author  paulson 
Thu, 19 May 2005 11:08:15 +0200  
changeset 16009  a6d480e6c5f0 
parent 15997  c71031d7988c 
child 16012  4ae42d8f2fea 
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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val setup : (theory > theory) 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 th = 
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((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN 

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REPEAT(Fast_tac 1); 
15347  41 

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(* This following version fails sometimes, need to investigate, do not use it now. *) 

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fun elimRule_tac' th = 
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((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN 

15347  46 
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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15390  51 
(* functions used to construct a formula *) 
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15347  53 
fun make_disjs [x] = x 
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 make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs) 
15347  55 

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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; 

15347  61 

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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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15347  67 

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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) 
15347  102 
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) 
15371  111 
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",[])) 

15956  121 
=> HOLogic.mk_Trueprop (elimR2Fol_aux prems concl) 
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 Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl) 

15347  123 
 _ => 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 th = 
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case (concl_of th) of (Const ("Trueprop", _) $ Var (idx,_)) => true 

15347  130 
 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 th = 
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if is_elimR th then 

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let val tm = elimR2Fol th 

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val ctm = cterm_of (sign_of_thm th) tm 

15997  139 
in 
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prove_goalw_cterm [] ctm (fn prems => [elimRule_tac th]) 
15997  141 
end 
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else th; 
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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 *) 
15347  150 
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) 

154 
end; 

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(*Convert a theorem into NNF and also skolemize it. Original version, using 
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Hilbert's epsilon in the resulting clauses.*) 

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fun skolem_axiom th = 

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if Term.is_first_order (prop_of th) then 

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let val th' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th 

15347  162 
in 
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repeat_RS th' someI_ex 
15872  164 
end 
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else raise THM ("skolem_axiom: not firstorder", 0, [th]); 
15347  166 

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fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)]; 
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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 th = 
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transfer recon_thy th handle THM _ => th; 

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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); 
15347  185 

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(* transform an Isabelle thm into CNF *) 

16009  187 
fun cnf_axiom_aux th = 
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map (zero_var_indexes o Thm.varifyT) 
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(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th))); 
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16009  192 
(**** SKOLEMIZATION BY INFERENCE (lcp) ****) 
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(*Traverse a term, accumulating Skolem function definitions.*) 

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fun declare_skofuns s t thy = 

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let fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, thy) = 

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(*Existential: declare a Skolem function, then insert into body and continue*) 

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let val cname = s ^ "_" ^ Int.toString n 

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val args = term_frees xtp 

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val Ts = map type_of args 

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val cT = Ts > T 

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val c = Const(NameSpace.append (PureThy.get_name thy) cname, cT) 

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val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp) 

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val def = equals cT $ c $ rhs 

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val thy' = Theory.add_consts_i [(cname, cT, NoSyn)] thy 

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val thy'' = Theory.add_defs_i false [(cname ^ "_def", def)] thy' 

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in dec_sko (subst_bound (list_comb(c,args), p)) (n+1, thy'') end 

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 dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thy) = 

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(*Universal: insert a free variable into body and continue*) 

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let val fname = variant (add_term_names (p,[])) a 

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in dec_sko (subst_bound (Free(fname,T), p)) (n+1, thy) end 

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 dec_sko (Const ("op &", _) $ p $ q) nthy = 

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dec_sko q (dec_sko p nthy) 

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 dec_sko (Const ("op ", _) $ p $ q) nthy = 

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dec_sko q (dec_sko p nthy) 

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 dec_sko (Const ("Trueprop", _) $ p) nthy = 

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dec_sko p nthy 

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 dec_sko t (n,thy) = (n,thy) (*Do nothing otherwise*) 

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in #2 (dec_sko t (1,thy)) end; 

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221 
(*cterms are used throughout for efficiency*) 

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val cTrueprop = Thm.cterm_of (Theory.sign_of HOL.thy) HOLogic.Trueprop; 

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224 
(*cterm version of mk_cTrueprop*) 

225 
fun c_mkTrueprop A = Thm.capply cTrueprop A; 

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227 
(*Given an abstraction over n variables, replace the bound variables by free 

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ones. Return the body, along with the list of free variables.*) 

229 
fun c_variant_abs_multi (ct0, vars) = 

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let val (cv,ct) = Thm.dest_abs NONE ct0 

