author | blanchet |
Thu, 13 Mar 2014 13:18:14 +0100 | |
changeset 56101 | 6d9fe46429e6 |
parent 56090 | 34bd10a9a2ad |
child 56102 | 439dda276b3f |
permissions | -rw-r--r-- |
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(* Title: HOL/SMT2.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 based on SMT-LIB 2 *} |
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theory SMT2 |
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imports Record |
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keywords "smt2_status" :: diag |
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begin |
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ML_file "Tools/SMT2/smt2_util.ML" |
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ML_file "Tools/SMT2/smt2_failure.ML" |
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ML_file "Tools/SMT2/smt2_config.ML" |
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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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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 {* 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 nonnegative. 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 :: "'a \<Rightarrow> 'a" 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 fun_upd_upd fun_app_def |
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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_as_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 & 0 < l) \<or> (k < 0 & 0 < l) then z3div k l |
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else z3div (-k) (-l))" |
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by (simp add: z3div_def) |
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lemma mod_as_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 & 0 < l) \<or> (k < 0 & 0 < l) then z3mod k l |
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else - z3mod (-k) (-l))" |
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by (simp add: z3mod_def) |
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subsection {* Setup *} |
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ML_file "Tools/SMT2/smt2_builtin.ML" |
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ML_file "Tools/SMT2/smt2_datatypes.ML" |
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ML_file "Tools/SMT2/smt2_normalize.ML" |
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ML_file "Tools/SMT2/smt2_translate.ML" |
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ML_file "Tools/SMT2/smtlib2.ML" |
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ML_file "Tools/SMT2/smtlib2_interface.ML" |
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ML_file "Tools/SMT2/z3_new_model.ML" |
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ML_file "Tools/SMT2/z3_new_proof.ML" |
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ML_file "Tools/SMT2/smt2_solver.ML" |
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ML_file "Tools/SMT2/z3_new_isar.ML" |
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ML_file "Tools/SMT2/z3_new_interface.ML" |
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ML_file "Tools/SMT2/z3_new_replay_util.ML" |
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ML_file "Tools/SMT2/z3_new_replay_literals.ML" |
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ML_file "Tools/SMT2/z3_new_replay_rules.ML" |
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ML_file "Tools/SMT2/z3_new_replay_methods.ML" |
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ML_file "Tools/SMT2/z3_new_replay.ML" |
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ML_file "Tools/SMT2/smt2_systems.ML" |
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method_setup smt2 = {* |
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Scan.optional Attrib.thms [] >> |
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(fn thms => fn ctxt => |
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METHOD (fn facts => HEADGOAL (SMT2_Solver.smt2_tac ctxt (thms @ facts)))) |
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*} "apply an SMT solver to the current goal (based on SMT-LIB 2)" |
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|
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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 smt2_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 smt2_solver} can be used to change the target SMT |
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solver. The possible values can be obtained from the @{text smt2_status} |
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command. |
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|
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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 Isabelle system option @{text z3_non_commercial} to @{text yes}. |
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*} |
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declare [[ smt2_solver = z3_new ]] |
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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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|
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declare [[ smt2_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 [[ smt2_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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|
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declare [[ smt2_oracle = false ]] |
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|
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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 = "" ]] TODO *) |
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(* declare [[ yices_options = "" ]] TODO *) |
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(* declare [[ z3_options = "" ]] TODO *) |
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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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|
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declare [[ smt2_infer_triggers = false ]] |
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|
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text {* |
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Enable the following option to use built-in support for div/mod, datatypes, |
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and records in Z3. Currently, this is implemented only in oracle mode. |
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*} |
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|
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declare [[ z3_new_extensions = false ]] |
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|
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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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|
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declare [[ monomorph_max_rounds = 5 ]] |
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|
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text {* |
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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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*} |
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|
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declare [[ monomorph_max_new_instances = 500 ]] |
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|
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subsection {* Certificates *} |
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|
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text {* |
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By setting the option @{text smt2_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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|
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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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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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*} |
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|
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declare [[ smt2_certificates = "" ]] |
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|
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text {* |
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The option @{text smt2_read_only_certificates} 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 @{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 [[ smt2_read_only_certificates = false ]] |
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|
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|
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|
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subsection {* Tracing *} |
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|
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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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|
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declare [[ smt2_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 smt2_trace} should be set to @{text true}. |
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*} |
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|
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declare [[ smt2_trace = false ]] |
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|
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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 smt2_solver} and @{text smt2_oracle} above). |
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*} |
