(*  Title:      Pure/axclass.ML

     Author:     Markus Wenzel, TU Muenchen

 Type classes defined as predicates, associated with a record of

 parameters.  Proven class relations and type arities.

 *)

 signature AX_CLASS =

 sig

   type info = {def: thm, intro: thm, axioms: thm list, params: (string * typ) list}

   val get_info: theory -> class -> info

   val class_of_param: theory -> string -> class option

   val instance_name: string * class -> string

   val thynames_of_arity: theory -> class * string -> string list

   val param_of_inst: theory -> string * string -> string

   val inst_of_param: theory -> string -> (string * string) option

   val unoverload: theory -> thm -> thm

   val overload: theory -> thm -> thm

   val unoverload_conv: theory -> conv

   val overload_conv: theory -> conv

   val lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option

   val unoverload_const: theory -> string * typ -> string

   val cert_classrel: theory -> class * class -> class * class

   val read_classrel: theory -> xstring * xstring -> class * class

   val declare_overloaded: string * typ -> theory -> term * theory

   val define_overloaded: binding -> string * term -> theory -> thm * theory

   val add_classrel: bool -> thm -> theory -> theory

   val add_arity: bool -> thm -> theory -> theory

   val prove_classrel: class * class -> tactic -> theory -> theory

   val prove_arity: string * sort list * sort -> tactic -> theory -> theory

   val define_class: binding * class list -> string list ->

     (Thm.binding * term list) list -> theory -> class * theory

   val axiomatize_class: binding * class list -> theory -> theory

   val axiomatize_class_cmd: binding * xstring list -> theory -> theory

   val axiomatize_classrel: (class * class) list -> theory -> theory

   val axiomatize_classrel_cmd: (xstring * xstring) list -> theory -> theory

   val axiomatize_arity: arity -> theory -> theory

   val axiomatize_arity_cmd: xstring * string list * string -> theory -> theory

 end;

 structure AxClass: AX_CLASS =

 struct

 (** theory data **)

 (* axclass info *)

 type info =

  {def: thm,

   intro: thm,

   axioms: thm list,

   params: (string * typ) list};

 fun make_axclass (def, intro, axioms, params): info =

   {def = def, intro = intro, axioms = axioms, params = params};

 (* class parameters (canonical order) *)

 type param = string * class;

 fun add_param pp ((x, c): param) params =

   (case AList.lookup (op =) params x of

     NONE => (x, c) :: params

   | SOME c' => error ("Duplicate class parameter " ^ quote x ^

       " for " ^ Pretty.string_of_sort pp [c] ^

       (if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));

 (* setup data *)

 datatype data = Data of

  {axclasses: info Symtab.table,

   params: param list,

   proven_classrels: thm Symreltab.table,

   proven_arities: ((class * sort list) * (thm * string)) list Symtab.table,

     (*arity theorems with theory name*)

   inst_params:

     (string * thm) Symtab.table Symtab.table *

       (*constant name ~> type constructor ~> (constant name, equation)*)

     (string * string) Symtab.table (*constant name ~> (constant name, type constructor)*),

   diff_classrels: (class * class) list};

 fun make_data

     (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =

   Data {axclasses = axclasses, params = params, proven_classrels = proven_classrels,

     proven_arities = proven_arities, inst_params = inst_params,

     diff_classrels = diff_classrels};

 fun diff_table tab1 tab2 =

   Symreltab.fold (fn (x, _) => if Symreltab.defined tab2 x then I else cons x) tab1 [];

 structure Data = Theory_Data_PP

 (

   type T = data;

   val empty =

     make_data (Symtab.empty, [], Symreltab.empty, Symtab.empty, (Symtab.empty, Symtab.empty), []);

   val extend = I;

   fun merge pp

       (Data {axclasses = axclasses1, params = params1, proven_classrels = proven_classrels1,

         proven_arities = proven_arities1, inst_params = inst_params1,

         diff_classrels = diff_classrels1},

        Data {axclasses = axclasses2, params = params2, proven_classrels = proven_classrels2,

         proven_arities = proven_arities2, inst_params = inst_params2,

         diff_classrels = diff_classrels2}) =

     let

       val axclasses' = Symtab.merge (K true) (axclasses1, axclasses2);

       val params' =

         if null params1 then params2

         else fold_rev (fn p => if member (op =) params1 p then I else add_param pp p) params2 params1;

