src/HOL/Tools/metis_tools.ML

declare cpair_strict [simp]

(*  Title:      HOL/Tools/metis_tools.ML
    Author:     Kong W. Susanto and Lawrence C. Paulson, CU Computer Laboratory
    Copyright   Cambridge University 2007

HOL setup for the Metis prover (version 2.0 from 2007).
*)

signature METIS_TOOLS =
sig
  val type_lits: bool Config.T
  val metis_tac: Proof.context -> thm list -> int -> tactic
  val metisF_tac: Proof.context -> thm list -> int -> tactic
  val metisH_tac: Proof.context -> thm list -> int -> tactic
  val setup: theory -> theory
end

structure MetisTools: METIS_TOOLS =
struct

  structure Recon = ResReconstruct;

  val (type_lits, type_lits_setup) = Attrib.config_bool "metis_type_lits" true;

  (* ------------------------------------------------------------------------- *)
  (* Useful Theorems                                                           *)
  (* ------------------------------------------------------------------------- *)
  val EXCLUDED_MIDDLE  = rotate_prems 1 (read_instantiate [("R","False")] notE);
  val REFL_THM         = incr_indexes 2 (Meson.make_meta_clause refl);  (*Rename from 0,1*)
  val subst_em  = zero_var_indexes (subst RS EXCLUDED_MIDDLE);
  val ssubst_em  = read_instantiate [("t","?s"),("s","?t")] (sym RS subst_em);

  (* ------------------------------------------------------------------------- *)
  (* Useful Functions                                                          *)
  (* ------------------------------------------------------------------------- *)

  (* match untyped terms*)
  fun untyped_aconv (Const(a,_))   (Const(b,_))   = (a=b)
    | untyped_aconv (Free(a,_))    (Free(b,_))    = (a=b)
    | untyped_aconv (Var((a,_),_)) (Var((b,_),_)) = (a=b)   (*the index is ignored!*)
    | untyped_aconv (Bound i)      (Bound j)      = (i=j)
    | untyped_aconv (Abs(a,_,t))  (Abs(b,_,u))    = (a=b) andalso untyped_aconv t u
    | untyped_aconv (t1$t2) (u1$u2)  = untyped_aconv t1 u1 andalso untyped_aconv t2 u2
    | untyped_aconv _              _              = false;

  (* Finding the relative location of an untyped term within a list of terms *)
  fun get_index lit =
    let val lit = Envir.eta_contract lit
        fun get n [] = raise Empty
          | get n (x::xs) = if untyped_aconv lit (Envir.eta_contract (HOLogic.dest_Trueprop x))
                            then n  else get (n+1) xs
    in get 1 end;

  (* ------------------------------------------------------------------------- *)
  (* HOL to FOL  (Isabelle to Metis)                                           *)
  (* ------------------------------------------------------------------------- *)

  fun fn_isa_to_met "equal" = "="
    | fn_isa_to_met x       = x;

  fun metis_lit b c args = (b, (c, args));

  fun hol_type_to_fol (ResClause.AtomV x) = Metis.Term.Var x
    | hol_type_to_fol (ResClause.AtomF x) = Metis.Term.Fn(x,[])
    | hol_type_to_fol (ResClause.Comp(tc,tps)) = Metis.Term.Fn(tc, map hol_type_to_fol tps);

  (*These two functions insert type literals before the real literals. That is the
    opposite order from TPTP linkup, but maybe OK.*)

  fun hol_term_to_fol_FO tm =
    case ResHolClause.strip_comb tm of
        (ResHolClause.CombConst(c,_,tys), tms) =>
          let val tyargs = map hol_type_to_fol tys
              val args   = map hol_term_to_fol_FO tms
          in Metis.Term.Fn (c, tyargs @ args) end
      | (ResHolClause.CombVar(v,_), []) => Metis.Term.Var v
      | _ => error "hol_term_to_fol_FO";

  fun hol_term_to_fol_HO (ResHolClause.CombVar(a, ty)) = Metis.Term.Var a
    | hol_term_to_fol_HO (ResHolClause.CombConst(a, ty, tylist)) =
        Metis.Term.Fn(fn_isa_to_met a, map hol_type_to_fol tylist)
    | hol_term_to_fol_HO (ResHolClause.CombApp(tm1,tm2)) =
         Metis.Term.Fn(".", map hol_term_to_fol_HO [tm1,tm2]);

