summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
file |
latest |
revisions |
annotate |
diff |
comparison |
raw |
help

src/HOL/SMT.thy

author | boehmes |

Mon, 22 Nov 2010 23:37:00 +0100 | |

changeset 40664 | e023788a91a1 |

parent 40662 | 798aad2229c0 |

child 40681 | 872b08416fb4 |

permissions | -rw-r--r-- |

added support for quantifier weight annotations

(* Title: HOL/SMT.thy Author: Sascha Boehme, TU Muenchen *) header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *} theory SMT imports List uses "Tools/Datatype/datatype_selectors.ML" "Tools/SMT/smt_failure.ML" "Tools/SMT/smt_config.ML" "Tools/SMT/smt_utils.ML" "Tools/SMT/smt_monomorph.ML" ("Tools/SMT/smt_builtin.ML") ("Tools/SMT/smt_normalize.ML") ("Tools/SMT/smt_translate.ML") ("Tools/SMT/smt_solver.ML") ("Tools/SMT/smtlib_interface.ML") ("Tools/SMT/z3_proof_parser.ML") ("Tools/SMT/z3_proof_tools.ML") ("Tools/SMT/z3_proof_literals.ML") ("Tools/SMT/z3_proof_methods.ML") ("Tools/SMT/z3_proof_reconstruction.ML") ("Tools/SMT/z3_model.ML") ("Tools/SMT/z3_interface.ML") ("Tools/SMT/smt_setup_solvers.ML") begin subsection {* Triggers for quantifier instantiation *} text {* Some SMT solvers support triggers for quantifier instantiation. Each trigger consists of one ore more patterns. A pattern may either be a list of positive subterms (each being tagged by "pat"), or a list of negative subterms (each being tagged by "nopat"). When an SMT solver finds a term matching a positive pattern (a pattern with positive subterms only), it instantiates the corresponding quantifier accordingly. Negative patterns inhibit quantifier instantiations. Each pattern should mention all preceding bound variables. *} datatype pattern = Pattern definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern" definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = 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 {* Distinctness *} text {* As an abbreviation for a quadratic number of inequalities, SMT solvers provide a built-in @{text distinct}. To avoid confusion with the already defined (and more general) @{term List.distinct}, a separate constant is defined. *} definition distinct :: "'a list \<Rightarrow> bool" where "distinct xs = List.distinct xs" 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 x = f x" 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 require a strict separation between formulas and terms. When translating higher-order into first-order problems, all uninterpreted constants (those not builtin 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) is turned into terms by equating such atoms with @{term True} using the following term-level equation symbol. *} definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)" 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))" lemma div_by_z3div: "k div l = ( if k = 0 \<or> l = 0 then 0 else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l else z3div (-k) (-l))" by (auto simp add: z3div_def) lemma mod_by_z3mod: "k mod l = ( if l = 0 then k else if k = 0 then 0 else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l else - z3mod (-k) (-l))" by (auto simp add: z3mod_def) subsection {* Setup *} use "Tools/SMT/smt_builtin.ML" use "Tools/SMT/smt_normalize.ML" use "Tools/SMT/smt_translate.ML" use "Tools/SMT/smt_solver.ML" use "Tools/SMT/smtlib_interface.ML" use "Tools/SMT/z3_interface.ML" use "Tools/SMT/z3_proof_parser.ML" use "Tools/SMT/z3_proof_tools.ML" use "Tools/SMT/z3_proof_literals.ML" use "Tools/SMT/z3_proof_methods.ML" use "Tools/SMT/z3_proof_reconstruction.ML" use "Tools/SMT/z3_model.ML" use "Tools/SMT/smt_setup_solvers.ML" setup {* SMT_Config.setup #> SMT_Solver.setup #> Z3_Proof_Reconstruction.setup #> SMT_Setup_Solvers.setup *} 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 are @{text cvc3}, @{text yices}, and @{text z3}. It is advisable to locally install the selected solver, although this is not necessary for @{text cvc3} and @{text z3}, which can also be used over an Internet-based service. When using local SMT solvers, the path to their binaries should be declared by setting the following environment variables: @{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}. *} 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 {* 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 = "", yices_options = "", 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 ]] 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. *} declare [[ smt_certificates = "" ]] text {* The option @{text smt_fixed} 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_fixed = 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 \<longrightarrow> Q) = (Q \<or> \<not>P)" "(\<not>P \<longrightarrow> Q) = (P \<or> Q)" "(\<not>P \<longrightarrow> Q) = (Q \<or> P)" by auto lemma [z3_rule]: "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))" by auto lemma [z3_rule]: "((\<not>P) = P) = False" "(P = (\<not>P)) = False" "(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 \<not>P then x else y) = (if P then y else x)" by auto lemma [z3_rule]: "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" by auto lemma [z3_rule]: "0 + (x::int) = x" "x + 0 = x" "0 * x = 0" "1 * x = x" "x + y = y + x" by auto hide_type (open) pattern hide_const Pattern term_eq hide_const (open) trigger pat nopat weight distinct fun_app z3div z3mod subsection {* Selectors for datatypes *} setup {* Datatype_Selectors.setup *} declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]] declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]] end