
boehmes@36898: (*  Title:      HOL/SMT.thy
boehmes@36898:     Author:     Sascha Boehme, TU Muenchen
boehmes@36898: *)
boehmes@36898: 
boehmes@36898: header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *}
boehmes@36898: 
boehmes@36898: theory SMT
boehmes@41426: imports Record
wenzelm@46950: keywords "smt_status" :: diag
boehmes@36898: begin
boehmes@36898: 
wenzelm@48892: ML_file "Tools/SMT/smt_utils.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_failure.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_config.ML"
boehmes@36898: 
boehmes@36898: 
huffman@36902: subsection {* Triggers for quantifier instantiation *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@41125: Some SMT solvers support patterns as a quantifier instantiation
boehmes@41125: heuristics.  Patterns may either be positive terms (tagged by "pat")
boehmes@41125: triggering quantifier instantiations -- when the solver finds a
boehmes@41125: term matching a positive pattern, it instantiates the corresponding
boehmes@41125: quantifier accordingly -- or negative terms (tagged by "nopat")
boehmes@41125: inhibiting quantifier instantiations.  A list of patterns
boehmes@41125: of the same kind is called a multipattern, and all patterns in a
boehmes@41125: multipattern are considered conjunctively for quantifier instantiation.
boehmes@41125: A list of multipatterns is called a trigger, and their multipatterns
boehmes@41125: act disjunctively during quantifier instantiation.  Each multipattern
boehmes@41125: should mention at least all quantified variables of the preceding
boehmes@41125: quantifier block.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: datatype pattern = Pattern
boehmes@36898: 
boehmes@37124: definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
boehmes@37124: definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
boehmes@36898: 
boehmes@37124: definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
boehmes@36898: where "trigger _ P = P"
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@40664: subsection {* Quantifier weights *}
boehmes@40664: 
boehmes@40664: text {*
boehmes@40664: Weight annotations to quantifiers influence the priority of quantifier
boehmes@40664: instantiations.  They should be handled with care for solvers, which support
boehmes@40664: them, because incorrect choices of weights might render a problem unsolvable.
boehmes@40664: *}
boehmes@40664: 
boehmes@40664: definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
boehmes@40664: 
boehmes@40664: text {*
boehmes@40664: Weights must be non-negative.  The value @{text 0} is equivalent to providing
boehmes@40664: no weight at all.
boehmes@40664: 
boehmes@40664: Weights should only be used at quantifiers and only inside triggers (if the
boehmes@40664: quantifier has triggers).  Valid usages of weights are as follows:
boehmes@40664: 
boehmes@40664: \begin{itemize}
boehmes@40664: \item
boehmes@40664: @{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}
boehmes@40664: \item
boehmes@40664: @{term "\<forall>x. weight 3 (P x)"}
boehmes@40664: \end{itemize}
boehmes@40664: *}
boehmes@40664: 
boehmes@40664: 
boehmes@40664: 
huffman@36902: subsection {* Higher-order encoding *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: Application is made explicit for constants occurring with varying
boehmes@36898: numbers of arguments.  This is achieved by the introduction of the
boehmes@36898: following constant.
boehmes@36898: *}
boehmes@36898: 
boehmes@41127: definition fun_app where "fun_app f = f"
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: Some solvers support a theory of arrays which can be used to encode
boehmes@36898: higher-order functions.  The following set of lemmas specifies the
boehmes@36898: properties of such (extensional) arrays.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
boehmes@37157:   fun_upd_upd fun_app_def
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
huffman@36902: subsection {* First-order logic *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@41059: Some SMT solvers only accept problems in first-order logic, i.e.,
boehmes@41059: where formulas and terms are syntactically separated. When
boehmes@41059: translating higher-order into first-order problems, all
boehmes@41059: uninterpreted constants (those not built-in in the target solver)
boehmes@36898: are treated as function symbols in the first-order sense.  Their
boehmes@41059: occurrences as head symbols in atoms (i.e., as predicate symbols) are
boehmes@41281: turned into terms by logically equating such atoms with @{term True}.
