author  paulson 
Wed, 18 May 2005 10:23:47 +0200  
changeset 15997  c71031d7988c 
parent 15956  0da64b5a9a00 
child 16009  a6d480e6c5f0 
permissions  rwrr 
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_AXIOMS = 
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sig 

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exception ELIMR2FOL of string 

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val elimRule_tac : thm > Tactical.tactic 

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val elimR2Fol : thm > Term.term 

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val transform_elim : thm > thm 

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val clausify_axiom : thm > ResClause.clause list 

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val cnf_axiom : (string * thm) > thm list 

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val meta_cnf_axiom : thm > thm list 

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val cnf_rule : thm > thm list 

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val cnf_classical_rules_thy : theory > thm list list * thm list 

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val clausify_classical_rules_thy : theory > ResClause.clause list list * thm list 

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val cnf_simpset_rules_thy : theory > thm list list * thm list 

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val clausify_simpset_rules_thy : theory > ResClause.clause list list * thm list 

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val rm_Eps 

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: (Term.term * Term.term) list > thm list > Term.term list 

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val claset_rules_of_thy : theory > (string * thm) list 

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val simpset_rules_of_thy : theory > (string * thm) list 

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val clausify_rules : thm list > thm list > ResClause.clause list list * thm list 

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end; 

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structure ResAxioms : RES_AXIOMS = 
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struct 

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(**** Transformation of Elimination Rules into FirstOrder Formulas****) 
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(* a tactic used to prove an elimrule. *) 
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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); 
15347  42 

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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_disjs [x] = x 
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 make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs) 
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fun make_conjs [x] = x 

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 make_conjs (x :: xs) = HOLogic.mk_conj(x, make_conjs xs) 
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fun add_EX tm [] = tm 

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 add_EX tm ((x,xtp)::xs) = add_EX (HOLogic.exists_const xtp $ Abs(x,xtp,tm)) xs; 

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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' = HOLogic.dest_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' = HOLogic.Not $ (HOLogic.dest_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' = HOLogic.dest_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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HOLogic.mk_imp (HOLogic.dest_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 HOLogic.Not $ (HOLogic.dest_Trueprop main) 
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else trans_elim (main, tl prems, concl) 
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end; 
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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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=> HOLogic.mk_Trueprop (elimR2Fol_aux prems concl) 
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 Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl) 

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 _ => raise ELIMR2FOL("Not an elimination rule!") 
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end; 

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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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(* convert an elimrule into an equivalent theorem that does not have the 
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predicate variable. Leave other theorems unchanged.*) 

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fun transform_elim thm = 

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if is_elimR thm then 

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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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else thm; 

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(**** Transformation of Clasets and Simpsets into FirstOrder Axioms ****) 

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(* to be fixed: cnf_intro, cnf_rule, is_introR *) 

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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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(* convert a theorem into NNF and also skolemize it. *) 
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fun skolem_axiom thm = 
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if Term.is_first_order (prop_of thm) then 
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let val thm' = (skolemize 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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else raise THM ("skolem_axiom: not firstorder", 0, [thm]); 

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fun cnf_rule thm = make_clauses [skolem_axiom (transform_elim thm)]; 
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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 *) 
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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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fun is_taut th = 
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case (prop_of th) of 
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(Const ("Trueprop", _) $ Const ("True", _)) => true 
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 _ => false; 
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(* remove tautologous clauses *) 
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val rm_redundant_cls = List.filter (not o is_taut); 
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(* transform an Isabelle thm into CNF *) 

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fun cnf_axiom_aux thm = 
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map (zero_var_indexes o Thm.varifyT) 
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(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction thm))); 

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(*Cache for clauses: could be a hash table if we provided them.*) 
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val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table) 
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fun cnf_axiom (name,th) = 
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case name of 

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"" => cnf_axiom_aux th (*no name, so can't cache*) 
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 s => case Symtab.lookup (!clause_cache,s) of 
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NONE => 
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let val cls = cnf_axiom_aux th 
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in clause_cache := Symtab.update ((s, (th,cls)), !clause_cache); cls 
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end 
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 SOME(th',cls) => 
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if eq_thm(th,th') then cls 
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else (*New theorem stored under the same name? Possible??*) 
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let val cls = cnf_axiom_aux th 
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in clause_cache := Symtab.update ((s, (th,cls)), !clause_cache); cls 
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end; 
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fun pairname th = (Thm.name_of_thm th, th); 
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fun meta_cnf_axiom th = 

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map Meson.make_meta_clause (cnf_axiom (pairname th)); 

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15347  215 

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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 = 
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univ_vars_of_aux body vars 

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 univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = 

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univ_vars_of_aux body vars (* EX x. body *) 

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 univ_vars_of_aux (P $ Q) vars = 
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univ_vars_of_aux Q (univ_vars_of_aux P vars) 
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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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case (sk_lookup epss t) of NONE => get_new_skolem epss t 
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 SOME sk => (sk,epss); 

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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 = rm_Eps_cls_aux epss (prop_of thm); 
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(* 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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(* convert a theorem into CNF and then into Clause.clause format. *) 
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fun clausify_axiom thm = 
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let val name = Thm.name_of_thm thm 
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val isa_clauses = cnf_axiom (name, thm) 

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(*"isa_clauses" are already in "standard" form. *) 
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val isa_clauses' = rm_Eps [] isa_clauses 
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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) = 
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(ResClause.make_axiom_clause cls (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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(**** Extract and Clausify theorems from a theory's claset and simpset ****) 
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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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map pairname (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 rules = #rules(fst (rep_ss (simpset_of thy))) 
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in 
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map (fn (_,r) => (#name r, #thm r)) (Net.dest rules) 
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end; 
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(**** Translate a set of classical/simplifier rules into CNF (still as type "thm") ****) 
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(* classical rules *) 

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fun cnf_rules [] err_list = ([],err_list) 
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 cnf_rules ((name,thm) :: thms) err_list = 
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let val (ts,es) = cnf_rules thms err_list 
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in (cnf_axiom (name,thm) :: ts,es) handle _ => (ts, (thm::es)) end; 
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(* CNF all rules from a given theory's classical reasoner *) 

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fun cnf_classical_rules_thy thy = 

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cnf_rules (claset_rules_of_thy thy) []; 
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(* CNF all simplifier rules from a given theory's simpset *) 

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fun cnf_simpset_rules_thy thy = 

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cnf_rules (simpset_rules_of_thy thy) []; 
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(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****) 
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(* classical rules *) 

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fun clausify_rules [] err_list = ([],err_list) 
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 clausify_rules (thm::thms) err_list = 

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let val (ts,es) = clausify_rules thms err_list 

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in 
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((clausify_axiom thm)::ts,es) handle _ => (ts,(thm::es)) 

330 
end; 

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(* convert all classical rules from a given theory into Clause.clause format. *) 
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fun clausify_classical_rules_thy thy = 
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clausify_rules (map #2 (claset_rules_of_thy thy)) []; 
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(* convert all simplifier rules from a given theory into Clause.clause format. *) 
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fun clausify_simpset_rules_thy thy = 
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clausify_rules (map #2 (simpset_rules_of_thy thy)) []; 
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end; 