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(* Title: HOL/SMT.thy |
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Author: Sascha Boehme, TU Muenchen |
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*) |
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header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *} |
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theory SMT |
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imports List |
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uses |
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"Tools/Datatype/datatype_selectors.ML" |
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"Tools/SMT/smt_utils.ML" |
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"Tools/SMT/smt_failure.ML" |
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"Tools/SMT/smt_config.ML" |
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"Tools/SMT/smt_monomorph.ML" |
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("Tools/SMT/smt_builtin.ML") |
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("Tools/SMT/smt_normalize.ML") |
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("Tools/SMT/smt_translate.ML") |
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("Tools/SMT/smt_solver.ML") |
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("Tools/SMT/smtlib_interface.ML") |
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("Tools/SMT/z3_proof_parser.ML") |
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("Tools/SMT/z3_proof_tools.ML") |
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("Tools/SMT/z3_proof_literals.ML") |
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("Tools/SMT/z3_proof_methods.ML") |
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("Tools/SMT/z3_proof_reconstruction.ML") |
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("Tools/SMT/z3_model.ML") |
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("Tools/SMT/z3_interface.ML") |
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("Tools/SMT/smt_setup_solvers.ML") |
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begin |
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subsection {* Triggers for quantifier instantiation *} |
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text {* |
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Some SMT solvers support patterns as a quantifier instantiation |
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heuristics. Patterns may either be positive terms (tagged by "pat") |
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triggering quantifier instantiations -- when the solver finds a |
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term matching a positive pattern, it instantiates the corresponding |
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quantifier accordingly -- or negative terms (tagged by "nopat") |
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inhibiting quantifier instantiations. A list of patterns |
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of the same kind is called a multipattern, and all patterns in a |
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multipattern are considered conjunctively for quantifier instantiation. |
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A list of multipatterns is called a trigger, and their multipatterns |
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act disjunctively during quantifier instantiation. Each multipattern |
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should mention at least all quantified variables of the preceding |
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quantifier block. |
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*} |
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datatype pattern = Pattern |
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definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern" |
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definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern" |
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definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool" |
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where "trigger _ P = P" |
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subsection {* Quantifier weights *} |
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text {* |
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Weight annotations to quantifiers influence the priority of quantifier |
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instantiations. They should be handled with care for solvers, which support |
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them, because incorrect choices of weights might render a problem unsolvable. |
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*} |
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definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P" |
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text {* |
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Weights must be non-negative. The value @{text 0} is equivalent to providing |
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no weight at all. |
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Weights should only be used at quantifiers and only inside triggers (if the |
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quantifier has triggers). Valid usages of weights are as follows: |
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\begin{itemize} |
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\item |
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@{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"} |
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\item |
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@{term "\<forall>x. weight 3 (P x)"} |
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\end{itemize} |
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*} |
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subsection {* Higher-order encoding *} |
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text {* |
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Application is made explicit for constants occurring with varying |
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numbers of arguments. This is achieved by the introduction of the |
