author | paulson |
Thu, 09 Dec 2004 15:49:40 +0100 | |
changeset 15390 | 87f78411f7c9 |
parent 15371 | 40f5045c5985 |
child 15495 | 50fde6f04e4c |
permissions | -rw-r--r-- |
15347 | 1 |
(* Author: Jia Meng, Cambridge University Computer Laboratory |
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ID: $Id$ |
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Copyright 2004 University of Cambridge |
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Transformation of axiom rules (elim/intro/etc) into CNF forms. |
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*) |
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signature RES_ELIM_RULE = |
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sig |
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exception ELIMR2FOL of string |
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val elimRule_tac : Thm.thm -> Tactical.tactic |
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val elimR2Fol : Thm.thm -> Term.term |
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val transform_elim : Thm.thm -> Thm.thm |
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end; |
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structure ResElimRule: RES_ELIM_RULE = |
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struct |
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(* a tactic used to prove an elim-rule. *) |
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fun elimRule_tac thm = |
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((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN |
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REPEAT(Fast_tac 1); |
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(* This following version fails sometimes, need to investigate, do not use it now. *) |
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fun elimRule_tac' thm = |
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((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN |
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REPEAT(SOLVE((etac exI 1) ORELSE (rtac conjI 1) ORELSE (rtac disjI1 1) ORELSE (rtac disjI2 1))); |
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exception ELIMR2FOL of string; |
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(* functions used to construct a formula *) |
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fun make_imp (prem,concl) = Const("op -->", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ prem $ concl; |
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fun make_disjs [x] = x |
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| make_disjs (x :: xs) = Const("op |",Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_disjs xs) |
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fun make_conjs [x] = x |
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| make_conjs (x :: xs) = Const("op &", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_conjs xs) |
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fun add_EX term [] = term |
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| add_EX term ((x,xtp)::xs) = add_EX (Const ("Ex",Type("fun",[Type("fun",[xtp,Type("bool",[])]),Type("bool",[])])) $ Abs (x,xtp,term)) xs; |
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exception TRUEPROP of string; |
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fun strip_trueprop (Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ P) = P |
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| strip_trueprop _ = raise TRUEPROP("not a prop!"); |
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fun neg P = Const ("Not", Type("fun",[Type("bool",[]),Type("bool",[])])) $ P; |
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fun is_neg (Const("Trueprop",_) $ (Const("Not",_) $ Free(p,_))) (Const("Trueprop",_) $ Free(q,_))= (p = q) |
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| is_neg _ _ = false; |
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exception STRIP_CONCL; |
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fun strip_concl' prems bvs (Const ("==>",_) $ P $ Q) = |
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let val P' = strip_trueprop P |
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val prems' = P'::prems |
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in |
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strip_concl' prems' bvs Q |
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end |
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| strip_concl' prems bvs P = |
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let val P' = neg (strip_trueprop P) |
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in |
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add_EX (make_conjs (P'::prems)) bvs |
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end; |
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fun strip_concl prems bvs concl (Const ("all", _) $ Abs (x,xtp,body)) = strip_concl prems ((x,xtp)::bvs) concl body |
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| strip_concl prems bvs concl (Const ("==>",_) $ P $ Q) = |
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if (is_neg P concl) then (strip_concl' prems bvs Q) |
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else |
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(let val P' = strip_trueprop P |
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val prems' = P'::prems |
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in |
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strip_concl prems' bvs concl Q |
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end) |
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| strip_concl prems bvs concl _ = add_EX (make_conjs prems) bvs; |
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fun trans_elim (main,others,concl) = |
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let val others' = map (strip_concl [] [] concl) others |
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val disjs = make_disjs others' |
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in |
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make_imp(strip_trueprop main,disjs) |
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end; |
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(* aux function of elim2Fol, take away predicate variable. *) |
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fun elimR2Fol_aux prems concl = |
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let val nprems = length prems |
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val main = hd prems |
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in |
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if (nprems = 1) then neg (strip_trueprop main) |
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else trans_elim (main, tl prems, concl) |
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end; |