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|
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declare [[ smt2_trace_used_facts = false ]] |
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|
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|
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subsection {* Schematic rules for Z3 proof reconstruction *} |
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|
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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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|
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lemmas [z3_new_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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|
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lemma [z3_new_rule]: |
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"(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))" |
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"(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))" |
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"(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))" |
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"(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))" |
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"(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))" |
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"(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))" |
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"(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))" |
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"(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))" |
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by auto |
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|
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lemma [z3_new_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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"(True \<longrightarrow> P) = P" |
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"(P \<longrightarrow> True) = True" |
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"(False \<longrightarrow> P) = True" |
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"(P \<longrightarrow> P) = True" |
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by auto |
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|
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lemma [z3_new_rule]: |
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"((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))" |
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by auto |
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329 |
|
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lemma [z3_new_rule]: |
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"(\<not>True) = False" |
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"(\<not>False) = True" |
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"(x = x) = True" |
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"(P = True) = P" |
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"(True = P) = P" |
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"(P = False) = (\<not>P)" |
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"(False = P) = (\<not>P)" |
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"((\<not>P) = P) = False" |
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"(P = (\<not>P)) = False" |
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"((\<not>P) = (\<not>Q)) = (P = Q)" |
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"\<not>(P = (\<not>Q)) = (P = Q)" |
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"\<not>((\<not>P) = Q) = (P = Q)" |
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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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347 |
|
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lemma [z3_new_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 P then Q else True) = ((\<not>P) \<or> Q)" |
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"(if P then Q else True) = (Q \<or> (\<not>P))" |
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"(if P then Q else \<not>Q) = (P = Q)" |
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"(if P then Q else \<not>Q) = (Q = P)" |
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"(if P then \<not>Q else Q) = (P = (\<not>Q))" |
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"(if P then \<not>Q else Q) = ((\<not>Q) = P)" |
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"(if \<not>P then x else y) = (if P then y else x)" |
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"(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)" |
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"(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)" |
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"(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)" |
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"(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)" |
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"(if P then x else if P then y else z) = (if P then x else z)" |
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"(if P then x else if Q then x else y) = (if P \<or> Q then x else y)" |
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"(if P then x else if Q then x else y) = (if Q \<or> P then x else y)" |
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"(if P then x = y else x = z) = (x = (if P then y else z))" |
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"(if P then x = y else y = z) = (y = (if P then x else z))" |
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"(if P then x = y else z = y) = (y = (if P then x else z))" |
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by auto |
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371 |
|
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
372 |
lemma [z3_new_rule]: |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
373 |
"0 + (x::int) = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
374 |
"x + 0 = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
375 |
"x + x = 2 * x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
376 |
"0 * x = 0" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
377 |
"1 * x = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
378 |
"x + y = y + x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
379 |
by auto |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
380 |
|
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
381 |
lemma [z3_new_rule]: (* for def-axiom *) |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
382 |
"P = Q \<or> P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
383 |
"P = Q \<or> \<not>P \<or> \<not>Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
384 |
"(\<not>P) = Q \<or> \<not>P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
385 |
"(\<not>P) = Q \<or> P \<or> \<not>Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
386 |
"P = (\<not>Q) \<or> \<not>P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
387 |
"P = (\<not>Q) \<or> P \<or> \<not>Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
388 |
"P \<noteq> Q \<or> P \<or> \<not>Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
389 |
"P \<noteq> Q \<or> \<not>P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
390 |
"P \<noteq> (\<not>Q) \<or> P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
391 |
"(\<not>P) \<noteq> Q \<or> P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
392 |
"P \<or> Q \<or> P \<noteq> (\<not>Q)" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
393 |
"P \<or> Q \<or> (\<not>P) \<noteq> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
394 |
"P \<or> \<not>Q \<or> P \<noteq> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
395 |
"\<not>P \<or> Q \<or> P \<noteq> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
396 |
"P \<or> y = (if P then x else y)" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
397 |
"P \<or> (if P then x else y) = y" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
398 |
"\<not>P \<or> x = (if P then x else y)" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
399 |
"\<not>P \<or> (if P then x else y) = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
400 |
"P \<or> R \<or> \<not>(if P then Q else R)" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
401 |
"\<not>P \<or> Q \<or> \<not>(if P then Q else R)" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
402 |
"\<not>(if P then Q else R) \<or> \<not>P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
403 |
"\<not>(if P then Q else R) \<or> P \<or> R" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
404 |
"(if P then Q else R) \<or> \<not>P \<or> \<not>Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
405 |
"(if P then Q else R) \<or> P \<or> \<not>R" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
406 |
"(if P then \<not>Q else R) \<or> \<not>P \<or> Q" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
407 |
"(if P then Q else \<not>R) \<or> P \<or> R" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
408 |
by auto |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
409 |
|
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
410 |
hide_type (open) pattern |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
411 |
hide_const fun_app z3div z3mod |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
412 |
hide_const (open) trigger pat nopat weight |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
413 |
|
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
diff
changeset
|
414 |
end |