  fun hol_literal_to_fol true (ResHolClause.Literal (pol, tm)) =  (*first-order*)
        let val (ResHolClause.CombConst(p,_,tys), tms) = ResHolClause.strip_comb tm
            val tylits = if p = "equal" then [] else map hol_type_to_fol tys
            val lits = map hol_term_to_fol_FO tms
        in metis_lit pol (fn_isa_to_met p) (tylits @ lits) end
    | hol_literal_to_fol false (ResHolClause.Literal (pol, tm)) =    (*higher-order*)
        case ResHolClause.strip_comb tm of
            (ResHolClause.CombConst("equal",_,_), tms) =>
              metis_lit pol "=" (map hol_term_to_fol_HO tms)
          | _ => metis_lit pol "{}" [hol_term_to_fol_HO tm];

  fun literals_of_hol_thm thy isFO t =
        let val (lits, types_sorts) = ResHolClause.literals_of_term thy t
        in  (map (hol_literal_to_fol isFO) lits, types_sorts) end;

  (*Sign should be "true" for conjecture type constraints, "false" for type lits in clauses.*)
  fun metis_of_typeLit pos (ResClause.LTVar (s,x))  = metis_lit pos s [Metis.Term.Var x]
    | metis_of_typeLit pos (ResClause.LTFree (s,x)) = metis_lit pos s [Metis.Term.Fn(x,[])];

  fun default_sort ctxt (TVar _) = false
    | default_sort ctxt (TFree(x,s)) = (s = Option.getOpt (Variable.def_sort ctxt (x,~1), []));

  fun metis_of_tfree tf =
    Metis.Thm.axiom (Metis.LiteralSet.singleton (metis_of_typeLit true tf));

  fun hol_thm_to_fol is_conjecture ctxt isFO th =
    let val thy = ProofContext.theory_of ctxt
        val (mlits, types_sorts) =
               (literals_of_hol_thm thy isFO o HOLogic.dest_Trueprop o prop_of) th
    in
        if is_conjecture then
            (Metis.Thm.axiom (Metis.LiteralSet.fromList mlits), ResClause.add_typs types_sorts)
        else
          let val tylits = ResClause.add_typs
                             (filter (not o default_sort ctxt) types_sorts)
              val mtylits = if Config.get ctxt type_lits
                            then map (metis_of_typeLit false) tylits else []
          in
            (Metis.Thm.axiom (Metis.LiteralSet.fromList(mtylits @ mlits)), [])
          end
    end;

  (* ARITY CLAUSE *)

  fun m_arity_cls (ResClause.TConsLit (c,t,args)) =
        metis_lit true (ResClause.make_type_class c) [Metis.Term.Fn(t, map Metis.Term.Var args)]
    | m_arity_cls (ResClause.TVarLit (c,str))     =
        metis_lit false (ResClause.make_type_class c) [Metis.Term.Var str];

  (*TrueI is returned as the Isabelle counterpart because there isn't any.*)
  fun arity_cls (ResClause.ArityClause{conclLit,premLits,...}) =
    (TrueI,
     Metis.Thm.axiom (Metis.LiteralSet.fromList (map m_arity_cls (conclLit :: premLits))));

  (* CLASSREL CLAUSE *)

  fun m_classrel_cls subclass superclass =
    [metis_lit false subclass [Metis.Term.Var "T"], metis_lit true superclass [Metis.Term.Var "T"]];

  fun classrel_cls (ResClause.ClassrelClause {axiom_name,subclass,superclass,...}) =
    (TrueI, Metis.Thm.axiom (Metis.LiteralSet.fromList (m_classrel_cls subclass superclass)));

  (* ------------------------------------------------------------------------- *)
  (* FOL to HOL  (Metis to Isabelle)                                           *)
  (* ------------------------------------------------------------------------- *)

 datatype term_or_type = Term of Term.term | Type of Term.typ;

  fun terms_of [] = []
    | terms_of (Term t :: tts) = t :: terms_of tts
    | terms_of (Type _ :: tts) = terms_of tts;

  fun types_of [] = []
    | types_of (Term (Term.Var((a,idx), T)) :: tts) =
        if String.isPrefix "_" a then
            (*Variable generated by Metis, which might have been a type variable.*)
            TVar(("'" ^ a, idx), HOLogic.typeS) :: types_of tts
        else types_of tts
    | types_of (Term _ :: tts) = types_of tts
    | types_of (Type T :: tts) = T :: types_of tts;

  fun apply_list rator nargs rands =
    let val trands = terms_of rands
    in  if length trands = nargs then Term (list_comb(rator, trands))
        else error ("apply_list: wrong number of arguments: " ^ Display.raw_string_of_term rator ^
                    " expected " ^
                    Int.toString nargs ^ " received " ^ commas (map Display.raw_string_of_term trands))
    end;