boehmes@41281: For technical reasons, @{term True} and @{term False} occurring inside
boehmes@41281: terms are replaced by the following constants.
boehmes@36898: *}
boehmes@36898: 
boehmes@41281: definition term_true where "term_true = True"
boehmes@41281: definition term_false where "term_false = False"
boehmes@41281: 
boehmes@36898: 
boehmes@36898: 
boehmes@37151: subsection {* Integer division and modulo for Z3 *}
boehmes@37151: 
boehmes@37151: definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
boehmes@37151:   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
boehmes@37151: 
boehmes@37151: definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
boehmes@37151:   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
boehmes@37151: 
boehmes@37151: 
boehmes@37151: 
huffman@36902: subsection {* Setup *}
boehmes@36898: 
wenzelm@48892: ML_file "Tools/SMT/smt_builtin.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_datatypes.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_normalize.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_translate.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_solver.ML"
wenzelm@48892: ML_file "Tools/SMT/smtlib_interface.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_interface.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_proof_parser.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_proof_tools.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_proof_literals.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_proof_methods.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_proof_reconstruction.ML"
wenzelm@48892: ML_file "Tools/SMT/z3_model.ML"
wenzelm@48892: ML_file "Tools/SMT/smt_setup_solvers.ML"
boehmes@36898: 
boehmes@36898: setup {*
boehmes@40424:   SMT_Config.setup #>
boehmes@41059:   SMT_Normalize.setup #>
boehmes@41059:   SMTLIB_Interface.setup #>
boehmes@41059:   Z3_Interface.setup #>
boehmes@36898:   Z3_Proof_Reconstruction.setup #>
boehmes@40162:   SMT_Setup_Solvers.setup
boehmes@36898: *}
boehmes@36898: 
wenzelm@47701: method_setup smt = {*
wenzelm@47701:   Scan.optional Attrib.thms [] >>
wenzelm@47701:     (fn thms => fn ctxt =>
wenzelm@47701:       METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))
wenzelm@47701: *} "apply an SMT solver to the current goal"
boehmes@36898: 
boehmes@36898: 
huffman@36902: subsection {* Configuration *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@36899: The current configuration can be printed by the command
boehmes@36899: @{text smt_status}, which shows the values of most options.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: subsection {* General configuration options *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: The option @{text smt_solver} can be used to change the target SMT
boehmes@41432: solver.  The possible values can be obtained from the @{text smt_status}
boehmes@41432: command.
boehmes@41459: 
boehmes@41459: Due to licensing restrictions, Yices and Z3 are not installed/enabled
boehmes@41459: by default.  Z3 is free for non-commercial applications and can be enabled
boehmes@41462: by simply setting the environment variable @{text Z3_NON_COMMERCIAL} to
boehmes@41462: @{text yes}.
boehmes@36898: *}
boehmes@36898: 
boehmes@41601: declare [[ smt_solver = z3 ]]
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: Since SMT solvers are potentially non-terminating, there is a timeout
boehmes@36898: (given in seconds) to restrict their runtime.  A value greater than
boehmes@36898: 120 (seconds) is in most cases not advisable.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: declare [[ smt_timeout = 20 ]]
boehmes@36898: 
boehmes@40162: text {*
boehmes@41121: SMT solvers apply randomized heuristics.  In case a problem is not
boehmes@41121: solvable by an SMT solver, changing the following option might help.
boehmes@41121: *}
boehmes@41121: 
boehmes@41121: declare [[ smt_random_seed = 1 ]]
boehmes@41121: 
boehmes@41121: text {*
boehmes@40162: In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
boehmes@40162: solvers are fully trusted without additional checks.  The following
boehmes@40162: option can cause the SMT solver to run in proof-producing mode, giving
boehmes@40162: a checkable certificate.  This is currently only implemented for Z3.
