(*  Title:      Pure/variable.ML

     Author:     Makarius

 Fixed type/term variables and polymorphic term abbreviations.

 *)

 signature VARIABLE =

 sig

   val is_body: Proof.context -> bool

   val set_body: bool -> Proof.context -> Proof.context

   val restore_body: Proof.context -> Proof.context -> Proof.context

   val names_of: Proof.context -> Name.context

   val binds_of: Proof.context -> (typ * term) Vartab.table

   val maxidx_of: Proof.context -> int

   val sorts_of: Proof.context -> sort list

   val constraints_of: Proof.context -> typ Vartab.table * sort Vartab.table

   val is_declared: Proof.context -> string -> bool

   val check_name: binding -> string

   val default_type: Proof.context -> string -> typ option

   val def_type: Proof.context -> bool -> indexname -> typ option

   val def_sort: Proof.context -> indexname -> sort option

   val declare_names: term -> Proof.context -> Proof.context

   val declare_constraints: term -> Proof.context -> Proof.context

   val declare_term: term -> Proof.context -> Proof.context

   val declare_typ: typ -> Proof.context -> Proof.context

   val declare_prf: Proofterm.proof -> Proof.context -> Proof.context

   val declare_thm: thm -> Proof.context -> Proof.context

   val global_thm_context: thm -> Proof.context

   val variant_frees: Proof.context -> term list -> (string * 'a) list -> (string * 'a) list

   val bind_term: indexname * term option -> Proof.context -> Proof.context

   val expand_binds: Proof.context -> term -> term

   val lookup_const: Proof.context -> string -> string option

   val is_const: Proof.context -> string -> bool

   val declare_const: string * string -> Proof.context -> Proof.context

   val is_fixed: Proof.context -> string -> bool

   val newly_fixed: Proof.context -> Proof.context -> string -> bool

   val fixed_ord: Proof.context -> string * string -> order

   val intern_fixed: Proof.context -> string -> string

   val markup_fixed: Proof.context -> string -> Markup.T

   val lookup_fixed: Proof.context -> string -> string option

   val revert_fixed: Proof.context -> string -> string

   val add_fixed_names: Proof.context -> term -> string list -> string list

   val add_fixed: Proof.context -> term -> (string * typ) list -> (string * typ) list

   val add_free_names: Proof.context -> term -> string list -> string list

   val add_frees: Proof.context -> term -> (string * typ) list -> (string * typ) list

   val add_fixes_binding: binding list -> Proof.context -> string list * Proof.context

   val add_fixes: string list -> Proof.context -> string list * Proof.context

   val add_fixes_direct: string list -> Proof.context -> Proof.context

   val auto_fixes: term -> Proof.context -> Proof.context

   val variant_fixes: string list -> Proof.context -> string list * Proof.context

   val dest_fixes: Proof.context -> (string * string) list

   val invent_types: sort list -> Proof.context -> (string * sort) list * Proof.context

   val export_terms: Proof.context -> Proof.context -> term list -> term list

   val exportT_terms: Proof.context -> Proof.context -> term list -> term list

   val exportT: Proof.context -> Proof.context -> thm list -> thm list

   val export_prf: Proof.context -> Proof.context -> Proofterm.proof -> Proofterm.proof

   val export: Proof.context -> Proof.context -> thm list -> thm list

   val export_morphism: Proof.context -> Proof.context -> morphism

   val importT_inst: term list -> Proof.context -> ((indexname * sort) * typ) list * Proof.context

   val import_inst: bool -> term list -> Proof.context ->

     (((indexname * sort) * typ) list * ((indexname * typ) * term) list) * Proof.context

   val importT_terms: term list -> Proof.context -> term list * Proof.context

   val import_terms: bool -> term list -> Proof.context -> term list * Proof.context

   val importT: thm list -> Proof.context -> ((ctyp * ctyp) list * thm list) * Proof.context

   val import_prf: bool -> Proofterm.proof -> Proof.context -> Proofterm.proof * Proof.context