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fun trueprop term = Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ term; |
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(* convert an elim rule into an equivalent formula, of type Term.term. *) |
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fun elimR2Fol elimR = |
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let val elimR' = Drule.freeze_all elimR |
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val (prems,concl) = (prems_of elimR', concl_of elimR') |
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in |
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case concl of Const("Trueprop",_) $ Free(_,Type("bool",[])) |
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=> trueprop (elimR2Fol_aux prems concl) |
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| Free(x,Type("prop",[])) => trueprop(elimR2Fol_aux prems concl) |
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| _ => raise ELIMR2FOL("Not an elimination rule!") |
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end; |
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(**** use prove_goalw_cterm to prove ****) |
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(* convert an elim-rule into an equivalent theorem that does not have the predicate variable. *) |
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fun transform_elim thm = |
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let val tm = elimR2Fol thm |
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val ctm = cterm_of (sign_of_thm thm) tm |
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in |
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prove_goalw_cterm [] ctm (fn prems => [elimRule_tac thm]) |
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end; |
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end; |
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(* some lemmas *) |
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Goal "(P==True) ==> P"; |
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by(Blast_tac 1); |
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qed "Eq_TrueD1"; |
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Goal "(P=True) ==> P"; |
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by(Blast_tac 1); |
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qed "Eq_TrueD2"; |
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Goal "(P==False) ==> ~P"; |
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by(Blast_tac 1); |
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qed "Eq_FalseD1"; |
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Goal "(P=False) ==> ~P"; |
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by(Blast_tac 1); |
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qed "Eq_FalseD2"; |
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15359
8bad1f42fec0
new CLAUSIFY attribute for proof reconstruction with lemmas
paulson
parents:
15347
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local |
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8bad1f42fec0
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fun prove s = prove_goal (the_context()) s (fn _ => [Simp_tac 1]); |
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val small_simps = |
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map prove |
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["(P | True) == True", "(True | P) == True", |
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"(P & True) == P", "(True & P) == P", |
8bad1f42fec0
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"(False | P) == P", "(P | False) == P", |
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"(False & P) == False", "(P & False) == False", |
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"~True == False", "~False == True"]; |
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in |
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val small_simpset = empty_ss addsimps small_simps |
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end; |
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signature RES_AXIOMS = |
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sig |
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val clausify_axiom : Thm.thm -> ResClause.clause list |
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val cnf_axiom : Thm.thm -> Thm.thm list |
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val cnf_elim : Thm.thm -> Thm.thm list |
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val cnf_intro : Thm.thm -> Thm.thm list |
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val cnf_rule : Thm.thm -> Thm.thm list |
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val cnf_classical_rules_thy : Theory.theory -> Thm.thm list list * Thm.thm list |
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val clausify_classical_rules_thy |
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: Theory.theory -> ResClause.clause list list * Thm.thm list |
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val cnf_simpset_rules_thy |
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: Theory.theory -> Thm.thm list list * Thm.thm list |
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val clausify_simpset_rules_thy |
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: Theory.theory -> ResClause.clause list list * Thm.thm list |
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val rm_Eps |
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: (Term.term * Term.term) list -> Thm.thm list -> Term.term list |
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end; |
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structure ResAxioms : RES_AXIOMS = |
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struct |
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open ResElimRule; |
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(* to be fixed: cnf_intro, cnf_rule, is_introR *) |
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(* check if a rule is an elim rule *) |
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fun is_elimR thm = |
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case (concl_of thm) of (Const ("Trueprop", _) $ Var (idx,_)) => true |
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| Var(indx,Type("prop",[])) => true |
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| _ => false; |
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(* repeated resolution *) |
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fun repeat_RS thm1 thm2 = |
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let val thm1' = thm1 RS thm2 handle THM _ => thm1 |
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in |
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if eq_thm(thm1,thm1') then thm1' else (repeat_RS thm1' thm2) |
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end; |
