src/HOL/SMT2.thy
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(*  Title:      HOL/SMT2.thy
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    Author:     Sascha Boehme, TU Muenchen
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*)
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header {* Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2 *}
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theory SMT2
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imports Record
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keywords "smt2_status" :: diag
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begin
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ML_file "Tools/SMT2/smt2_util.ML"
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ML_file "Tools/SMT2/smt2_failure.ML"
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ML_file "Tools/SMT2/smt2_config.ML"
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subsection {* Triggers for quantifier instantiation *}
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text {*
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Some SMT solvers support patterns as a quantifier instantiation
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heuristics.  Patterns may either be positive terms (tagged by "pat")
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triggering quantifier instantiations -- when the solver finds a
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term matching a positive pattern, it instantiates the corresponding
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quantifier accordingly -- or negative terms (tagged by "nopat")
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inhibiting quantifier instantiations.  A list of patterns
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of the same kind is called a multipattern, and all patterns in a
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multipattern are considered conjunctively for quantifier instantiation.
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A list of multipatterns is called a trigger, and their multipatterns
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act disjunctively during quantifier instantiation.  Each multipattern
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should mention at least all quantified variables of the preceding
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quantifier block.
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*}
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typedecl pattern
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consts
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  pat :: "'a \<Rightarrow> pattern"
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  nopat :: "'a \<Rightarrow> pattern"
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definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool" where "trigger _ P = P"
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subsection {* Higher-order encoding *}
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text {*
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Application is made explicit for constants occurring with varying
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numbers of arguments.  This is achieved by the introduction of the
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following constant.
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*}
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definition fun_app :: "'a \<Rightarrow> 'a" where "fun_app f = f"
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text {*
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Some solvers support a theory of arrays which can be used to encode
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higher-order functions.  The following set of lemmas specifies the
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properties of such (extensional) arrays.
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*}
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lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def
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subsection {* Normalization *}
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lemma case_bool_if[abs_def]: "case_bool x y P = (if P then x else y)"
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  by simp
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lemma nat_int': "\<forall>n. nat (int n) = n" by simp
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lemma int_nat_nneg: "\<forall>i. i \<ge> 0 \<longrightarrow> int (nat i) = i" by simp
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lemma int_nat_neg: "\<forall>i. i < 0 \<longrightarrow> int (nat i) = 0" by simp
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lemma nat_zero_as_int: "0 = nat 0" by (rule transfer_nat_int_numerals(1))
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lemma nat_one_as_int: "1 = nat 1" by (rule transfer_nat_int_numerals(2))
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lemma nat_numeral_as_int: "numeral = (\<lambda>i. nat (numeral i))" by simp
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lemma nat_less_as_int: "op < = (\<lambda>a b. int a < int b)" by simp
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lemma nat_leq_as_int: "op \<le> = (\<lambda>a b. int a <= int b)" by simp
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lemma Suc_as_int: "Suc = (\<lambda>a. nat (int a + 1))" by (rule ext) simp
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lemma nat_plus_as_int: "op + = (\<lambda>a b. nat (int a + int b))" by (rule ext)+ simp
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lemma nat_minus_as_int: "op - = (\<lambda>a b. nat (int a - int b))" by (rule ext)+ simp
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lemma nat_times_as_int: "op * = (\<lambda>a b. nat (int a * int b))" by (simp add: nat_mult_distrib)
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lemma nat_div_as_int: "op div = (\<lambda>a b. nat (int a div int b))" by (simp add: nat_div_distrib)
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lemma nat_mod_as_int: "op mod = (\<lambda>a b. nat (int a mod int b))" by (simp add: nat_mod_distrib)
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lemma int_Suc: "int (Suc n) = int n + 1" by simp
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lemma int_plus: "int (n + m) = int n + int m" by (rule of_nat_add)
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lemma int_minus: "int (n - m) = int (nat (int n - int m))" by auto
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lemmas Ex1_def_raw = Ex1_def[abs_def]
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lemmas Ball_def_raw = Ball_def[abs_def]
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lemmas Bex_def_raw = Bex_def[abs_def]
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lemmas abs_if_raw = abs_if[abs_def]
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lemmas min_def_raw = min_def[abs_def]
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lemmas max_def_raw = max_def[abs_def]
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subsection {* Integer division and modulo for Z3 *}
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text {*
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The following Z3-inspired definitions are overspecified for the case where @{text "l = 0"}. This
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Schönheitsfehler is corrected in the @{text div_as_z3div} and @{text mod_as_z3mod} theorems.