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in c_variant_abs_multi (ct, cv::vars) end 

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handle CTERM _ => (ct0, rev vars); 

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(*Given the definition of a Skolem function, return a theorem to replace 

235 
an existential formula by a use of that function.*) 

236 
fun skolem_of_def def = 

237 
let val (c,rhs) = Drule.dest_equals (cprop_of (Drule.freeze_all def)) 

238 
val (ch, frees) = c_variant_abs_multi (rhs, []) 

239 
val (chil,cabs) = Thm.dest_comb ch 

240 
val {sign, t, ...} = rep_cterm chil 

241 
val (Const ("Hilbert_Choice.Eps", Type("fun",[_,T]))) = t 

242 
val cex = Thm.cterm_of sign (HOLogic.exists_const T) 

243 
val ex_tm = c_mkTrueprop (Thm.capply cex cabs) 

244 
and conc = c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees))); 

245 
in prove_goalw_cterm [def] (Drule.mk_implies (ex_tm, conc)) 

246 
(fn [prem] => [ rtac (prem RS someI_ex) 1 ]) 

247 
end; 

248 

249 

250 
(*Converts an Isabelle theorem (intro, elim or simp format) into nnf.*) 

251 
fun to_nnf thy th = 

252 
if Term.is_first_order (prop_of th) then 

253 
th > Thm.transfer thy > transform_elim > Drule.freeze_all 

254 
> ObjectLogic.atomize_thm > make_nnf 

255 
else raise THM ("to_nnf: not firstorder", 0, [th]); 

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(*The cache prevents repeated clausification of a theorem, 

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and also repeated declaration of Skolem functions*) 

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val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table) 
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(*Declare Skolem functions for a theorem, supplied in nnf and with its name*) 
262 
fun skolem thy (name,th) = 

263 
let val cname = (case name of 

264 
"" => gensym "sko"  s => Sign.base_name s) 

265 
val thy' = declare_skofuns cname (#prop (rep_thm th)) thy 

266 
in (map (skolem_of_def o #2) (axioms_of thy'), thy') end; 

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268 
(*Populate the clause cache using the supplied theorems*) 

269 
fun skolemlist [] thy = thy 

270 
 skolemlist ((name,th)::nths) thy = 

271 
(case Symtab.lookup (!clause_cache,name) of 

272 
NONE => 

273 
let val nnfth = to_nnf thy th 

274 
val (skoths,thy') = skolem thy (name, nnfth) 

275 
val cls = Meson.make_cnf skoths nnfth 

276 
in clause_cache := Symtab.update ((name, (th,cls)), !clause_cache); 

277 
skolemlist nths thy' 

278 
end 

279 
 SOME _ => skolemlist nths thy) (*FIXME: check for duplicate names?*) 

280 
handle THM _ => skolemlist nths thy; 

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282 
(*Exported function to convert Isabelle theorems into axiom clauses*) 

15956  283 
fun cnf_axiom (name,th) = 
284 
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; 
15347  297 

15956  298 
fun pairname th = (Thm.name_of_thm th, th); 
299 

300 
fun meta_cnf_axiom th = 

301 
map Meson.make_meta_clause (cnf_axiom (pairname th)); 

15499  302 

15347  303 

304 
(* changed: with one extra case added *) 

15956  305 
fun univ_vars_of_aux (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,body)) vars = 
306 
univ_vars_of_aux body vars 

307 
 univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = 

308 
univ_vars_of_aux body vars (* EX x. body *) 

15347  309 
 univ_vars_of_aux (P $ Q) vars = 
15956  310 
univ_vars_of_aux Q (univ_vars_of_aux P vars) 
15347  311 
 univ_vars_of_aux (t as Var(_,_)) vars = 
15956  312 
if (t mem vars) then vars else (t::vars) 
15347  313 
 univ_vars_of_aux _ vars = vars; 
314 

315 
fun univ_vars_of t = univ_vars_of_aux t []; 

316 

317 

318 
fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_))) = 

319 
let val all_vars = univ_vars_of t 

320 
val sk_term = ResSkolemFunction.gen_skolem all_vars tp 

321 
in 

322 
(sk_term,(t,sk_term)::epss) 

323 
end; 

