
blanchet@56078: (*  Title:      HOL/SMT2.thy
blanchet@56078:     Author:     Sascha Boehme, TU Muenchen
blanchet@56078: *)
blanchet@56078: 
blanchet@56078: header {* Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2 *}
blanchet@56078: 
blanchet@56078: theory SMT2
blanchet@57230: imports Divides
blanchet@56078: keywords "smt2_status" :: diag
blanchet@56078: begin
blanchet@56078: 
blanchet@56078: subsection {* Triggers for quantifier instantiation *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Some SMT solvers support patterns as a quantifier instantiation
blanchet@56078: heuristics.  Patterns may either be positive terms (tagged by "pat")
blanchet@56078: triggering quantifier instantiations -- when the solver finds a
blanchet@56078: term matching a positive pattern, it instantiates the corresponding
blanchet@56078: quantifier accordingly -- or negative terms (tagged by "nopat")
blanchet@56078: inhibiting quantifier instantiations.  A list of patterns
blanchet@56078: of the same kind is called a multipattern, and all patterns in a
blanchet@56078: multipattern are considered conjunctively for quantifier instantiation.
blanchet@56078: A list of multipatterns is called a trigger, and their multipatterns
blanchet@56078: act disjunctively during quantifier instantiation.  Each multipattern
blanchet@56078: should mention at least all quantified variables of the preceding
blanchet@56078: quantifier block.
blanchet@56078: *}
blanchet@56078: 
blanchet@57230: typedecl 'a symb_list
blanchet@57230: 
blanchet@57230: consts
blanchet@57230:   Symb_Nil :: "'a symb_list"
blanchet@57230:   Symb_Cons :: "'a \<Rightarrow> 'a symb_list \<Rightarrow> 'a symb_list"
blanchet@57230: 
blanchet@56078: typedecl pattern
blanchet@56078: 
blanchet@56078: consts
blanchet@56078:   pat :: "'a \<Rightarrow> pattern"
blanchet@56078:   nopat :: "'a \<Rightarrow> pattern"
blanchet@56078: 
blanchet@57230: definition trigger :: "pattern symb_list symb_list \<Rightarrow> bool \<Rightarrow> bool" where
blanchet@57230:   "trigger _ P = P"
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* Higher-order encoding *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Application is made explicit for constants occurring with varying
blanchet@56078: numbers of arguments.  This is achieved by the introduction of the
blanchet@56078: following constant.
blanchet@56078: *}
blanchet@56078: 
blanchet@56078: definition fun_app :: "'a \<Rightarrow> 'a" where "fun_app f = f"
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Some solvers support a theory of arrays which can be used to encode
blanchet@56078: higher-order functions.  The following set of lemmas specifies the
blanchet@56078: properties of such (extensional) arrays.
blanchet@56078: *}
blanchet@56078: 
blanchet@56078: lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def
blanchet@56078: 
blanchet@56078: 
blanchet@56103: subsection {* Normalization *}
blanchet@56103: 
blanchet@56103: lemma case_bool_if[abs_def]: "case_bool x y P = (if P then x else y)"
blanchet@56103:   by simp
blanchet@56103: 
blanchet@56107: lemma nat_int': "\<forall>n. nat (int n) = n" by simp
blanchet@56103: lemma int_nat_nneg: "\<forall>i. i \<ge> 0 \<longrightarrow> int (nat i) = i" by simp
blanchet@56103: lemma int_nat_neg: "\<forall>i. i < 0 \<longrightarrow> int (nat i) = 0" by simp
blanchet@56103: 
blanchet@57230: lemma nat_zero_as_int: "0 = nat 0" by simp
blanchet@57230: lemma nat_one_as_int: "1 = nat 1" by simp
blanchet@56103: lemma nat_numeral_as_int: "numeral = (\<lambda>i. nat (numeral i))" by simp
blanchet@56103: lemma nat_less_as_int: "op < = (\<lambda>a b. int a < int b)" by simp
blanchet@56103: lemma nat_leq_as_int: "op \<le> = (\<lambda>a b. int a <= int b)" by simp
blanchet@56103: lemma Suc_as_int: "Suc = (\<lambda>a. nat (int a + 1))" by (rule ext) simp
blanchet@56103: lemma nat_plus_as_int: "op + = (\<lambda>a b. nat (int a + int b))" by (rule ext)+ simp