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following constant. |
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*} |
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definition fun_app where "fun_app f = f" |
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text {* |
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Some solvers support a theory of arrays which can be used to encode |
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higher-order functions. The following set of lemmas specifies the |
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properties of such (extensional) arrays. |
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*} |
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lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other |
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fun_upd_upd fun_app_def |
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subsection {* First-order logic *} |
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text {* |
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Some SMT solvers only accept problems in first-order logic, i.e., |
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where formulas and terms are syntactically separated. When |
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translating higher-order into first-order problems, all |
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uninterpreted constants (those not built-in in the target solver) |
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are treated as function symbols in the first-order sense. Their |
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occurrences as head symbols in atoms (i.e., as predicate symbols) are |
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turned into terms by equating such atoms with @{term True}. |
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Whenever the boolean type occurs in first-order terms, it is replaced |
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by the following type. |
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*} |
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typedecl term_bool |
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subsection {* Integer division and modulo for Z3 *} |
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definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where |
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"z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))" |
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definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where |
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"z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))" |
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lemma div_by_z3div: |
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"\<forall>k l. k div l = ( |
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if k = 0 \<or> l = 0 then 0 |
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else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l |
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else z3div (-k) (-l))" |
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by (auto simp add: z3div_def trigger_def) |
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lemma mod_by_z3mod: |
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"\<forall>k l. k mod l = ( |
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if l = 0 then k |
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else if k = 0 then 0 |
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else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l |
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else - z3mod (-k) (-l))" |
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by (auto simp add: z3mod_def trigger_def) |
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subsection {* Setup *} |
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use "Tools/SMT/smt_builtin.ML" |
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use "Tools/SMT/smt_normalize.ML" |
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use "Tools/SMT/smt_translate.ML" |
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use "Tools/SMT/smt_solver.ML" |
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use "Tools/SMT/smtlib_interface.ML" |
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use "Tools/SMT/z3_interface.ML" |
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use "Tools/SMT/z3_proof_parser.ML" |
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use "Tools/SMT/z3_proof_tools.ML" |
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use "Tools/SMT/z3_proof_literals.ML" |
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use "Tools/SMT/z3_proof_methods.ML" |
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use "Tools/SMT/z3_proof_reconstruction.ML" |
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use "Tools/SMT/z3_model.ML" |
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use "Tools/SMT/smt_setup_solvers.ML" |
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setup {* |
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SMT_Config.setup #> |
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SMT_Normalize.setup #> |
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SMT_Solver.setup #> |
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SMTLIB_Interface.setup #> |
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Z3_Interface.setup #> |
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Z3_Proof_Reconstruction.setup #> |
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SMT_Setup_Solvers.setup |
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*} |
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subsection {* Configuration *} |
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text {* |
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The current configuration can be printed by the command |
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@{text smt_status}, which shows the values of most options. |
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*} |
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subsection {* General configuration options *} |
