(*  Title:      HOL/SMT.thy

     Author:     Sascha Boehme, TU Muenchen

 *)

 header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *}

 theory SMT

 imports Record

 uses

   "Tools/SMT/smt_utils.ML"

   "Tools/SMT/smt_failure.ML"

   "Tools/SMT/smt_config.ML"

   ("Tools/SMT/smt_monomorph.ML")

   ("Tools/SMT/smt_builtin.ML")

   ("Tools/SMT/smt_datatypes.ML")

   ("Tools/SMT/smt_normalize.ML")

   ("Tools/SMT/smt_translate.ML")

   ("Tools/SMT/smt_solver.ML")

   ("Tools/SMT/smtlib_interface.ML")

   ("Tools/SMT/z3_interface.ML")

   ("Tools/SMT/z3_proof_parser.ML")

   ("Tools/SMT/z3_proof_tools.ML")

   ("Tools/SMT/z3_proof_literals.ML")

   ("Tools/SMT/z3_proof_methods.ML")

   ("Tools/SMT/z3_proof_reconstruction.ML")

   ("Tools/SMT/z3_model.ML")

   ("Tools/SMT/smt_setup_solvers.ML")

 begin

 subsection {* Triggers for quantifier instantiation *}

 text {*

 Some SMT solvers support patterns as a quantifier instantiation

 heuristics.  Patterns may either be positive terms (tagged by "pat")

 triggering quantifier instantiations -- when the solver finds a

 term matching a positive pattern, it instantiates the corresponding

 quantifier accordingly -- or negative terms (tagged by "nopat")

 inhibiting quantifier instantiations.  A list of patterns

 of the same kind is called a multipattern, and all patterns in a

 multipattern are considered conjunctively for quantifier instantiation.

 A list of multipatterns is called a trigger, and their multipatterns

 act disjunctively during quantifier instantiation.  Each multipattern

 should mention at least all quantified variables of the preceding

 quantifier block.

 *}

 datatype pattern = Pattern

 definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"

 definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"

 definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"

 where "trigger _ P = P"

 subsection {* Quantifier weights *}

 text {*

 Weight annotations to quantifiers influence the priority of quantifier

 instantiations.  They should be handled with care for solvers, which support

 them, because incorrect choices of weights might render a problem unsolvable.

 *}

 definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"

 text {*

 Weights must be non-negative.  The value @{text 0} is equivalent to providing

 no weight at all.

 Weights should only be used at quantifiers and only inside triggers (if the

 quantifier has triggers).  Valid usages of weights are as follows:

 \begin{itemize}

 \item

 @{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}

 \item

 @{term "\<forall>x. weight 3 (P x)"}

 \end{itemize}

 *}

 subsection {* Higher-order encoding *}

 text {*

 Application is made explicit for constants occurring with varying

 numbers of arguments.  This is achieved by the introduction of the

 following constant.

 *}

 definition fun_app where "fun_app f = f"

 text {*

 Some solvers support a theory of arrays which can be used to encode

 higher-order functions.  The following set of lemmas specifies the

 properties of such (extensional) arrays.

 *}

 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other

   fun_upd_upd fun_app_def

 subsection {* First-order logic *}

 text {*

 Some SMT solvers only accept problems in first-order logic, i.e.,

 where formulas and terms are syntactically separated. When

 translating higher-order into first-order problems, all

 uninterpreted constants (those not built-in in the target solver)

 are treated as function symbols in the first-order sense.  Their

 occurrences as head symbols in atoms (i.e., as predicate symbols) are

 turned into terms by logically equating such atoms with @{term True}.

 For technical reasons, @{term True} and @{term False} occurring inside

 terms are replaced by the following constants.

