src/Pure/Isar/class.ML

more robust future_proof result with specific error message (e.g. relevant for incomplete proof of non-registered theorem);

(*  Title:      Pure/Isar/class.ML
    Author:     Florian Haftmann, TU Muenchen

Type classes derived from primitive axclasses and locales.
*)

signature CLASS =
sig
  (*classes*)
  val is_class: theory -> class -> bool
  val these_params: theory -> sort -> (string * (class * (string * typ))) list
  val base_sort: theory -> class -> sort
  val rules: theory -> class -> thm option * thm
  val these_defs: theory -> sort -> thm list
  val these_operations: theory -> sort
    -> (string * (class * (typ * term))) list
  val print_classes: Proof.context -> unit
  val init: class -> theory -> Proof.context
  val begin: class list -> sort -> Proof.context -> Proof.context
  val const: class -> (binding * mixfix) * (term list * term list * term) -> local_theory -> local_theory
  val abbrev: class -> Syntax.mode -> (binding * mixfix) * term -> local_theory -> local_theory
  val redeclare_operations: theory -> sort -> Proof.context -> Proof.context
  val class_prefix: string -> string
  val register: class -> class list -> ((string * typ) * (string * typ)) list
    -> sort -> morphism -> morphism -> thm option -> thm option -> thm
    -> theory -> theory

  (*instances*)
  val instantiation: string list * (string * sort) list * sort -> theory -> local_theory
  val instantiation_instance: (local_theory -> local_theory)
    -> local_theory -> Proof.state
  val prove_instantiation_instance: (Proof.context -> tactic)
    -> local_theory -> local_theory
  val prove_instantiation_exit: (Proof.context -> tactic)
    -> local_theory -> theory
  val prove_instantiation_exit_result: (morphism -> 'a -> 'b)
    -> (Proof.context -> 'b -> tactic) -> 'a -> local_theory -> 'b * theory
  val read_multi_arity: theory -> xstring list * xstring list * xstring
    -> string list * (string * sort) list * sort
  val instantiation_cmd: xstring list * xstring list * xstring -> theory -> local_theory
  val instance_arity_cmd: xstring list * xstring list * xstring -> theory -> Proof.state

  (*subclasses*)
  val classrel: class * class -> theory -> Proof.state
  val classrel_cmd: xstring * xstring -> theory -> Proof.state
  val register_subclass: class * class -> morphism option -> Element.witness option
    -> morphism -> theory -> theory

  (*tactics*)
  val intro_classes_tac: thm list -> tactic
  val default_intro_tac: Proof.context -> thm list -> tactic
end;

structure Class: CLASS =
struct

(** class data **)

datatype class_data = Class_Data of {

  (* static part *)
  consts: (string * string) list
    (*locale parameter ~> constant name*),
  base_sort: sort,
  base_morph: morphism
    (*static part of canonical morphism*),
  export_morph: morphism,
  assm_intro: thm option,
  of_class: thm,
  axiom: thm option,

  (* dynamic part *)
  defs: thm list,
  operations: (string * (class * (typ * term))) list

  (* n.b.
    params = logical parameters of class
    operations = operations participating in user-space type system
  *)
};

fun make_class_data ((consts, base_sort, base_morph, export_morph, assm_intro, of_class, axiom),
    (defs, operations)) =
  Class_Data {consts = consts, base_sort = base_sort,
    base_morph = base_morph, export_morph = export_morph, assm_intro = assm_intro,
    of_class = of_class, axiom = axiom, defs = defs, operations = operations};
fun map_class_data f (Class_Data {consts, base_sort, base_morph, export_morph, assm_intro,
    of_class, axiom, defs, operations}) =
  make_class_data (f ((consts, base_sort, base_morph, export_morph, assm_intro, of_class, axiom),
    (defs, operations)));
fun merge_class_data _ (Class_Data {consts = consts,
    base_sort = base_sort, base_morph = base_morph, export_morph = export_morph, assm_intro = assm_intro,
    of_class = of_class, axiom = axiom, defs = defs1, operations = operations1},
  Class_Data {consts = _, base_sort = _, base_morph = _, export_morph = _, assm_intro = _,
    of_class = _, axiom = _, defs = defs2, operations = operations2}) =
  make_class_data ((consts, base_sort, base_morph, export_morph, assm_intro, of_class, axiom),
    (Thm.merge_thms (defs1, defs2),
      AList.merge (op =) (K true) (operations1, operations2)));

