src/HOL/SMT.thy
author boehmes
Wed Dec 15 10:12:44 2010 +0100 (2010-12-15)
changeset 41127 2ea84c8535c6
parent 41126 e0bd443c0fdd
child 41174 10eb369f8c01
permissions -rw-r--r--
re-implemented eta-expansion, lambda-lifting, and explicit application on terms (exploiting the control over the term structure);
abolished SMT interface concept in favor of solver classes (now also the translation configuration is stored in the context);
proof reconstruction is now expected to return a theorem stating False (and hence needs to discharge all hypothetical definitions);
built-in functions carry additionally their arity and their most general type;
slightly generalized the definition of fun_app
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(*  Title:      HOL/SMT.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 *}
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theory SMT
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imports List
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uses
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  "Tools/Datatype/datatype_selectors.ML"
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  "Tools/SMT/smt_utils.ML"
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  "Tools/SMT/smt_failure.ML"
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  "Tools/SMT/smt_config.ML"
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  "Tools/SMT/smt_monomorph.ML"
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  ("Tools/SMT/smt_builtin.ML")
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  ("Tools/SMT/smt_normalize.ML")
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  ("Tools/SMT/smt_translate.ML")
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  ("Tools/SMT/smt_solver.ML")
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  ("Tools/SMT/smtlib_interface.ML")
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  ("Tools/SMT/z3_proof_parser.ML")
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  ("Tools/SMT/z3_proof_tools.ML")
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  ("Tools/SMT/z3_proof_literals.ML")
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  ("Tools/SMT/z3_proof_methods.ML")
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  ("Tools/SMT/z3_proof_reconstruction.ML")
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  ("Tools/SMT/z3_model.ML")
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  ("Tools/SMT/z3_interface.ML")
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  ("Tools/SMT/smt_setup_solvers.ML")
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begin
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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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datatype pattern = Pattern
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definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
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definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
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definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
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where "trigger _ P = P"
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subsection {* Quantifier weights *}
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text {*
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Weight annotations to quantifiers influence the priority of quantifier
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instantiations.  They should be handled with care for solvers, which support
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them, because incorrect choices of weights might render a problem unsolvable.
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*}
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definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
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text {*
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Weights must be non-negative.  The value @{text 0} is equivalent to providing
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no weight at all.
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Weights should only be used at quantifiers and only inside triggers (if the
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quantifier has triggers).  Valid usages of weights are as follows:
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\begin{itemize}
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\item
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@{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}
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\item
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@{term "\<forall>x. weight 3 (P x)"}
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\end{itemize}
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*}
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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 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
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  fun_upd_upd fun_app_def
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subsection {* First-order logic *}
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text {*
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Some SMT solvers only accept problems in first-order logic, i.e.,
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where formulas and terms are syntactically separated. When
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translating higher-order into first-order problems, all
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uninterpreted constants (those not built-in in the target solver)
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are treated as function symbols in the first-order sense.  Their
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occurrences as head symbols in atoms (i.e., as predicate symbols) are
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turned into terms by equating such atoms with @{term True}.
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Whenever the boolean type occurs in first-order terms, it is replaced
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by the following type.