 *}

 definition term_true where "term_true = True"

 definition term_false where "term_false = False"

 subsection {* Integer division and modulo for Z3 *}

 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where

   "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"

 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where

   "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"

 subsection {* Setup *}

 use "Tools/SMT/smt_monomorph.ML"

 use "Tools/SMT/smt_builtin.ML"

 use "Tools/SMT/smt_datatypes.ML"

 use "Tools/SMT/smt_normalize.ML"

 use "Tools/SMT/smt_translate.ML"

 use "Tools/SMT/smt_solver.ML"

 use "Tools/SMT/smtlib_interface.ML"

 use "Tools/SMT/z3_interface.ML"

 use "Tools/SMT/z3_proof_parser.ML"

 use "Tools/SMT/z3_proof_tools.ML"

 use "Tools/SMT/z3_proof_literals.ML"

 use "Tools/SMT/z3_proof_methods.ML"

 use "Tools/SMT/z3_proof_reconstruction.ML"

 use "Tools/SMT/z3_model.ML"

 use "Tools/SMT/smt_setup_solvers.ML"

 setup {*

   SMT_Config.setup #>

   SMT_Normalize.setup #>

   SMT_Solver.setup #>

   SMTLIB_Interface.setup #>

   Z3_Interface.setup #>

   Z3_Proof_Reconstruction.setup #>

   SMT_Setup_Solvers.setup

 *}

 subsection {* Configuration *}

 text {*

 The current configuration can be printed by the command

 @{text smt_status}, which shows the values of most options.

 *}

 subsection {* General configuration options *}

 text {*

 The option @{text smt_solver} can be used to change the target SMT

 solver.  The possible values can be obtained from the @{text smt_status}

 command.

 Due to licensing restrictions, Yices and Z3 are not installed/enabled

 by default.  Z3 is free for non-commercial applications and can be enabled

 by simply setting the environment variable @{text Z3_NON_COMMERCIAL} to

 @{text yes}.

 *}

 declare [[ smt_solver = z3 ]]

 text {*

 Since SMT solvers are potentially non-terminating, there is a timeout

 (given in seconds) to restrict their runtime.  A value greater than

 120 (seconds) is in most cases not advisable.

 *}

 declare [[ smt_timeout = 20 ]]

 text {*

 SMT solvers apply randomized heuristics.  In case a problem is not

 solvable by an SMT solver, changing the following option might help.

 *}

 declare [[ smt_random_seed = 1 ]]

 text {*

 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT

 solvers are fully trusted without additional checks.  The following

 option can cause the SMT solver to run in proof-producing mode, giving

 a checkable certificate.  This is currently only implemented for Z3.

 *}

 declare [[ smt_oracle = false ]]

 text {*

 Each SMT solver provides several commandline options to tweak its

 behaviour.  They can be passed to the solver by setting the following

 options.

 *}

 declare [[ cvc3_options = "", remote_cvc3_options = "" ]]

 declare [[ yices_options = "" ]]

 declare [[ z3_options = "", remote_z3_options = "" ]]

 text {*

 Enable the following option to use built-in support for datatypes and

 records.  Currently, this is only implemented for Z3 running in oracle

 mode.

 *}

 declare [[ smt_datatypes = false ]]

 text {*

 The SMT method provides an inference mechanism to detect simple triggers

 in quantified formulas, which might increase the number of problems

 solvable by SMT solvers (note: triggers guide quantifier instantiations

 in the SMT solver).  To turn it on, set the following option.

 *}

 declare [[ smt_infer_triggers = false ]]

 text {*

 The SMT method monomorphizes the given facts, that is, it tries to

 instantiate all schematic type variables with fixed types occurring

 in the problem.  This is a (possibly nonterminating) fixed-point

 construction whose cycles are limited by the following option.

 *}

 declare [[ monomorph_max_rounds = 5 ]]

 text {*

 In addition, the number of generated monomorphic instances is limited

 by the following option.

 *}

 declare [[ monomorph_max_new_instances = 500 ]]

 subsection {* Certificates *}

 text {*

 By setting the option @{text smt_certificates} to the name of a file,

 all following applications of an SMT solver a cached in that file.

 Any further application of the same SMT solver (using the very same

 configuration) re-uses the cached certificate instead of invoking the

 solver.  An empty string disables caching certificates.

 The filename should be given as an explicit path.  It is good

 practice to use the name of the current theory (with ending

 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.

