(*  Author: Jia Meng, Cambridge University Computer Laboratory

    ID: $Id$

    Copyright 2004 University of Cambridge

Transformation of axiom rules (elim/intro/etc) into CNF forms.    

*)

signature RES_ELIM_RULE =

sig

exception ELIMR2FOL of string

val elimRule_tac : Thm.thm -> Tactical.tactic

val elimR2Fol : Thm.thm -> Term.term

val transform_elim : Thm.thm -> Thm.thm

end;

structure ResElimRule: RES_ELIM_RULE =

struct

(* a tactic used to prove an elim-rule. *)

fun elimRule_tac thm =

    ((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN

    REPEAT(Fast_tac 1);

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

fun elimRule_tac' thm =

   ((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac thm 1) THEN

   REPEAT(SOLVE((etac exI 1) ORELSE (rtac conjI 1) ORELSE (rtac disjI1 1) ORELSE (rtac disjI2 1))); 

exception ELIMR2FOL of string;

(* functions used to construct a formula *)

fun make_imp (prem,concl) = Const("op -->", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ prem $ concl;

fun make_disjs [x] = x

  | make_disjs (x :: xs) = Const("op |",Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_disjs xs)

fun make_conjs [x] = x

  | make_conjs (x :: xs) = Const("op &", Type("fun",[Type("bool",[]),Type("fun",[Type("bool",[]),Type("bool",[])])])) $ x $ (make_conjs xs)

fun add_EX term [] = term

  | add_EX term ((x,xtp)::xs) = add_EX (Const ("Ex",Type("fun",[Type("fun",[xtp,Type("bool",[])]),Type("bool",[])])) $ Abs (x,xtp,term)) xs;

exception TRUEPROP of string; 

fun strip_trueprop (Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ P) = P

  | strip_trueprop _ = raise TRUEPROP("not a prop!");

fun neg P = Const ("Not", Type("fun",[Type("bool",[]),Type("bool",[])])) $ P;

fun is_neg (Const("Trueprop",_) $ (Const("Not",_) $ Free(p,_))) (Const("Trueprop",_) $ Free(q,_))= (p = q)

  | is_neg _ _ = false;

exception STRIP_CONCL;

fun strip_concl' prems bvs (Const ("==>",_) $ P $ Q) =

    let val P' = strip_trueprop P

	val prems' = P'::prems

    in

	strip_concl' prems' bvs  Q

    end

  | strip_concl' prems bvs P = 

    let val P' = neg (strip_trueprop P)

    in

	add_EX (make_conjs (P'::prems)) bvs

    end;

fun strip_concl prems bvs concl (Const ("all", _) $ Abs (x,xtp,body))  = strip_concl prems ((x,xtp)::bvs) concl body

  | strip_concl prems bvs concl (Const ("==>",_) $ P $ Q) =

    if (is_neg P concl) then (strip_concl' prems bvs Q)

    else

	(let val P' = strip_trueprop P

	     val prems' = P'::prems

	 in

	     strip_concl prems' bvs  concl Q

	 end)

  | strip_concl prems bvs concl _ = add_EX (make_conjs prems) bvs;

fun trans_elim (main,others,concl) =

    let val others' = map (strip_concl [] [] concl) others

	val disjs = make_disjs others'

    in

	make_imp(strip_trueprop main,disjs)

    end;

(* aux function of elim2Fol, take away predicate variable. *)

fun elimR2Fol_aux prems concl = 

    let val nprems = length prems

	val main = hd prems

    in

	if (nprems = 1) then neg (strip_trueprop main)

        else trans_elim (main, tl prems, concl)

    end;

fun trueprop term = Const ("Trueprop", Type("fun",[Type("bool",[]),Type("prop",[])])) $ term; 

(* convert an elim rule into an equivalent formula, of type Term.term. *)

fun elimR2Fol elimR = 

    let val elimR' = Drule.freeze_all elimR

	val (prems,concl) = (prems_of elimR', concl_of elimR')

    in

	case concl of Const("Trueprop",_) $ Free(_,Type("bool",[])) 

