--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_hol_clause.ML Wed Mar 17 18:16:31 2010 +0100
@@ -0,0 +1,531 @@
+(* Title: HOL/Tools/res_hol_clause.ML
+ Author: Jia Meng, NICTA
+
+FOL clauses translated from HOL formulae.
+*)
+
+signature RES_HOL_CLAUSE =
+sig
+ val ext: thm
+ val comb_I: thm
+ val comb_K: thm
+ val comb_B: thm
+ val comb_C: thm
+ val comb_S: thm
+ val minimize_applies: bool
+ type axiom_name = string
+ type polarity = bool
+ type clause_id = int
+ datatype combterm =
+ CombConst of string * Res_Clause.fol_type * Res_Clause.fol_type list (*Const and Free*)
+ | CombVar of string * Res_Clause.fol_type
+ | CombApp of combterm * combterm
+ datatype literal = Literal of polarity * combterm
+ datatype clause = Clause of {clause_id: clause_id, axiom_name: axiom_name, th: thm,
+ kind: Res_Clause.kind,literals: literal list, ctypes_sorts: typ list}
+ val type_of_combterm: combterm -> Res_Clause.fol_type
+ val strip_comb: combterm -> combterm * combterm list
+ val literals_of_term: theory -> term -> literal list * typ list
+ exception TOO_TRIVIAL
+ val make_conjecture_clauses: bool -> theory -> thm list -> clause list
+ val make_axiom_clauses: bool ->
+ theory ->
+ (thm * (axiom_name * clause_id)) list -> (axiom_name * clause) list
+ val get_helper_clauses: bool ->
+ theory ->
+ bool ->
+ clause list * (thm * (axiom_name * clause_id)) list * string list ->
+ clause list
+ val tptp_write_file: bool -> Path.T ->
+ clause list * clause list * clause list * clause list *
+ Res_Clause.classrelClause list * Res_Clause.arityClause list ->
+ int * int
+ val dfg_write_file: bool -> Path.T ->
+ clause list * clause list * clause list * clause list *
+ Res_Clause.classrelClause list * Res_Clause.arityClause list ->
+ int * int
+end
+
+structure Res_HOL_Clause: RES_HOL_CLAUSE =
+struct
+
+structure RC = Res_Clause; (* FIXME avoid structure alias *)
+
+(* theorems for combinators and function extensionality *)
+val ext = thm "HOL.ext";
+val comb_I = thm "ATP_Linkup.COMBI_def";
+val comb_K = thm "ATP_Linkup.COMBK_def";
+val comb_B = thm "ATP_Linkup.COMBB_def";
+val comb_C = thm "ATP_Linkup.COMBC_def";
+val comb_S = thm "ATP_Linkup.COMBS_def";
+val fequal_imp_equal = thm "ATP_Linkup.fequal_imp_equal";
+val equal_imp_fequal = thm "ATP_Linkup.equal_imp_fequal";
+
+
+(* Parameter t_full below indicates that full type information is to be
+exported *)
+
+(*If true, each function will be directly applied to as many arguments as possible, avoiding
+ use of the "apply" operator. Use of hBOOL is also minimized.*)
+val minimize_applies = true;
+
+fun min_arity_of const_min_arity c = the_default 0 (Symtab.lookup const_min_arity c);
+
+(*True if the constant ever appears outside of the top-level position in literals.