blanchet@56103: lemma nat_minus_as_int: "op - = (\<lambda>a b. nat (int a - int b))" by (rule ext)+ simp
blanchet@56103: lemma nat_times_as_int: "op * = (\<lambda>a b. nat (int a * int b))" by (simp add: nat_mult_distrib)
blanchet@56103: lemma nat_div_as_int: "op div = (\<lambda>a b. nat (int a div int b))" by (simp add: nat_div_distrib)
blanchet@56103: lemma nat_mod_as_int: "op mod = (\<lambda>a b. nat (int a mod int b))" by (simp add: nat_mod_distrib)
blanchet@56103: 
blanchet@56103: lemma int_Suc: "int (Suc n) = int n + 1" by simp
blanchet@56103: lemma int_plus: "int (n + m) = int n + int m" by (rule of_nat_add)
blanchet@56103: lemma int_minus: "int (n - m) = int (nat (int n - int m))" by auto
blanchet@56103: 
blanchet@56103: lemmas Ex1_def_raw = Ex1_def[abs_def]
blanchet@56103: lemmas Ball_def_raw = Ball_def[abs_def]
blanchet@56103: lemmas Bex_def_raw = Bex_def[abs_def]
blanchet@56103: lemmas abs_if_raw = abs_if[abs_def]
blanchet@56103: lemmas min_def_raw = min_def[abs_def]
blanchet@56103: lemmas max_def_raw = max_def[abs_def]
blanchet@56103: 
blanchet@56103: 
blanchet@56078: subsection {* Integer division and modulo for Z3 *}
blanchet@56078: 
blanchet@56102: text {*
blanchet@56102: The following Z3-inspired definitions are overspecified for the case where @{text "l = 0"}. This
blanchet@56102: Schönheitsfehler is corrected in the @{text div_as_z3div} and @{text mod_as_z3mod} theorems.
blanchet@56102: *}
blanchet@56102: 
blanchet@56078: definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@56102:   "z3div k l = (if l \<ge> 0 then k div l else - (k div - l))"
blanchet@56078: 
blanchet@56078: definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@56102:   "z3mod k l = k mod (if l \<ge> 0 then l else - l)"
blanchet@56078: 
blanchet@56101: lemma div_as_z3div:
blanchet@56102:   "\<forall>k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"
blanchet@56101:   by (simp add: z3div_def)
blanchet@56101: 
blanchet@56101: lemma mod_as_z3mod:
blanchet@56102:   "\<forall>k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"
blanchet@56101:   by (simp add: z3mod_def)
blanchet@56101: 
blanchet@56078: 
blanchet@56078: subsection {* Setup *}
blanchet@56078: 
blanchet@57230: ML_file "Tools/SMT2/smt2_util.ML"
blanchet@57230: ML_file "Tools/SMT2/smt2_failure.ML"
blanchet@57230: ML_file "Tools/SMT2/smt2_config.ML"
blanchet@56078: ML_file "Tools/SMT2/smt2_builtin.ML"
blanchet@56078: ML_file "Tools/SMT2/smt2_datatypes.ML"
blanchet@56078: ML_file "Tools/SMT2/smt2_normalize.ML"
blanchet@56078: ML_file "Tools/SMT2/smt2_translate.ML"
blanchet@56078: ML_file "Tools/SMT2/smtlib2.ML"
blanchet@56078: ML_file "Tools/SMT2/smtlib2_interface.ML"
blanchet@57219: ML_file "Tools/SMT2/smtlib2_proof.ML"
blanchet@56083: ML_file "Tools/SMT2/z3_new_proof.ML"
blanchet@57159: ML_file "Tools/SMT2/z3_new_isar.ML"
blanchet@56083: ML_file "Tools/SMT2/smt2_solver.ML"
blanchet@56078: ML_file "Tools/SMT2/z3_new_interface.ML"
blanchet@56090: ML_file "Tools/SMT2/z3_new_replay_util.ML"
blanchet@56090: ML_file "Tools/SMT2/z3_new_replay_literals.ML"
blanchet@56090: ML_file "Tools/SMT2/z3_new_replay_rules.ML"
blanchet@56090: ML_file "Tools/SMT2/z3_new_replay_methods.ML"
blanchet@56090: ML_file "Tools/SMT2/z3_new_replay.ML"
blanchet@56090: ML_file "Tools/SMT2/smt2_systems.ML"
blanchet@56078: 
blanchet@56078: method_setup smt2 = {*
blanchet@56078:   Scan.optional Attrib.thms [] >>
blanchet@56078:     (fn thms => fn ctxt =>
blanchet@56078:       METHOD (fn facts => HEADGOAL (SMT2_Solver.smt2_tac ctxt (thms @ facts))))
blanchet@56078: *} "apply an SMT solver to the current goal (based on SMT-LIB 2)"
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* Configuration *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: The current configuration can be printed by the command
blanchet@56078: @{text smt2_status}, which shows the values of most options.