324 

325 

15531  326 
fun sk_lookup [] t = NONE 
327 
 sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then SOME (sk_tm) else (sk_lookup tms t); 

15347  328 

329 

15390  330 

331 
(* get the proper skolem term to replace epsilon term *) 

15347  332 
fun get_skolem epss t = 
15956  333 
case (sk_lookup epss t) of NONE => get_new_skolem epss t 
334 
 SOME sk => (sk,epss); 

15347  335 

336 

16009  337 
fun rm_Eps_cls_aux epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,_))) = 
338 
get_skolem epss t 

15347  339 
 rm_Eps_cls_aux epss (P $ Q) = 
16009  340 
let val (P',epss') = rm_Eps_cls_aux epss P 
341 
val (Q',epss'') = rm_Eps_cls_aux epss' Q 

342 
in (P' $ Q',epss'') end 

15347  343 
 rm_Eps_cls_aux epss t = (t,epss); 
344 

345 

16009  346 
fun rm_Eps_cls epss th = rm_Eps_cls_aux epss (prop_of th); 
15347  347 

348 

15390  349 
(* remove the epsilon terms in a formula, by skolem terms. *) 
15347  350 
fun rm_Eps _ [] = [] 
16009  351 
 rm_Eps epss (th::thms) = 
352 
let val (th',epss') = rm_Eps_cls epss th 

353 
in th' :: (rm_Eps epss' thms) end; 

15347  354 

355 

15390  356 
(* convert a theorem into CNF and then into Clause.clause format. *) 
16009  357 
fun clausify_axiom th = 
358 
let val name = Thm.name_of_thm th 

359 
val isa_clauses = cnf_axiom (name, th) 

15997  360 
(*"isa_clauses" are already in "standard" form. *) 
15347  361 
val isa_clauses' = rm_Eps [] isa_clauses 
15956  362 
val clauses_n = length isa_clauses 
15347  363 
fun make_axiom_clauses _ [] = [] 
15997  364 
 make_axiom_clauses i (cls::clss) = 
365 
(ResClause.make_axiom_clause cls (name,i)) :: make_axiom_clauses (i+1) clss 

15347  366 
in 
15872  367 
make_axiom_clauses 0 isa_clauses' 
15347  368 
end; 
369 

370 

15872  371 
(**** Extract and Clausify theorems from a theory's claset and simpset ****) 
15347  372 

373 
fun claset_rules_of_thy thy = 

374 
let val clsset = rep_cs (claset_of thy) 

375 
val safeEs = #safeEs clsset 

376 
val safeIs = #safeIs clsset 

377 
val hazEs = #hazEs clsset 

378 
val hazIs = #hazIs clsset 

379 
in 

15956  380 
map pairname (safeEs @ safeIs @ hazEs @ hazIs) 
15347  381 
end; 
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383 
fun simpset_rules_of_thy thy = 

15872  384 
let val rules = #rules(fst (rep_ss (simpset_of thy))) 
15347  385 
in 
15872  386 
map (fn (_,r) => (#name r, #thm r)) (Net.dest rules) 
15347  387 
end; 
388 

389 

15872  390 
(**** Translate a set of classical/simplifier rules into CNF (still as type "thm") ****) 
15347  391 

392 
(* classical rules *) 

15872  393 
fun cnf_rules [] err_list = ([],err_list) 
16009  394 
 cnf_rules ((name,th) :: thms) err_list = 
15872  395 
let val (ts,es) = cnf_rules thms err_list 
16009  396 
in (cnf_axiom (name,th) :: ts,es) handle _ => (ts, (th::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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402 
(* 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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406 

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(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****) 
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409 
(* classical rules *) 

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fun clausify_rules [] err_list = ([],err_list) 
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 clausify_rules (th::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 th)::ts,es) handle _ => (ts,(th::es)) 
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end; 
416 

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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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(*Setup function: takes a theory and installs ALL simprules and claset rules 
426 
into the clause cache*) 

427 
fun clause_cache_setup thy = 

428 
let val simps = simpset_rules_of_thy thy 

429 
and clas = claset_rules_of_thy thy 

430 
in skolemlist clas (skolemlist simps thy) end; 

431 

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val setup = [clause_cache_setup]; 

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434 
end; 