       (*transitive closure of classrels and arity completion is done in Theory.at_begin hook*)

       val proven_classrels' = Symreltab.join (K #1) (proven_classrels1, proven_classrels2);

       val proven_arities' =

         Symtab.join (K (Library.merge (eq_fst op =))) (proven_arities1, proven_arities2);

       val diff_classrels' =

         diff_table proven_classrels1 proven_classrels2 @

         diff_table proven_classrels2 proven_classrels1 @

         diff_classrels1 @ diff_classrels2;

       val inst_params' =

         (Symtab.join (K (Symtab.merge (K true))) (#1 inst_params1, #1 inst_params2),

           Symtab.merge (K true) (#2 inst_params1, #2 inst_params2));

     in

       make_data

         (axclasses', params', proven_classrels', proven_arities', inst_params', diff_classrels')

     end;

 );

 fun map_data f =

   Data.map (fn Data {axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels} =>

     make_data (f (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels)));

 fun map_axclasses f =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>

     (f axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels));

 fun map_params f =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>

     (axclasses, f params, proven_classrels, proven_arities, inst_params, diff_classrels));

 fun map_proven_classrels f =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>

     (axclasses, params, f proven_classrels, proven_arities, inst_params, diff_classrels));

 fun map_proven_arities f =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>

     (axclasses, params, proven_classrels, f proven_arities, inst_params, diff_classrels));

 fun map_inst_params f =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>

     (axclasses, params, proven_classrels, proven_arities, f inst_params, diff_classrels));

 val clear_diff_classrels =

   map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, _) =>

     (axclasses, params, proven_classrels, proven_arities, inst_params, []));

 val rep_data = Data.get #> (fn Data args => args);

 val axclasses_of = #axclasses o rep_data;

 val params_of = #params o rep_data;

 val proven_classrels_of = #proven_classrels o rep_data;

 val proven_arities_of = #proven_arities o rep_data;

 val inst_params_of = #inst_params o rep_data;

 val diff_classrels_of = #diff_classrels o rep_data;

 (* axclasses with parameters *)

 fun get_info thy c =

   (case Symtab.lookup (axclasses_of thy) c of

     SOME info => info

   | NONE => error ("No such axclass: " ^ quote c));

 fun all_params_of thy S =

   let val params = params_of thy;

   in fold (fn (x, c) => if Sign.subsort thy (S, [c]) then cons x else I) params [] end;

 fun class_of_param thy = AList.lookup (op =) (params_of thy);

 (* maintain instances *)

 val classrel_prefix = "classrel_";

 val arity_prefix = "arity_";

 fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;

 infix 0 RSO;

 fun (SOME a) RSO (SOME b) = SOME (a RS b)

   | x RSO NONE = x

   | NONE RSO y = y;

 fun the_classrel thy (c1, c2) =

   (case Symreltab.lookup (proven_classrels_of thy) (c1, c2) of

     SOME thm => thm

   | NONE => error ("Unproven class relation " ^

       Syntax.string_of_classrel (ProofContext.init_global thy) [c1, c2]));  (* FIXME stale thy (!?) *)

 fun put_trancl_classrel ((c1, c2), th) thy =

   let

     val classes = Sorts.classes_of (Sign.classes_of thy);

     val classrels = proven_classrels_of thy;

     fun reflcl_classrel (c1', c2') =

       if c1' = c2' then NONE else SOME (Thm.transfer thy (the_classrel thy (c1', c2')));

     fun gen_classrel (c1_pred, c2_succ) =

       let

         val th' =

           the ((reflcl_classrel (c1_pred, c1) RSO SOME th) RSO reflcl_classrel (c2, c2_succ))

           |> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [])))] []

           |> Thm.close_derivation;

       in ((c1_pred, c2_succ), th') end;