structure Class_Data = Theory_Data
(
  type T = class_data Graph.T
  val empty = Graph.empty;
  val extend = I;
  val merge = Graph.join merge_class_data;
);


(* queries *)

fun lookup_class_data thy class =
  (case try (Graph.get_node (Class_Data.get thy)) class of
    SOME (Class_Data data) => SOME data
  | NONE => NONE);

fun the_class_data thy class =
  (case lookup_class_data thy class of
    NONE => error ("Undeclared class " ^ quote class)
  | SOME data => data);

val is_class = is_some oo lookup_class_data;

val ancestry = Graph.all_succs o Class_Data.get;
val heritage = Graph.all_preds o Class_Data.get;

fun these_params thy =
  let
    fun params class =
      let
        val const_typs = (#params o AxClass.get_info thy) class;
        val const_names = (#consts o the_class_data thy) class;
      in
        (map o apsnd)
          (fn c => (class, (c, (the o AList.lookup (op =) const_typs) c))) const_names
      end;
  in maps params o ancestry thy end;

val base_sort = #base_sort oo the_class_data;

fun rules thy class =
  let val {axiom, of_class, ...} = the_class_data thy class
  in (axiom, of_class) end;

fun all_assm_intros thy =
  Graph.fold (fn (_, (Class_Data {assm_intro, ...}, _)) => fold (insert Thm.eq_thm)
    (the_list assm_intro)) (Class_Data.get thy) [];

fun these_defs thy = maps (#defs o the_class_data thy) o ancestry thy;
fun these_operations thy = maps (#operations o the_class_data thy) o ancestry thy;

val base_morphism = #base_morph oo the_class_data;

fun morphism thy class =
  (case Element.eq_morphism thy (these_defs thy [class]) of
    SOME eq_morph => base_morphism thy class $> eq_morph
  | NONE => base_morphism thy class);

val export_morphism = #export_morph oo the_class_data;

fun print_classes ctxt =
  let
    val thy = Proof_Context.theory_of ctxt;
    val algebra = Sign.classes_of thy;
    val arities =
      Symtab.empty
      |> Symtab.fold (fn (tyco, arities) => fold (fn (class, _) =>
           Symtab.map_default (class, []) (insert (op =) tyco)) arities)
             (Sorts.arities_of algebra);
    val the_arities = these o Symtab.lookup arities;
    fun mk_arity class tyco =
      let
        val Ss = Sorts.mg_domain algebra tyco [class];
      in Syntax.pretty_arity ctxt (tyco, Ss, [class]) end;
    fun mk_param (c, ty) =
      Pretty.str (Proof_Context.extern_const ctxt c ^ " :: " ^
        Syntax.string_of_typ ctxt (Type.strip_sorts_dummy ty));
    fun mk_entry class = (Pretty.block o Pretty.fbreaks o map_filter I) [
      (SOME o Pretty.str) ("class " ^ Proof_Context.extern_class ctxt class ^ ":"),
      (SOME o Pretty.block) [Pretty.str "supersort: ",
        (Syntax.pretty_sort ctxt o Sign.minimize_sort thy o Sign.super_classes thy) class],
      ((fn [] => NONE | ps => (SOME o Pretty.block o Pretty.fbreaks)
          (Pretty.str "parameters:" :: ps)) o map mk_param
        o these o Option.map #params o try (AxClass.get_info thy)) class,
      (SOME o Pretty.block o Pretty.breaks) [
        Pretty.str "instances:",
        Pretty.list "" "" (map (mk_arity class) (the_arities class))
      ]
    ]
  in
    (Pretty.writeln o Pretty.chunks o separate (Pretty.str "")
      o map mk_entry o Sorts.all_classes) algebra
  end;


