diff options
Diffstat (limited to 'src/expr')
-rw-r--r-- | src/expr/proof_rule.cpp | 23 | ||||
-rw-r--r-- | src/expr/proof_rule.h | 301 |
2 files changed, 181 insertions, 143 deletions
diff --git a/src/expr/proof_rule.cpp b/src/expr/proof_rule.cpp index 595e1d5f7..339b4f582 100644 --- a/src/expr/proof_rule.cpp +++ b/src/expr/proof_rule.cpp @@ -31,15 +31,6 @@ const char* toString(PfRule id) case PfRule::MACRO_SR_PRED_INTRO: return "MACRO_SR_PRED_INTRO"; case PfRule::MACRO_SR_PRED_ELIM: return "MACRO_SR_PRED_ELIM"; case PfRule::MACRO_SR_PRED_TRANSFORM: return "MACRO_SR_PRED_TRANSFORM"; - //================================================= Equality rules - case PfRule::REFL: return "REFL"; - case PfRule::SYMM: return "SYMM"; - case PfRule::TRANS: return "TRANS"; - case PfRule::CONG: return "CONG"; - case PfRule::TRUE_INTRO: return "TRUE_INTRO"; - case PfRule::TRUE_ELIM: return "TRUE_ELIM"; - case PfRule::FALSE_INTRO: return "FALSE_INTRO"; - case PfRule::FALSE_ELIM: return "FALSE_ELIM"; //================================================= Boolean rules case PfRule::SPLIT: return "SPLIT"; case PfRule::AND_ELIM: return "AND_ELIM"; @@ -85,6 +76,20 @@ const char* toString(PfRule id) case PfRule::CNF_ITE_NEG1: return "CNF_ITE_NEG1"; case PfRule::CNF_ITE_NEG2: return "CNF_ITE_NEG2"; case PfRule::CNF_ITE_NEG3: return "CNF_ITE_NEG3"; + //================================================= Equality rules + case PfRule::REFL: return "REFL"; + case PfRule::SYMM: return "SYMM"; + case PfRule::TRANS: return "TRANS"; + case PfRule::CONG: return "CONG"; + case PfRule::TRUE_INTRO: return "TRUE_INTRO"; + case PfRule::TRUE_ELIM: return "TRUE_ELIM"; + case PfRule::FALSE_INTRO: return "FALSE_INTRO"; + case PfRule::FALSE_ELIM: return "FALSE_ELIM"; + //================================================= Quantifiers rules + case PfRule::WITNESS_INTRO: return "WITNESS_INTRO"; + case PfRule::EXISTS_INTRO: return "EXISTS_INTRO"; + case PfRule::SKOLEMIZE: return "SKOLEMIZE"; + case PfRule::INSTANTIATE: return "INSTANTIATE"; //================================================= Unknown rule case PfRule::UNKNOWN: return "UNKNOWN"; default: return "?"; diff --git a/src/expr/proof_rule.h b/src/expr/proof_rule.h index 6acccfffd..daded84d4 100644 --- a/src/expr/proof_rule.h +++ b/src/expr/proof_rule.h @@ -71,55 +71,102 @@ enum class PfRule : uint32_t // has the conclusion (=> F F) and has no free assumptions. More generally, a // proof with no free assumptions always concludes a valid formula. SCOPE, - //================================================= Equality rules - // ======== Reflexive + + //======================== Builtin theory (common node operations) + // ======== Substitution + // Children: (P1:F1, ..., Pn:Fn) + // Arguments: (t, (ids)?) + // --------------------------------------------------------------- + // Conclusion: (= t t*sigma{ids}(Fn)*...*sigma{ids}(F1)) + // where sigma{ids}(Fi) are substitutions, which notice are applied in + // reverse order. + // Notice that ids is a MethodId identifier, which determines how to convert + // the formulas F1, ..., Fn into substitutions. + SUBS, + // ======== Rewrite // Children: none - // Arguments: (t) - // --------------------- - // Conclusion: (= t t) - REFL, - // ======== Symmetric - // Children: (P:(= t1 t2)) or (P:(not (= t1 t2))) - // Arguments: none - // ----------------------- - // Conclusion: (= t2 t1) or (not (= t2 t1)) - SYMM, - // ======== Transitivity - // Children: (P1:(= t1 t2), ..., Pn:(= t{n-1} tn)) - // Arguments: none - // ----------------------- - // Conclusion: (= t1 tn) - TRANS, - // ======== Congruence (subsumed by Substitute?) - // Children: (P1:(= t1 s1), ..., Pn:(= tn sn)) - // Arguments: (f) - // --------------------------------------------- - // Conclusion: (= (f t1 ... tn) (f s1 ... sn)) - CONG, - // ======== True intro - // Children: (P:F) - // Arguments: none - // ---------------------------------------- - // Conclusion: (= F true) - TRUE_INTRO, - // ======== True elim - // Children: (P:(= F true) - // Arguments: none + // Arguments: (t, (idr)?) // ---------------------------------------- + // Conclusion: (= t Rewriter{idr}(t)) + // where idr is a MethodId identifier, which determines the kind of rewriter + // to apply, e.g. Rewriter::rewrite. + REWRITE, + // ======== Substitution + Rewriting equality introduction + // + // In this rule, we provide a term t and conclude that it is equal to its + // rewritten form under a (proven) substitution. + // + // Children: (P1:F1, ..., Pn:Fn) + // Arguments: (t, (ids (idr)?)