   val import: bool -> thm list -> Proof.context ->

     (((ctyp * ctyp) list * (cterm * cterm) list) * thm list) * Proof.context

   val tradeT: (Proof.context -> thm list -> thm list) -> Proof.context -> thm list -> thm list

   val trade: (Proof.context -> thm list -> thm list) -> Proof.context -> thm list -> thm list

   val focus: term -> Proof.context -> ((string * (string * typ)) list * term) * Proof.context

   val focus_cterm: cterm -> Proof.context -> ((string * cterm) list * cterm) * Proof.context

   val focus_subgoal: int -> thm -> Proof.context -> ((string * cterm) list * cterm) * Proof.context

   val warn_extra_tfrees: Proof.context -> Proof.context -> unit

   val polymorphic_types: Proof.context -> term list -> (indexname * sort) list * term list

   val polymorphic: Proof.context -> term list -> term list

 end;

 structure Variable: VARIABLE =

 struct

 (** local context data **)

 type fixes = string Name_Space.table;

 val empty_fixes: fixes = Name_Space.empty_table Isabelle_Markup.fixedN;

 datatype data = Data of

  {is_body: bool,                        (*inner body mode*)

   names: Name.context,                  (*type/term variable names*)

   consts: string Symtab.table,          (*consts within the local scope*)

   fixes: fixes,                         (*term fixes -- global name space, intern ~> extern*)

   binds: (typ * term) Vartab.table,     (*term bindings*)

   type_occs: string list Symtab.table,  (*type variables -- possibly within term variables*)

   maxidx: int,                          (*maximum var index*)

   sorts: sort Ord_List.T,               (*declared sort occurrences*)

   constraints:

     typ Vartab.table *                  (*type constraints*)

     sort Vartab.table};                 (*default sorts*)

 fun make_data (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =

   Data {is_body = is_body, names = names, consts = consts, fixes = fixes, binds = binds,

     type_occs = type_occs, maxidx = maxidx, sorts = sorts, constraints = constraints};

 structure Data = Proof_Data

 (

   type T = data;

   fun init _ =

     make_data (false, Name.context, Symtab.empty, empty_fixes, Vartab.empty,

       Symtab.empty, ~1, [], (Vartab.empty, Vartab.empty));

 );

 fun map_data f =

   Data.map (fn Data {is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints} =>

     make_data (f (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints)));

 fun map_names f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, f names, consts, fixes, binds, type_occs, maxidx, sorts, constraints));

 fun map_consts f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, f consts, fixes, binds, type_occs, maxidx, sorts, constraints));

 fun map_fixes f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, f fixes, binds, type_occs, maxidx, sorts, constraints));

 fun map_binds f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, fixes, f binds, type_occs, maxidx, sorts, constraints));

 fun map_type_occs f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, fixes, binds, f type_occs, maxidx, sorts, constraints));

 fun map_maxidx f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, fixes, binds, type_occs, f maxidx, sorts, constraints));

 fun map_sorts f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, fixes, binds, type_occs, maxidx, f sorts, constraints));

 fun map_constraints f =

   map_data (fn (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (is_body, names, consts, fixes, binds, type_occs, maxidx, sorts, f constraints));

 fun rep_data ctxt = Data.get ctxt |> (fn Data rep => rep);

 val is_body = #is_body o rep_data;

 fun set_body b =

   map_data (fn (_, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints) =>

     (b, names, consts, fixes, binds, type_occs, maxidx, sorts, constraints));

 fun restore_body ctxt = set_body (is_body ctxt);

 val names_of = #names o rep_data;

 val fixes_of = #fixes o rep_data;

 val fixes_space = #1 o fixes_of;

 val binds_of = #binds o rep_data;

 val type_occs_of = #type_occs o rep_data;

 val maxidx_of = #maxidx o rep_data;

 val sorts_of = #sorts o rep_data;

 val constraints_of = #constraints o rep_data;

 val is_declared = Name.is_declared o names_of;

 val check_name =

   Long_Name.base_name o Name_Space.full_name Name_Space.default_naming o tap Binding.check;

 (** declarations **)

