File res_axioms.ML (Isabelle2005: October 2005)

# File res_axioms.ML

    (*  Author: Jia Meng, Cambridge University Computer Laboratory
        ID: $Id: res_axioms.ML,v 1.33 2005/09/19 13:12:14 paulson Exp $
        Copyright 2004 University of Cambridge

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

    signature RES_AXIOMS =
      sig
      exception ELIMR2FOL of string
      val tagging_enabled : bool
      val elimRule_tac : thm -> Tactical.tactic
      val elimR2Fol : thm -> term
      val transform_elim : thm -> thm
      val clausify_axiom_pairs : (string*thm) -> (ResClause.clause*thm) list
      val cnf_axiom : (string * thm) -> thm list
      val meta_cnf_axiom : thm -> thm list
      val rm_Eps : (term * term) list -> thm list -> term list
      val claset_rules_of_thy : theory -> (string * thm) list
      val simpset_rules_of_thy : theory -> (string * thm) list
      val claset_rules_of_ctxt: Proof.context -> (string * thm) list
      val simpset_rules_of_ctxt : Proof.context -> (string * thm) list
      val clausify_rules_pairs :
            (string*thm) list -> thm list -> (ResClause.clause*thm) list list * thm list
      val clause_setup : (theory -> theory) list
      val meson_method_setup : (theory -> theory) list
      end;

    structure ResAxioms : RES_AXIOMS =

    struct

    val tagging_enabled = false (*compile_time option*)

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

    (* a tactic used to prove an elim-rule. *)
    fun elimRule_tac th =
        ((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN
        REPEAT(fast_tac HOL_cs 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. *)
    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 th =
        case (concl_of th) 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 th =
      if is_elimR th then
        let val tm = elimR2Fol th
    	val ctm = cterm_of (sign_of_thm th) tm
        in
    	prove_goalw_cterm [] ctm (fn prems => [elimRule_tac th])
        end
     else th;

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

    (* 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. Original version, using
      Hilbert's epsilon in the resulting clauses.*)
    fun skolem_axiom th =
      let val th' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th
      in  repeat_RS th' someI_ex
      end;

    fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)];

    (*Transfer a theorem into theory Reconstruction.thy if it is not already
      inside that theory -- because it's needed for Skolemization *)

    (*This will refer to the final version of theory Reconstruction.*)
    val recon_thy_ref = Theory.self_ref (the_context ());

    (*If called while Reconstruction is being created, it will transfer to the
      current version. If called afterward, it will transfer to the final version.*)
    fun transfer_to_Reconstruction th =
        transfer (Theory.deref recon_thy_ref) th handle THM _ => th;

    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 th =
        map zero_var_indexes
            (rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th)));

    (**** SKOLEMIZATION BY INFERENCE (lcp) ****)

    (*Traverse a term, accumulating Skolem function definitions.*)
    fun declare_skofuns s t thy =
      let fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, (thy, axs)) =
    	    (*Existential: declare a Skolem function, then insert into body and continue*)
    	    let val cname = s ^ "_" ^ Int.toString n
    		val args = term_frees xtp  (*get the formal parameter list*)
    		val Ts = map type_of args
    		val cT = Ts ---> T
    		val c = Const (Sign.full_name (Theory.sign_of thy) cname, cT)
    		val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp)
    		        (*Forms a lambda-abstraction over the formal parameters*)
    		val def = equals cT $ c $ rhs
    		val thy' = Theory.add_consts_i [(cname, cT, NoSyn)] thy
    		           (*Theory is augmented with the constant, then its def*)
    		val cdef = cname ^ "_def"
    		val thy'' = Theory.add_defs_i false [(cdef, def)] thy'
    	    in dec_sko (subst_bound (list_comb(c,args), p))
    	               (n+1, (thy'', get_axiom thy'' cdef :: axs))
    	    end
    	| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thx) =
    	    (*Universal quant: insert a free variable into body and continue*)
    	    let val fname = variant (add_term_names (p,[])) a
    	    in dec_sko (subst_bound (Free(fname,T), p)) (n, thx) end
    	| dec_sko (Const ("op &", _) $ p $ q) nthy =
    	    dec_sko q (dec_sko p nthy)
    	| dec_sko (Const ("op |", _) $ p $ q) nthy =
    	    dec_sko q (dec_sko p nthy)
    	| dec_sko (Const ("HOL.tag", _) $ p) nthy =
    	    dec_sko p nthy
    	| dec_sko (Const ("Trueprop", _) $ p) nthy =
    	    dec_sko p nthy
    	| dec_sko t nthx = nthx (*Do nothing otherwise*)
      in  #2 (dec_sko t (1, (thy,[])))  end;