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(* added this function to remove True/False in a theorem that is in NNF form. *) |
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fun rm_TF_nnf thm = simplify small_simpset thm; |
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(* convert a theorem into NNF and also skolemize it. *) |
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fun skolem_axiom thm = |
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let val thm' = (skolemize o rm_TF_nnf o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) thm |
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in |
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repeat_RS thm' someI_ex |
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end; |
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fun isa_cls thm = |
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let val thm' = skolem_axiom thm |
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in |
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map standard (make_clauses [thm']) |
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end; |
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fun cnf_elim thm = |
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let val thm' = transform_elim thm; |
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in |
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isa_cls thm' |
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end; |
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val cnf_intro = isa_cls; |
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val cnf_rule = isa_cls; |
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fun is_introR thm = true; |
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(*Transfer a theorem in to theory Reconstruction.thy if it is not already |
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inside that theory -- because it's needed for Skolemization *) |
8bad1f42fec0
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parents:
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val recon_thy = ThyInfo.get_theory"Reconstruction"; |
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fun transfer_to_Reconstruction thm = |
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transfer recon_thy thm handle THM _ => thm; |
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(* remove "True" clause *) |
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fun rm_redundant_cls [] = [] |
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| rm_redundant_cls (thm::thms) = |
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let val t = prop_of thm |
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in |
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case t of (Const ("Trueprop", _) $ Const ("True", _)) => rm_redundant_cls thms |
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| _ => thm::(rm_redundant_cls thms) |
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end; |
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(* transform an Isabelle thm into CNF *) |
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fun cnf_axiom thm = |
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let val thm' = transfer_to_Reconstruction thm |
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val thm'' = if (is_elimR thm') then (cnf_elim thm') |
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else (if (is_introR thm') then cnf_intro thm' else cnf_rule thm') |
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in |
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rm_redundant_cls thm'' |
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end; |
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(* changed: with one extra case added *) |
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fun univ_vars_of_aux (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,body)) vars = univ_vars_of_aux body vars |
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| univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = univ_vars_of_aux body vars (* EX x. body *) |
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| univ_vars_of_aux (P $ Q) vars = |
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let val vars' = univ_vars_of_aux P vars |
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in |
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univ_vars_of_aux Q vars' |
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end |
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| univ_vars_of_aux (t as Var(_,_)) vars = |
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if (t mem vars) then vars else (t::vars) |
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| univ_vars_of_aux _ vars = vars; |
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fun univ_vars_of t = univ_vars_of_aux t []; |
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fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_))) = |
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let val all_vars = univ_vars_of t |
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val sk_term = ResSkolemFunction.gen_skolem all_vars tp |
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in |
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(sk_term,(t,sk_term)::epss) |
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end; |
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fun sk_lookup [] t = None |
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| sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then Some (sk_tm) else (sk_lookup tms t); |
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(* get the proper skolem term to replace epsilon term *) |
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fun get_skolem epss t = |
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let val sk_fun = sk_lookup epss t |
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in |
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case sk_fun of None => get_new_skolem epss t |
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| Some sk => (sk,epss) |
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end; |
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fun rm_Eps_cls_aux epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,_))) = get_skolem epss t |
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| rm_Eps_cls_aux epss (P $ Q) = |
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let val (P',epss') = rm_Eps_cls_aux epss P |
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val (Q',epss'') = rm_Eps_cls_aux epss' Q |
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in |
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(P' $ Q',epss'') |
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end |
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| rm_Eps_cls_aux epss t = (t,epss); |
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fun rm_Eps_cls epss thm = |