 *}

 declare [[ smt_certificates = "" ]]

 text {*

 The option @{text smt_fixed} controls whether only stored

 certificates are should be used or invocation of an SMT solver is

 allowed.  When set to @{text true}, no SMT solver will ever be

 invoked and only the existing certificates found in the configured

 cache are used;  when set to @{text false} and there is no cached

 certificate for some proposition, then the configured SMT solver is

 invoked.

 *}

 declare [[ smt_fixed = false ]]

 subsection {* Tracing *}

 text {*

 The SMT method, when applied, traces important information.  To

 make it entirely silent, set the following option to @{text false}.

 *}

 declare [[ smt_verbose = true ]]

 text {*

 For tracing the generated problem file given to the SMT solver as

 well as the returned result of the solver, the option

 @{text smt_trace} should be set to @{text true}.

 *}

 declare [[ smt_trace = false ]]

 text {*

 From the set of assumptions given to the SMT solver, those assumptions

 used in the proof are traced when the following option is set to

 @{term true}.  This only works for Z3 when it runs in non-oracle mode

 (see options @{text smt_solver} and @{text smt_oracle} above).

 *}

 declare [[ smt_trace_used_facts = false ]]

 subsection {* Schematic rules for Z3 proof reconstruction *}

 text {*

 Several prof rules of Z3 are not very well documented.  There are two

 lemma groups which can turn failing Z3 proof reconstruction attempts

 into succeeding ones: the facts in @{text z3_rule} are tried prior to

 any implemented reconstruction procedure for all uncertain Z3 proof

 rules;  the facts in @{text z3_simp} are only fed to invocations of

 the simplifier when reconstructing theory-specific proof steps.

 *}

 lemmas [z3_rule] =

   refl eq_commute conj_commute disj_commute simp_thms nnf_simps

   ring_distribs field_simps times_divide_eq_right times_divide_eq_left

   if_True if_False not_not

 lemma [z3_rule]:

   "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"

   "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"

   "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"

   by auto

 lemma [z3_rule]:

   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"

   by auto

 lemma [z3_rule]:

   "((\<not>P) = P) = False"

   "(P = (\<not>P)) = False"

   "(P \<noteq> Q) = (Q = (\<not>P))"

   "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"

   "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"

   by auto

 lemma [z3_rule]:

   "(if P then P else \<not>P) = True"

   "(if \<not>P then \<not>P else P) = True"

   "(if P then True else False) = P"

   "(if P then False else True) = (\<not>P)"

   "(if \<not>P then x else y) = (if P then y else x)"

   "f (if P then x else y) = (if P then f x else f y)"

   by auto

 lemma [z3_rule]:

   "P = Q \<or> P \<or> Q"

   "P = Q \<or> \<not>P \<or> \<not>Q"

   "(\<not>P) = Q \<or> \<not>P \<or> Q"

   "(\<not>P) = Q \<or> P \<or> \<not>Q"

   "P = (\<not>Q) \<or> \<not>P \<or> Q"

   "P = (\<not>Q) \<or> P \<or> \<not>Q"

   "P \<noteq> Q \<or> P \<or> \<not>Q"

   "P \<noteq> Q \<or> \<not>P \<or> Q"

   "P \<noteq> (\<not>Q) \<or> P \<or> Q"

   "(\<not>P) \<noteq> Q \<or> P \<or> Q"

   "P \<or> Q \<or> P \<noteq> (\<not>Q)"

   "P \<or> Q \<or> (\<not>P) \<noteq> Q"

   "P \<or> \<not>Q \<or> P \<noteq> Q"

   "\<not>P \<or> Q \<or> P \<noteq> Q"

   by auto

 lemma [z3_rule]:

   "0 + (x::int) = x"

   "x + 0 = x"

   "0 * x = 0"

   "1 * x = x"

   "x + y = y + x"

   by auto

 hide_type (open) pattern

 hide_const Pattern fun_app term_true term_false z3div z3mod

 hide_const (open) trigger pat nopat weight

 end