?) + // --------------------------------------------------------------- + // Conclusion: (= t t') + // where + // t' is + // toWitness(Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1))) + // toSkolem(...) converts terms from witness form to Skolem form, + // toWitness(...) converts terms from Skolem form to witness form. + // + // Notice that: + // toSkolem(t')=Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1)) + // In other words, from the point of view of Skolem forms, this rule + // transforms t to t' by standard substitution + rewriting. + // + // The argument ids and idr is optional and specify the identifier of the + // substitution and rewriter respectively to be used. For details, see + // theory/builtin/proof_checker.h. + MACRO_SR_EQ_INTRO, + // ======== Substitution + Rewriting predicate introduction + // + // In this rule, we provide a formula F and conclude it, under the condition + // that it rewrites to true under a proven substitution. + // + // Children: (P1:F1, ..., Pn:Fn) + // Arguments: (F, (ids (idr)?)?) + // --------------------------------------------------------------- // Conclusion: F - TRUE_ELIM, - // ======== False intro - // Children: (P:(not F)) - // Arguments: none + // where + // Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) == true + // where ids and idr are method identifiers. + // + // Notice that we apply rewriting on the witness form of F, meaning that this + // rule may conclude an F whose Skolem form is justified by the definition of + // its (fresh) Skolem variables. Furthermore, notice that the rewriting and + // substitution is applied only within the side condition, meaning the + // rewritten form of the witness form of F does not escape this rule. + MACRO_SR_PRED_INTRO, + // ======== Substitution + Rewriting predicate elimination + // + // In this rule, if we have proven a formula F, then we may conclude its + // rewritten form under a proven substitution. + // + // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn) + // Arguments: ((ids (idr)?)?) // ---------------------------------------- - // Conclusion: (= F false) - FALSE_INTRO, - // ======== False elim - // Children: (P:(= F false) - // Arguments: none + // Conclusion: F' + // where + // F' is + // toWitness(Rewriter{idr}(toSkolem(F)*sigma{ids}(Fn)*...*sigma{ids}(F1)). + // where ids and idr are method identifiers. + // + // We rewrite only on the Skolem form of F, similar to MACRO_SR_EQ_INTRO. + MACRO_SR_PRED_ELIM, + // ======== Substitution + Rewriting predicate transform + // + // In this rule, if we have proven a formula F, then we may provide a formula + // G and conclude it if F and G are equivalent after rewriting under a proven + // substitution. + // + // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn) + // Arguments: (G, (ids (idr)?)?) // ---------------------------------------- - // Conclusion: (not F) - FALSE_ELIM, + // Conclusion: G + // where + // Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) == + // Rewriter{idr}(G*sigma{ids}(Fn)*...*sigma{ids}(F1)) + // + // Notice that we apply rewriting on the witness form of F and G, similar to + // MACRO_SR_PRED_INTRO. + MACRO_SR_PRED_TRANSFORM, //================================================= Boolean rules // ======== Split @@ -378,101 +425,87 @@ enum class PfRule : uint32_t // Conclusion: (or (ite C F1 F2) (not F1) (not F2)) CNF_ITE_NEG3, - //======================== Builtin theory (common node operations) - // ======== Substitution - // Children: (P1:F1, ..., Pn:Fn) - // Arguments: (t, (ids)?) - // --------------------------------------------------------------- - // Conclusion: (= t t*sigma{ids}(Fn)*...*sigma{ids}(F1)) - // where sigma{ids}(Fi) are substitutions, which notice are applied in - // reverse order. - // Notice that ids is a MethodId identifier, which determines how to convert - // the formulas F1, ..., Fn into substitutions. - SUBS, - // ======== Rewrite + //================================================= Equality rules + // ======== Reflexive // Children: none - // Arguments: (t, (idr)?) + // Arguments: (t) + // --------------------- + // Conclusion: (= t t) + REFL, + // ======== Symmetric + // Children: (P:(= t1 t2)) or (P:(not (= t1 t2))) + // Arguments: none + // ----------------------- + // Conclusion: (= t2 t1) or (not (= t2 t1)) + SYMM, + // ======== Transitivity + // Children: (P1:(= t1 t2), ..., Pn:(= t{n-1} tn)) + // Arguments: none + // ----------------------- + // Conclusion: (= t1 tn) + TRANS, + // ======== Congruence (subsumed by Substitute?) + // Children: (P1:(= t1 s1), ..., Pn:(= tn sn)) + // Arguments: (f) + // --------------------------------------------- + // Conclusion: (= (f t1 ... tn) (f s1 ... sn)) + CONG, + // ======== True intro + // Children: (P:F) + // Arguments: none + // ---------------------------------------- + // Conclusion: (= F true) + TRUE_INTRO, + // ======== True elim + // Children: (P:(= F true) + // Arguments: none // ---------------------------------------- - // Conclusion: (= t Rewriter{idr}(t)) - // where idr is a MethodId identifier, which determines the kind of rewriter - // to apply, e.g. Rewriter::rewrite. - REWRITE, - // ======== Substitution + Rewriting equality introduction - // - // In this rule, we provide a term t and conclude that it is equal to its - // rewritten form under a (proven) substitution. - // - // Children: (P1:F1, ..., Pn:Fn) - // Arguments: (t, (ids (idr)?)