(*Instantiate constant c with the supplied types, but if they don't match its declared
  sort constraints, replace by a general type.*)
fun const_of ctxt (c,Ts) =  Const (c, dummyT)

fun infer_types ctxt =
  Syntax.check_terms (ProofContext.set_mode ProofContext.mode_pattern ctxt);

  (*We use 1 rather than 0 because variable references in clauses may otherwise conflict
    with variable constraints in the goal...at least, type inference often fails otherwise.
    SEE ALSO axiom_inf below.*)
  fun mk_var w = Term.Var((w,1), HOLogic.typeT);

  (*include the default sort, if available*)
  fun mk_tfree ctxt w =
    let val ww = "'" ^ w
    in  TFree(ww, getOpt (Variable.def_sort ctxt (ww,~1), HOLogic.typeS))  end;

  (*Remove the "apply" operator from an HO term*)
  fun strip_happ args (Metis.Term.Fn(".",[t,u])) = strip_happ (u::args) t
    | strip_happ args x = (x, args);

  (*Maps metis terms to isabelle terms*)
  fun fol_term_to_hol_RAW ctxt fol_tm =
    let val thy = ProofContext.theory_of ctxt
        val _ = Output.debug (fn () => "fol_term_to_hol: " ^ Metis.Term.toString fol_tm)
        fun tm_to_tt (Metis.Term.Var v) =
               (case Recon.strip_prefix ResClause.tvar_prefix v of
                    SOME w => Type (Recon.make_tvar w)
                  | NONE =>
                case Recon.strip_prefix ResClause.schematic_var_prefix v of
                    SOME w => Term (mk_var w)
                  | NONE   => Term (mk_var v) )
                      (*Var from Metis with a name like _nnn; possibly a type variable*)
          | tm_to_tt (Metis.Term.Fn ("{}", [arg])) = tm_to_tt arg   (*hBOOL*)
          | tm_to_tt (t as Metis.Term.Fn (".",_)) =
              let val (rator,rands) = strip_happ [] t
              in  case rator of
                      Metis.Term.Fn(fname,ts) => applic_to_tt (fname, ts @ rands)
                    | _ => case tm_to_tt rator of
                               Term t => Term (list_comb(t, terms_of (map tm_to_tt rands)))
                             | _ => error "tm_to_tt: HO application"
              end
          | tm_to_tt (Metis.Term.Fn (fname, args)) = applic_to_tt (fname,args)
        and applic_to_tt ("=",ts) =
              Term (list_comb(Const ("op =", HOLogic.typeT), terms_of (map tm_to_tt ts)))
          | applic_to_tt (a,ts) =
              case Recon.strip_prefix ResClause.const_prefix a of
                  SOME b =>
                    let val c = Recon.invert_const b
                        val ntypes = Recon.num_typargs thy c
                        val nterms = length ts - ntypes
                        val tts = map tm_to_tt ts
                        val tys = types_of (List.take(tts,ntypes))
                        val ntyargs = Recon.num_typargs thy c
                    in if length tys = ntyargs then
                           apply_list (const_of ctxt (c, tys)) nterms (List.drop(tts,ntypes))
                       else error ("Constant " ^ c ^ " expects " ^ Int.toString ntyargs ^
                                   " but gets " ^ Int.toString (length tys) ^
                                   " type arguments\n" ^
                                   space_implode "\n" (map (Syntax.string_of_typ ctxt) tys) ^
                                   " the terms are \n" ^
                                   space_implode "\n" (map (Syntax.string_of_term ctxt) (terms_of tts)))
                       end
                | NONE => (*Not a constant. Is it a type constructor?*)
              case Recon.strip_prefix ResClause.tconst_prefix a of
                  SOME b => Type (Term.Type(Recon.invert_type_const b, types_of (map tm_to_tt ts)))
                | NONE => (*Maybe a TFree. Should then check that ts=[].*)
              case Recon.strip_prefix ResClause.tfree_prefix a of
                  SOME b => Type (mk_tfree ctxt b)
                | NONE => (*a fixed variable? They are Skolem functions.*)
              case Recon.strip_prefix ResClause.fixed_var_prefix a of
                  SOME b =>
                    let val opr = Term.Free(b, HOLogic.typeT)
                    in  apply_list opr (length ts) (map tm_to_tt ts)  end
                | NONE => error ("unexpected metis function: " ^ a)
    in  case tm_to_tt fol_tm of Term t => t | _ => error "fol_tm_to_tt: Term expected"  end;

