--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/SMT.thy Thu Aug 28 00:40:37 2014 +0200
@@ -0,0 +1,429 @@
+(* Title: HOL/Library/SMT.thy
+ Author: Sascha Boehme, TU Muenchen
+*)
+
+header {* Old Version of Bindings to Satisfiability Modulo Theories (SMT) solvers *}
+
+theory SMT
+imports "../Real"
+keywords "smt_status" :: diag
+begin
+
+ML_file "SMT/smt_utils.ML"
+ML_file "SMT/smt_failure.ML"
+ML_file "SMT/smt_config.ML"
+
+
+subsection {* Triggers for quantifier instantiation *}
+
+text {*
+Some SMT solvers support patterns as a quantifier instantiation
+heuristics. Patterns may either be positive terms (tagged by "pat")
+triggering quantifier instantiations -- when the solver finds a
+term matching a positive pattern, it instantiates the corresponding
+quantifier accordingly -- or negative terms (tagged by "nopat")
+inhibiting quantifier instantiations. A list of patterns
+of the same kind is called a multipattern, and all patterns in a
+multipattern are considered conjunctively for quantifier instantiation.
+A list of multipatterns is called a trigger, and their multipatterns
+act disjunctively during quantifier instantiation. Each multipattern
+should mention at least all quantified variables of the preceding
+quantifier block.
+*}
+
+typedecl pattern
+
+consts
+ pat :: "'a \<Rightarrow> pattern"
+ nopat :: "'a \<Rightarrow> pattern"
+
+definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool" where "trigger _ P = P"
+
+
+subsection {* Quantifier weights *}
+
+text {*
+Weight annotations to quantifiers influence the priority of quantifier
+instantiations. They should be handled with care for solvers, which support
+them, because incorrect choices of weights might render a problem unsolvable.
+*}
+
+definition weight :: "int \<Rightarrow> bool \<Rightarrow> bool" where "weight _ P = P"
+
+text {*
+Weights must be non-negative. The value @{text 0} is equivalent to providing
+no weight at all.
+
+Weights should only be used at quantifiers and only inside triggers (if the
+quantifier has triggers). Valid usages of weights are as follows:
+
+\begin{itemize}
+\item
+@{term "\<forall>x. trigger [[pat (P x)]] (weight 2 (P x))"}
+\item
+@{term "\<forall>x. weight 3 (P x)"}
+\end{itemize}
+*}
+
+
+subsection {* Higher-order encoding *}
+
+text {*
+Application is made explicit for constants occurring with varying
+numbers of arguments. This is achieved by the introduction of the
+following constant.
+*}
+
+definition fun_app where "fun_app f = f"
+
+text {*
+Some solvers support a theory of arrays which can be used to encode
+higher-order functions. The following set of lemmas specifies the
+properties of such (extensional) arrays.
+*}
+
+lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
+ fun_upd_upd fun_app_def
+
+
+subsection {* First-order logic *}
+
+text {*
+Some SMT solvers only accept problems in first-order logic, i.e.,
+where formulas and terms are syntactically separated. When
+translating higher-order into first-order problems, all
+uninterpreted constants (those not built-in in the target solver)
+are treated as function symbols in the first-order sense. Their
+occurrences as head symbols in atoms (i.e., as predicate symbols) are
+turned into terms by logically equating such atoms with @{term True}.
+For technical reasons, @{term True} and @{term False} occurring inside
+terms are replaced by the following constants.