?) - // --------------------------------------------------------------- - // Conclusion: (= t t') - // where - // t' is - // toWitness(Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1))) - // toSkolem(...) converts terms from witness form to Skolem form, - // toWitness(...) converts terms from Skolem form to witness form. - // - // Notice that: - // toSkolem(t')=Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1)) - // In other words, from the point of view of Skolem forms, this rule - // transforms t to t' by standard substitution + rewriting. - // - // The argument ids and idr is optional and specify the identifier of the - // substitution and rewriter respectively to be used. For details, see - // theory/builtin/proof_checker.h. - MACRO_SR_EQ_INTRO, - // ======== Substitution + Rewriting predicate introduction - // - // In this rule, we provide a formula F and conclude it, under the condition - // that it rewrites to true under a proven substitution. - // - // Children: (P1:F1, ..., Pn:Fn) - // Arguments: (F, (ids (idr)?)?) - // --------------------------------------------------------------- // Conclusion: F - // where - // Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) == true - // where ids and idr are method identifiers. - // - // Notice that we apply rewriting on the witness form of F, meaning that this - // rule may conclude an F whose Skolem form is justified by the definition of - // its (fresh) Skolem variables. Furthermore, notice that the rewriting and - // substitution is applied only within the side condition, meaning the - // rewritten form of the witness form of F does not escape this rule. - MACRO_SR_PRED_INTRO, - // ======== Substitution + Rewriting predicate elimination - // - // In this rule, if we have proven a formula F, then we may conclude its - // rewritten form under a proven substitution. - // - // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn) - // Arguments: ((ids (idr)?)?) + TRUE_ELIM, + // ======== False intro + // Children: (P:(not F)) + // Arguments: none // ---------------------------------------- - // Conclusion: F' - // where - // F' is - // toWitness(Rewriter{idr}(toSkolem(F)*sigma{ids}(Fn)*...*sigma{ids}(F1)). - // where ids and idr are method identifiers. - // - // We rewrite only on the Skolem form of F, similar to MACRO_SR_EQ_INTRO. - MACRO_SR_PRED_ELIM, - // ======== Substitution + Rewriting predicate transform - // - // In this rule, if we have proven a formula F, then we may provide a formula - // G and conclude it if F and G are equivalent after rewriting under a proven - // substitution. - // - // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn) - // Arguments: (G, (ids (idr)?)?) + // Conclusion: (= F false) + FALSE_INTRO, + // ======== False elim + // Children: (P:(= F false) + // Arguments: none // ---------------------------------------- - // Conclusion: G - // where - // Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) == - // Rewriter{idr}(G*sigma{ids}(Fn)*...*sigma{ids}(F1)) - // - // Notice that we apply rewriting on the witness form of F and G, similar to - // MACRO_SR_PRED_INTRO. - MACRO_SR_PRED_TRANSFORM, + // Conclusion: (not F) + FALSE_ELIM, + + //================================================= Quantifiers rules + // ======== Witness intro + // Children: (P:F[t]) + // Arguments: (t) + // ---------------------------------------- + // Conclusion: (= t (witness ((x T)) F[x])) + // where x is a BOUND_VARIABLE unique to the pair F,t. + WITNESS_INTRO, + // ======== Exists intro + // Children: (P:F[t]) + // Arguments: (t) + // ---------------------------------------- + // Conclusion: (exists ((x T)) F[x]) + // where x is a BOUND_VARIABLE unique to the pair F,t. + EXISTS_INTRO, + // ======== Skolemize + // Children: (P:(exists ((x1 T1) ... (xn Tn)) F)) + // Arguments: none + // ---------------------------------------- + // Conclusion: F*sigma + // sigma maps x1 ... xn to their representative skolems obtained by + // SkolemManager::mkSkolemize, returned in the skolems argument of that + // method. + SKOLEMIZE, + // ======== Instantiate + // Children: (P:(forall ((x1 T1) ... (xn Tn)) F)) + // Arguments: (t1 ... tn) + // ---------------------------------------- + // Conclusion: F*sigma + // sigma maps x1 ... xn to t1 ... tn. + INSTANTIATE, //================================================= Unknown rule UNKNOWN, |