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let val tm = prop_of thm |
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in |
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rm_Eps_cls_aux epss tm |
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end; |
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15390 | 339 |
(* remove the epsilon terms in a formula, by skolem terms. *) |
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fun rm_Eps _ [] = [] |
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| rm_Eps epss (thm::thms) = |
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let val (thm',epss') = rm_Eps_cls epss thm |
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in |
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thm' :: (rm_Eps epss' thms) |
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end; |
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(* changed, now it also finds out the name of the theorem. *) |
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15390 | 350 |
(* convert a theorem into CNF and then into Clause.clause format. *) |
15347 | 351 |
fun clausify_axiom thm = |
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let val isa_clauses = cnf_axiom thm (*"isa_clauses" are already "standard"ed. *) |
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val isa_clauses' = rm_Eps [] isa_clauses |
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val thm_name = Thm.name_of_thm thm |
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val clauses_n = length isa_clauses |
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fun make_axiom_clauses _ [] = [] |
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| make_axiom_clauses i (cls::clss) = (ResClause.make_axiom_clause cls (thm_name,i)) :: make_axiom_clauses (i+1) clss |
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in |
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make_axiom_clauses 0 isa_clauses' |
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end; |
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(******** Extracting and CNF/Clausify theorems from a classical reasoner and simpset of a given theory ******) |
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local |
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fun retr_thms ([]:MetaSimplifier.rrule list) = [] |
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| retr_thms (r::rs) = (#thm r)::(retr_thms rs); |
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fun snds [] = [] |
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| snds ((x,y)::l) = y::(snds l); |
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in |
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fun claset_rules_of_thy thy = |
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let val clsset = rep_cs (claset_of thy) |
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val safeEs = #safeEs clsset |
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val safeIs = #safeIs clsset |
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val hazEs = #hazEs clsset |
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val hazIs = #hazIs clsset |
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in |
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safeEs @ safeIs @ hazEs @ hazIs |
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end; |
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fun simpset_rules_of_thy thy = |
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let val simpset = simpset_of thy |
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val rules = #rules(fst (rep_ss simpset)) |
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val thms = retr_thms (snds(Net.dest rules)) |
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in |
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thms |
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end; |
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end; |
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(**** Translate a set of classical rules or simplifier rules into CNF (still as type "thm") from a given theory ****) |
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(* classical rules *) |
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fun cnf_classical_rules [] err_list = ([],err_list) |
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| cnf_classical_rules (thm::thms) err_list = |
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let val (ts,es) = cnf_classical_rules thms err_list |
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in |
|
407 |
((cnf_axiom thm)::ts,es) handle _ => (ts,(thm::es)) |
|
408 |
end; |
|
409 |
||
410 |
||
411 |
(* CNF all rules from a given theory's classical reasoner *) |
|
412 |
fun cnf_classical_rules_thy thy = |
|
413 |
let val rules = claset_rules_of_thy thy |
|
414 |
in |
|
415 |
cnf_classical_rules rules [] |
|
416 |
end; |
|
417 |
||
418 |
||
419 |
(* simplifier rules *) |
|
420 |
fun cnf_simpset_rules [] err_list = ([],err_list) |
|
421 |
| cnf_simpset_rules (thm::thms) err_list = |
|
422 |
let val (ts,es) = cnf_simpset_rules thms err_list |
|
423 |
in |
|
424 |
((cnf_axiom thm)::ts,es) handle _ => (ts,(thm::es)) |
|
425 |
end; |
|
426 |
||
427 |
||
428 |
(* CNF all simplifier rules from a given theory's simpset *) |
|
429 |
fun cnf_simpset_rules_thy thy = |
|
430 |
let val thms = simpset_rules_of_thy thy |
|
431 |
in |
|
432 |
cnf_simpset_rules thms [] |
|
433 |
end; |
|
434 |
||
435 |
||
436 |
||
437 |
(**** Convert all theorems of a classical reason/simpset into clauses (ResClause.clause) ****) |
|
438 |
||
439 |
(* classical rules *) |
|
440 |
fun clausify_classical_rules [] err_list = ([],err_list) |
|
441 |
| clausify_classical_rules (thm::thms) err_list = |
|
442 |
let val (ts,es) = clausify_classical_rules thms err_list |
|
443 |
in |
|
444 |
((clausify_axiom thm)::ts,es) handle _ => (ts,(thm::es)) |
|
445 |
end; |
|
446 |
||
15390 | 447 |
|
448 |
||
449 |
(* convert all classical rules from a given theory's classical reasoner into Clause.clause format. *) |
|
15347 | 450 |
fun clausify_classical_rules_thy thy = |
451 |
let val rules = claset_rules_of_thy thy |
|
452 |
in |
|
453 |
clausify_classical_rules rules [] |
|
454 |
end; |
|
455 |
||
456 |
||
457 |
(* simplifier rules *) |
|
458 |
fun clausify_simpset_rules [] err_list = ([],err_list) |
|
459 |
| clausify_simpset_rules (thm::thms) err_list = |
|
460 |
let val (ts,es) = clausify_simpset_rules thms err_list |
|
461 |
in |
|
462 |
((clausify_axiom thm)::ts,es) handle _ => (ts,(thm::es)) |
|
463 |
end; |
|
464 |
||
465 |
||
15390 | 466 |
(* convert all simplifier rules from a given theory's simplifier into Clause.clause format. *) |
15347 | 467 |
fun clausify_simpset_rules_thy thy = |
468 |
let val thms = simpset_rules_of_thy thy |
|
469 |
in |
|
470 |
clausify_simpset_rules thms [] |
|
471 |
end; |
|
472 |
||
473 |
||
474 |
||
475 |
||
476 |
end; |