src/HOL/Library/Old_SMT.thy
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(*  Title:      HOL/Library/Old_SMT.thy
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    Author:     Sascha Boehme, TU Muenchen
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*)
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section \<open>Old Version of Bindings to Satisfiability Modulo Theories (SMT) solvers\<close>
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theory Old_SMT
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imports "../Real" "../Word/Word"
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keywords "old_smt_status" :: diag
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begin
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ML_file "Old_SMT/old_smt_utils.ML"
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ML_file "Old_SMT/old_smt_failure.ML"
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ML_file "Old_SMT/old_smt_config.ML"
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subsection \<open>Triggers for quantifier instantiation\<close>
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text \<open>
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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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\<close>
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typedecl pattern
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consts
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  pat :: "'a \<Rightarrow> pattern"
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  nopat :: "'a \<Rightarrow> pattern"
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definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool" where "trigger _ P = P"
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subsection \<open>Quantifier weights\<close>
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text \<open>
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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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\<close>
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definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
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text \<open>
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Weights must be non-negative.  The value \<open>0\<close> 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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\<close>
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subsection \<open>Higher-order encoding\<close>
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text \<open>
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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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\<close>
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definition fun_app where "fun_app f = f"
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text \<open>
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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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\<close>
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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 \<open>First-order logic\<close>
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text \<open>
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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 logically equating such atoms with @{term True}.
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For technical reasons, @{term True} and @{term False} occurring inside
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terms are replaced by the following constants.
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\<close>
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definition term_true where "term_true = True"
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definition term_false where "term_false = False"
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subsection \<open>Integer division and modulo for Z3\<close>
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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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subsection \<open>Setup\<close>
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ML_file "Old_SMT/old_smt_builtin.ML"
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ML_file "Old_SMT/old_smt_datatypes.ML"
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ML_file "Old_SMT/old_smt_normalize.ML"
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ML_file "Old_SMT/old_smt_translate.ML"
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ML_file "Old_SMT/old_smt_solver.ML"
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ML_file "Old_SMT/old_smtlib_interface.ML"
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ML_file "Old_SMT/old_z3_interface.ML"
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ML_file "Old_SMT/old_z3_proof_parser.ML"
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ML_file "Old_SMT/old_z3_proof_tools.ML"
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ML_file "Old_SMT/old_z3_proof_literals.ML"
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ML_file "Old_SMT/old_z3_proof_methods.ML"
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named_theorems old_z3_simp "simplification rules for Z3 proof reconstruction"
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ML_file "Old_SMT/old_z3_proof_reconstruction.ML"
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ML_file "Old_SMT/old_z3_model.ML"
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ML_file "Old_SMT/old_smt_setup_solvers.ML"
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setup \<open>
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  Old_SMT_Config.setup #>
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  Old_SMT_Normalize.setup #>
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  Old_SMTLIB_Interface.setup #>
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  Old_Z3_Interface.setup #>
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  Old_SMT_Setup_Solvers.setup
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\<close>
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method_setup old_smt = \<open>
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  Scan.optional Attrib.thms [] >>
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    (fn thms => fn ctxt =>
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      (legacy_feature "Proof method \"old_smt\" will be discontinued soon -- use \"smt\" instead";
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       METHOD (fn facts => HEADGOAL (Old_SMT_Solver.smt_tac ctxt (thms @ facts)))))
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\<close> "apply an SMT solver to the current goal"
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subsection \<open>Configuration\<close>
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text \<open>
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The current configuration can be printed by the command
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\<open>old_smt_status\<close>, which shows the values of most options.
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\<close>
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subsection \<open>General configuration options\<close>
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text \<open>
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The option \<open>old_smt_solver\<close> can be used to change the target SMT
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solver.  The possible values can be obtained from the \<open>old_smt_status\<close>
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command.
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Due to licensing restrictions, Yices and Z3 are not installed/enabled
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by default.  Z3 is free for non-commercial applications and can be enabled
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by setting the \<open>OLD_Z3_NON_COMMERCIAL\<close> environment variable to
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\<open>yes\<close>.
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\<close>
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declare [[ old_smt_solver = z3 ]]
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text \<open>
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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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\<close>
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declare [[ old_smt_timeout = 20 ]]
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text \<open>
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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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\<close>
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declare [[ old_smt_random_seed = 1 ]]
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text \<open>
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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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\<close>
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declare [[ old_smt_oracle = false ]]
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text \<open>
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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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\<close>
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declare [[ old_cvc3_options = "" ]]
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declare [[ old_yices_options = "" ]]
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declare [[ old_z3_options = "" ]]
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text \<open>
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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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\<close>
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declare [[ old_smt_datatypes = false ]]
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text \<open>
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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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\<close>
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declare [[ old_smt_infer_triggers = false ]]
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text \<open>
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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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\<close>
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declare [[ monomorph_max_rounds = 5 ]]
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text \<open>
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In addition, the number of generated monomorphic instances is limited
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by the following option.