     val new_classrels =

       Library.map_product pair (c1 :: Graph.imm_preds classes c1) (c2 :: Graph.imm_succs classes c2)

       |> filter_out ((op =) orf Symreltab.defined classrels)

       |> map gen_classrel;

     val needed = not (null new_classrels);

   in

     (needed,

       if needed then map_proven_classrels (fold Symreltab.update new_classrels) thy

       else thy)

   end;

 fun complete_classrels thy =

   let

     val classrels = proven_classrels_of thy;

     val diff_classrels = diff_classrels_of thy;

     val (needed, thy') = (false, thy) |>

       fold (fn rel => fn (needed, thy) =>

           put_trancl_classrel (rel, Symreltab.lookup classrels rel |> the) thy

           |>> (fn b => needed orelse b))

         diff_classrels;

   in

     if null diff_classrels then NONE

     else SOME (clear_diff_classrels thy')

   end;

 fun the_arity thy (a, Ss, c) =

   (case AList.lookup (op =) (Symtab.lookup_list (proven_arities_of thy) a) (c, Ss) of

     SOME (thm, _) => thm

   | NONE => error ("Unproven type arity " ^

       Syntax.string_of_arity (ProofContext.init_global thy) (a, Ss, [c])));  (* FIXME stale thy (!?) *)

 fun thynames_of_arity thy (c, a) =

   Symtab.lookup_list (proven_arities_of thy) a

   |> map_filter (fn ((c', _), (_, name)) => if c = c' then SOME name else NONE)

   |> rev;

 fun insert_arity_completions thy t ((c, Ss), ((th, thy_name))) (finished, arities) =

   let

     val algebra = Sign.classes_of thy;

     val ars = Symtab.lookup_list arities t;

     val super_class_completions =

       Sign.super_classes thy c

       |> filter_out (fn c1 => exists (fn ((c2, Ss2), _) =>

             c1 = c2 andalso Sorts.sorts_le algebra (Ss2, Ss)) ars);

     val names = Name.invents Name.context Name.aT (length Ss);

     val std_vars = map (fn a => SOME (ctyp_of thy (TVar ((a, 0), [])))) names;

     val completions = super_class_completions |> map (fn c1 =>

       let

         val th1 =

           (th RS Thm.transfer thy (the_classrel thy (c, c1)))

           |> Drule.instantiate' std_vars []

           |> Thm.close_derivation;

       in ((th1, thy_name), c1) end);

     val finished' = finished andalso null completions;

     val arities' = fold (fn (th, c1) => Symtab.cons_list (t, ((c1, Ss), th))) completions arities;

   in (finished', arities') end;

 fun put_arity ((t, Ss, c), th) thy =

   let val ar = ((c, Ss), (th, Context.theory_name thy)) in

     thy

     |> map_proven_arities

       (Symtab.insert_list (eq_fst op =) (t, ar) #>

        curry (insert_arity_completions thy t ar) true #> #2)

   end;

 fun complete_arities thy =

   let

     val arities = proven_arities_of thy;

     val (finished, arities') =

       Symtab.fold (fn (t, ars) => fold (insert_arity_completions thy t) ars) arities (true, arities);

   in

     if finished then NONE

     else SOME (map_proven_arities (K arities') thy)

   end;

 val _ = Context.>> (Context.map_theory

   (Theory.at_begin complete_classrels #>

    Theory.at_begin complete_arities));

 val _ = Proofterm.install_axclass_proofs

   {classrel_proof = Thm.proof_of oo the_classrel,

    arity_proof = Thm.proof_of oo the_arity};

 (* maintain instance parameters *)

 fun get_inst_param thy (c, tyco) =

   (case Symtab.lookup (the_default Symtab.empty (Symtab.lookup (#1 (inst_params_of thy)) c)) tyco of

     SOME c' => c'

   | NONE => error ("No instance parameter for constant " ^ quote c ^ " on type " ^ quote tyco));

 fun add_inst_param (c, tyco) inst =

   (map_inst_params o apfst o Symtab.map_default (c, Symtab.empty)) (Symtab.update_new (tyco, inst))