  fun fol_terms_to_hol ctxt fol_tms =
    let val ts = map (fol_term_to_hol_RAW ctxt) fol_tms
        val _ = Output.debug (fn () => "  calling type inference:")
        val _ = app (fn t => Output.debug (fn () => Syntax.string_of_term ctxt t)) ts
        val ts' = infer_types ctxt ts;
        val _ = app (fn t => Output.debug
                      (fn () => "  final term: " ^ Syntax.string_of_term ctxt t ^
                                "  of type  " ^ Syntax.string_of_typ ctxt (type_of t)))
                    ts'
    in  ts'  end;

  fun mk_not (Const ("Not", _) $ b) = b
    | mk_not b = HOLogic.mk_not b;

  (* ------------------------------------------------------------------------- *)
  (* FOL step Inference Rules                                                  *)
  (* ------------------------------------------------------------------------- *)

  (*for debugging only*)
  fun print_thpair (fth,th) =
    (Output.debug (fn () => "=============================================");
     Output.debug (fn () => "Metis: " ^ Metis.Thm.toString fth);
     Output.debug (fn () => "Isabelle: " ^ string_of_thm th));

  fun lookth thpairs (fth : Metis.Thm.thm) =
    valOf (AList.lookup (uncurry Metis.Thm.equal) thpairs fth)
    handle Option => error ("Failed to find a Metis theorem " ^ Metis.Thm.toString fth);

  fun is_TrueI th = Thm.eq_thm(TrueI,th);

  fun cterm_incr_types thy idx = cterm_of thy o (map_types (Logic.incr_tvar idx));

  fun inst_excluded_middle th thy i_atm =
    let val vx = hd (term_vars (prop_of th))
        val substs = [(cterm_of thy vx, cterm_of thy i_atm)]
    in  cterm_instantiate substs th  end;

  (* INFERENCE RULE: AXIOM *)
  fun axiom_inf ctxt thpairs th = incr_indexes 1 (lookth thpairs th);
      (*This causes variables to have an index of 1 by default. SEE ALSO mk_var above.*)

  (* INFERENCE RULE: ASSUME *)
  fun assume_inf ctxt atm =
    inst_excluded_middle EXCLUDED_MIDDLE
      (ProofContext.theory_of ctxt)
      (singleton (fol_terms_to_hol ctxt) (Metis.Term.Fn atm));

  (* INFERENCE RULE: INSTANTIATE. Type instantiations are ignored. Attempting to reconstruct
     them admits new possibilities of errors, e.g. concerning sorts. Instead we try to arrange
     that new TVars are distinct and that types can be inferred from terms.*)
  fun inst_inf ctxt thpairs fsubst th =
    let val thy = ProofContext.theory_of ctxt
        val i_th   = lookth thpairs th
        val i_th_vars = term_vars (prop_of i_th)
        fun find_var x = valOf (List.find (fn Term.Var((a,_),_) => a=x) i_th_vars)
        fun subst_translation (x,y) =
              let val v = find_var x
                  val t = fol_term_to_hol_RAW ctxt y
              in  SOME (cterm_of thy v, t)  end
              handle Option =>
                  (Output.debug (fn() => "List.find failed for the variable " ^ x ^
                                         " in " ^ string_of_thm i_th);
                   NONE)
        fun remove_typeinst (a, t) =
              case Recon.strip_prefix ResClause.schematic_var_prefix a of
                  SOME b => SOME (b, t)
                | NONE   => case Recon.strip_prefix ResClause.tvar_prefix a of
                  SOME _ => NONE          (*type instantiations are forbidden!*)
                | NONE   => SOME (a,t)    (*internal Metis var?*)
        val _ = Output.debug (fn () => "  isa th: " ^ string_of_thm i_th)
        val substs = List.mapPartial remove_typeinst (Metis.Subst.toList fsubst)
        val (vars,rawtms) = ListPair.unzip (List.mapPartial subst_translation substs)
        val tms = infer_types ctxt rawtms;
        val ctm_of = cterm_incr_types thy (1 + Thm.maxidx_of i_th)
        val substs' = ListPair.zip (vars, map ctm_of tms)
        val _ = Output.debug (fn() => "subst_translations:")
        val _ = app (fn (x,y) => Output.debug (fn () => string_of_cterm x ^ " |-> " ^
                                                        string_of_cterm y))
                  substs'
    in  cterm_instantiate substs' i_th  end;

  (* INFERENCE RULE: RESOLVE *)