    (*cterms are used throughout for efficiency*)
    val cTrueprop = Thm.cterm_of (Theory.sign_of HOL.thy) HOLogic.Trueprop;

    (*cterm version of mk_cTrueprop*)
    fun c_mkTrueprop A = Thm.capply cTrueprop A;

    (*Given an abstraction over n variables, replace the bound variables by free
      ones. Return the body, along with the list of free variables.*)
    fun c_variant_abs_multi (ct0, vars) =
          let val (cv,ct) = Thm.dest_abs NONE ct0
          in  c_variant_abs_multi (ct, cv::vars)  end
          handle CTERM _ => (ct0, rev vars);

    (*Given the definition of a Skolem function, return a theorem to replace
      an existential formula by a use of that function.*)
    fun skolem_of_def def =
      let val (c,rhs) = Drule.dest_equals (cprop_of (Drule.freeze_all def))
          val (ch, frees) = c_variant_abs_multi (rhs, [])
          val (chil,cabs) = Thm.dest_comb ch
          val {sign,t, ...} = rep_cterm chil
          val (Const ("Hilbert_Choice.Eps", Type("fun",[_,T]))) = t
          val cex = Thm.cterm_of sign (HOLogic.exists_const T)
          val ex_tm = c_mkTrueprop (Thm.capply cex cabs)
          and conc =  c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees)));
      in  prove_goalw_cterm [def] (Drule.mk_implies (ex_tm, conc))
    	    (fn [prem] => [ rtac (prem RS someI_ex) 1 ])
      end;

    (*Converts an Isabelle theorem (intro, elim or simp format) into nnf.*)
    fun to_nnf thy th =
        th |> Thm.transfer thy
           |> transform_elim |> Drule.freeze_all
           |> ObjectLogic.atomize_thm |> make_nnf;

    (*The cache prevents repeated clausification of a theorem,
      and also repeated declaration of Skolem functions*)  (* FIXME better use Termtab!? *)
    val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)

    (*Declare Skolem functions for a theorem, supplied in nnf and with its name*)
    fun skolem thy (name,th) =
      let val cname = (case name of "" => gensym "sko" | s => Sign.base_name s)
          val (thy',axs) = declare_skofuns cname (#prop (rep_thm th)) thy
      in (map skolem_of_def axs, thy') end;

    (*Populate the clause cache using the supplied theorems*)
    fun skolemlist [] thy = thy
      | skolemlist ((name,th)::nths) thy =
          (case Symtab.lookup (!clause_cache) name of
    	  NONE =>
    	    let val (nnfth,ok) = (to_nnf thy th, true)
    	                 handle THM _ => (asm_rl, false)
                in
                    if ok then
                        let val (skoths,thy') = skolem thy (name, nnfth)
    			val cls = Meson.make_cnf skoths nnfth
    		    in change clause_cache (Symtab.update (name, (th, cls)));
    			skolemlist nths thy'
    		    end
    		else skolemlist nths thy
                end
    	| SOME _ => skolemlist nths thy) (*FIXME: check for duplicate names?*)

    (*Exported function to convert Isabelle theorems into axiom clauses*)
    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 change clause_cache (Symtab.update (s, (th, cls))); 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 change clause_cache (Symtab.update (s, (th, cls))); 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;

    (*FIXME: use a-list lookup!!*)
    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 th = rm_Eps_cls_aux epss (prop_of th);

    (* replace the epsilon terms in a formula by skolem terms. *)
    fun rm_Eps _ [] = []
      | rm_Eps epss (th::thms) =
          let val (th',epss') = rm_Eps_cls epss th
          in th' :: (rm_Eps epss' thms) end;

