 lemma [z3_rule]:
   291   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
   292   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
   293   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
   294   by auto
   295 
   296 lemma [z3_rule]:
   297   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
   298   by auto
   299 
   300 lemma [z3_rule]:
   301   "((\<not>P) = P) = False"
   302   "(P = (\<not>P)) = False"
   303   "(P \<noteq> Q) = (Q = (\<not>P))"
   304   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
   305   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
   306   by auto
   307 
   308 lemma [z3_rule]:
   309   "(if P then P else \<not>P) = True"
   310   "(if \<not>P then \<not>P else P) = True"
   311   "(if P then True else False) = P"
   312   "(if P then False else True) = (\<not>P)"
   313   "(if \<not>P then x else y) = (if P then y else x)"
   314   by auto
   315 
   316 lemma [z3_rule]:
   317   "P = Q \<or> P \<or> Q"
   318   "P = Q \<or> \<not>P \<or> \<not>Q"
   319   "(\<not>P) = Q \<or> \<not>P \<or> Q"
   320   "(\<not>P) = Q \<or> P \<or> \<not>Q"
   321   "P = (\<not>Q) \<or> \<not>P \<or> Q"
   322   "P = (\<not>Q) \<or> P \<or> \<not>Q"
   323   "P \<noteq> Q \<or> P \<or> \<not>Q"
   324   "P \<noteq> Q \<or> \<not>P \<or> Q"
   325   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
   326   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
   327   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
   328   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
   329   "P \<or> \<not>Q \<or> P \<noteq> Q"
   330   "\<not>P \<or> Q \<or> P \<noteq> Q"
   331   by auto
   332 
   333 lemma [z3_rule]:
   334   "0 + (x::int) = x"
   335   "x + 0 = x"
   336   "0 * x = 0"
   337   "1 * x = x"
   338   "x + y = y + x"
   339   by auto
   340 
   341 
   342 
   343 hide_type (open) pattern
   344 hide_const Pattern term_eq
   345 hide_const (open) trigger pat nopat distinct fun_app z3div z3mod
   346 
   347 
   348 
   349 subsection {* Selectors for datatypes *}
   350 
   351 setup {* Datatype_Selectors.setup *}
   352 
   353 declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
   354 declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
   355 
   356 end