  (*Like RSN, but we rename apart only the type variables. Vars here typically have an index
    of 1, and the use of RSN would increase this typically to 3. Instantiations of those Vars
    could then fail. See comment on mk_var.*)
  fun resolve_inc_tyvars(tha,i,thb) =
    let val tha = Drule.incr_type_indexes (1 + Thm.maxidx_of thb) tha
	val ths = Seq.list_of (bicompose false (false,tha,nprems_of tha) i thb)
    in
	case distinct Thm.eq_thm ths of
	  [th] => th
	| _ => raise THM ("resolve_inc_tyvars: unique result expected", i, [tha,thb])
    end;

  fun resolve_inf ctxt thpairs atm th1 th2 =
    let
      val thy = ProofContext.theory_of ctxt
      val i_th1 = lookth thpairs th1 and i_th2 = lookth thpairs th2
      val _ = Output.debug (fn () => "  isa th1 (pos): " ^ string_of_thm i_th1)
      val _ = Output.debug (fn () => "  isa th2 (neg): " ^ string_of_thm i_th2)
    in
      if is_TrueI i_th1 then i_th2 (*Trivial cases where one operand is type info*)
      else if is_TrueI i_th2 then i_th1
      else
        let
          val i_atm = singleton (fol_terms_to_hol ctxt) (Metis.Term.Fn atm)
          val _ = Output.debug (fn () => "  atom: " ^ Syntax.string_of_term ctxt i_atm)
          val prems_th1 = prems_of i_th1
          val prems_th2 = prems_of i_th2
          val index_th1 = get_index (mk_not i_atm) prems_th1
                handle Empty => error "Failed to find literal in th1"
          val _ = Output.debug (fn () => "  index_th1: " ^ Int.toString index_th1)
          val index_th2 = get_index i_atm prems_th2
                handle Empty => error "Failed to find literal in th2"
          val _ = Output.debug (fn () => "  index_th2: " ^ Int.toString index_th2)
      in  resolve_inc_tyvars (Meson.select_literal index_th1 i_th1, index_th2, i_th2)  end
    end;

  (* INFERENCE RULE: REFL *)
  val refl_x = cterm_of @{theory} (hd (term_vars (prop_of REFL_THM)));
  val refl_idx = 1 + Thm.maxidx_of REFL_THM;

  fun refl_inf ctxt t =
    let val thy = ProofContext.theory_of ctxt
        val i_t = singleton (fol_terms_to_hol ctxt) t
        val _ = Output.debug (fn () => "  term: " ^ Syntax.string_of_term ctxt i_t)
        val c_t = cterm_incr_types thy refl_idx i_t
    in  cterm_instantiate [(refl_x, c_t)] REFL_THM  end;

  fun get_ty_arg_size thy (Const("op =",_)) = 0  (*equality has no type arguments*)
    | get_ty_arg_size thy (Const(c,_))      = (Recon.num_typargs thy c handle TYPE _ => 0)
    | get_ty_arg_size thy _      = 0;