(* updaters *)

fun register class sups params base_sort base_morph export_morph
    axiom assm_intro of_class thy =
  let
    val operations = map (fn (v_ty as (_, ty), (c, _)) =>
      (c, (class, (ty, Free v_ty)))) params;
    val add_class = Graph.new_node (class,
        make_class_data (((map o pairself) fst params, base_sort,
          base_morph, export_morph, assm_intro, of_class, axiom), ([], operations)))
      #> fold (curry Graph.add_edge class) sups;
  in Class_Data.map add_class thy end;

fun activate_defs class thms thy =
  (case Element.eq_morphism thy thms of
    SOME eq_morph => fold (fn cls => fn thy =>
      Context.theory_map (Locale.amend_registration (cls, base_morphism thy cls)
        (eq_morph, true) (export_morphism thy cls)) thy) (heritage thy [class]) thy
  | NONE => thy);

fun register_operation class (c, (t, some_def)) thy =
  let
    val base_sort = base_sort thy class;
    val prep_typ = map_type_tfree
      (fn (v, sort) => if Name.aT = v
        then TFree (v, base_sort) else TVar ((v, 0), sort));
    val t' = map_types prep_typ t;
    val ty' = Term.fastype_of t';
  in
    thy
    |> (Class_Data.map o Graph.map_node class o map_class_data o apsnd)
      (fn (defs, operations) =>
        (fold cons (the_list some_def) defs,
          (c, (class, (ty', t'))) :: operations))
    |> activate_defs class (the_list some_def)
  end;

fun register_subclass (sub, sup) some_dep_morph some_wit export thy =
  let
    val intros = (snd o rules thy) sup :: map_filter I
      [Option.map (Drule.export_without_context_open o Element.conclude_witness) some_wit,
        (fst o rules thy) sub];
    val classrel =
      Skip_Proof.prove_global thy [] [] (Logic.mk_classrel (sub, sup))
        (K (EVERY (map (TRYALL o Tactic.rtac) intros)));
    val diff_sort = Sign.complete_sort thy [sup]
      |> subtract (op =) (Sign.complete_sort thy [sub])
      |> filter (is_class thy);
    val add_dependency =
      (case some_dep_morph of
        SOME dep_morph => Locale.add_dependency sub
          (sup, dep_morph $> Element.satisfy_morphism (the_list some_wit)) NONE export
      | NONE => I);
  in
    thy
    |> AxClass.add_classrel classrel
    |> Class_Data.map (Graph.add_edge (sub, sup))
    |> activate_defs sub (these_defs thy diff_sort)
    |> add_dependency
  end;


(** classes and class target **)

(* class context syntax *)

fun these_unchecks thy = map (fn (c, (_, (ty, t))) => (t, Const (c, ty)))
  o these_operations thy;

fun redeclare_const thy c =
  let val b = Long_Name.base_name c
  in Sign.intern_const thy b = c ? Variable.declare_const (b, c) end;