		      => trueprop (elimR2Fol_aux prems concl)

                    | Free(x,Type("prop",[])) => trueprop(elimR2Fol_aux prems concl) 

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

    end;

(**** use prove_goalw_cterm to prove ****)

(* convert an elim-rule into an equivalent theorem that does not have the predicate variable. *) 

fun transform_elim thm =

    let val tm = elimR2Fol thm

	val ctm = cterm_of (sign_of_thm thm) tm	

    in

	prove_goalw_cterm [] ctm (fn prems => [elimRule_tac thm])

    end;	

end;

signature RES_AXIOMS =

sig

val clausify_axiom : Thm.thm -> ResClause.clause list

val cnf_axiom : Thm.thm -> Thm.thm list

val meta_cnf_axiom : Thm.thm -> Thm.thm list

val cnf_elim : Thm.thm -> Thm.thm list

val cnf_rule : Thm.thm -> Thm.thm list

val cnf_classical_rules_thy : Theory.theory -> Thm.thm list list * Thm.thm list

val clausify_classical_rules_thy 

: Theory.theory -> ResClause.clause list list * Thm.thm list

val cnf_simpset_rules_thy 

: Theory.theory -> Thm.thm list list * Thm.thm list

val clausify_simpset_rules_thy 

: Theory.theory -> ResClause.clause list list * Thm.thm list

val rm_Eps 

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

val claset_rules_of_thy : Theory.theory -> Thm.thm list

val simpset_rules_of_thy : Theory.theory -> (string * Thm.thm) list

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

end;

structure ResAxioms : RES_AXIOMS =

struct

open ResElimRule;

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

(* check if a rule is an elim rule *)

fun is_elimR thm = 

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

			 | Var(indx,Type("prop",[])) => true

			 | _ => false;

(* repeated resolution *)

fun repeat_RS thm1 thm2 =

    let val thm1' =  thm1 RS thm2 handle THM _ => thm1

    in

	if eq_thm(thm1,thm1') then thm1' else (repeat_RS thm1' thm2)

    end;

(* convert a theorem into NNF and also skolemize it. *)

fun skolem_axiom thm = 

  if Term.is_first_order (prop_of thm) then

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

    in 

	repeat_RS thm' someI_ex

    end

  else raise THM ("skolem_axiom: not first-order", 0, [thm]);

fun cnf_rule thm = make_clauses [skolem_axiom thm]

fun cnf_elim thm = cnf_rule (transform_elim thm);

(*Transfer a theorem in to theory Reconstruction.thy if it is not already

  inside that theory -- because it's needed for Skolemization *)

val recon_thy = ThyInfo.get_theory"Reconstruction";

fun transfer_to_Reconstruction thm =

    transfer recon_thy thm handle THM _ => thm;

(* remove "True" clause *)

fun rm_redundant_cls [] = []

  | rm_redundant_cls (thm::thms) =

    let val t = prop_of thm

    in

	case t of (Const ("Trueprop", _) $ Const ("True", _)) => rm_redundant_cls thms

		| _ => thm::(rm_redundant_cls thms)

    end;

(* transform an Isabelle thm into CNF *)

fun cnf_axiom thm =

    let val thm' = transfer_to_Reconstruction thm

	val thm'' = if (is_elimR thm') then (cnf_elim thm')  else cnf_rule thm'

    in

	map Thm.varifyT (rm_redundant_cls thm'')

    end;

fun meta_cnf_axiom thm = 

    map (zero_var_indexes o Meson.make_meta_clause) (cnf_axiom thm);

(* changed: with one extra case added *)

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

  | univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = univ_vars_of_aux body vars (* EX x. body *)

  | univ_vars_of_aux (P $ Q) vars =

    let val vars' = univ_vars_of_aux P vars

    in

	univ_vars_of_aux Q vars'

    end

  | univ_vars_of_aux (t as Var(_,_)) vars = 

    if (t mem vars) then vars else (t::vars)

  | univ_vars_of_aux _ vars = vars;

fun univ_vars_of t = univ_vars_of_aux t [];