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\<close>
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declare [[ monomorph_max_new_instances = 500 ]]
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subsection \<open>Certificates\<close>
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text \<open>
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By setting the option \<open>old_smt_certificates\<close> 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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\<open>.certs\<close> instead of \<open>.thy\<close>) as the certificates file.
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Certificate files should be used at most once in a certain theory context,
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to avoid race conditions with other concurrent accesses.
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\<close>
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declare [[ old_smt_certificates = "" ]]
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text \<open>
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The option \<open>old_smt_read_only_certificates\<close> controls whether only
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stored certificates are should be used or invocation of an SMT solver
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is allowed.  When set to \<open>true\<close>, 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 \<open>false\<close> 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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\<close>
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declare [[ old_smt_read_only_certificates = false ]]
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subsection \<open>Tracing\<close>
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text \<open>
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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 \<open>false\<close>.
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\<close>
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declare [[ old_smt_verbose = true ]]
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text \<open>
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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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\<open>old_smt_trace\<close> should be set to \<open>true\<close>.
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\<close>
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declare [[ old_smt_trace = false ]]
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text \<open>
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   293
From the set of assumptions given to the SMT solver, those assumptions
7f58a9a843c2 joined setup of SMT solvers in one place; turned Z3-specific options into SMT options (renamed configuration options from z3_* to smt_*); more detailed SMT exception; improved SMT filter interface
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diff changeset
   294
used in the proof are traced when the following option is set to
7f58a9a843c2 joined setup of SMT solvers in one place; turned Z3-specific options into SMT options (renamed configuration options from z3_* to smt_*); more detailed SMT exception; improved SMT filter interface
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diff changeset
   295
@{term true}.  This only works for Z3 when it runs in non-oracle mode
61585
a9599d3d7610 isabelle update_cartouches -c -t;
wenzelm
parents: 60500
diff changeset
   296
(see options \<open>old_smt_solver\<close> and \<open>old_smt_oracle\<close> above).
60500
903bb1495239 isabelle update_cartouches;
wenzelm
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diff changeset
   297
\<close>
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diff changeset
   298
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   299
declare [[ old_smt_trace_used_facts = false ]]
39298
5aefb5bc8a93 added preliminary support for SMT datatypes (for now restricted to tuples and lists); only the Z3 interface (in oracle mode) makes use of it, there is especially no Z3 proof reconstruction support for datatypes yet
boehmes
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diff changeset
   300
36898
8e55aa1306c5 integrated SMT into the HOL image
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diff changeset
   301
8e55aa1306c5 integrated SMT into the HOL image
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diff changeset
   302
60500
903bb1495239 isabelle update_cartouches;
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diff changeset
   303
subsection \<open>Schematic rules for Z3 proof reconstruction\<close>
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8e55aa1306c5 integrated SMT into the HOL image
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diff changeset
   304
60500
903bb1495239 isabelle update_cartouches;
wenzelm
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diff changeset
   305
text \<open>
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
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diff changeset
   306
Several prof rules of Z3 are not very well documented.  There are two
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   307
lemma groups which can turn failing Z3 proof reconstruction attempts
61585
a9599d3d7610 isabelle update_cartouches -c -t;
wenzelm
parents: 60500
diff changeset
   308
into succeeding ones: the facts in \<open>z3_rule\<close> are tried prior to
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
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diff changeset
   309
any implemented reconstruction procedure for all uncertain Z3 proof
61585
a9599d3d7610 isabelle update_cartouches -c -t;
wenzelm
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diff changeset
   310
rules;  the facts in \<open>z3_simp\<close> are only fed to invocations of
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8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   311
the simplifier when reconstructing theory-specific proof steps.