+ If false, the constant always receives all of its arguments and is used as a predicate.*)
+fun needs_hBOOL const_needs_hBOOL c =
+ not minimize_applies orelse
+ the_default false (Symtab.lookup const_needs_hBOOL c);
+
+
+(******************************************************)
+(* data types for typed combinator expressions *)
+(******************************************************)
+
+type axiom_name = string;
+type polarity = bool;
+type clause_id = int;
+
+datatype combterm = CombConst of string * RC.fol_type * RC.fol_type list (*Const and Free*)
+ | CombVar of string * RC.fol_type
+ | CombApp of combterm * combterm
+
+datatype literal = Literal of polarity * combterm;
+
+datatype clause =
+ Clause of {clause_id: clause_id,
+ axiom_name: axiom_name,
+ th: thm,
+ kind: RC.kind,
+ literals: literal list,
+ ctypes_sorts: typ list};
+
+
+(*********************************************************************)
+(* convert a clause with type Term.term to a clause with type clause *)
+(*********************************************************************)
+
+fun isFalse (Literal(pol, CombConst(c,_,_))) =
+ (pol andalso c = "c_False") orelse
+ (not pol andalso c = "c_True")
+ | isFalse _ = false;
+
+fun isTrue (Literal (pol, CombConst(c,_,_))) =
+ (pol andalso c = "c_True") orelse
+ (not pol andalso c = "c_False")
+ | isTrue _ = false;
+
+fun isTaut (Clause {literals,...}) = exists isTrue literals;
+
+fun type_of dfg (Type (a, Ts)) =
+ let val (folTypes,ts) = types_of dfg Ts
+ in (RC.Comp(RC.make_fixed_type_const dfg a, folTypes), ts) end
+ | type_of _ (tp as TFree (a, _)) =
+ (RC.AtomF (RC.make_fixed_type_var a), [tp])
+ | type_of _ (tp as TVar (v, _)) =
+ (RC.AtomV (RC.make_schematic_type_var v), [tp])
+and types_of dfg Ts =
+ let val (folTyps,ts) = ListPair.unzip (map (type_of dfg) Ts)
+ in (folTyps, RC.union_all ts) end;
+
+(* same as above, but no gathering of sort information *)
+fun simp_type_of dfg (Type (a, Ts)) =
+ RC.Comp(RC.make_fixed_type_const dfg a, map (simp_type_of dfg) Ts)
+ | simp_type_of _ (TFree (a, _)) = RC.AtomF(RC.make_fixed_type_var a)
+ | simp_type_of _ (TVar (v, _)) = RC.AtomV(RC.make_schematic_type_var v);
+
+
+fun const_type_of dfg thy (c,t) =
+ let val (tp,ts) = type_of dfg t
+ in (tp, ts, map (simp_type_of dfg) (Sign.const_typargs thy (c,t))) end;
+
+(* convert a Term.term (with combinators) into a combterm, also accummulate sort info *)
+fun combterm_of dfg thy (Const(c,t)) =
+ let val (tp,ts,tvar_list) = const_type_of dfg thy (c,t)
+ val c' = CombConst(RC.make_fixed_const dfg c, tp, tvar_list)
+ in (c',ts) end
+ | combterm_of dfg _ (Free(v,t)) =
+ let val (tp,ts) = type_of dfg t
+ val v' = CombConst(RC.make_fixed_var v, tp, [])
+ in (v',ts) end
+ | combterm_of dfg _ (Var(v,t)) =
+ let val (tp,ts) = type_of dfg t
+ val v' = CombVar(RC.make_schematic_var v,tp)
+ in (v',ts) end
+ | combterm_of dfg thy (P $ Q) =
+ let val (P',tsP) = combterm_of dfg thy P
+ val (Q',tsQ) = combterm_of dfg thy Q
+ in (CombApp(P',Q'), union (op =) tsP tsQ) end
+ | combterm_of _ _ (t as Abs _) = raise RC.CLAUSE ("HOL CLAUSE", t);
+
+fun predicate_of dfg thy ((Const("Not",_) $ P), polarity) = predicate_of dfg thy (P, not polarity)
+ | predicate_of dfg thy (t,polarity) = (combterm_of dfg thy (Envir.eta_contract t), polarity);
+
+fun literals_of_term1 dfg thy args (Const("Trueprop",_) $ P) = literals_of_term1 dfg thy args P
+ | literals_of_term1 dfg thy args (Const("op |",_) $ P $ Q) =
+ literals_of_term1 dfg thy (literals_of_term1 dfg thy args P) Q