+*}
+
+definition term_true where "term_true = True"
+definition term_false where "term_false = False"
+
+
+subsection {* Integer division and modulo for Z3 *}
+
+definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
+
+definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
+
+
+subsection {* Setup *}
+
+ML_file "SMT/smt_builtin.ML"
+ML_file "SMT/smt_datatypes.ML"
+ML_file "SMT/smt_normalize.ML"
+ML_file "SMT/smt_translate.ML"
+ML_file "SMT/smt_solver.ML"
+ML_file "SMT/smtlib_interface.ML"
+ML_file "SMT/z3_interface.ML"
+ML_file "SMT/z3_proof_parser.ML"
+ML_file "SMT/z3_proof_tools.ML"
+ML_file "SMT/z3_proof_literals.ML"
+ML_file "SMT/z3_proof_methods.ML"
+named_theorems z3_simp "simplification rules for Z3 proof reconstruction"
+ML_file "SMT/z3_proof_reconstruction.ML"
+ML_file "SMT/z3_model.ML"
+ML_file "SMT/smt_setup_solvers.ML"
+
+setup {*
+ SMT_Config.setup #>
+ SMT_Normalize.setup #>
+ SMTLIB_Interface.setup #>
+ Z3_Interface.setup #>
+ SMT_Setup_Solvers.setup
+*}
+
+method_setup smt = {*
+ Scan.optional Attrib.thms [] >>
+ (fn thms => fn ctxt =>
+ METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))
+*} "apply an SMT solver to the current goal"
+
+
+subsection {* Configuration *}
+
+text {*
+The current configuration can be printed by the command
+@{text smt_status}, which shows the values of most options.
+*}
+
+
+
+subsection {* General configuration options *}
+
+text {*
+The option @{text smt_solver} can be used to change the target SMT
+solver. The possible values can be obtained from the @{text smt_status}
+command.
+
+Due to licensing restrictions, Yices and Z3 are not installed/enabled
+by default. Z3 is free for non-commercial applications and can be enabled
+by setting Isabelle system option @{text z3_non_commercial} to @{text yes}.
+*}
+
+declare [[ smt_solver = z3 ]]
+
+text {*
+Since SMT solvers are potentially non-terminating, there is a timeout
+(given in seconds) to restrict their runtime. A value greater than
+120 (seconds) is in most cases not advisable.
+*}
+
+declare [[ smt_timeout = 20 ]]
+
+text {*
+SMT solvers apply randomized heuristics. In case a problem is not
+solvable by an SMT solver, changing the following option might help.
+*}
+
+declare [[ smt_random_seed = 1 ]]
+
+text {*
+In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
+solvers are fully trusted without additional checks. The following
+option can cause the SMT solver to run in proof-producing mode, giving
+a checkable certificate. This is currently only implemented for Z3.
+*}
+
+declare [[ smt_oracle = false ]]
+
+text {*
+Each SMT solver provides several commandline options to tweak its
+behaviour. They can be passed to the solver by setting the following
+options.
+*}
+
+declare [[ cvc3_options = "" ]]
+declare [[ yices_options = "" ]]
+declare [[ z3_options = "" ]]
+
+text {*
+Enable the following option to use built-in support for datatypes and
+records. Currently, this is only implemented for Z3 running in oracle
+mode.
+*}
+
+declare [[ smt_datatypes = false ]]
+
+text {*
+The SMT method provides an inference mechanism to detect simple triggers
+in quantified formulas, which might increase the number of problems
+solvable by SMT solvers (note: triggers guide quantifier instantiations
+in the SMT solver). To turn it on, set the following option.
+*}
+
+declare [[ smt_infer_triggers = false ]]
+
+text {*
+The SMT method monomorphizes the given facts, that is, it tries to
+instantiate all schematic type variables with fixed types occurring
+in the problem. This is a (possibly nonterminating) fixed-point
+construction whose cycles are limited by the following option.
+*}
+
+declare [[ monomorph_max_rounds = 5 ]]
+
+text {*
+In addition, the number of generated monomorphic instances is limited
+by the following option.
+*}
+
+declare [[ monomorph_max_new_instances = 500 ]]
+
+
+
+subsection {* Certificates *}
+
+text {*
+By setting the option @{text smt_certificates} to the name of a file,
+all following applications of an SMT solver a cached in that file.
+Any further application of the same SMT solver (using the very same
+configuration) re-uses the cached certificate instead of invoking the
+solver. An empty string disables caching certificates.