boehmes@40162: *}
boehmes@40162: 
boehmes@40162: declare [[ smt_oracle = false ]]
boehmes@40162: 
boehmes@40162: text {*
boehmes@40162: Each SMT solver provides several commandline options to tweak its
boehmes@40162: behaviour.  They can be passed to the solver by setting the following
boehmes@40162: options.
boehmes@40162: *}
boehmes@40162: 
boehmes@41432: declare [[ cvc3_options = "", remote_cvc3_options = "" ]]
boehmes@41432: declare [[ yices_options = "" ]]
boehmes@41432: declare [[ z3_options = "", remote_z3_options = "" ]]
boehmes@40162: 
boehmes@40162: text {*
boehmes@40162: Enable the following option to use built-in support for datatypes and
boehmes@40162: records.  Currently, this is only implemented for Z3 running in oracle
boehmes@40162: mode.
boehmes@40162: *}
boehmes@40162: 
boehmes@40162: declare [[ smt_datatypes = false ]]
boehmes@40162: 
boehmes@41125: text {*
boehmes@41125: The SMT method provides an inference mechanism to detect simple triggers
boehmes@41125: in quantified formulas, which might increase the number of problems
boehmes@41125: solvable by SMT solvers (note: triggers guide quantifier instantiations
boehmes@41125: in the SMT solver).  To turn it on, set the following option.
boehmes@41125: *}
boehmes@41125: 
boehmes@41125: declare [[ smt_infer_triggers = false ]]
boehmes@41125: 
boehmes@41125: text {*
boehmes@41125: The SMT method monomorphizes the given facts, that is, it tries to
boehmes@41125: instantiate all schematic type variables with fixed types occurring
boehmes@41125: in the problem.  This is a (possibly nonterminating) fixed-point
boehmes@41125: construction whose cycles are limited by the following option.
boehmes@41125: *}
boehmes@41125: 
boehmes@43230: declare [[ monomorph_max_rounds = 5 ]]
boehmes@41125: 
boehmes@41762: text {*
boehmes@41762: In addition, the number of generated monomorphic instances is limited
boehmes@41762: by the following option.
boehmes@41762: *}
boehmes@41762: 
boehmes@43230: declare [[ monomorph_max_new_instances = 500 ]]
boehmes@41762: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: subsection {* Certificates *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: By setting the option @{text smt_certificates} to the name of a file,
boehmes@36898: all following applications of an SMT solver a cached in that file.
boehmes@36898: Any further application of the same SMT solver (using the very same
boehmes@36898: configuration) re-uses the cached certificate instead of invoking the
boehmes@36898: solver.  An empty string disables caching certificates.
boehmes@36898: 
boehmes@36898: The filename should be given as an explicit path.  It is good
boehmes@36898: practice to use the name of the current theory (with ending
boehmes@36898: @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: declare [[ smt_certificates = "" ]]
boehmes@36898: 
boehmes@36898: text {*
blanchet@47152: The option @{text smt_read_only_certificates} controls whether only
blanchet@47152: stored certificates are should be used or invocation of an SMT solver
blanchet@47152: is allowed.  When set to @{text true}, no SMT solver will ever be
boehmes@36898: invoked and only the existing certificates found in the configured
boehmes@36898: cache are used;  when set to @{text false} and there is no cached
boehmes@36898: certificate for some proposition, then the configured SMT solver is
boehmes@36898: invoked.
boehmes@36898: *}
boehmes@36898: 
blanchet@47152: declare [[ smt_read_only_certificates = false ]]
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: subsection {* Tracing *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@40424: The SMT method, when applied, traces important information.  To
boehmes@40424: make it entirely silent, set the following option to @{text false}.