   #> (map_inst_params o apsnd) (Symtab.update_new (#1 inst, (c, tyco)));

 val inst_of_param = Symtab.lookup o #2 o inst_params_of;

 val param_of_inst = #1 oo get_inst_param;

 fun inst_thms thy =

   Symtab.fold (Symtab.fold (cons o #2 o #2) o #2) (#1 (inst_params_of thy)) [];

 fun get_inst_tyco consts = try (#1 o dest_Type o the_single o Consts.typargs consts);

 fun unoverload thy = MetaSimplifier.simplify true (inst_thms thy);

 fun overload thy = MetaSimplifier.simplify true (map Thm.symmetric (inst_thms thy));

 fun unoverload_conv thy = MetaSimplifier.rewrite true (inst_thms thy);

 fun overload_conv thy = MetaSimplifier.rewrite true (map Thm.symmetric (inst_thms thy));

 fun lookup_inst_param consts params (c, T) =

   (case get_inst_tyco consts (c, T) of

     SOME tyco => AList.lookup (op =) params (c, tyco)

   | NONE => NONE);

 fun unoverload_const thy (c_ty as (c, _)) =

   if is_some (class_of_param thy c) then

     (case get_inst_tyco (Sign.consts_of thy) c_ty of

       SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c

     | NONE => c)

   else c;

 (** instances **)

 (* class relations *)

 fun cert_classrel thy raw_rel =

   let

     val string_of_sort = Syntax.string_of_sort_global thy;

     val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;

     val _ = Sign.primitive_classrel (c1, c2) (Theory.copy thy);

     val _ =

       (case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of

         [] => ()

       | xs => raise TYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^

           commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));

   in (c1, c2) end;

 fun read_classrel thy raw_rel =

   cert_classrel thy (pairself (ProofContext.read_class (ProofContext.init_global thy)) raw_rel)

     handle TYPE (msg, _, _) => error msg;

 (* declaration and definition of instances of overloaded constants *)

 fun inst_tyco_of thy (c, T) =

   (case get_inst_tyco (Sign.consts_of thy) (c, T) of

     SOME tyco => tyco

   | NONE => error ("Illegal type for instantiation of class parameter: " ^

       quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T)));

 fun declare_overloaded (c, T) thy =

   let

     val class =

       (case class_of_param thy c of

         SOME class => class

       | NONE => error ("Not a class parameter: " ^ quote c));

     val tyco = inst_tyco_of thy (c, T);

     val name_inst = instance_name (tyco, class) ^ "_inst";

     val c' = instance_name (tyco, c);

     val T' = Type.strip_sorts T;

   in

     thy

     |> Sign.qualified_path true (Binding.name name_inst)

     |> Sign.declare_const ((Binding.name c', T'), NoSyn)

     |-> (fn const' as Const (c'', _) =>

       Thm.add_def false true

         (Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))

       #>> apsnd Thm.varifyT_global

       #-> (fn (_, thm) => add_inst_param (c, tyco) (c'', thm)

         #> PureThy.add_thm ((Binding.conceal (Binding.name c'), thm), [])

         #> #2

         #> pair (Const (c, T))))

     ||> Sign.restore_naming thy

   end;

 fun define_overloaded b (c, t) thy =

   let

     val T = Term.fastype_of t;

     val tyco = inst_tyco_of thy (c, T);

     val (c', eq) = get_inst_param thy (c, tyco);

     val prop = Logic.mk_equals (Const (c', T), t);

     val b' = Thm.def_binding_optional (Binding.name (instance_name (tyco, c))) b;

   in

     thy

     |> Thm.add_def false false (b', prop)

     |>> (fn (_, thm) =>  Drule.transitive_thm OF [eq, thm])

   end;