+ | literals_of_term1 dfg thy (lits,ts) P =
+ let val ((pred,ts'),pol) = predicate_of dfg thy (P,true)
+ in
+ (Literal(pol,pred)::lits, union (op =) ts ts')
+ end;
+
+fun literals_of_term_dfg dfg thy P = literals_of_term1 dfg thy ([],[]) P;
+val literals_of_term = literals_of_term_dfg false;
+
+(* Problem too trivial for resolution (empty clause) *)
+exception TOO_TRIVIAL;
+
+(* making axiom and conjecture clauses *)
+fun make_clause dfg thy (clause_id,axiom_name,kind,th) =
+ let val (lits,ctypes_sorts) = literals_of_term_dfg dfg thy (prop_of th)
+ in
+ if forall isFalse lits
+ then raise TOO_TRIVIAL
+ else
+ Clause {clause_id = clause_id, axiom_name = axiom_name, th = th, kind = kind,
+ literals = lits, ctypes_sorts = ctypes_sorts}
+ end;
+
+
+fun add_axiom_clause dfg thy ((th,(name,id)), pairs) =
+ let val cls = make_clause dfg thy (id, name, RC.Axiom, th)
+ in
+ if isTaut cls then pairs else (name,cls)::pairs
+ end;
+
+fun make_axiom_clauses dfg thy = List.foldl (add_axiom_clause dfg thy) [];
+
+fun make_conjecture_clauses_aux _ _ _ [] = []
+ | make_conjecture_clauses_aux dfg thy n (th::ths) =
+ make_clause dfg thy (n,"conjecture", RC.Conjecture, th) ::
+ make_conjecture_clauses_aux dfg thy (n+1) ths;
+
+fun make_conjecture_clauses dfg thy = make_conjecture_clauses_aux dfg thy 0;
+
+
+(**********************************************************************)
+(* convert clause into ATP specific formats: *)
+(* TPTP used by Vampire and E *)
+(* DFG used by SPASS *)
+(**********************************************************************)
+
+(*Result of a function type; no need to check that the argument type matches.*)
+fun result_type (RC.Comp ("tc_fun", [_, tp2])) = tp2
+ | result_type _ = error "result_type"
+
+fun type_of_combterm (CombConst (_, tp, _)) = tp
+ | type_of_combterm (CombVar (_, tp)) = tp
+ | type_of_combterm (CombApp (t1, _)) = result_type (type_of_combterm t1);
+
+(*gets the head of a combinator application, along with the list of arguments*)
+fun strip_comb u =
+ let fun stripc (CombApp(t,u), ts) = stripc (t, u::ts)
+ | stripc x = x
+ in stripc(u,[]) end;
+
+val type_wrapper = "ti";
+
+fun head_needs_hBOOL const_needs_hBOOL (CombConst(c,_,_)) = needs_hBOOL const_needs_hBOOL c
+ | head_needs_hBOOL _ _ = true;
+
+fun wrap_type t_full (s, tp) =
+ if t_full then
+ type_wrapper ^ RC.paren_pack [s, RC.string_of_fol_type tp]
+ else s;
+
+fun apply ss = "hAPP" ^ RC.paren_pack ss;
+
+fun rev_apply (v, []) = v
+ | rev_apply (v, arg::args) = apply [rev_apply (v, args), arg];
+
+fun string_apply (v, args) = rev_apply (v, rev args);
+
+(*Apply an operator to the argument strings, using either the "apply" operator or
+ direct function application.*)
+fun string_of_applic t_full cma (CombConst (c, _, tvars), args) =
+ let val c = if c = "equal" then "c_fequal" else c
+ val nargs = min_arity_of cma c
+ val args1 = List.take(args, nargs)
+ handle Subscript => error ("string_of_applic: " ^ c ^ " has arity " ^
+ Int.toString nargs ^ " but is applied to " ^
+ space_implode ", " args)
+ val args2 = List.drop(args, nargs)
+ val targs = if not t_full then map RC.string_of_fol_type tvars
+ else []
+ in
+ string_apply (c ^ RC.paren_pack (args1@targs), args2)
+ end
+ | string_of_applic _ _ (CombVar (v, _), args) = string_apply (v, args)
+ | string_of_applic _ _ _ = error "string_of_applic";
+
+fun wrap_type_if t_full cnh (head, s, tp) =
+ if head_needs_hBOOL cnh head then wrap_type t_full (s, tp) else s;
+
+fun string_of_combterm (params as (t_full, cma, cnh)) t =