  (* INFERENCE RULE: EQUALITY *)
  fun equality_inf ctxt isFO thpairs (pos,atm) fp fr =
    let val thy = ProofContext.theory_of ctxt
        val [i_atm,i_tm] = fol_terms_to_hol ctxt [Metis.Term.Fn atm, fr]
        val _ = Output.debug (fn () => "sign of the literal: " ^ Bool.toString pos)
        fun replace_item_list lx 0 (l::ls) = lx::ls
          | replace_item_list lx i (l::ls) = l :: replace_item_list lx (i-1) ls
        fun path_finder_FO tm (p::ps) =
              let val (tm1,args) = Term.strip_comb tm
                  val adjustment = get_ty_arg_size thy tm1
                  val p' = if adjustment > p then p else p-adjustment
                  val tm_p = List.nth(args,p')
                    handle Subscript => error ("equality_inf: " ^ Int.toString p ^ " adj " ^
                      Int.toString adjustment  ^ " term " ^  Syntax.string_of_term ctxt tm)
              in
                  Output.debug (fn () => "path_finder: " ^ Int.toString p ^
                                        "  " ^ Syntax.string_of_term ctxt tm_p);
                  if null ps   (*FIXME: why not use pattern-matching and avoid repetition*)
                  then (tm_p, list_comb (tm1, replace_item_list (Term.Bound 0) p' args))
                  else let val (r,t) = path_finder_FO tm_p ps
                       in (r, list_comb (tm1, replace_item_list t p' args)) end
              end
        fun path_finder_HO tm [] = (tm, Term.Bound 0)
          | path_finder_HO (t$u) (0::ps) = (fn(x,y) => (x, y$u)) (path_finder_HO t ps)
          | path_finder_HO (t$u) (p::ps) = (fn(x,y) => (x, t$y)) (path_finder_HO u ps)
        fun path_finder true tm ps = path_finder_FO tm ps
          | path_finder false (tm as Const("op =",_) $ _ $ _) (p::ps) =
               (*equality: not curried, as other predicates are*)
               if p=0 then path_finder_HO tm (0::1::ps)  (*select first operand*)
               else path_finder_HO tm (p::ps)        (*1 selects second operand*)
          | path_finder false tm (p::ps) =
               path_finder_HO tm ps      (*if not equality, ignore head to skip hBOOL*)
        fun path_finder_lit ((nt as Term.Const ("Not", _)) $ tm_a) idx =
              let val (tm, tm_rslt) = path_finder isFO tm_a idx
              in (tm, nt $ tm_rslt) end
          | path_finder_lit tm_a idx = path_finder isFO tm_a idx
        val (tm_subst, body) = path_finder_lit i_atm fp
        val tm_abs = Term.Abs("x", Term.type_of tm_subst, body)
        val _ = Output.debug (fn () => "abstraction: " ^ Syntax.string_of_term ctxt tm_abs)
        val _ = Output.debug (fn () => "i_tm: " ^ Syntax.string_of_term ctxt i_tm)
        val _ = Output.debug (fn () => "located term: " ^ Syntax.string_of_term ctxt tm_subst)
        val imax = maxidx_of_term (i_tm $ tm_abs $ tm_subst)  (*ill typed but gives right max*)
        val subst' = incr_indexes (imax+1) (if pos then subst_em else ssubst_em)
        val _ = Output.debug (fn () => "subst' " ^ string_of_thm subst')
        val eq_terms = map (pairself (cterm_of thy))
                           (ListPair.zip (term_vars (prop_of subst'), [tm_abs, tm_subst, i_tm]))
    in  cterm_instantiate eq_terms subst'  end;

  fun step ctxt isFO thpairs (fol_th, Metis.Proof.Axiom _)                        =
        axiom_inf ctxt thpairs fol_th
    | step ctxt isFO thpairs (_, Metis.Proof.Assume f_atm)                        =
        assume_inf ctxt f_atm
    | step ctxt isFO thpairs (_, Metis.Proof.Subst(f_subst, f_th1))                =
        inst_inf ctxt thpairs f_subst f_th1
    | step ctxt isFO thpairs (_, Metis.Proof.Resolve(f_atm, f_th1, f_th2))        =
        resolve_inf ctxt thpairs f_atm f_th1 f_th2
    | step ctxt isFO thpairs (_, Metis.Proof.Refl f_tm)                           =
        refl_inf ctxt f_tm
    | step ctxt isFO thpairs (_, Metis.Proof.Equality(f_lit, f_p, f_r)) =
        equality_inf ctxt isFO thpairs f_lit f_p f_r;

  val factor = Seq.hd o distinct_subgoals_tac;

  fun real_literal (b, (c, _)) = not (String.isPrefix ResClause.class_prefix c);

  fun translate isFO _    thpairs [] = thpairs
    | translate isFO ctxt thpairs ((fol_th, inf) :: infpairs) =
        let val _ = Output.debug (fn () => "=============================================")
            val _ = Output.debug (fn () => "METIS THM: " ^ Metis.Thm.toString fol_th)
            val _ = Output.debug (fn () => "INFERENCE: " ^ Metis.Proof.inferenceToString inf)
            val th = Meson.flexflex_first_order (factor (step ctxt isFO thpairs (fol_th, inf)))
            val _ = Output.debug (fn () => "ISABELLE THM: " ^ string_of_thm th)
            val _ = Output.debug (fn () => "=============================================")
            val n_metis_lits = length (filter real_literal (Metis.LiteralSet.toList (Metis.Thm.clause fol_th)))
        in
            if nprems_of th = n_metis_lits then ()
            else error "Metis: proof reconstruction has gone wrong";
            translate isFO ctxt ((fol_th, th) :: thpairs) infpairs
        end;

  (*Determining which axiom clauses are actually used*)
  fun used_axioms axioms (th, Metis.Proof.Axiom _) = SOME (lookth axioms th)
    | used_axioms axioms _                         = NONE;

