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src/HOL/SMT.thy

author | huffman |

Wed, 12 May 2010 22:33:10 -0700 | |

changeset 36902 | c6bae4456741 |

parent 36899 | bcd6fce5bf06 |

child 37124 | fe22fc54b876 |

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

use 'subsection' instead of 'section', to maintain 1 chapter per file in generated document

(* Title: HOL/SMT.thy Author: Sascha Boehme, TU Muenchen *) header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *} theory SMT imports List uses "~~/src/Tools/cache_io.ML" ("Tools/SMT/smt_monomorph.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_reconstruction.ML") ("Tools/SMT/z3_model.ML") ("Tools/SMT/z3_interface.ML") ("Tools/SMT/z3_solver.ML") ("Tools/SMT/cvc3_solver.ML") ("Tools/SMT/yices_solver.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 (the first being tagged by "pat" and the consecutive subterms tagged by "andpat"), or a list of negative subterms (the first being tagged by "nopat" and the consecutive subterms tagged by "andpat"). *} datatype pattern = Pattern definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern" definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern" definition andpat :: "pattern \<Rightarrow> 'a \<Rightarrow> pattern" (infixl "andpat" 60) where "_ andpat _ = Pattern" definition trigger :: "pattern list \<Rightarrow> bool \<Rightarrow> bool" where "trigger _ P = P" 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 "apply" where "apply 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 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" (infix "term'_eq" 50) where "(x term_eq y) = (x = y)" subsection {* Setup *} use "Tools/SMT/smt_monomorph.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_reconstruction.ML" use "Tools/SMT/z3_model.ML" use "Tools/SMT/z3_solver.ML" use "Tools/SMT/cvc3_solver.ML" use "Tools/SMT/yices_solver.ML" setup {* SMT_Solver.setup #> Z3_Proof_Reconstruction.setup #> Z3_Solver.setup #> CVC3_Solver.setup #> Yices_Solver.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 ]] 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 {* 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 ]] subsection {* Z3-specific options *} text {* Z3 is the only SMT solver whose proofs are checked (or reconstructed) in Isabelle (all other solvers are implemented as oracles). Enabling or disabling proof reconstruction for Z3 is controlled by the option @{text z3_proofs}. *} declare [[ z3_proofs = true ]] text {* From the set of assumptions given to Z3, those assumptions used in the proof are traced when the option @{text z3_trace_assms} is set to @{term true}. *} declare [[ z3_trace_assms = false ]] text {* Z3 provides several commandline options to tweak its behaviour. They can be configured by writing them literally as value for the option @{text z3_options}. *} declare [[ z3_options = "" ]] 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 end