blanchet@56078: *}
blanchet@56078: 
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* General configuration options *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: The option @{text smt2_solver} can be used to change the target SMT
blanchet@56078: solver.  The possible values can be obtained from the @{text smt2_status}
blanchet@56078: command.
blanchet@56078: 
blanchet@57237: Due to licensing restrictions, Z3 is not enabled by default.  Z3 is free
blanchet@57237: for non-commercial applications and can be enabled by setting Isabelle
blanchet@57237: system option @{text z3_non_commercial} to @{text yes}.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_solver = z3]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Since SMT solvers are potentially non-terminating, there is a timeout
blanchet@56078: (given in seconds) to restrict their runtime.  A value greater than
blanchet@56078: 120 (seconds) is in most cases not advisable.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_timeout = 20]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: SMT solvers apply randomized heuristics.  In case a problem is not
blanchet@56078: solvable by an SMT solver, changing the following option might help.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_random_seed = 1]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
blanchet@56078: solvers are fully trusted without additional checks.  The following
blanchet@56078: option can cause the SMT solver to run in proof-producing mode, giving
blanchet@56078: a checkable certificate.  This is currently only implemented for Z3.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_oracle = false]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Each SMT solver provides several commandline options to tweak its
blanchet@56078: behaviour.  They can be passed to the solver by setting the following
blanchet@56078: options.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[cvc3_new_options = ""]]
blanchet@57246: declare [[cvc4_new_options = ""]]
blanchet@57209: declare [[z3_new_options = ""]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: The SMT method provides an inference mechanism to detect simple triggers
blanchet@56078: in quantified formulas, which might increase the number of problems
blanchet@56078: solvable by SMT solvers (note: triggers guide quantifier instantiations
blanchet@56078: in the SMT solver).  To turn it on, set the following option.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_infer_triggers = false]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Enable the following option to use built-in support for div/mod, datatypes,
blanchet@56078: and records in Z3.  Currently, this is implemented only in oracle mode.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[z3_new_extensions = false]]
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* Certificates *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: By setting the option @{text smt2_certificates} to the name of a file,
blanchet@56078: all following applications of an SMT solver a cached in that file.
blanchet@56078: Any further application of the same SMT solver (using the very same
blanchet@56078: configuration) re-uses the cached certificate instead of invoking the
blanchet@56078: solver.  An empty string disables caching certificates.
blanchet@56078: 
blanchet@56078: The filename should be given as an explicit path.  It is good
blanchet@56078: practice to use the name of the current theory (with ending
blanchet@56078: @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
blanchet@56078: Certificate files should be used at most once in a certain theory context,
blanchet@56078: to avoid race conditions with other concurrent accesses.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_certificates = ""]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: The option @{text smt2_read_only_certificates} controls whether only
blanchet@56078: stored certificates are should be used or invocation of an SMT solver
blanchet@56078: is allowed.  When set to @{text true}, no SMT solver will ever be
blanchet@56078: invoked and only the existing certificates found in the configured
blanchet@56078: cache are used;  when set to @{text false} and there is no cached
blanchet@56078: certificate for some proposition, then the configured SMT solver is
blanchet@56078: invoked.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_read_only_certificates = false]]
blanchet@56078: 
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* Tracing *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: The SMT method, when applied, traces important information.  To
blanchet@56078: make it entirely silent, set the following option to @{text false}.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_verbose = true]]
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: For tracing the generated problem file given to the SMT solver as
blanchet@56078: well as the returned result of the solver, the option
blanchet@56078: @{text smt2_trace} should be set to @{text true}.