fun synchronize_class_syntax sort base_sort ctxt =
  let
    val thy = Proof_Context.theory_of ctxt;
    val algebra = Sign.classes_of thy;
    val operations = these_operations thy sort;
    fun subst_class_typ sort = map_type_tfree (K (TVar ((Name.aT, 0), sort)));
    val primary_constraints =
      (map o apsnd) (subst_class_typ base_sort o fst o snd) operations;
    val secondary_constraints =
      (map o apsnd) (fn (class, (ty, _)) => subst_class_typ [class] ty) operations;
    fun improve (c, ty) =
      (case AList.lookup (op =) primary_constraints c of
        SOME ty' =>
          (case try (Type.raw_match (ty', ty)) Vartab.empty of
            SOME tyenv =>
              (case Vartab.lookup tyenv (Name.aT, 0) of
                SOME (_, ty' as TVar (vi, sort)) =>
                  if Type_Infer.is_param vi andalso Sorts.sort_le algebra (base_sort, sort)
                  then SOME (ty', TFree (Name.aT, base_sort))
                  else NONE
              | _ => NONE)
          | NONE => NONE)
      | NONE => NONE);
    fun subst (c, _) = Option.map snd (AList.lookup (op =) operations c);
    val unchecks = these_unchecks thy sort;
  in
    ctxt
    |> fold (redeclare_const thy o fst) primary_constraints
    |> Overloading.map_improvable_syntax (K (((primary_constraints, secondary_constraints),
        (((improve, subst), true), unchecks)), false))
    |> Overloading.set_primary_constraints
  end;

fun redeclare_operations thy sort =
  fold (redeclare_const thy o fst) (these_operations thy sort);

fun begin sort base_sort ctxt =
  ctxt
  |> Variable.declare_term
      (Logic.mk_type (TFree (Name.aT, base_sort)))
  |> synchronize_class_syntax sort base_sort
  |> Overloading.activate_improvable_syntax;

fun init class thy =
  thy
  |> Locale.init class
  |> begin [class] (base_sort thy class);


(* class target *)

val class_prefix = Logic.const_of_class o Long_Name.base_name;

local

fun target_extension f class lthy =
  lthy
  |> Local_Theory.raw_theory f
  |> Local_Theory.map_contexts
      (K (synchronize_class_syntax [class] (base_sort (Proof_Context.theory_of lthy) class)));

fun target_const class ((c, mx), (type_params, term_params, dict)) thy =
  let
    val morph = morphism thy class;
    val class_params = map fst (these_params thy [class]);
    val additional_params =
      subtract (fn (v, Free (w, _)) => v = w | _ => false) class_params term_params;
    val context_params = map (Morphism.term morph) (type_params @ additional_params);
    val b = Morphism.binding morph c;
    val b_def = Morphism.binding morph (Binding.suffix_name "_dict" b);
    val c' = Sign.full_name thy b;
    val dict' = Morphism.term morph dict;
    val ty' = map Term.fastype_of context_params ---> Term.fastype_of dict';
    val def_eq = Logic.mk_equals (list_comb (Const (c', ty'), context_params), dict')
      |> map_types Type.strip_sorts;
  in
    thy
    |> Sign.declare_const_global ((b, Type.strip_sorts ty'), mx)
    |> snd
    |> Thm.add_def_global false false (b_def, def_eq)
    |>> apsnd Thm.varifyT_global
    |-> (fn (_, def_thm) => Global_Theory.store_thm (b_def, def_thm)
      #> snd
      #> null context_params ? register_operation class (c', (dict', SOME (Thm.symmetric def_thm))))
    |> Sign.add_const_constraint (c', SOME ty')
  end;

fun target_abbrev class prmode ((c, mx), rhs) thy =
  let
    val morph = morphism thy class;
    val unchecks = these_unchecks thy [class];
    val b = Morphism.binding morph c;
    val c' = Sign.full_name thy b;
    val rhs' = Pattern.rewrite_term thy unchecks [] rhs;
    val ty' = Term.fastype_of rhs';
    val rhs'' = Logic.varify_types_global rhs';
  in
    thy
    |> Sign.add_abbrev (#1 prmode) (b, rhs'')
    |> snd
    |> Sign.add_const_constraint (c', SOME ty')
    |> Sign.notation true prmode [(Const (c', ty'), mx)]
    |> not (#1 prmode = Print_Mode.input) ? register_operation class (c', (rhs', NONE))
  end;

in

fun const class arg = target_extension (target_const class arg) class;
fun abbrev class prmode arg = target_extension (target_abbrev class prmode arg) class;

end;