 (* default sorts and types *)

 fun default_type ctxt x = Vartab.lookup (#1 (constraints_of ctxt)) (x, ~1);

 fun def_type ctxt pattern xi =

   let val {binds, constraints = (types, _), ...} = rep_data ctxt in

     (case Vartab.lookup types xi of

       NONE =>

         if pattern then NONE

         else Vartab.lookup binds xi |> Option.map (Type.mark_polymorphic o #1)

     | some => some)

   end;

 val def_sort = Vartab.lookup o #2 o constraints_of;

 (* names *)

 fun declare_type_names t =

   map_names (fold_types (fold_atyps Term.declare_typ_names) t) #>

   map_maxidx (fold_types Term.maxidx_typ t);

 fun declare_names t =

   declare_type_names t #>

   map_names (fold_aterms Term.declare_term_frees t) #>

   map_maxidx (Term.maxidx_term t);

 (* type occurrences *)

 fun decl_type_occsT T = fold_atyps (fn TFree (a, _) => Symtab.default (a, []) | _ => I) T;

 val decl_type_occs = fold_term_types

   (fn Free (x, _) => fold_atyps (fn TFree (a, _) => Symtab.insert_list (op =) (a, x) | _ => I)

     | _ => decl_type_occsT);

 val declare_type_occsT = map_type_occs o fold_types decl_type_occsT;

 val declare_type_occs = map_type_occs o decl_type_occs;

 (* constraints *)

 fun constrain_tvar (xi, S) =

   if S = dummyS then Vartab.delete_safe xi else Vartab.update (xi, S);

 fun declare_constraints t = map_constraints (fn (types, sorts) =>

   let

     val types' = fold_aterms

       (fn Free (x, T) => Vartab.update ((x, ~1), T)

         | Var v => Vartab.update v

         | _ => I) t types;

     val sorts' = (fold_types o fold_atyps)

       (fn TFree (x, S) => constrain_tvar ((x, ~1), S)

         | TVar v => constrain_tvar v

         | _ => I) t sorts;

   in (types', sorts') end)

   #> declare_type_occsT t

   #> declare_type_names t;

 (* common declarations *)

 fun declare_internal t =

   declare_names t #>

   declare_type_occs t #>

   map_sorts (Sorts.insert_term t);

 fun declare_term t =

   declare_internal t #>

   declare_constraints t;

 val declare_typ = declare_term o Logic.mk_type;

 val declare_prf = Proofterm.fold_proof_terms declare_internal (declare_internal o Logic.mk_type);

 val declare_thm = Thm.fold_terms declare_internal;

 fun global_thm_context th = declare_thm th (Proof_Context.init_global (Thm.theory_of_thm th));

 (* renaming term/type frees *)

 fun variant_frees ctxt ts frees =

   let

     val names = names_of (fold declare_names ts ctxt);

     val xs = fst (fold_map Name.variant (map #1 frees) names);

   in xs ~~ map snd frees end;

 (** term bindings **)

 fun bind_term (xi, NONE) = map_binds (Vartab.delete_safe xi)

   | bind_term ((x, i), SOME t) =

       let

         val u = Term.close_schematic_term t;

         val U = Term.fastype_of u;

       in declare_term u #> map_binds (Vartab.update ((x, i), (U, u))) end;

 fun expand_binds ctxt =

   let

     val binds = binds_of ctxt;

     val get = fn Var (xi, _) => Vartab.lookup binds xi | _ => NONE;

   in Envir.beta_norm o Envir.expand_term get end;

 (** consts **)

 val lookup_const = Symtab.lookup o #consts o rep_data;

 val is_const = is_some oo lookup_const;

 val declare_fixed = map_consts o Symtab.delete_safe;

 val declare_const = map_consts o Symtab.update;

 (** fixes **)