blanchet@56078: *}
blanchet@56078: 
blanchet@57209: declare [[smt2_trace = false]]
blanchet@56078: 
blanchet@56078: 
blanchet@56078: subsection {* Schematic rules for Z3 proof reconstruction *}
blanchet@56078: 
blanchet@56078: text {*
blanchet@56078: Several prof rules of Z3 are not very well documented.  There are two
blanchet@56078: lemma groups which can turn failing Z3 proof reconstruction attempts
blanchet@56078: into succeeding ones: the facts in @{text z3_rule} are tried prior to
blanchet@56078: any implemented reconstruction procedure for all uncertain Z3 proof
blanchet@56078: rules;  the facts in @{text z3_simp} are only fed to invocations of
blanchet@56078: the simplifier when reconstructing theory-specific proof steps.
blanchet@56078: *}
blanchet@56078: 
blanchet@56078: lemmas [z3_new_rule] =
blanchet@56078:   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
blanchet@56078:   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
blanchet@56078:   if_True if_False not_not
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@57169:   "(P \<and> Q) = (\<not> (\<not> P \<or> \<not> Q))"
blanchet@57169:   "(P \<and> Q) = (\<not> (\<not> Q \<or> \<not> P))"
blanchet@57169:   "(\<not> P \<and> Q) = (\<not> (P \<or> \<not> Q))"
blanchet@57169:   "(\<not> P \<and> Q) = (\<not> (\<not> Q \<or> P))"
blanchet@57169:   "(P \<and> \<not> Q) = (\<not> (\<not> P \<or> Q))"
blanchet@57169:   "(P \<and> \<not> Q) = (\<not> (Q \<or> \<not> P))"
blanchet@57169:   "(\<not> P \<and> \<not> Q) = (\<not> (P \<or> Q))"
blanchet@57169:   "(\<not> P \<and> \<not> Q) = (\<not> (Q \<or> P))"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@57169:   "(P \<longrightarrow> Q) = (Q \<or> \<not> P)"
blanchet@57169:   "(\<not> P \<longrightarrow> Q) = (P \<or> Q)"
blanchet@57169:   "(\<not> P \<longrightarrow> Q) = (Q \<or> P)"
blanchet@56078:   "(True \<longrightarrow> P) = P"
blanchet@56078:   "(P \<longrightarrow> True) = True"
blanchet@56078:   "(False \<longrightarrow> P) = True"
blanchet@56078:   "(P \<longrightarrow> P) = True"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@57169:   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not> P)))"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@57169:   "(\<not> True) = False"
blanchet@57169:   "(\<not> False) = True"
blanchet@56078:   "(x = x) = True"
blanchet@56078:   "(P = True) = P"
blanchet@56078:   "(True = P) = P"
blanchet@57169:   "(P = False) = (\<not> P)"
blanchet@57169:   "(False = P) = (\<not> P)"
blanchet@57169:   "((\<not> P) = P) = False"
blanchet@57169:   "(P = (\<not> P)) = False"
blanchet@57169:   "((\<not> P) = (\<not> Q)) = (P = Q)"
blanchet@57169:   "\<not> (P = (\<not> Q)) = (P = Q)"
blanchet@57169:   "\<not> ((\<not> P) = Q) = (P = Q)"
blanchet@57169:   "(P \<noteq> Q) = (Q = (\<not> P))"
blanchet@57169:   "(P = Q) = ((\<not> P \<or> Q) \<and> (P \<or> \<not> Q))"
blanchet@57169:   "(P \<noteq> Q) = ((\<not> P \<or> \<not> Q) \<and> (P \<or> Q))"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@57169:   "(if P then P else \<not> P) = True"
blanchet@57169:   "(if \<not> P then \<not> P else P) = True"
blanchet@56078:   "(if P then True else False) = P"
blanchet@57169:   "(if P then False else True) = (\<not> P)"
blanchet@57169:   "(if P then Q else True) = ((\<not> P) \<or> Q)"