+ let val (head, args) = strip_comb t
+ in wrap_type_if t_full cnh (head,
+ string_of_applic t_full cma (head, map (string_of_combterm (params)) args),
+ type_of_combterm t)
+ end;
+
+(*Boolean-valued terms are here converted to literals.*)
+fun boolify params t =
+ "hBOOL" ^ RC.paren_pack [string_of_combterm params t];
+
+fun string_of_predicate (params as (_,_,cnh)) t =
+ case t of
+ (CombApp(CombApp(CombConst("equal",_,_), t1), t2)) =>
+ (*DFG only: new TPTP prefers infix equality*)
+ ("equal" ^ RC.paren_pack [string_of_combterm params t1, string_of_combterm params t2])
+ | _ =>
+ case #1 (strip_comb t) of
+ CombConst(c,_,_) => if needs_hBOOL cnh c then boolify params t else string_of_combterm params t
+ | _ => boolify params t;
+
+
+(*** tptp format ***)
+
+fun tptp_of_equality params pol (t1,t2) =
+ let val eqop = if pol then " = " else " != "
+ in string_of_combterm params t1 ^ eqop ^ string_of_combterm params t2 end;
+
+fun tptp_literal params (Literal(pol, CombApp(CombApp(CombConst("equal",_,_), t1), t2))) =
+ tptp_of_equality params pol (t1,t2)
+ | tptp_literal params (Literal(pol,pred)) =
+ RC.tptp_sign pol (string_of_predicate params pred);
+
+(*Given a clause, returns its literals paired with a list of literals concerning TFrees;
+ the latter should only occur in conjecture clauses.*)
+fun tptp_type_lits params pos (Clause{literals, ctypes_sorts, ...}) =
+ (map (tptp_literal params) literals,
+ map (RC.tptp_of_typeLit pos) (RC.add_typs ctypes_sorts));
+
+fun clause2tptp params (cls as Clause {axiom_name, clause_id, kind, ...}) =
+ let val (lits,tylits) = tptp_type_lits params (kind = RC.Conjecture) cls
+ in
+ (RC.gen_tptp_cls(clause_id,axiom_name,kind,lits,tylits), tylits)
+ end;
+
+
+(*** dfg format ***)
+
+fun dfg_literal params (Literal(pol,pred)) = RC.dfg_sign pol (string_of_predicate params pred);
+
+fun dfg_type_lits params pos (Clause{literals, ctypes_sorts, ...}) =
+ (map (dfg_literal params) literals,
+ map (RC.dfg_of_typeLit pos) (RC.add_typs ctypes_sorts));
+
+fun get_uvars (CombConst _) vars = vars
+ | get_uvars (CombVar(v,_)) vars = (v::vars)
+ | get_uvars (CombApp(P,Q)) vars = get_uvars P (get_uvars Q vars);
+
+fun get_uvars_l (Literal(_,c)) = get_uvars c [];
+
+fun dfg_vars (Clause {literals,...}) = RC.union_all (map get_uvars_l literals);
+
+fun clause2dfg params (cls as Clause{axiom_name,clause_id,kind,ctypes_sorts,...}) =
+ let val (lits,tylits) = dfg_type_lits params (kind = RC.Conjecture) cls
+ val vars = dfg_vars cls
+ val tvars = RC.get_tvar_strs ctypes_sorts
+ in
+ (RC.gen_dfg_cls(clause_id, axiom_name, kind, lits, tylits, tvars@vars), tylits)
+ end;
+
+
+(** For DFG format: accumulate function and predicate declarations **)
+
+fun addtypes tvars tab = List.foldl RC.add_foltype_funcs tab tvars;
+
+fun add_decls (t_full, cma, cnh) (CombConst (c, _, tvars), (funcs, preds)) =
+ if c = "equal" then (addtypes tvars funcs, preds)
+ else
+ let val arity = min_arity_of cma c
+ val ntys = if not t_full then length tvars else 0
+ val addit = Symtab.update(c, arity+ntys)
+ in
+ if needs_hBOOL cnh c then (addtypes tvars (addit funcs), preds)
+ else (addtypes tvars funcs, addit preds)
+ end
+ | add_decls _ (CombVar(_,ctp), (funcs,preds)) =
+ (RC.add_foltype_funcs (ctp,funcs), preds)
+ | add_decls params (CombApp(P,Q),decls) = add_decls params (P,add_decls params (Q,decls));
+
+fun add_literal_decls params (Literal(_,c), decls) = add_decls params (c,decls);
+
+fun add_clause_decls params (Clause {literals, ...}, decls) =