boehmes@40424: *}
boehmes@40424: 
boehmes@40424: declare [[ smt_verbose = true ]]
boehmes@40424: 
boehmes@40424: text {*
boehmes@36898: For tracing the generated problem file given to the SMT solver as
boehmes@36898: well as the returned result of the solver, the option
boehmes@36898: @{text smt_trace} should be set to @{text true}.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: declare [[ smt_trace = false ]]
boehmes@36898: 
boehmes@36898: text {*
boehmes@40162: From the set of assumptions given to the SMT solver, those assumptions
boehmes@40162: used in the proof are traced when the following option is set to
boehmes@40162: @{term true}.  This only works for Z3 when it runs in non-oracle mode
boehmes@40162: (see options @{text smt_solver} and @{text smt_oracle} above).
boehmes@36898: *}
boehmes@36898: 
boehmes@40162: declare [[ smt_trace_used_facts = false ]]
boehmes@39298: 
boehmes@36898: 
boehmes@36898: 
huffman@36902: subsection {* Schematic rules for Z3 proof reconstruction *}
boehmes@36898: 
boehmes@36898: text {*
boehmes@36898: Several prof rules of Z3 are not very well documented.  There are two
boehmes@36898: lemma groups which can turn failing Z3 proof reconstruction attempts
boehmes@36898: into succeeding ones: the facts in @{text z3_rule} are tried prior to
boehmes@36898: any implemented reconstruction procedure for all uncertain Z3 proof
boehmes@36898: rules;  the facts in @{text z3_simp} are only fed to invocations of
boehmes@36898: the simplifier when reconstructing theory-specific proof steps.
boehmes@36898: *}
boehmes@36898: 
boehmes@36898: lemmas [z3_rule] =
boehmes@36898:   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
boehmes@36898:   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
boehmes@36898:   if_True if_False not_not
boehmes@36898: 
boehmes@36898: lemma [z3_rule]:
boehmes@44488:   "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
boehmes@44488:   "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
boehmes@44488:   "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
boehmes@44488:   "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
boehmes@44488:   "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
boehmes@44488:   "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
boehmes@44488:   "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
boehmes@44488:   "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
boehmes@44488:   by auto
boehmes@44488: 
boehmes@44488: lemma [z3_rule]:
boehmes@36898:   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
boehmes@36898:   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
boehmes@36898:   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
boehmes@44488:   "(True \<longrightarrow> P) = P"
boehmes@44488:   "(P \<longrightarrow> True) = True"
boehmes@44488:   "(False \<longrightarrow> P) = True"
boehmes@44488:   "(P \<longrightarrow> P) = True"
boehmes@36898:   by auto
boehmes@36898: 
boehmes@36898: lemma [z3_rule]:
boehmes@36898:   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
boehmes@36898:   by auto
boehmes@36898: 
boehmes@36898: lemma [z3_rule]:
boehmes@44488:   "(\<not>True) = False"
boehmes@44488:   "(\<not>False) = True"
boehmes@44488:   "(x = x) = True"
boehmes@44488:   "(P = True) = P"
boehmes@44488:   "(True = P) = P"
boehmes@44488:   "(P = False) = (\<not>P)"
boehmes@44488:   "(False = P) = (\<not>P)"
boehmes@36898:   "((\<not>P) = P) = False"
boehmes@36898:   "(P = (\<not>P)) = False"
boehmes@44488:   "((\<not>P) = (\<not>Q)) = (P = Q)"
boehmes@44488:   "\<not>(P = (\<not>Q)) = (P = Q)"
boehmes@44488:   "\<not>((\<not>P) = Q) = (P = Q)"
boehmes@36898:   "(P \<noteq> Q) = (Q = (\<not>P))"
boehmes@36898:   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