blanchet@57169:   "(if P then Q else True) = (Q \<or> (\<not> P))"
blanchet@57169:   "(if P then Q else \<not> Q) = (P = Q)"
blanchet@57169:   "(if P then Q else \<not> Q) = (Q = P)"
blanchet@57169:   "(if P then \<not> Q else Q) = (P = (\<not> Q))"
blanchet@57169:   "(if P then \<not> Q else Q) = ((\<not> Q) = P)"
blanchet@57169:   "(if \<not> P then x else y) = (if P then y else x)"
blanchet@57169:   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not> Q) then y else x)"
blanchet@57169:   "(if P then (if Q then x else y) else x) = (if (\<not> Q) \<and> P then y else x)"
blanchet@56078:   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
blanchet@56078:   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
blanchet@56078:   "(if P then x else if P then y else z) = (if P then x else z)"
blanchet@56078:   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
blanchet@56078:   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
blanchet@56078:   "(if P then x = y else x = z) = (x = (if P then y else z))"
blanchet@56078:   "(if P then x = y else y = z) = (y = (if P then x else z))"
blanchet@56078:   "(if P then x = y else z = y) = (y = (if P then x else z))"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:
blanchet@56078:   "0 + (x::int) = x"
blanchet@56078:   "x + 0 = x"
blanchet@56078:   "x + x = 2 * x"
blanchet@56078:   "0 * x = 0"
blanchet@56078:   "1 * x = x"
blanchet@56078:   "x + y = y + x"
blanchet@57230:   by (auto simp add: mult_2)
blanchet@56078: 
blanchet@56078: lemma [z3_new_rule]:  (* for def-axiom *)
blanchet@56078:   "P = Q \<or> P \<or> Q"
blanchet@57169:   "P = Q \<or> \<not> P \<or> \<not> Q"
blanchet@57169:   "(\<not> P) = Q \<or> \<not> P \<or> Q"
blanchet@57169:   "(\<not> P) = Q \<or> P \<or> \<not> Q"
blanchet@57169:   "P = (\<not> Q) \<or> \<not> P \<or> Q"
blanchet@57169:   "P = (\<not> Q) \<or> P \<or> \<not> Q"
blanchet@57169:   "P \<noteq> Q \<or> P \<or> \<not> Q"
blanchet@57169:   "P \<noteq> Q \<or> \<not> P \<or> Q"
blanchet@57169:   "P \<noteq> (\<not> Q) \<or> P \<or> Q"
blanchet@57169:   "(\<not> P) \<noteq> Q \<or> P \<or> Q"
blanchet@57169:   "P \<or> Q \<or> P \<noteq> (\<not> Q)"
blanchet@57169:   "P \<or> Q \<or> (\<not> P) \<noteq> Q"
blanchet@57169:   "P \<or> \<not> Q \<or> P \<noteq> Q"
blanchet@57169:   "\<not> P \<or> Q \<or> P \<noteq> Q"
blanchet@56078:   "P \<or> y = (if P then x else y)"
blanchet@56078:   "P \<or> (if P then x else y) = y"
blanchet@57169:   "\<not> P \<or> x = (if P then x else y)"
blanchet@57169:   "\<not> P \<or> (if P then x else y) = x"
blanchet@57169:   "P \<or> R \<or> \<not> (if P then Q else R)"
blanchet@57169:   "\<not> P \<or> Q \<or> \<not> (if P then Q else R)"
blanchet@57169:   "\<not> (if P then Q else R) \<or> \<not> P \<or> Q"
blanchet@57169:   "\<not> (if P then Q else R) \<or> P \<or> R"
blanchet@57169:   "(if P then Q else R) \<or> \<not> P \<or> \<not> Q"
blanchet@57169:   "(if P then Q else R) \<or> P \<or> \<not> R"
blanchet@57169:   "(if P then \<not> Q else R) \<or> \<not> P \<or> Q"
blanchet@57169:   "(if P then Q else \<not> R) \<or> P \<or> R"
blanchet@56078:   by auto
blanchet@56078: 
blanchet@57230: hide_type (open) symb_list pattern
blanchet@57230: hide_const (open) Symb_Nil Symb_Cons trigger pat nopat fun_app z3div z3mod
blanchet@56078: 
blanchet@56078: end