60500
903bb1495239 isabelle update_cartouches;
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parents: 59966
diff changeset
   312
\<close>
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8e55aa1306c5 integrated SMT into the HOL image
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diff changeset
   313
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4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
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parents: 58058
diff changeset
   314
lemmas [old_z3_rule] =
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   315
  refl eq_commute conj_commute disj_commute simp_thms nnf_simps
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   316
  ring_distribs field_simps times_divide_eq_right times_divide_eq_left
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   317
  if_True if_False not_not
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   318
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4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   319
lemma [old_z3_rule]:
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
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diff changeset
   320
  "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   321
  "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   322
  "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   323
  "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   324
  "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   325
  "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   326
  "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   327
  "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   328
  by auto
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   329
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   330
lemma [old_z3_rule]:
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   331
  "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   332
  "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   333
  "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   334
  "(True \<longrightarrow> P) = P"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   335
  "(P \<longrightarrow> True) = True"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   336
  "(False \<longrightarrow> P) = True"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   337
  "(P \<longrightarrow> P) = True"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   338
  by auto
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   339
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   340
lemma [old_z3_rule]:
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   341
  "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   342
  by auto
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   343
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   344
lemma [old_z3_rule]:
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   345
  "(\<not>True) = False"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   346
  "(\<not>False) = True"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   347
  "(x = x) = True"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   348
  "(P = True) = P"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   349
  "(True = P) = P"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   350
  "(P = False) = (\<not>P)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   351
  "(False = P) = (\<not>P)"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   352
  "((\<not>P) = P) = False"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   353
  "(P = (\<not>P)) = False"
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   354
  "((\<not>P) = (\<not>Q)) = (P = Q)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   355
  "\<not>(P = (\<not>Q)) = (P = Q)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   356
  "\<not>((\<not>P) = Q) = (P = Q)"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   357
  "(P \<noteq> Q) = (Q = (\<not>P))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   358
  "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   359
  "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   360
  by auto
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   361
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   362
lemma [old_z3_rule]:
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   363
  "(if P then P else \<not>P) = True"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   364
  "(if \<not>P then \<not>P else P) = True"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   365
  "(if P then True else False) = P"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   366
  "(if P then False else True) = (\<not>P)"
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   367
  "(if P then Q else True) = ((\<not>P) \<or> Q)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   368
  "(if P then Q else True) = (Q \<or> (\<not>P))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   369
  "(if P then Q else \<not>Q) = (P = Q)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   370
  "(if P then Q else \<not>Q) = (Q = P)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   371
  "(if P then \<not>Q else Q) = (P = (\<not>Q))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   372
  "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   373
  "(if \<not>P then x else y) = (if P then y else x)"
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   374
  "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   375
  "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   376
  "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   377
  "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   378
  "(if P then x else if P then y else z) = (if P then x else z)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   379
  "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   380
  "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   381
  "(if P then x = y else x = z) = (x = (if P then y else z))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   382
  "(if P then x = y else y = z) = (y = (if P then x else z))"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   383
  "(if P then x = y else z = y) = (y = (if P then x else z))"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   384
  by auto
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   385
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   386
lemma [old_z3_rule]:
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   387
  "0 + (x::int) = x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   388
  "x + 0 = x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   389
  "x + x = 2 * x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   390
  "0 * x = 0"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   391
  "1 * x = x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   392
  "x + y = y + x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   393
  by auto
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   394
58059
4e477dcd050a prefixed all old SMT commands, attributes, etc., with 'old_'
blanchet
parents: 58058
diff changeset
   395
lemma [old_z3_rule]:  (* for def-axiom *)
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   396
  "P = Q \<or> P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   397
  "P = Q \<or> \<not>P \<or> \<not>Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   398
  "(\<not>P) = Q \<or> \<not>P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   399
  "(\<not>P) = Q \<or> P \<or> \<not>Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   400
  "P = (\<not>Q) \<or> \<not>P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   401
  "P = (\<not>Q) \<or> P \<or> \<not>Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   402
  "P \<noteq> Q \<or> P \<or> \<not>Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   403
  "P \<noteq> Q \<or> \<not>P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   404
  "P \<noteq> (\<not>Q) \<or> P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   405
  "(\<not>P) \<noteq> Q \<or> P \<or> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   406
  "P \<or> Q \<or> P \<noteq> (\<not>Q)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   407
  "P \<or> Q \<or> (\<not>P) \<noteq> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   408
  "P \<or> \<not>Q \<or> P \<noteq> Q"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   409
  "\<not>P \<or> Q \<or> P \<noteq> Q"
44488
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   410
  "P \<or> y = (if P then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   411
  "P \<or> (if P then x else y) = y"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   412
  "\<not>P \<or> x = (if P then x else y)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   413
  "\<not>P \<or>  (if P then x else y) = x"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   414
  "P \<or> R \<or> \<not>(if P then Q else R)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   415
  "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   416
  "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   417
  "\<not>(if P then Q else R) \<or> P \<or> R"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   418
  "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   419
  "(if P then Q else R) \<or> P \<or> \<not>R"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   420
  "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"
587bf61a00a1 improved completeness and efficiency of Z3 proof reconstruction, especially by an improved handling of Skolemization
boehmes
parents: 44087
diff changeset
   421
  "(if P then Q else \<not>R) \<or> P \<or> R"
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   422
  by auto
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   423
58058
1a0b18176548 add 'old_' prefix to SMT file names; add 'Old_' to ML module names;
blanchet
parents: 58057
diff changeset
   424
ML_file "Old_SMT/old_smt_real.ML"
1a0b18176548 add 'old_' prefix to SMT file names; add 'Old_' to ML module names;
blanchet
parents: 58057
diff changeset
   425
ML_file "Old_SMT/old_smt_word.ML"
58056
fc6dd578d506 moved old setup for SMT out
blanchet
parents: 58055
diff changeset
   426
37124
fe22fc54b876 hide constants and types introduced by SMT,
boehmes
parents: 36902
diff changeset
   427
hide_type (open) pattern
56078
624faeda77b5 moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents: 56046
diff changeset
   428
hide_const fun_app term_true term_false z3div z3mod
41280
a7de9d36f4f2 only linear occurrences of multiplication are treated as built-in (SMT solvers only support linear arithmetic in general);
boehmes
parents: 41174
diff changeset
   429
hide_const (open) trigger pat nopat weight
37124
fe22fc54b876 hide constants and types introduced by SMT,
boehmes
parents: 36902
diff changeset
   430
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   431
end