+ List.foldl (add_literal_decls params) decls literals
+ handle Symtab.DUP a => error ("function " ^ a ^ " has multiple arities")
+
+fun decls_of_clauses params clauses arity_clauses =
+ let val init_functab = Symtab.update (type_wrapper,2) (Symtab.update ("hAPP",2) RC.init_functab)
+ val init_predtab = Symtab.update ("hBOOL",1) Symtab.empty
+ val (functab,predtab) = (List.foldl (add_clause_decls params) (init_functab, init_predtab) clauses)
+ in
+ (Symtab.dest (List.foldl RC.add_arityClause_funcs functab arity_clauses),
+ Symtab.dest predtab)
+ end;
+
+fun add_clause_preds (Clause {ctypes_sorts, ...}, preds) =
+ List.foldl RC.add_type_sort_preds preds ctypes_sorts
+ handle Symtab.DUP a => error ("predicate " ^ a ^ " has multiple arities")
+
+(*Higher-order clauses have only the predicates hBOOL and type classes.*)
+fun preds_of_clauses clauses clsrel_clauses arity_clauses =
+ Symtab.dest
+ (List.foldl RC.add_classrelClause_preds
+ (List.foldl RC.add_arityClause_preds
+ (List.foldl add_clause_preds Symtab.empty clauses)
+ arity_clauses)
+ clsrel_clauses)
+
+
+(**********************************************************************)
+(* write clauses to files *)
+(**********************************************************************)
+
+val init_counters =
+ Symtab.make [("c_COMBI", 0), ("c_COMBK", 0),
+ ("c_COMBB", 0), ("c_COMBC", 0),
+ ("c_COMBS", 0)];
+
+fun count_combterm (CombConst (c, _, _), ct) =
+ (case Symtab.lookup ct c of NONE => ct (*no counter*)
+ | SOME n => Symtab.update (c,n+1) ct)
+ | count_combterm (CombVar _, ct) = ct
+ | count_combterm (CombApp(t1,t2), ct) = count_combterm(t1, count_combterm(t2, ct));
+
+fun count_literal (Literal(_,t), ct) = count_combterm(t,ct);
+
+fun count_clause (Clause{literals,...}, ct) = List.foldl count_literal ct literals;
+
+fun count_user_clause user_lemmas (Clause{axiom_name,literals,...}, ct) =
+ if axiom_name mem_string user_lemmas then List.foldl count_literal ct literals
+ else ct;
+
+fun cnf_helper_thms thy =
+ Res_Axioms.cnf_rules_pairs thy o map Res_Axioms.pairname
+
+fun get_helper_clauses dfg thy isFO (conjectures, axcls, user_lemmas) =
+ if isFO then [] (*first-order*)
+ else
+ let
+ val axclauses = map #2 (make_axiom_clauses dfg thy axcls)
+ val ct0 = List.foldl count_clause init_counters conjectures
+ val ct = List.foldl (count_user_clause user_lemmas) ct0 axclauses
+ fun needed c = the (Symtab.lookup ct c) > 0
+ val IK = if needed "c_COMBI" orelse needed "c_COMBK"
+ then cnf_helper_thms thy [comb_I,comb_K]
+ else []
+ val BC = if needed "c_COMBB" orelse needed "c_COMBC"
+ then cnf_helper_thms thy [comb_B,comb_C]
+ else []
+ val S = if needed "c_COMBS"
+ then cnf_helper_thms thy [comb_S]
+ else []
+ val other = cnf_helper_thms thy [fequal_imp_equal,equal_imp_fequal]
+ in
+ map #2 (make_axiom_clauses dfg thy (other @ IK @ BC @ S))
+ end;
+
+(*Find the minimal arity of each function mentioned in the term. Also, note which uses
+ are not at top level, to see if hBOOL is needed.*)
+fun count_constants_term toplev t (const_min_arity, const_needs_hBOOL) =
+ let val (head, args) = strip_comb t
+ val n = length args
+ val (const_min_arity, const_needs_hBOOL) = fold (count_constants_term false) args (const_min_arity, const_needs_hBOOL)
+ in
+ case head of
+ CombConst (a,_,_) => (*predicate or function version of "equal"?*)
+ let val a = if a="equal" andalso not toplev then "c_fequal" else a
+ val const_min_arity = Symtab.map_default (a, n) (Integer.min n) const_min_arity
+ in
+ if toplev then (const_min_arity, const_needs_hBOOL)