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*}
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definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "z3div k l = (if l \<ge> 0 then k div l else - (k div - l))"
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definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "z3mod k l = k mod (if l \<ge> 0 then l else - l)"
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lemma div_as_z3div:
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  "\<forall>k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"
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  by (simp add: z3div_def)
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lemma mod_as_z3mod:
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  "\<forall>k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"
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  by (simp add: z3mod_def)
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subsection {* Setup *}
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ML_file "Tools/SMT2/smt2_builtin.ML"
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ML_file "Tools/SMT2/smt2_datatypes.ML"
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ML_file "Tools/SMT2/smt2_normalize.ML"
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ML_file "Tools/SMT2/smt2_translate.ML"
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ML_file "Tools/SMT2/smtlib2.ML"
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ML_file "Tools/SMT2/smtlib2_interface.ML"
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ML_file "Tools/SMT2/z3_new_proof.ML"
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ML_file "Tools/SMT2/z3_new_isar.ML"
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ML_file "Tools/SMT2/smt2_solver.ML"
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ML_file "Tools/SMT2/z3_new_interface.ML"
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ML_file "Tools/SMT2/z3_new_replay_util.ML"
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ML_file "Tools/SMT2/z3_new_replay_literals.ML"
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ML_file "Tools/SMT2/z3_new_replay_rules.ML"
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ML_file "Tools/SMT2/z3_new_replay_methods.ML"
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ML_file "Tools/SMT2/z3_new_replay.ML"
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ML_file "Tools/SMT2/smt2_systems.ML"
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method_setup smt2 = {*
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  Scan.optional Attrib.thms [] >>
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    (fn thms => fn ctxt =>
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      METHOD (fn facts => HEADGOAL (SMT2_Solver.smt2_tac ctxt (thms @ facts))))
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*} "apply an SMT solver to the current goal (based on SMT-LIB 2)"
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subsection {* Configuration *}
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text {*
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The current configuration can be printed by the command
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@{text smt2_status}, which shows the values of most options.
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*}
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subsection {* General configuration options *}
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text {*
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The option @{text smt2_solver} can be used to change the target SMT
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solver.  The possible values can be obtained from the @{text smt2_status}
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command.
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Due to licensing restrictions, Yices and Z3 are not installed/enabled
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by default.  Z3 is free for non-commercial applications and can be enabled
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by setting Isabelle system option @{text z3_non_commercial} to @{text yes}.
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*}
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declare [[ smt2_solver = z3_new ]]
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text {*
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Since SMT solvers are potentially non-terminating, there is a timeout
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(given in seconds) to restrict their runtime.  A value greater than
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120 (seconds) is in most cases not advisable.
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*}
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declare [[ smt2_timeout = 20 ]]
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text {*
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SMT solvers apply randomized heuristics.  In case a problem is not
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solvable by an SMT solver, changing the following option might help.
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*}
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declare [[ smt2_random_seed = 1 ]]
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text {*
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In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
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solvers are fully trusted without additional checks.  The following
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option can cause the SMT solver to run in proof-producing mode, giving
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a checkable certificate.  This is currently only implemented for Z3.
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*}
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declare [[ smt2_oracle = false ]]
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text {*
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Each SMT solver provides several commandline options to tweak its
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behaviour.  They can be passed to the solver by setting the following
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options.
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*}
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(* declare [[ cvc3_options = "" ]] TODO *)
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(* declare [[ yices_options = "" ]] TODO *)
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(* declare [[ z3_options = "" ]] TODO *)
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text {*
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The SMT method provides an inference mechanism to detect simple triggers
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in quantified formulas, which might increase the number of problems
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solvable by SMT solvers (note: triggers guide quantifier instantiations
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in the SMT solver).  To turn it on, set the following option.
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*}
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declare [[ smt2_infer_triggers = false ]]
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text {*
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Enable the following option to use built-in support for div/mod, datatypes,
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and records in Z3.  Currently, this is implemented only in oracle mode.
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*}
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declare [[ z3_new_extensions = false ]]
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text {*
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The SMT method monomorphizes the given facts, that is, it tries to
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instantiate all schematic type variables with fixed types occurring
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in the problem.  This is a (possibly nonterminating) fixed-point
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construction whose cycles are limited by the following option.