+
+The filename should be given as an explicit path. It is good
+practice to use the name of the current theory (with ending
+@{text ".certs"} instead of @{text ".thy"}) as the certificates file.
+Certificate files should be used at most once in a certain theory context,
+to avoid race conditions with other concurrent accesses.
+*}
+
+declare [[ smt_certificates = "" ]]
+
+text {*
+The option @{text smt_read_only_certificates} controls whether only
+stored certificates are should be used or invocation of an SMT solver
+is allowed. When set to @{text true}, no SMT solver will ever be
+invoked and only the existing certificates found in the configured
+cache are used; when set to @{text false} and there is no cached
+certificate for some proposition, then the configured SMT solver is
+invoked.
+*}
+
+declare [[ smt_read_only_certificates = false ]]
+
+
+
+subsection {* Tracing *}
+
+text {*
+The SMT method, when applied, traces important information. To
+make it entirely silent, set the following option to @{text false}.
+*}
+
+declare [[ smt_verbose = true ]]
+
+text {*
+For tracing the generated problem file given to the SMT solver as
+well as the returned result of the solver, the option
+@{text smt_trace} should be set to @{text true}.
+*}
+
+declare [[ smt_trace = false ]]
+
+text {*
+From the set of assumptions given to the SMT solver, those assumptions
+used in the proof are traced when the following option is set to
+@{term true}. This only works for Z3 when it runs in non-oracle mode
+(see options @{text smt_solver} and @{text smt_oracle} above).
+*}
+
+declare [[ smt_trace_used_facts = false ]]
+
+
+
+subsection {* Schematic rules for Z3 proof reconstruction *}
+
+text {*
+Several prof rules of Z3 are not very well documented. There are two
+lemma groups which can turn failing Z3 proof reconstruction attempts
+into succeeding ones: the facts in @{text z3_rule} are tried prior to
+any implemented reconstruction procedure for all uncertain Z3 proof
+rules; the facts in @{text z3_simp} are only fed to invocations of
+the simplifier when reconstructing theory-specific proof steps.
+*}
+
+lemmas [z3_rule] =
+ refl eq_commute conj_commute disj_commute simp_thms nnf_simps
+ ring_distribs field_simps times_divide_eq_right times_divide_eq_left
+ if_True if_False not_not
+
+lemma [z3_rule]:
+ "(P \<and> Q) = (\<not>(\<not>P \<or> \<not>Q))"
+ "(P \<and> Q) = (\<not>(\<not>Q \<or> \<not>P))"
+ "(\<not>P \<and> Q) = (\<not>(P \<or> \<not>Q))"
+ "(\<not>P \<and> Q) = (\<not>(\<not>Q \<or> P))"
+ "(P \<and> \<not>Q) = (\<not>(\<not>P \<or> Q))"
+ "(P \<and> \<not>Q) = (\<not>(Q \<or> \<not>P))"
+ "(\<not>P \<and> \<not>Q) = (\<not>(P \<or> Q))"
+ "(\<not>P \<and> \<not>Q) = (\<not>(Q \<or> P))"
+ by auto
+
+lemma [z3_rule]:
+ "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
+ "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
+ "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
+ "(True \<longrightarrow> P) = P"
+ "(P \<longrightarrow> True) = True"
+ "(False \<longrightarrow> P) = True"
+ "(P \<longrightarrow> P) = True"
+ by auto
+
+lemma [z3_rule]:
+ "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
