src/HOL/SMT.thy
author boehmes
Mon Nov 29 23:41:09 2010 +0100 (2010-11-29)
changeset 40806 59d96f777da3
parent 40681 872b08416fb4
child 41059 d2b1fc1b8e19
permissions -rw-r--r--
also support higher-order rules for Z3 proof reconstruction
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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_failure.ML"
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  "Tools/SMT/smt_config.ML"
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  "Tools/SMT/smt_utils.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 triggers for quantifier instantiation.
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Each trigger consists of one ore more patterns.  A pattern may either
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be a list of positive subterms (each being tagged by "pat"), or a
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list of negative subterms (each being tagged by "nopat").
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When an SMT solver finds a term matching a positive pattern (a
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pattern with positive subterms only), it instantiates the
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corresponding quantifier accordingly.  Negative patterns inhibit
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quantifier instantiations.  Each pattern should mention all preceding
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bound variables.
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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 x = f x"
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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 require a strict separation between formulas and
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terms.  When translating higher-order into first-order problems,
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all uninterpreted constants (those not builtin 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) is
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turned into terms by equating such atoms with @{term True} using the
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following term-level equation symbol.
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*}
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definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)"
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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: "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)
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lemma mod_by_z3mod: "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)
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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_Solver.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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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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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 (open) pattern
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hide_const Pattern term_eq
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hide_const (open) trigger pat nopat weight fun_app 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