(* simple subclasses *)

local

fun gen_classrel mk_prop classrel thy =
  let
    fun after_qed results =
      Proof_Context.background_theory ((fold o fold) AxClass.add_classrel results);
  in
    thy
    |> Proof_Context.init_global
    |> Proof.theorem NONE after_qed [[(mk_prop thy classrel, [])]]
  end;

in

val classrel =
  gen_classrel (Logic.mk_classrel oo AxClass.cert_classrel);
val classrel_cmd =
  gen_classrel (Logic.mk_classrel oo AxClass.read_classrel);

end; (*local*)


(** instantiation target **)

(* bookkeeping *)

datatype instantiation = Instantiation of {
  arities: string list * (string * sort) list * sort,
  params: ((string * string) * (string * typ)) list
    (*(instantiation parameter, type constructor), (local instantiation parameter, typ)*)
}

structure Instantiation = Proof_Data
(
  type T = instantiation;
  fun init _ = Instantiation {arities = ([], [], []), params = []};
);

fun mk_instantiation (arities, params) =
  Instantiation {arities = arities, params = params};

val get_instantiation =
  (fn Instantiation data => data) o Instantiation.get o Local_Theory.target_of;

fun map_instantiation f =
  (Local_Theory.target o Instantiation.map)
    (fn Instantiation {arities, params} => mk_instantiation (f (arities, params)));

fun the_instantiation lthy =
  (case get_instantiation lthy of
    {arities = ([], [], []), ...} => error "No instantiation target"
  | data => data);

val instantiation_params = #params o get_instantiation;

fun instantiation_param lthy b = instantiation_params lthy
  |> find_first (fn (_, (v, _)) => Binding.name_of b = v)
  |> Option.map (fst o fst);

fun read_multi_arity thy (raw_tycos, raw_sorts, raw_sort) =
  let
    val ctxt = Proof_Context.init_global thy;
    val all_arities = map (fn raw_tyco => Proof_Context.read_arity ctxt
      (raw_tyco, raw_sorts, raw_sort)) raw_tycos;
    val tycos = map #1 all_arities;
    val (_, sorts, sort) = hd all_arities;
    val vs = Name.invent_names Name.context Name.aT sorts;
  in (tycos, vs, sort) end;


(* syntax *)

fun synchronize_inst_syntax ctxt =
  let
    val Instantiation {params, ...} = Instantiation.get ctxt;

    val lookup_inst_param = AxClass.lookup_inst_param
      (Sign.consts_of (Proof_Context.theory_of ctxt)) params;
    fun subst (c, ty) =
      (case lookup_inst_param (c, ty) of
        SOME (v_ty as (_, ty)) => SOME (ty, Free v_ty)
      | NONE => NONE);
    val unchecks =
      map (fn ((c, _), v_ty as (_, ty)) => (Free v_ty, Const (c, ty))) params;
  in
    ctxt
    |> Overloading.map_improvable_syntax
      (fn (((primary_constraints, _), (((improve, _), _), _)), _) =>
          (((primary_constraints, []), (((improve, subst), false), unchecks)), false))
  end;

fun resort_terms ctxt algebra consts constraints ts =
  let
    fun matchings (Const (c_ty as (c, _))) =
          (case constraints c of
            NONE => I
          | SOME sorts =>
              fold2 (curry (Sorts.meet_sort algebra)) (Consts.typargs consts c_ty) sorts)
      | matchings _ = I;
    val tvartab = (fold o fold_aterms) matchings ts Vartab.empty
      handle Sorts.CLASS_ERROR e => error (Sorts.class_error (Context.pretty ctxt) e);
    val inst = map_type_tvar
      (fn (vi, sort) => TVar (vi, the_default sort (Vartab.lookup tvartab vi)));
  in if Vartab.is_empty tvartab then ts else (map o map_types) inst ts end;