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*}
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declare [[ monomorph_max_rounds = 5 ]]
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text {*
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In addition, the number of generated monomorphic instances is limited
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by the following option.
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*}
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declare [[ monomorph_max_new_instances = 500 ]]
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subsection {* Certificates *}
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text {*
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By setting the option @{text smt2_certificates} to the name of a file,
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all following applications of an SMT solver a cached in that file.
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Any further application of the same SMT solver (using the very same
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configuration) re-uses the cached certificate instead of invoking the
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solver.  An empty string disables caching certificates.
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The filename should be given as an explicit path.  It is good
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practice to use the name of the current theory (with ending
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@{text ".certs"} instead of @{text ".thy"}) as the certificates file.
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Certificate files should be used at most once in a certain theory context,
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to avoid race conditions with other concurrent accesses.
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*}
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declare [[ smt2_certificates = "" ]]
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text {*
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The option @{text smt2_read_only_certificates} controls whether only
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stored certificates are should be used or invocation of an SMT solver
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is allowed.  When set to @{text true}, no SMT solver will ever be
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invoked and only the existing certificates found in the configured
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cache are used;  when set to @{text false} and there is no cached
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certificate for some proposition, then the configured SMT solver is
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invoked.
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*}
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declare [[ smt2_read_only_certificates = false ]]
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subsection {* Tracing *}
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text {*
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The SMT method, when applied, traces important information.  To
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make it entirely silent, set the following option to @{text false}.
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*}
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declare [[ smt2_verbose = true ]]
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text {*
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For tracing the generated problem file given to the SMT solver as
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well as the returned result of the solver, the option
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@{text smt2_trace} should be set to @{text true}.
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*}
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declare [[ smt2_trace = false ]]
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subsection {* Schematic rules for Z3 proof reconstruction *}
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text {*
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Several prof rules of Z3 are not very well documented.  There are two
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lemma groups which can turn failing Z3 proof reconstruction attempts
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into succeeding ones: the facts in @{text z3_rule} are tried prior to
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any implemented reconstruction procedure for all uncertain Z3 proof
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rules;  the facts in @{text z3_simp} are only fed to invocations of
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the simplifier when reconstructing theory-specific proof steps.
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*}
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lemmas [z3_new_rule] =
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  refl eq_commute conj_commute disj_commute simp_thms nnf_simps
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  ring_distribs field_simps times_divide_eq_right times_divide_eq_left
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  if_True if_False not_not
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lemma [z3_new_rule]:
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  "(P \<and> Q) = (\<not> (\<not> P \<or> \<not> Q))"
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  "(P \<and> Q) = (\<not> (\<not> Q \<or> \<not> P))"
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  "(\<not> P \<and> Q) = (\<not> (P \<or> \<not> Q))"
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  "(\<not> P \<and> Q) = (\<not> (\<not> Q \<or> P))"
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  "(P \<and> \<not> Q) = (\<not> (\<not> P \<or> Q))"
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  "(P \<and> \<not> Q) = (\<not> (Q \<or> \<not> P))"
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  "(\<not> P \<and> \<not> Q) = (\<not> (P \<or> Q))"
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  "(\<not> P \<and> \<not> Q) = (\<not> (Q \<or> P))"
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  by auto
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lemma [z3_new_rule]:
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  "(P \<longrightarrow> Q) = (Q \<or> \<not> P)"
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  "(\<not> P \<longrightarrow> Q) = (P \<or> Q)"
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  "(\<not> P \<longrightarrow> Q) = (Q \<or> P)"
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  "(True \<longrightarrow> P) = P"
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  "(P \<longrightarrow> True) = True"
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  "(False \<longrightarrow> P) = True"
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  "(P \<longrightarrow> P) = True"
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  by auto
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lemma [z3_new_rule]:
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  "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not> P)))"
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  by auto
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lemma [z3_new_rule]:
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  "(\<not> True) = False"