 (* specialized name space *)

 val is_fixed = Name_Space.defined_entry o fixes_space;

 fun newly_fixed inner outer = is_fixed inner andf (not o is_fixed outer);

 val fixed_ord = Name_Space.entry_ord o fixes_space;

 val intern_fixed = Name_Space.intern o fixes_space;

 fun lookup_fixed ctxt x =

   let val x' = intern_fixed ctxt x

   in if is_fixed ctxt x' then SOME x' else NONE end;

 fun revert_fixed ctxt x =

   (case Symtab.lookup (#2 (fixes_of ctxt)) x of

     SOME x' => if intern_fixed ctxt x' = x then x' else x

   | NONE => x);

 fun markup_fixed ctxt x =

   Name_Space.markup (fixes_space ctxt) x

   |> Markup.name (revert_fixed ctxt x);

 fun dest_fixes ctxt =

   let val (space, tab) = fixes_of ctxt

   in sort (Name_Space.entry_ord space o pairself #2) (map swap (Symtab.dest tab)) end;

 (* collect variables *)

 fun add_free_names ctxt =

   fold_aterms (fn Free (x, _) => not (is_fixed ctxt x) ? insert (op =) x | _ => I);

 fun add_frees ctxt =

   fold_aterms (fn Free (x, T) => not (is_fixed ctxt x) ? insert (op =) (x, T) | _ => I);

 fun add_fixed_names ctxt =

   fold_aterms (fn Free (x, _) => is_fixed ctxt x ? insert (op =) x | _ => I);

 fun add_fixed ctxt =

   fold_aterms (fn Free (x, T) => is_fixed ctxt x ? insert (op =) (x, T) | _ => I);

 (* declarations *)

 local

 fun err_dups dups =

   error ("Duplicate fixed variable(s): " ^ commas (map Binding.print dups));

 fun new_fixed ((x, x'), pos) ctxt =

   if is_some (lookup_fixed ctxt x') then err_dups [Binding.make (x, pos)]

   else

     let val context = Context.Proof ctxt |> Name_Space.map_naming (K Name_Space.default_naming) in

       ctxt

       |> map_fixes

         (Name_Space.define context true (Binding.make (x', pos), x) #> snd #>>

           Name_Space.alias Name_Space.default_naming (Binding.make (x, pos)) x')

       |> declare_fixed x

       |> declare_constraints (Syntax.free x')

   end;

 fun new_fixes names' xs xs' ps =

   map_names (K names') #>

   fold new_fixed ((xs ~~ xs') ~~ ps) #>

   pair xs';

 in

 fun add_fixes_binding bs ctxt =

   let

     val _ =

       (case filter (can Name.dest_skolem o Binding.name_of) bs of

         [] => ()

       | bads => error ("Illegal internal Skolem constant(s): " ^ commas (map Binding.print bads)));

     val _ =

       (case duplicates (op = o pairself Binding.name_of) bs of

         [] => ()

       | dups => err_dups dups);

     val xs = map check_name bs;

     val names = names_of ctxt;

     val (xs', names') =

       if is_body ctxt then fold_map Name.variant xs names |>> map Name.skolem

       else (xs, fold Name.declare xs names);

   in ctxt |> new_fixes names' xs xs' (map Binding.pos_of bs) end;

 fun variant_fixes raw_xs ctxt =

   let

     val names = names_of ctxt;

     val xs = map (fn x => Name.clean x |> can Name.dest_internal x ? Name.internal) raw_xs;

     val (xs', names') = fold_map Name.variant xs names |>> (is_body ctxt ? map Name.skolem);

   in ctxt |> new_fixes names' xs xs' (replicate (length xs) Position.none) end;

 end;

 val add_fixes = add_fixes_binding o map Binding.name;

 fun add_fixes_direct xs ctxt = ctxt

   |> set_body false

   |> (snd o add_fixes xs)

   |> restore_body ctxt;

 fun auto_fixes t ctxt = ctxt

   |> not (is_body ctxt) ? add_fixes_direct (rev (add_free_names ctxt t []))

   |> declare_term t;

 fun invent_types Ss ctxt =

   let

     val tfrees = Name.invent (names_of ctxt) Name.aT (length Ss) ~~ Ss;

     val ctxt' = fold (declare_constraints o Logic.mk_type o TFree) tfrees ctxt;

   in (tfrees, ctxt') end;

 (** export -- generalize type/term variables (beware of closure sizes) **)

 fun export_inst inner outer =

   let

     val declared_outer = is_declared outer;

     val still_fixed = not o newly_fixed inner outer;

     val gen_fixes =

       Symtab.fold (fn (y, _) => not (is_fixed outer y) ? cons y)