(* target *)

fun define_overloaded (c, U) v (b_def, rhs) =
  Local_Theory.background_theory_result (AxClass.declare_overloaded (c, U)
  ##>> AxClass.define_overloaded b_def (c, rhs))
  ##> (map_instantiation o apsnd) (filter_out (fn (_, (v', _)) => v' = v))
  ##> Local_Theory.map_contexts (K synchronize_inst_syntax);

fun foundation (((b, U), mx), (b_def, rhs)) params lthy =
  (case instantiation_param lthy b of
    SOME c =>
      if mx <> NoSyn then error ("Illegal mixfix syntax for overloaded constant " ^ quote c)
      else lthy |> define_overloaded (c, U) (Binding.name_of b) (b_def, rhs)
  | NONE => lthy |> Generic_Target.theory_foundation (((b, U), mx), (b_def, rhs)) params);

fun pretty lthy =
  let
    val {arities = (tycos, vs, sort), params} = the_instantiation lthy;
    fun pr_arity tyco = Syntax.pretty_arity lthy (tyco, map snd vs, sort);
    fun pr_param ((c, _), (v, ty)) =
      Pretty.block (Pretty.breaks
        [Pretty.str v, Pretty.str "==", Pretty.str (Proof_Context.extern_const lthy c),
          Pretty.str "::", Syntax.pretty_typ lthy ty]);
  in Pretty.command "instantiation" :: map pr_arity tycos @ map pr_param params end;

fun conclude lthy =
  let
    val (tycos, vs, sort) = #arities (the_instantiation lthy);
    val thy = Proof_Context.theory_of lthy;
    val _ = tycos |> List.app (fn tyco =>
      if Sign.of_sort thy (Type (tyco, map TFree vs), sort) then ()
      else error ("Missing instance proof for type " ^ quote (Proof_Context.extern_type lthy tyco)));
  in lthy end;

fun instantiation (tycos, vs, sort) thy =
  let
    val naming = Sign.naming_of thy;

    val _ = if null tycos then error "At least one arity must be given" else ();
    val class_params = these_params thy (filter (can (AxClass.get_info thy)) sort);
    fun get_param tyco (param, (_, (c, ty))) =
      if can (AxClass.param_of_inst thy) (c, tyco)
      then NONE else SOME ((c, tyco),
        (param ^ "_" ^ Long_Name.base_name tyco, map_atyps (K (Type (tyco, map TFree vs))) ty));
    val params = map_product get_param tycos class_params |> map_filter I;
    val primary_constraints = map (apsnd
      (map_atyps (K (TVar ((Name.aT, 0), [])))) o snd o snd) class_params;
    val algebra = Sign.classes_of thy
      |> fold (fn tyco => Sorts.add_arities (Context.pretty_global thy)
            (tyco, map (fn class => (class, map snd vs)) sort)) tycos;
    val consts = Sign.consts_of thy;
    val improve_constraints = AList.lookup (op =)
      (map (fn (_, (class, (c, _))) => (c, [[class]])) class_params);
    fun resort_check ctxt ts = resort_terms ctxt algebra consts improve_constraints ts;
    val lookup_inst_param = AxClass.lookup_inst_param consts params;
    fun improve (c, ty) =
      (case lookup_inst_param (c, ty) of
        SOME (_, ty') => if Sign.typ_instance thy (ty', ty) then SOME (ty, ty') else NONE
      | NONE => NONE);
  in
    thy
    |> Theory.checkpoint
    |> Proof_Context.init_global
    |> Instantiation.put (mk_instantiation ((tycos, vs, sort), params))
    |> fold (Variable.declare_typ o TFree) vs
    |> fold (Variable.declare_names o Free o snd) params
    |> (Overloading.map_improvable_syntax o apfst)
         (K ((primary_constraints, []), (((improve, K NONE), false), [])))
    |> Overloading.activate_improvable_syntax
    |> Context.proof_map (Syntax_Phases.term_check 0 "resorting" resort_check)
    |> synchronize_inst_syntax
    |> Local_Theory.init naming
       {define = Generic_Target.define foundation,
        notes = Generic_Target.notes Generic_Target.theory_notes,
        abbrev = Generic_Target.abbrev Generic_Target.theory_abbrev,
        declaration = K Generic_Target.theory_declaration,
        pretty = pretty,
        exit = Local_Theory.target_of o conclude}
  end;

