src/HOL/Tools/res_axioms.ML

consolidation and simplification

(*  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_AXIOMS =
  sig
  exception ELIMR2FOL of string
  val elimRule_tac : thm -> Tactical.tactic
  val elimR2Fol : thm -> Term.term
  val transform_elim : thm -> thm
  
  val clausify_axiom : thm -> ResClause.clause list
  val cnf_axiom : (string * thm) -> thm list
  val meta_cnf_axiom : thm -> thm list
  val cnf_rule : thm -> thm list
  val cnf_classical_rules_thy : theory -> thm list list * thm list
  val clausify_classical_rules_thy : theory -> ResClause.clause list list * thm list
  val cnf_simpset_rules_thy : theory -> thm list list * thm list
  val clausify_simpset_rules_thy : theory -> ResClause.clause list list * thm list
  val rm_Eps 
  : (Term.term * Term.term) list -> thm list -> Term.term list
  val claset_rules_of_thy : theory -> (string * thm) list
  val simpset_rules_of_thy : theory -> (string * thm) list
  val clausify_rules : thm list -> thm list -> ResClause.clause list list * thm list
  end;

structure ResAxioms : RES_AXIOMS =
 
struct

(**** Transformation of Elimination Rules into First-Order Formulas****)

(* 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_disjs [x] = x
  | make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs)

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

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



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' = HOLogic.dest_Trueprop P
  	  val prems' = P'::prems
      in
	strip_concl' prems' bvs  Q
      end
  | strip_concl' prems bvs P = 
      let val P' = HOLogic.Not $ (HOLogic.dest_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' = HOLogic.dest_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
	HOLogic.mk_imp (HOLogic.dest_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 HOLogic.Not $ (HOLogic.dest_Trueprop main)
        else trans_elim (main, tl prems, concl)
    end;

    
(* 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",[])) 
		      => HOLogic.mk_Trueprop (elimR2Fol_aux prems concl)
                    | Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl) 
		    | _ => raise ELIMR2FOL("Not an elimination rule!")
    end;


(* 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;

(* convert an elim-rule into an equivalent theorem that does not have the 
   predicate variable.  Leave other theorems unchanged.*) 
fun transform_elim thm =
  if is_elimR thm then
    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
  else thm;


(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)

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

(* 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 (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;

fun is_taut th =
      case (prop_of th) of
           (Const ("Trueprop", _) $ Const ("True", _)) => true
         | _ => false;

(* remove tautologous clauses *)
val rm_redundant_cls = List.filter (not o is_taut);

(* transform an Isabelle thm into CNF *)
fun cnf_axiom_aux thm =
    map (zero_var_indexes o Thm.varifyT) 
        (rm_redundant_cls (cnf_rule (transfer_to_Reconstruction thm)));
       
       
(*Cache for clauses: could be a hash table if we provided them.*)
val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)

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

fun pairname th = (Thm.name_of_thm th, th);

fun meta_cnf_axiom th = 
    map Meson.make_meta_clause (cnf_axiom (pairname th));


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


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 = rm_Eps_cls_aux epss (prop_of thm);


(* 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;


(* convert a theorem into CNF and then into Clause.clause format. *)
fun clausify_axiom thm =
    let val name = Thm.name_of_thm thm
	val isa_clauses = cnf_axiom (name, thm)
	      (*"isa_clauses" are already in "standard" form. *)
        val isa_clauses' = rm_Eps [] isa_clauses
        val clauses_n = length isa_clauses
	fun make_axiom_clauses _ [] = []
	  | make_axiom_clauses i (cls::clss) = 
	      (ResClause.make_axiom_clause cls (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
	map pairname (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 ((name,thm) :: thms) err_list = 
      let val (ts,es) = cnf_rules thms err_list
      in  (cnf_axiom (name,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 (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 (map #2 (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;