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  "(\<not> False) = True"
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  "(x = x) = True"
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  "(P = True) = P"
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  "(True = P) = P"
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  "(P = False) = (\<not> P)"
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  "(False = P) = (\<not> P)"
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  "((\<not> P) = P) = False"
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  "(P = (\<not> P)) = False"
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  "((\<not> P) = (\<not> Q)) = (P = Q)"
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  "\<not> (P = (\<not> Q)) = (P = Q)"
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  "\<not> ((\<not> P) = Q) = (P = Q)"
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  "(P \<noteq> Q) = (Q = (\<not> P))"
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  "(P = Q) = ((\<not> P \<or> Q) \<and> (P \<or> \<not> Q))"
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  "(P \<noteq> Q) = ((\<not> P \<or> \<not> Q) \<and> (P \<or> Q))"
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  by auto
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lemma [z3_new_rule]:
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  "(if P then P else \<not> P) = True"
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  "(if \<not> P then \<not> P else P) = True"
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  "(if P then True else False) = P"
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  "(if P then False else True) = (\<not> P)"
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  "(if P then Q else True) = ((\<not> P) \<or> Q)"
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  "(if P then Q else True) = (Q \<or> (\<not> P))"
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  "(if P then Q else \<not> Q) = (P = Q)"
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  "(if P then Q else \<not> Q) = (Q = P)"
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  "(if P then \<not> Q else Q) = (P = (\<not> Q))"
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  "(if P then \<not> Q else Q) = ((\<not> Q) = P)"
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  "(if \<not> P then x else y) = (if P then y else x)"
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  "(if P then (if Q then x else y) else x) = (if P \<and> (\<not> Q) then y else x)"
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  "(if P then (if Q then x else y) else x) = (if (\<not> Q) \<and> P then y else x)"
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  "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
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  "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
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  "(if P then x else if P then y else z) = (if P then x else z)"
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  "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
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  "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
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  "(if P then x = y else x = z) = (x = (if P then y else z))"
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  "(if P then x = y else y = z) = (y = (if P then x else z))"
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  "(if P then x = y else z = y) = (y = (if P then x else z))"
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   365
  by auto
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lemma [z3_new_rule]:
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  "0 + (x::int) = x"
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  "x + 0 = x"
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  "x + x = 2 * x"
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  "0 * x = 0"
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  "1 * x = x"
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  "x + y = y + x"
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   374
  by auto
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   375
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lemma [z3_new_rule]:  (* for def-axiom *)
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  "P = Q \<or> P \<or> Q"
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   378
  "P = Q \<or> \<not> P \<or> \<not> Q"
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   379
  "(\<not> P) = Q \<or> \<not> P \<or> Q"
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   380
  "(\<not> P) = Q \<or> P \<or> \<not> Q"
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   381
  "P = (\<not> Q) \<or> \<not> P \<or> Q"
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   382
  "P = (\<not> Q) \<or> P \<or> \<not> Q"
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   383
  "P \<noteq> Q \<or> P \<or> \<not> Q"
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   384
  "P \<noteq> Q \<or> \<not> P \<or> Q"
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   385
  "P \<noteq> (\<not> Q) \<or> P \<or> Q"
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   386
  "(\<not> P) \<noteq> Q \<or> P \<or> Q"
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   387
  "P \<or> Q \<or> P \<noteq> (\<not> Q)"
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   388
  "P \<or> Q \<or> (\<not> P) \<noteq> Q"
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   389
  "P \<or> \<not> Q \<or> P \<noteq> Q"
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   390
  "\<not> P \<or> Q \<or> P \<noteq> Q"
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  "P \<or> y = (if P then x else y)"
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  "P \<or> (if P then x else y) = y"
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   393
  "\<not> P \<or> x = (if P then x else y)"
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   394
  "\<not> P \<or> (if P then x else y) = x"
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   395
  "P \<or> R \<or> \<not> (if P then Q else R)"
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   396
  "\<not> P \<or> Q \<or> \<not> (if P then Q else R)"
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diff changeset
   397
  "\<not> (if P then Q else R) \<or> \<not> P \<or> Q"
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diff changeset
   398
  "\<not> (if P then Q else R) \<or> P \<or> R"
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diff changeset
   399
  "(if P then Q else R) \<or> \<not> P \<or> \<not> Q"
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   400
  "(if P then Q else R) \<or> P \<or> \<not> R"
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diff changeset
   401
  "(if P then \<not> Q else R) \<or> \<not> P \<or> Q"
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   402
  "(if P then Q else \<not> R) \<or> P \<or> R"
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   403
  by auto
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diff changeset
   404
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   405
hide_type (open) pattern
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   406
hide_const fun_app z3div z3mod
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   407
hide_const (open) trigger pat nopat
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   408
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end