 (* primitive rules *)

 fun add_classrel store raw_th thy =

   let

     val th = Thm.strip_shyps (Thm.transfer thy raw_th);

     val prop = Thm.plain_prop_of th;

     fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);

     val rel = Logic.dest_classrel prop handle TERM _ => err ();

     val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();

     val (th', thy') =

       if store then PureThy.store_thm

         (Binding.name (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th) thy

       else (th, thy);

     val th'' = th'

       |> Thm.unconstrainT

       |> Drule.instantiate' [SOME (ctyp_of thy' (TVar ((Name.aT, 0), [])))] [];

   in

     thy'

     |> Sign.primitive_classrel (c1, c2)

     |> (#2 oo put_trancl_classrel) ((c1, c2), th'')

     |> perhaps complete_arities

   end;

 fun add_arity store raw_th thy =

   let

     val th = Thm.strip_shyps (Thm.transfer thy raw_th);

     val prop = Thm.plain_prop_of th;

     fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);

     val arity as (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();

     val (th', thy') =

       if store then PureThy.store_thm

         (Binding.name (prefix arity_prefix (Logic.name_arity arity)), th) thy

       else (th, thy);

     val args = Name.names Name.context Name.aT Ss;

     val T = Type (t, map TFree args);

     val std_vars = map (fn (a, S) => SOME (ctyp_of thy' (TVar ((a, 0), [])))) args;

     val missing_params = Sign.complete_sort thy' [c]

       |> maps (these o Option.map #params o try (get_info thy'))

       |> filter_out (fn (const, _) => can (get_inst_param thy') (const, t))

       |> (map o apsnd o map_atyps) (K T);

     val th'' = th'

       |> Thm.unconstrainT

       |> Drule.instantiate' std_vars [];

   in

     thy'

     |> fold (#2 oo declare_overloaded) missing_params

     |> Sign.primitive_arity (t, Ss, [c])

     |> put_arity ((t, Ss, c), th'')

   end;

 (* tactical proofs *)

 fun prove_classrel raw_rel tac thy =

   let

     val ctxt = ProofContext.init_global thy;

     val (c1, c2) = cert_classrel thy raw_rel;

     val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>

       cat_error msg ("The error(s) above occurred while trying to prove class relation " ^

         quote (Syntax.string_of_classrel ctxt [c1, c2]));

   in

     thy

     |> PureThy.add_thms [((Binding.name

         (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th), [])]

     |-> (fn [th'] => add_classrel false th')

   end;

 fun prove_arity raw_arity tac thy =

   let

     val ctxt = ProofContext.init_global thy;

     val arity = ProofContext.cert_arity ctxt raw_arity;

     val names = map (prefix arity_prefix) (Logic.name_arities arity);

     val props = Logic.mk_arities arity;

     val ths = Goal.prove_multi ctxt [] [] props

       (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>

         cat_error msg ("The error(s) above occurred while trying to prove type arity " ^

           quote (Syntax.string_of_arity ctxt arity));

   in

     thy

     |> PureThy.add_thms (map (rpair []) (map Binding.name names ~~ ths))

     |-> fold (add_arity false)

   end;

 (** class definitions **)

 fun split_defined n eq =

   let

     val intro =

       (eq RS Drule.equal_elim_rule2)

       |> Conjunction.curry_balanced n

       |> n = 0 ? Thm.eq_assumption 1;

     val dests =

       if n = 0 then []

       else

         (eq RS Drule.equal_elim_rule1)

         |> Balanced_Tree.dest (fn th =>

           (th RS Conjunction.conjunctionD1, th RS Conjunction.conjunctionD2)) n;

   in (intro, dests) end;

 fun define_class (bclass, raw_super) raw_params raw_specs thy =

   let

     val ctxt = ProofContext.init_global thy;

     val pp = Syntax.pp ctxt;

     (* class *)

     val bconst = Binding.map_name Logic.const_of_class bclass;

     val class = Sign.full_name thy bclass;

     val super = Sign.minimize_sort thy (Sign.certify_sort thy raw_super);

     fun check_constraint (a, S) =

       if Sign.subsort thy (super, S) then ()

       else error ("Sort constraint of type variable " ^

         setmp_CRITICAL show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^

         " needs to be weaker than " ^ Pretty.string_of_sort pp super);