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text {* |
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The option @{text smt_solver} can be used to change the target SMT |
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solver. The possible values are @{text cvc3}, @{text yices}, and |
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@{text z3}. It is advisable to locally install the selected solver, |
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although this is not necessary for @{text cvc3} and @{text z3}, which |
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can also be used over an Internet-based service. |
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When using local SMT solvers, the path to their binaries should be |
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declared by setting the following environment variables: |
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@{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}. |
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*} |
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declare [[ smt_solver = z3 ]] |
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text {* |
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Since SMT solvers are potentially non-terminating, there is a timeout |
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(given in seconds) to restrict their runtime. A value greater than |
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120 (seconds) is in most cases not advisable. |
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*} |
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declare [[ smt_timeout = 20 ]] |
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text {* |
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SMT solvers apply randomized heuristics. In case a problem is not |
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solvable by an SMT solver, changing the following option might help. |
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*} |
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declare [[ smt_random_seed = 1 ]] |
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text {* |
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In general, the binding to SMT solvers runs as an oracle, i.e, the SMT |
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solvers are fully trusted without additional checks. The following |
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option can cause the SMT solver to run in proof-producing mode, giving |
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a checkable certificate. This is currently only implemented for Z3. |
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*} |
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declare [[ smt_oracle = false ]] |
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text {* |
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Each SMT solver provides several commandline options to tweak its |
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behaviour. They can be passed to the solver by setting the following |
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options. |
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*} |
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|
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declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]] |
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234 |
|
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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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240 |
|
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declare [[ smt_datatypes = false ]] |
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|
41125 | 243 |
text {* |
244 |
The SMT method provides an inference mechanism to detect simple triggers |
|
245 |
in quantified formulas, which might increase the number of problems |
|
246 |
solvable by SMT solvers (note: triggers guide quantifier instantiations |
|
247 |
in the SMT solver). To turn it on, set the following option. |
|
248 |
*} |
|
249 |
||
250 |
declare [[ smt_infer_triggers = false ]] |
|
251 |
||
252 |
text {* |
|
253 |
The SMT method monomorphizes the given facts, that is, it tries to |
|
254 |
instantiate all schematic type variables with fixed types occurring |
|
255 |
in the problem. This is a (possibly nonterminating) fixed-point |
|
256 |
construction whose cycles are limited by the following option. |
|
257 |
*} |
|
258 |
||
259 |
declare [[ smt_monomorph_limit = 10 ]] |
|
260 |
||
36898 | 261 |
|
262 |
||
263 |
subsection {* Certificates *} |
|
264 |
||
265 |
text {* |
|
266 |
By setting the option @{text smt_certificates} to the name of a file, |
|
267 |
all following applications of an SMT solver a cached in that file. |
|
268 |
Any further application of the same SMT solver (using the very same |
|
269 |
configuration) re-uses the cached certificate instead of invoking the |
|
270 |
solver. An empty string disables caching certificates. |
|
271 |
||
272 |
The filename should be given as an explicit path. It is good |
|
273 |
practice to use the name of the current theory (with ending |
|
274 |
@{text ".certs"} instead of @{text ".thy"}) as the certificates file. |
|
275 |
*} |
|
276 |
||
277 |
declare [[ smt_certificates = "" ]] |
|
278 |
||
279 |
text {* |
|
280 |
The option @{text smt_fixed} controls whether only stored |
|
281 |
certificates are should be used or invocation of an SMT solver is |
|
282 |
allowed. When set to @{text true}, no SMT solver will ever be |
|
283 |
invoked and only the existing certificates found in the configured |
|
284 |
cache are used; when set to @{text false} and there is no cached |
|
285 |
certificate for some proposition, then the configured SMT solver is |
|
286 |
invoked. |
|
287 |
*} |
|
288 |
||
289 |
declare [[ smt_fixed = false ]] |
|
290 |
||
291 |
||
292 |
||
293 |
subsection {* Tracing *} |
|
294 |
||
295 |
text {* |
|
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296 |
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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|
299 |
|
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declare [[ smt_verbose = true ]] |
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301 |
|