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*}
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typedecl term_bool
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subsection {* Integer division and modulo for Z3 *}
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definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "z3div k l = (if 0 \<le> l 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 = (if 0 \<le> l then k mod l else k mod (-l))"
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lemma div_by_z3div:
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  "\<forall>k l. k div l = (
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    if k = 0 \<or> l = 0 then 0
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    else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
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    else z3div (-k) (-l))"
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  by (auto simp add: z3div_def trigger_def)
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lemma mod_by_z3mod:
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  "\<forall>k l. k mod l = (
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    if l = 0 then k
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    else if k = 0 then 0
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    else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
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    else - z3mod (-k) (-l))"
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  by (auto simp add: z3mod_def trigger_def)
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subsection {* Setup *}
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use "Tools/SMT/smt_builtin.ML"
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use "Tools/SMT/smt_normalize.ML"
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use "Tools/SMT/smt_translate.ML"
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use "Tools/SMT/smt_solver.ML"
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use "Tools/SMT/smtlib_interface.ML"
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use "Tools/SMT/z3_interface.ML"
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use "Tools/SMT/z3_proof_parser.ML"
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use "Tools/SMT/z3_proof_tools.ML"
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use "Tools/SMT/z3_proof_literals.ML"
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use "Tools/SMT/z3_proof_methods.ML"
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use "Tools/SMT/z3_proof_reconstruction.ML"
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use "Tools/SMT/z3_model.ML"
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use "Tools/SMT/smt_setup_solvers.ML"
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setup {*
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  SMT_Config.setup #>
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  SMT_Normalize.setup #>
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  SMT_Solver.setup #>
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  SMTLIB_Interface.setup #>
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  Z3_Interface.setup #>
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  Z3_Proof_Reconstruction.setup #>
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  SMT_Setup_Solvers.setup
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*}
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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 smt_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 smt_solver} can be used to change the target SMT
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solver.  The possible values are @{text cvc3}, @{text yices}, and
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@{text z3}.  It is advisable to locally install the selected solver,
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although this is not necessary for @{text cvc3} and @{text z3}, which
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can also be used over an Internet-based service.
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When using local SMT solvers, the path to their binaries should be
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declared by setting the following environment variables:
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@{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
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*}
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declare [[ smt_solver = z3 ]]
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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 [[ smt_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 [[ smt_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 [[ smt_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 = "", yices_options = "", z3_options = "" ]]
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text {*
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Enable the following option to use built-in support for datatypes and
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records.  Currently, this is only implemented for Z3 running in oracle
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mode.
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*}
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declare [[ smt_datatypes = false ]]
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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 [[ smt_infer_triggers = 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 [[ smt_monomorph_limit = 10 ]]
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subsection {* Certificates *}
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text {*
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By setting the option @{text smt_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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*}
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declare [[ smt_certificates = "" ]]
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text {*
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The option @{text smt_fixed} controls whether only stored
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certificates are should be used or invocation of an SMT solver is
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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 [[ smt_fixed = 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 [[ smt_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 smt_trace} should be set to @{text true}.
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*}
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declare [[ smt_trace = false ]]
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text {*
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From the set of assumptions given to the SMT solver, those assumptions
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used in the proof are traced when the following option is set to
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@{term true}.  This only works for Z3 when it runs in non-oracle mode
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(see options @{text smt_solver} and @{text smt_oracle} above).
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*}
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declare [[ smt_trace_used_facts = 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_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_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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  by auto
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lemma [z3_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_rule]:
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  "((\<not>P) = P) = False"
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  "(P = (\<not>P)) = False"
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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_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 \<not>P then x else y) = (if P then y else x)"
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  "f (if P then x else y) = (if P then f x else f y)"
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  by auto
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lemma [z3_rule]:
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  "P = Q \<or> P \<or> Q"
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  "P = Q \<or> \<not>P \<or> \<not>Q"
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  "(\<not>P) = Q \<or> \<not>P \<or> Q"
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  "(\<not>P) = Q \<or> P \<or> \<not>Q"
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  "P = (\<not>Q) \<or> \<not>P \<or> Q"
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  "P = (\<not>Q) \<or> P \<or> \<not>Q"
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  "P \<noteq> Q \<or> P \<or> \<not>Q"
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  "P \<noteq> Q \<or> \<not>P \<or> Q"
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  "P \<noteq> (\<not>Q) \<or> P \<or> Q"
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  "(\<not>P) \<noteq> Q \<or> P \<or> Q"
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  "P \<or> Q \<or> P \<noteq> (\<not>Q)"
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  "P \<or> Q \<or> (\<not>P) \<noteq> Q"
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  "P \<or> \<not>Q \<or> P \<noteq> Q"
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  "\<not>P \<or> Q \<or> P \<noteq> Q"
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  by auto
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lemma [z3_rule]:
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  "0 + (x::int) = x"
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  "x + 0 = 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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  by auto
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hide_type term_bool
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hide_type (open) pattern
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hide_const Pattern fun_app
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hide_const (open) trigger pat nopat weight z3div z3mod
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subsection {* Selectors for datatypes *}
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setup {* Datatype_Selectors.setup *}
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declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
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declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
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end