+ else (const_min_arity, Symtab.update (a,true) (const_needs_hBOOL))
+ end
+ | _ => (const_min_arity, const_needs_hBOOL)
+ end;
+
+(*A literal is a top-level term*)
+fun count_constants_lit (Literal (_,t)) (const_min_arity, const_needs_hBOOL) =
+ count_constants_term true t (const_min_arity, const_needs_hBOOL);
+
+fun count_constants_clause (Clause{literals,...}) (const_min_arity, const_needs_hBOOL) =
+ fold count_constants_lit literals (const_min_arity, const_needs_hBOOL);
+
+fun display_arity const_needs_hBOOL (c,n) =
+ Res_Axioms.trace_msg (fn () => "Constant: " ^ c ^ " arity:\t" ^ Int.toString n ^
+ (if needs_hBOOL const_needs_hBOOL c then " needs hBOOL" else ""));
+
+fun count_constants (conjectures, _, extra_clauses, helper_clauses, _, _) =
+ if minimize_applies then
+ let val (const_min_arity, const_needs_hBOOL) =
+ fold count_constants_clause conjectures (Symtab.empty, Symtab.empty)
+ |> fold count_constants_clause extra_clauses
+ |> fold count_constants_clause helper_clauses
+ val _ = List.app (display_arity const_needs_hBOOL) (Symtab.dest (const_min_arity))
+ in (const_min_arity, const_needs_hBOOL) end
+ else (Symtab.empty, Symtab.empty);
+
+(* tptp format *)
+
+fun tptp_write_file t_full file clauses =
+ let
+ val (conjectures, axclauses, _, helper_clauses,
+ classrel_clauses, arity_clauses) = clauses
+ val (cma, cnh) = count_constants clauses
+ val params = (t_full, cma, cnh)
+ val (tptp_clss,tfree_litss) = ListPair.unzip (map (clause2tptp params) conjectures)
+ val tfree_clss = map RC.tptp_tfree_clause (List.foldl (uncurry (union (op =))) [] tfree_litss)
+ val _ =
+ File.write_list file (
+ map (#1 o (clause2tptp params)) axclauses @
+ tfree_clss @
+ tptp_clss @
+ map RC.tptp_classrelClause classrel_clauses @
+ map RC.tptp_arity_clause arity_clauses @
+ map (#1 o (clause2tptp params)) helper_clauses)
+ in (length axclauses + 1, length tfree_clss + length tptp_clss)
+ end;
+
+
+(* dfg format *)
+
+fun dfg_write_file t_full file clauses =
+ let
+ val (conjectures, axclauses, _, helper_clauses,
+ classrel_clauses, arity_clauses) = clauses
+ val (cma, cnh) = count_constants clauses
+ val params = (t_full, cma, cnh)
+ val (dfg_clss, tfree_litss) = ListPair.unzip (map (clause2dfg params) conjectures)
+ and probname = Path.implode (Path.base file)
+ val axstrs = map (#1 o (clause2dfg params)) axclauses
+ val tfree_clss = map RC.dfg_tfree_clause (RC.union_all tfree_litss)
+ val helper_clauses_strs = map (#1 o (clause2dfg params)) helper_clauses
+ val (funcs,cl_preds) = decls_of_clauses params (helper_clauses @ conjectures @ axclauses) arity_clauses
+ and ty_preds = preds_of_clauses axclauses classrel_clauses arity_clauses
+ val _ =
+ File.write_list file (
+ RC.string_of_start probname ::
+ RC.string_of_descrip probname ::
+ RC.string_of_symbols (RC.string_of_funcs funcs)
+ (RC.string_of_preds (cl_preds @ ty_preds)) ::
+ "list_of_clauses(axioms,cnf).\n" ::
+ axstrs @
+ map RC.dfg_classrelClause classrel_clauses @
+ map RC.dfg_arity_clause arity_clauses @
+ helper_clauses_strs @
+ ["end_of_list.\n\nlist_of_clauses(conjectures,cnf).\n"] @
+ tfree_clss @
+ dfg_clss @
+ ["end_of_list.\n\n",
+ (*VarWeight=3 helps the HO problems, probably by counteracting the presence of hAPP*)
+ "list_of_settings(SPASS).\n{*\nset_flag(VarWeight,3).\n*}\nend_of_list.\n\n",
+ "end_problem.\n"])
+
+ in (length axclauses + length classrel_clauses + length arity_clauses +
+ length helper_clauses + 1, length tfree_clss + length dfg_clss)
+ end;
+
+end;
+