+ by auto
+
+lemma [z3_rule]:
+ "(\<not>True) = False"
+ "(\<not>False) = True"
+ "(x = x) = True"
+ "(P = True) = P"
+ "(True = P) = P"
+ "(P = False) = (\<not>P)"
+ "(False = P) = (\<not>P)"
+ "((\<not>P) = P) = False"
+ "(P = (\<not>P)) = False"
+ "((\<not>P) = (\<not>Q)) = (P = Q)"
+ "\<not>(P = (\<not>Q)) = (P = Q)"
+ "\<not>((\<not>P) = Q) = (P = Q)"
+ "(P \<noteq> Q) = (Q = (\<not>P))"
+ "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
+ "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
+ by auto
+
+lemma [z3_rule]:
+ "(if P then P else \<not>P) = True"
+ "(if \<not>P then \<not>P else P) = True"
+ "(if P then True else False) = P"
+ "(if P then False else True) = (\<not>P)"
+ "(if P then Q else True) = ((\<not>P) \<or> Q)"
+ "(if P then Q else True) = (Q \<or> (\<not>P))"
+ "(if P then Q else \<not>Q) = (P = Q)"
+ "(if P then Q else \<not>Q) = (Q = P)"
+ "(if P then \<not>Q else Q) = (P = (\<not>Q))"
+ "(if P then \<not>Q else Q) = ((\<not>Q) = P)"
+ "(if \<not>P then x else y) = (if P then y else x)"
+ "(if P then (if Q then x else y) else x) = (if P \<and> (\<not>Q) then y else x)"
+ "(if P then (if Q then x else y) else x) = (if (\<not>Q) \<and> P then y else x)"
+ "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
+ "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
+ "(if P then x else if P then y else z) = (if P then x else z)"
+ "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
+ "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
+ "(if P then x = y else x = z) = (x = (if P then y else z))"
+ "(if P then x = y else y = z) = (y = (if P then x else z))"
+ "(if P then x = y else z = y) = (y = (if P then x else z))"
+ by auto
+
+lemma [z3_rule]:
+ "0 + (x::int) = x"
+ "x + 0 = x"
+ "x + x = 2 * x"
+ "0 * x = 0"
+ "1 * x = x"
+ "x + y = y + x"
+ by auto
+
+lemma [z3_rule]: (* for def-axiom *)
+ "P = Q \<or> P \<or> Q"
+ "P = Q \<or> \<not>P \<or> \<not>Q"
+ "(\<not>P) = Q \<or> \<not>P \<or> Q"
+ "(\<not>P) = Q \<or> P \<or> \<not>Q"
+ "P = (\<not>Q) \<or> \<not>P \<or> Q"
+ "P = (\<not>Q) \<or> P \<or> \<not>Q"
+ "P \<noteq> Q \<or> P \<or> \<not>Q"
+ "P \<noteq> Q \<or> \<not>P \<or> Q"
+ "P \<noteq> (\<not>Q) \<or> P \<or> Q"
+ "(\<not>P) \<noteq> Q \<or> P \<or> Q"
+ "P \<or> Q \<or> P \<noteq> (\<not>Q)"
+ "P \<or> Q \<or> (\<not>P) \<noteq> Q"
+ "P \<or> \<not>Q \<or> P \<noteq> Q"
+ "\<not>P \<or> Q \<or> P \<noteq> Q"
+ "P \<or> y = (if P then x else y)"
+ "P \<or> (if P then x else y) = y"
+ "\<not>P \<or> x = (if P then x else y)"
+ "\<not>P \<or> (if P then x else y) = x"
+ "P \<or> R \<or> \<not>(if P then Q else R)"
+ "\<not>P \<or> Q \<or> \<not>(if P then Q else R)"
+ "\<not>(if P then Q else R) \<or> \<not>P \<or> Q"
+ "\<not>(if P then Q else R) \<or> P \<or> R"
+ "(if P then Q else R) \<or> \<not>P \<or> \<not>Q"
+ "(if P then Q else R) \<or> P \<or> \<not>R"
+ "(if P then \<not>Q else R) \<or> \<not>P \<or> Q"
+ "(if P then Q else \<not>R) \<or> P \<or> R"
+ by auto
+
+ML_file "SMT/smt_real.ML"
+setup SMT_Real.setup
+
+hide_type (open) pattern
+hide_const fun_app term_true term_false z3div z3mod
+hide_const (open) trigger pat nopat weight
+
+end