  (* ------------------------------------------------------------------------- *)
  (* Translation of HO Clauses                                                 *)
  (* ------------------------------------------------------------------------- *)

  fun cnf_th th = hd (ResAxioms.cnf_axiom th);

  val equal_imp_fequal' = cnf_th (thm"equal_imp_fequal");
  val fequal_imp_equal' = cnf_th (thm"fequal_imp_equal");

  val comb_I = cnf_th ResHolClause.comb_I;
  val comb_K = cnf_th ResHolClause.comb_K;
  val comb_B = cnf_th ResHolClause.comb_B;
  val comb_C = cnf_th ResHolClause.comb_C;
  val comb_S = cnf_th ResHolClause.comb_S;

  fun type_ext thy tms =
    let val subs = ResAtp.tfree_classes_of_terms tms
        val supers = ResAtp.tvar_classes_of_terms tms
        and tycons = ResAtp.type_consts_of_terms thy tms
        val arity_clauses = ResClause.make_arity_clauses thy tycons supers
        val (supers',arity_clauses) = ResClause.make_arity_clauses thy tycons supers
        val classrel_clauses = ResClause.make_classrel_clauses thy subs supers'
    in  map classrel_cls classrel_clauses @ map arity_cls arity_clauses
    end;

  (* ------------------------------------------------------------------------- *)
  (* Logic maps manage the interface between HOL and first-order logic.        *)
  (* ------------------------------------------------------------------------- *)

  type logic_map =
    {isFO   : bool,
     axioms : (Metis.Thm.thm * Thm.thm) list,
     tfrees : ResClause.type_literal list};

  fun const_in_metis c (pol,(pred,tm_list)) =
    let
      fun in_mterm (Metis.Term.Var nm) = false
        | in_mterm (Metis.Term.Fn (".", tm_list)) = exists in_mterm tm_list
        | in_mterm (Metis.Term.Fn (nm, tm_list)) = c=nm orelse exists in_mterm tm_list
    in  c=pred orelse exists in_mterm tm_list  end;

  (*Extract TFree constraints from context to include as conjecture clauses*)
  fun init_tfrees ctxt =
    let fun add ((a,i),s) Ts = if i = ~1 then TFree(a,s) :: Ts else Ts
    in  ResClause.add_typs (Vartab.fold add (#2 (Variable.constraints_of ctxt)) []) end;

  (*transform isabelle clause to metis clause *)
  fun add_thm is_conjecture ctxt (ith, {isFO, axioms, tfrees}) : logic_map =
    let val (mth, tfree_lits) = hol_thm_to_fol is_conjecture ctxt isFO ith
    in
       {isFO = isFO,
        axioms = (mth, Meson.make_meta_clause ith) :: axioms,
        tfrees = tfree_lits union tfrees}
    end;

  (*transform isabelle type / arity clause to metis clause *)
  fun add_type_thm [] lmap = lmap
    | add_type_thm ((ith, mth) :: cls) {isFO, axioms, tfrees} =
        add_type_thm cls {isFO = isFO,
                          axioms = (mth, ith) :: axioms,
                          tfrees = tfrees}

  (*Insert non-logical axioms corresponding to all accumulated TFrees*)
  fun add_tfrees {isFO, axioms, tfrees} : logic_map =
       {isFO = isFO,
        axioms = (map (fn tf => (metis_of_tfree tf, TrueI)) (distinct op= tfrees)) @ axioms,
        tfrees = tfrees};

  (* Function to generate metis clauses, including comb and type clauses *)
  fun build_map mode ctxt cls ths =
    let val thy = ProofContext.theory_of ctxt
        val all_thms_FO = forall (Meson.is_fol_term thy o prop_of)
        val isFO = (mode = ResAtp.Fol) orelse
                   (mode <> ResAtp.Hol andalso all_thms_FO (cls @ ths))
        val lmap0 = foldl (add_thm true ctxt)
                          {isFO = isFO, axioms = [], tfrees = init_tfrees ctxt} cls
        val lmap = foldl (add_thm false ctxt) (add_tfrees lmap0) ths
        val clause_lists = map (Metis.Thm.clause o #1) (#axioms lmap)
        fun used c = exists (Metis.LiteralSet.exists (const_in_metis c)) clause_lists
        (*Now check for the existence of certain combinators*)
        val thI  = if used "c_COMBI" then [comb_I] else []
        val thK  = if used "c_COMBK" then [comb_K] else []
        val thB   = if used "c_COMBB" then [comb_B] else []
        val thC   = if used "c_COMBC" then [comb_C] else []
        val thS   = if used "c_COMBS" then [comb_S] else []
        val thEQ  = if used "c_fequal" then [fequal_imp_equal', equal_imp_fequal'] else []
        val lmap' = if isFO then lmap
                    else foldl (add_thm false ctxt) lmap (thEQ @ thS @ thC @ thB @ thK @ thI)
    in
        add_type_thm (type_ext thy (map prop_of (cls @ ths))) lmap'
    end;

  fun refute cls =
      Metis.Resolution.loop (Metis.Resolution.new Metis.Resolution.default cls);

  fun is_false t = t aconv (HOLogic.mk_Trueprop HOLogic.false_const);

  fun common_thm ths1 ths2 = exists (member Thm.eq_thm ths1) (map Meson.make_meta_clause ths2);