boehmes@36898:   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
boehmes@36898:   by auto
boehmes@36898: 
boehmes@36898: lemma [z3_rule]:
boehmes@36898:   "(if P then P else \<not>P) = True"
boehmes@36898:   "(if \<not>P then \<not>P else P) = True"
boehmes@36898:   "(if P then True else False) = P"
boehmes@36898:   "(if P then False else True) = (\<not>P)"
boehmes@44488:   "(if P then Q else True) = ((\<not>P) \<or> Q)"
boehmes@44488:   "(if P then Q else True) = (Q \<or> (\<not>P))"
boehmes@44488:   "(if P then Q else \<not>Q) = (P = Q)"
boehmes@44488:   "(if P then Q else \<not>Q) = (Q = P)"
boehmes@44488:   "(if P then \<not>Q else Q) = (P = (\<not>Q))"
boehmes@44488:   "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
boehmes@36898:   "(if \<not>P then x else y) = (if P then y else x)"
boehmes@44488:   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
boehmes@44488:   "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
boehmes@44488:   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
boehmes@44488:   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
boehmes@44488:   "(if P then x else if P then y else z) = (if P then x else z)"
boehmes@44488:   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
boehmes@44488:   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
boehmes@44488:   "(if P then x = y else x = z) = (x = (if P then y else z))"
boehmes@44488:   "(if P then x = y else y = z) = (y = (if P then x else z))"
boehmes@44488:   "(if P then x = y else z = y) = (y = (if P then x else z))"
boehmes@36898:   by auto
boehmes@36898: 
boehmes@36898: lemma [z3_rule]:
boehmes@44488:   "0 + (x::int) = x"
boehmes@44488:   "x + 0 = x"
boehmes@44488:   "x + x = 2 * x"
boehmes@44488:   "0 * x = 0"
boehmes@44488:   "1 * x = x"
boehmes@44488:   "x + y = y + x"
boehmes@44488:   by auto
boehmes@44488: 
boehmes@44488: lemma [z3_rule]:  (* for def-axiom *)
boehmes@36898:   "P = Q \<or> P \<or> Q"
boehmes@36898:   "P = Q \<or> \<not>P \<or> \<not>Q"
boehmes@36898:   "(\<not>P) = Q \<or> \<not>P \<or> Q"
boehmes@36898:   "(\<not>P) = Q \<or> P \<or> \<not>Q"
boehmes@36898:   "P = (\<not>Q) \<or> \<not>P \<or> Q"
boehmes@36898:   "P = (\<not>Q) \<or> P \<or> \<not>Q"
boehmes@36898:   "P \<noteq> Q \<or> P \<or> \<not>Q"
boehmes@36898:   "P \<noteq> Q \<or> \<not>P \<or> Q"
boehmes@36898:   "P \<noteq> (\<not>Q) \<or> P \<or> Q"
boehmes@36898:   "(\<not>P) \<noteq> Q \<or> P \<or> Q"
boehmes@36898:   "P \<or> Q \<or> P \<noteq> (\<not>Q)"
boehmes@36898:   "P \<or> Q \<or> (\<not>P) \<noteq> Q"
boehmes@36898:   "P \<or> \<not>Q \<or> P \<noteq> Q"
boehmes@36898:   "\<not>P \<or> Q \<or> P \<noteq> Q"
boehmes@44488:   "P \<or> y = (if P then x else y)"
boehmes@44488:   "P \<or> (if P then x else y) = y"
boehmes@44488:   "\<not>P \<or> x = (if P then x else y)"
boehmes@44488:   "\<not>P \<or>  (if P then x else y) = x"
boehmes@44488:   "P \<or> R \<or> \<not>(if P then Q else R)"
boehmes@44488:   "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"
boehmes@44488:   "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"
boehmes@44488:   "\<not>(if P then Q else R) \<or> P \<or> R"
boehmes@44488:   "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"
boehmes@44488:   "(if P then Q else R) \<or> P \<or> \<not>R"
boehmes@44488:   "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"
boehmes@44488:   "(if P then Q else \<not>R) \<or> P \<or> R"
boehmes@36898:   by auto
boehmes@36898: 
boehmes@37124: 
boehmes@37124: 
boehmes@37124: hide_type (open) pattern
boehmes@41281: hide_const Pattern fun_app term_true term_false z3div z3mod
boehmes@41280: hide_const (open) trigger pat nopat weight
boehmes@37124: 
boehmes@36898: end