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

    let val all_vars = univ_vars_of t

	val sk_term = ResSkolemFunction.gen_skolem all_vars tp

    in

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

    end;

fun sk_lookup [] t = NONE

  | sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then SOME (sk_tm) else (sk_lookup tms t);

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

fun get_skolem epss t = 

    let val sk_fun = sk_lookup epss t

    in

	case sk_fun of NONE => get_new_skolem epss t

		     | SOME sk => (sk,epss)

    end;

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

  | rm_Eps_cls_aux epss (P $ Q) =

    let val (P',epss') = rm_Eps_cls_aux epss P

	val (Q',epss'') = rm_Eps_cls_aux epss' Q

    in

	(P' $ Q',epss'')

    end

  | rm_Eps_cls_aux epss t = (t,epss);

fun rm_Eps_cls epss thm =

    let val tm = prop_of thm

    in

	rm_Eps_cls_aux epss tm

    end;

(* remove the epsilon terms in a formula, by skolem terms. *)

fun rm_Eps _ [] = []

  | rm_Eps epss (thm::thms) = 

    let val (thm',epss') = rm_Eps_cls epss thm

    in

	thm' :: (rm_Eps epss' thms)

    end;

(* changed, now it also finds out the name of the theorem. *)

(* convert a theorem into CNF and then into Clause.clause format. *)

fun clausify_axiom thm =

    let val isa_clauses = cnf_axiom thm (*"isa_clauses" are already "standard"ed. *)

        val isa_clauses' = rm_Eps [] isa_clauses

        val thm_name = Thm.name_of_thm thm

	val clauses_n = length isa_clauses

	fun make_axiom_clauses _ [] = []

	  | make_axiom_clauses i (cls::clss) = (ResClause.make_axiom_clause cls (thm_name,i)) :: make_axiom_clauses (i+1) clss 

    in

	make_axiom_clauses 0 isa_clauses'		

    end;

(**** Extract and Clausify theorems from a theory's claset and simpset ****)

fun claset_rules_of_thy thy =

    let val clsset = rep_cs (claset_of thy)

	val safeEs = #safeEs clsset

	val safeIs = #safeIs clsset

	val hazEs = #hazEs clsset

	val hazIs = #hazIs clsset

    in

	safeEs @ safeIs @ hazEs @ hazIs

    end;

fun simpset_rules_of_thy thy =

    let val rules = #rules(fst (rep_ss (simpset_of thy)))

    in

	map (fn (_,r) => (#name r, #thm r)) (Net.dest rules)

    end;

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

(* classical rules *)

fun cnf_rules [] err_list = ([],err_list)

  | cnf_rules (thm::thms) err_list = 

      let val (ts,es) = cnf_rules thms err_list

      in  (cnf_axiom thm :: ts,es) handle  _ => (ts,(thm::es))  end;

(* CNF all rules from a given theory's classical reasoner *)

fun cnf_classical_rules_thy thy = 

    cnf_rules (claset_rules_of_thy thy) [];

(* CNF all simplifier rules from a given theory's simpset *)

fun cnf_simpset_rules_thy thy =

    cnf_rules (map #2 (simpset_rules_of_thy thy)) [];

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

(* classical rules *)

fun clausify_rules [] err_list = ([],err_list)

  | clausify_rules (thm::thms) err_list =

    let val (ts,es) = clausify_rules thms err_list

    in

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

    end;

(* convert all classical rules from a given theory into Clause.clause format. *)

fun clausify_classical_rules_thy thy =

    clausify_rules (claset_rules_of_thy thy) [];

(* convert all simplifier rules from a given theory into Clause.clause format. *)

fun clausify_simpset_rules_thy thy =

    clausify_rules (map #2 (simpset_rules_of_thy thy)) [];

end;