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

    (*Preserve the name of "th" after the transformation "f"*)
    fun preserve_name f th = Thm.name_thm (Thm.name_of_thm th, f th);

    (*Tags identify the major premise or conclusion, as hints to resolution provers.
      However, they don't appear to help in recent tests, and they complicate the code.*)
    val tagI = thm "tagI";
    val tagD = thm "tagD";

    val tag_intro = preserve_name (fn th => th RS tagI);
    val tag_elim  = preserve_name (fn th => tagD RS th);

    fun rules_of_claset cs =
      let val {safeIs,safeEs,hazIs,hazEs,...} = rep_cs cs
          val intros = safeIs @ hazIs
          val elims  = safeEs @ hazEs
      in
         debug ("rules_of_claset intros: " ^ Int.toString(length intros) ^
                " elims: " ^ Int.toString(length elims));
         if tagging_enabled
         then map pairname (map tag_intro intros @ map tag_elim elims)
         else map pairname (intros @ elims)
      end;

    fun rules_of_simpset ss =
      let val ({rules,...}, _) = rep_ss ss
          val simps = Net.entries rules
      in
          debug ("rules_of_simpset: " ^ Int.toString(length simps));
          map (fn r => (#name r, #thm r)) simps
      end;

    fun claset_rules_of_thy thy = rules_of_claset (claset_of thy);
    fun simpset_rules_of_thy thy = rules_of_simpset (simpset_of thy);

    fun claset_rules_of_ctxt ctxt = rules_of_claset (local_claset_of ctxt);
    fun simpset_rules_of_ctxt ctxt = rules_of_simpset (local_simpset_of ctxt);

    (**** 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,th) :: thms) err_list =
          let val (ts,es) = cnf_rules thms err_list
          in  (cnf_axiom (name,th) :: ts,es) handle  _ => (ts, (th::es))  end;

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

    fun addclause (c,th) l =
      if ResClause.isTaut c then l else (c,th) :: l;

    (* outputs a list of (clause,thm) pairs *)
    fun clausify_axiom_pairs (thm_name,thm) =
        let val isa_clauses = cnf_axiom (thm_name,thm)
            val isa_clauses' = rm_Eps [] isa_clauses
            val clauses_n = length isa_clauses
    	fun make_axiom_clauses _ [] []= []
    	  | make_axiom_clauses i (cls::clss) (cls'::clss') =
    	      addclause (ResClause.make_axiom_clause cls (thm_name,i), cls')
    	                (make_axiom_clauses (i+1) clss clss')
        in
    	make_axiom_clauses 0 isa_clauses' isa_clauses
        end

    fun clausify_rules_pairs [] err_list = ([],err_list)
      | clausify_rules_pairs ((name,thm)::thms) err_list =
          let val (ts,es) = clausify_rules_pairs thms err_list
          in
    	  ((clausify_axiom_pairs (name,thm))::ts, es)
    	  handle THM (msg,_,_) =>
    		  (debug ("Cannot clausify " ^ name ^ ": " ^ msg);
    		   (ts, (thm::es)))
                 |  ResClause.CLAUSE (msg,t) =>
                      (debug ("Cannot clausify " ^ name ^ ": " ^ msg ^
                              ": " ^ TermLib.string_of_term t);
    		   (ts, (thm::es)))

          end;

    (*Setup function: takes a theory and installs ALL simprules and claset rules
      into the clause cache*)
    fun clause_cache_setup thy =
      let val simps = simpset_rules_of_thy thy
          and clas  = claset_rules_of_thy thy
      in skolemlist clas (skolemlist simps thy) end;

    val clause_setup = [clause_cache_setup];

    (*** meson proof methods ***)

    fun cnf_rules_of_ths ths = List.concat (#1 (cnf_rules (map pairname ths) []));

    fun meson_meth ths ctxt =
      Method.SIMPLE_METHOD' HEADGOAL
        (CHANGED_PROP o Meson.meson_claset_tac (cnf_rules_of_ths ths) (local_claset_of ctxt));

    val meson_method_setup =
     [Method.add_methods
      [("meson", Method.thms_ctxt_args meson_meth,
        "The MESON resolution proof procedure")]];

    end;