     (* params *)

     val params = raw_params |> map (fn p =>

       let

         val T = Sign.the_const_type thy p;

         val _ =

           (case Term.add_tvarsT T [] of

             [((a, _), S)] => check_constraint (a, S)

           | _ => error ("Exactly one type variable expected in class parameter " ^ quote p));

         val T' = Term.map_type_tvar (fn _ => TFree (Name.aT, [class])) T;

       in (p, T') end);

     (* axioms *)

     fun prep_axiom t =

       (case Term.add_tfrees t [] of

         [(a, S)] => check_constraint (a, S)

       | [] => ()

       | _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t);

       t

       |> Term.map_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))

       |> Logic.close_form);

     val axiomss = map (map (prep_axiom o Sign.cert_prop thy) o snd) raw_specs;

     val name_atts = map fst raw_specs;

     (* definition *)

     val conjs = Logic.mk_of_sort (Term.aT [], super) @ flat axiomss;

     val class_eq =

       Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);

     val ([def], def_thy) =

       thy

       |> Sign.primitive_class (bclass, super)

       |> PureThy.add_defs false [((Thm.def_binding bconst, class_eq), [])];

     val (raw_intro, (raw_classrel, raw_axioms)) =

       split_defined (length conjs) def ||> chop (length super);

     (* facts *)

     val class_triv = Thm.class_triv def_thy class;

     val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =

       def_thy

       |> Sign.qualified_path true bconst

       |> PureThy.note_thmss ""

         [((Binding.name "intro", []), [([Drule.export_without_context raw_intro], [])]),

          ((Binding.name "super", []), [(map Drule.export_without_context raw_classrel, [])]),

          ((Binding.name "axioms", []),

            [(map (fn th => Drule.export_without_context (class_triv RS th)) raw_axioms, [])])]

       ||> Sign.restore_naming def_thy;

     (* result *)

     val axclass = make_axclass (def, intro, axioms, params);

     val result_thy =

       facts_thy

       |> fold (#2 oo put_trancl_classrel) (map (pair class) super ~~ classrel)

       |> Sign.qualified_path false bconst

       |> PureThy.note_thmss "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> #2

       |> Sign.restore_naming facts_thy

       |> map_axclasses (Symtab.update (class, axclass))

       |> map_params (fold (fn (x, _) => add_param pp (x, class)) params);

   in (class, result_thy) end;

 (** axiomatizations **)

 local

 (*old-style axioms*)

 fun add_axiom (b, prop) =

   Thm.add_axiom (b, prop) #->

   (fn (_, thm) => PureThy.add_thm ((b, Drule.export_without_context thm), []));

 fun axiomatize prep mk name add raw_args thy =

   let

     val args = prep thy raw_args;

     val specs = mk args;

     val names = name args;

   in

     thy

     |> fold_map add_axiom (map Binding.name names ~~ specs)

     |-> fold add

   end;

 fun ax_classrel prep =

   axiomatize (map o prep) (map Logic.mk_classrel)

     (map (prefix classrel_prefix o Logic.name_classrel)) (add_classrel false);

 fun ax_arity prep =

   axiomatize (prep o ProofContext.init_global) Logic.mk_arities

     (map (prefix arity_prefix) o Logic.name_arities) (add_arity false);

 fun class_const c =

   (Logic.const_of_class c, Term.itselfT (Term.aT []) --> propT);

 fun ax_class prep_class prep_classrel (bclass, raw_super) thy =

   let

     val class = Sign.full_name thy bclass;

     val super = map (prep_class thy) raw_super |> Sign.minimize_sort thy;

   in

     thy

     |> Sign.primitive_class (bclass, super)

     |> ax_classrel prep_classrel (map (fn c => (class, c)) super)

     |> Theory.add_deps "" (class_const class) (map class_const super)

   end;

 in

 val axiomatize_class = ax_class Sign.certify_class cert_classrel;

 val axiomatize_class_cmd = ax_class (ProofContext.read_class o ProofContext.init_global) read_classrel;

 val axiomatize_classrel = ax_classrel cert_classrel;

 val axiomatize_classrel_cmd = ax_classrel read_classrel;

 val axiomatize_arity = ax_arity ProofContext.cert_arity;

 val axiomatize_arity_cmd = ax_arity ProofContext.read_arity;

 end;

 end;