  (* Main function to start metis prove and reconstruction *)
  fun FOL_SOLVE mode ctxt cls ths0 =
    let val th_cls_pairs = map (fn th => (PureThy.get_name_hint th, ResAxioms.cnf_axiom th)) ths0
        val ths = List.concat (map #2 th_cls_pairs)
        val _ = if exists(is_false o prop_of) cls then error "problem contains the empty clause"
                else ();
        val _ = Output.debug (fn () => "FOL_SOLVE: CONJECTURE CLAUSES")
        val _ = app (fn th => Output.debug (fn () => string_of_thm th)) cls
        val _ = Output.debug (fn () => "THEOREM CLAUSES")
        val _ = app (fn th => Output.debug (fn () => string_of_thm th)) ths
        val {isFO,axioms,tfrees} = build_map mode ctxt cls ths
        val _ = if null tfrees then ()
                else (Output.debug (fn () => "TFREE CLAUSES");
                      app (fn tf => Output.debug (fn _ => ResClause.tptp_of_typeLit true tf)) tfrees)
        val _ = Output.debug (fn () => "CLAUSES GIVEN TO METIS")
        val thms = map #1 axioms
        val _ = app (fn th => Output.debug (fn () => Metis.Thm.toString th)) thms
        val _ = if isFO
                then Output.debug (fn () => "goal is first-order")
                else Output.debug (fn () => "goal is higher-order")
        val _ = Output.debug (fn () => "START METIS PROVE PROCESS")
    in
        case refute thms of
            Metis.Resolution.Contradiction mth =>
              let val _ = Output.debug (fn () => "METIS RECONSTRUCTION START: " ^
                            Metis.Thm.toString mth)
                  val ctxt' = fold Variable.declare_constraints (map prop_of cls) ctxt
                               (*add constraints arising from converting goal to clause form*)
                  val proof = Metis.Proof.proof mth
                  val result = translate isFO ctxt' axioms proof
                  and used = List.mapPartial (used_axioms axioms) proof
                  val _ = Output.debug (fn () => "METIS COMPLETED...clauses actually used:")
	          val _ = app (fn th => Output.debug (fn () => string_of_thm th)) used
	          val unused = filter (fn (a,cls) => not (common_thm used cls)) th_cls_pairs
              in
                  if null unused then ()
                  else warning ("Unused theorems: " ^ commas (map #1 unused));
                  case result of
                      (_,ith)::_ => (Output.debug (fn () => "success: " ^ string_of_thm ith); ith)
                    | _ => error "METIS: no result"
              end
          | Metis.Resolution.Satisfiable _ => error "Metis finds the theorem to be invalid"
    end;

  fun metis_general_tac mode ctxt ths i st0 =
    let val _ = Output.debug (fn () =>
          "Metis called with theorems " ^ cat_lines (map string_of_thm ths))
    in
       if exists_type ResAxioms.type_has_empty_sort (prop_of st0)  
       then error "Proof state contains the empty sort" else ();
       (Meson.MESON ResAxioms.neg_clausify (fn cls => rtac (FOL_SOLVE mode ctxt cls ths) 1) i
        THEN ResAxioms.expand_defs_tac st0) st0
    end;

  val metis_tac = metis_general_tac ResAtp.Auto;
  val metisF_tac = metis_general_tac ResAtp.Fol;
  val metisH_tac = metis_general_tac ResAtp.Hol;

  fun method mode = Method.thms_ctxt_args (fn ths => fn ctxt =>
    Method.SIMPLE_METHOD' (CHANGED_PROP o metis_general_tac mode ctxt ths));

  val setup =
    type_lits_setup #>
    Method.add_methods
      [("metis", method ResAtp.Auto, "METIS for FOL & HOL problems"),
       ("metisF", method ResAtp.Fol, "METIS for FOL problems"),
       ("metisH", method ResAtp.Hol, "METIS for HOL problems"),
       ("finish_clausify",
         Method.no_args (Method.SIMPLE_METHOD' (K (ResAxioms.expand_defs_tac refl))),
    "cleanup after conversion to clauses")];

end;