fun instantiation_cmd arities thy =
  instantiation (read_multi_arity thy arities) thy;

fun gen_instantiation_instance do_proof after_qed lthy =
  let
    val (tycos, vs, sort) = (#arities o the_instantiation) lthy;
    val arities_proof = maps (fn tyco => Logic.mk_arities (tyco, map snd vs, sort)) tycos;
    fun after_qed' results =
      Local_Theory.background_theory (fold (AxClass.add_arity o Thm.varifyT_global) results)
      #> after_qed;
  in
    lthy
    |> do_proof after_qed' arities_proof
  end;

val instantiation_instance = gen_instantiation_instance (fn after_qed => fn ts =>
  Proof.theorem NONE (after_qed o map the_single) (map (fn t => [(t, [])]) ts));

fun prove_instantiation_instance tac = gen_instantiation_instance (fn after_qed =>
  fn ts => fn lthy => after_qed (map (fn t => Goal.prove lthy [] [] t
    (fn {context, ...} => tac context)) ts) lthy) I;

fun prove_instantiation_exit tac = prove_instantiation_instance tac
  #> Local_Theory.exit_global;

fun prove_instantiation_exit_result f tac x lthy =
  let
    val morph = Proof_Context.export_morphism lthy
      (Proof_Context.init_global (Proof_Context.theory_of lthy));
    val y = f morph x;
  in
    lthy
    |> prove_instantiation_exit (fn ctxt => tac ctxt y)
    |> pair y
  end;


(* simplified instantiation interface with no class parameter *)

fun instance_arity_cmd raw_arities thy =
  let
    val (tycos, vs, sort) = read_multi_arity thy raw_arities;
    val sorts = map snd vs;
    val arities = maps (fn tyco => Logic.mk_arities (tyco, sorts, sort)) tycos;
    fun after_qed results =
      Proof_Context.background_theory ((fold o fold) AxClass.add_arity results);
  in
    thy
    |> Proof_Context.init_global
    |> Proof.theorem NONE after_qed (map (fn t => [(t, [])]) arities)
  end;


(** tactics and methods **)

fun intro_classes_tac facts st =
  let
    val thy = Thm.theory_of_thm st;
    val classes = Sign.all_classes thy;
    val class_trivs = map (Thm.class_triv thy) classes;
    val class_intros = map_filter (try (#intro o AxClass.get_info thy)) classes;
    val assm_intros = all_assm_intros thy;
  in
    Method.intros_tac (class_trivs @ class_intros @ assm_intros) facts st
  end;

fun default_intro_tac ctxt [] =
      COND Thm.no_prems no_tac
        (intro_classes_tac [] ORELSE Locale.intro_locales_tac true ctxt [])
  | default_intro_tac _ _ = no_tac;

fun default_tac rules ctxt facts =
  HEADGOAL (Method.some_rule_tac rules ctxt facts) ORELSE
    default_intro_tac ctxt facts;

val _ = Context.>> (Context.map_theory
 (Method.setup (Binding.name "intro_classes") (Scan.succeed (K (METHOD intro_classes_tac)))
    "back-chain introduction rules of classes" #>
  Method.setup (Binding.name "default") (Attrib.thms >> (METHOD oo default_tac))
    "apply some intro/elim rule"));

end;