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302 |
text {* |
36898 | 303 |
For tracing the generated problem file given to the SMT solver as |
304 |
well as the returned result of the solver, the option |
|
305 |
@{text smt_trace} should be set to @{text true}. |
|
306 |
*} |
|
307 |
||
308 |
declare [[ smt_trace = false ]] |
|
309 |
||
310 |
text {* |
|
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|
311 |
From the set of assumptions given to the SMT solver, those assumptions |
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|
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used in the proof are traced when the following option is set to |
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|
313 |
@{term true}. This only works for Z3 when it runs in non-oracle mode |
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|
314 |
(see options @{text smt_solver} and @{text smt_oracle} above). |
36898 | 315 |
*} |
316 |
||
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|
317 |
declare [[ smt_trace_used_facts = false ]] |
39298
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|
318 |
|
36898 | 319 |
|
320 |
||
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subsection {* Schematic rules for Z3 proof reconstruction *} |
36898 | 322 |
|
323 |
text {* |
|
324 |
Several prof rules of Z3 are not very well documented. There are two |
|
325 |
lemma groups which can turn failing Z3 proof reconstruction attempts |
|
326 |
into succeeding ones: the facts in @{text z3_rule} are tried prior to |
|
327 |
any implemented reconstruction procedure for all uncertain Z3 proof |
|
328 |
rules; the facts in @{text z3_simp} are only fed to invocations of |
|
329 |
the simplifier when reconstructing theory-specific proof steps. |
|
330 |
*} |
|
331 |
||
332 |
lemmas [z3_rule] = |
|
333 |
refl eq_commute conj_commute disj_commute simp_thms nnf_simps |
|
334 |
ring_distribs field_simps times_divide_eq_right times_divide_eq_left |
|
335 |
if_True if_False not_not |
|
336 |
||
337 |
lemma [z3_rule]: |
|
338 |
"(P \<longrightarrow> Q) = (Q \<or> \<not>P)" |
|
339 |
"(\<not>P \<longrightarrow> Q) = (P \<or> Q)" |
|
340 |
"(\<not>P \<longrightarrow> Q) = (Q \<or> P)" |
|
341 |
by auto |
|
342 |
||
343 |
lemma [z3_rule]: |
|
344 |
"((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))" |
|
345 |
by auto |
|
346 |
||
347 |
lemma [z3_rule]: |
|
348 |
"((\<not>P) = P) = False" |
|
349 |
"(P = (\<not>P)) = False" |
|
350 |
"(P \<noteq> Q) = (Q = (\<not>P))" |
|
351 |
"(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))" |
|
352 |
"(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))" |
|
353 |
by auto |
|
354 |
||
355 |
lemma [z3_rule]: |
|
356 |
"(if P then P else \<not>P) = True" |
|
357 |
"(if \<not>P then \<not>P else P) = True" |
|
358 |
"(if P then True else False) = P" |
|
359 |
"(if P then False else True) = (\<not>P)" |
|
360 |
"(if \<not>P then x else y) = (if P then y else x)" |
|
40806
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|
361 |
"f (if P then x else y) = (if P then f x else f y)" |
36898 | 362 |
by auto |
363 |
||
364 |
lemma [z3_rule]: |
|
365 |
"P = Q \<or> P \<or> Q" |
|
366 |
"P = Q \<or> \<not>P \<or> \<not>Q" |
|
367 |
"(\<not>P) = Q \<or> \<not>P \<or> Q" |
|
368 |
"(\<not>P) = Q \<or> P \<or> \<not>Q" |
|
369 |
"P = (\<not>Q) \<or> \<not>P \<or> Q" |
|
370 |
"P = (\<not>Q) \<or> P \<or> \<not>Q" |
|
371 |
"P \<noteq> Q \<or> P \<or> \<not>Q" |
|
372 |
"P \<noteq> Q \<or> \<not>P \<or> Q" |
|
373 |
"P \<noteq> (\<not>Q) \<or> P \<or> Q" |
|
374 |
"(\<not>P) \<noteq> Q \<or> P \<or> Q" |
|
375 |
"P \<or> Q \<or> P \<noteq> (\<not>Q)" |
|
376 |
"P \<or> Q \<or> (\<not>P) \<noteq> Q" |
|
377 |
"P \<or> \<not>Q \<or> P \<noteq> Q" |
|
378 |
"\<not>P \<or> Q \<or> P \<noteq> Q" |
|
379 |
by auto |
|
380 |
||
381 |
lemma [z3_rule]: |
|
382 |
"0 + (x::int) = x" |
|
383 |
"x + 0 = x" |
|
384 |
"0 * x = 0" |
|
385 |
"1 * x = x" |
|
386 |
"x + y = y + x" |
|
387 |
by auto |
|
388 |
||
37124 | 389 |
|
390 |
||
41059
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centralized handling of built-in types and constants;
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|
391 |
hide_type term_bool |
37124 | 392 |
hide_type (open) pattern |
41059
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centralized handling of built-in types and constants;
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changeset
|
393 |
hide_const Pattern fun_app |
d2b1fc1b8e19
centralized handling of built-in types and constants;
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parents:
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diff
changeset
|
394 |
hide_const (open) trigger pat nopat weight z3div z3mod |
37124 | 395 |
|
39483
9f0e5684f04b
add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
boehmes
parents:
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diff
changeset
|
396 |
|
9f0e5684f04b
add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
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parents:
39298
diff
changeset
|
397 |
|
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add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
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diff
changeset
|
398 |
subsection {* Selectors for datatypes *} |
9f0e5684f04b
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diff
changeset
|
399 |
|
9f0e5684f04b
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changeset
|
400 |
setup {* Datatype_Selectors.setup *} |
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add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
boehmes
parents:
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diff
changeset
|
401 |
|
9f0e5684f04b
add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
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parents:
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diff
changeset
|
402 |
declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]] |
9f0e5684f04b
add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
boehmes
parents:
39298
diff
changeset
|
403 |
declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]] |
9f0e5684f04b
add full support for datatypes to the SMT interface (only used by Z3 in oracle mode so far); added store to keep track of datatype selector functions
boehmes
parents:
39298
diff
changeset
|
404 |
|
36898 | 405 |
end |