         (#2 (fixes_of inner)) [];

     val type_occs_inner = type_occs_of inner;

     fun gen_fixesT ts =

       Symtab.fold (fn (a, xs) =>

         if declared_outer a orelse exists still_fixed xs

         then I else cons a) (fold decl_type_occs ts type_occs_inner) [];

   in (gen_fixesT, gen_fixes) end;

 fun exportT_inst inner outer = #1 (export_inst inner outer);

 fun exportT_terms inner outer =

   let val mk_tfrees = exportT_inst inner outer in

     fn ts => ts |> map

       (Term_Subst.generalize (mk_tfrees ts, [])

         (fold (Term.fold_types Term.maxidx_typ) ts ~1 + 1))

   end;

 fun export_terms inner outer =

   let val (mk_tfrees, tfrees) = export_inst inner outer in

     fn ts => ts |> map

       (Term_Subst.generalize (mk_tfrees ts, tfrees)

         (fold Term.maxidx_term ts ~1 + 1))

   end;

 fun export_prf inner outer prf =

   let

     val (mk_tfrees, frees) = export_inst (declare_prf prf inner) outer;

     val tfrees = mk_tfrees [];

     val idx = Proofterm.maxidx_proof prf ~1 + 1;

     val gen_term = Term_Subst.generalize_same (tfrees, frees) idx;

     val gen_typ = Term_Subst.generalizeT_same tfrees idx;

   in Same.commit (Proofterm.map_proof_terms_same gen_term gen_typ) prf end;

 fun gen_export (mk_tfrees, frees) ths =

   let

     val tfrees = mk_tfrees (map Thm.full_prop_of ths);

     val maxidx = fold Thm.maxidx_thm ths ~1;

   in map (Thm.generalize (tfrees, frees) (maxidx + 1)) ths end;

 fun exportT inner outer = gen_export (exportT_inst inner outer, []);

 fun export inner outer = gen_export (export_inst inner outer);

 fun export_morphism inner outer =

   let

     val fact = export inner outer;

     val term = singleton (export_terms inner outer);

     val typ = Logic.type_map term;

   in Morphism.morphism {binding = [], typ = [typ], term = [term], fact = [fact]} end;

 (** import -- fix schematic type/term variables **)

 fun importT_inst ts ctxt =

   let

     val tvars = rev (fold Term.add_tvars ts []);

     val (tfrees, ctxt') = invent_types (map #2 tvars) ctxt;

   in (tvars ~~ map TFree tfrees, ctxt') end;

 fun import_inst is_open ts ctxt =

   let

     val ren = Name.clean #> (if is_open then I else Name.internal);

     val (instT, ctxt') = importT_inst ts ctxt;

     val vars = map (apsnd (Term_Subst.instantiateT instT)) (rev (fold Term.add_vars ts []));

     val (xs, ctxt'') = variant_fixes (map (ren o #1 o #1) vars) ctxt';

     val inst = vars ~~ map Free (xs ~~ map #2 vars);

   in ((instT, inst), ctxt'') end;

 fun importT_terms ts ctxt =

   let val (instT, ctxt') = importT_inst ts ctxt

   in (map (Term_Subst.instantiate (instT, [])) ts, ctxt') end;

 fun import_terms is_open ts ctxt =

   let val (inst, ctxt') = import_inst is_open ts ctxt

   in (map (Term_Subst.instantiate inst) ts, ctxt') end;

 fun importT ths ctxt =

   let

     val thy = Proof_Context.theory_of ctxt;

     val (instT, ctxt') = importT_inst (map Thm.full_prop_of ths) ctxt;

     val insts' as (instT', _) = Thm.certify_inst thy (instT, []);

     val ths' = map (Thm.instantiate insts') ths;

   in ((instT', ths'), ctxt') end;

 fun import_prf is_open prf ctxt =

   let

     val ts = rev (Proofterm.fold_proof_terms cons (cons o Logic.mk_type) prf []);

     val (insts, ctxt') = import_inst is_open ts ctxt;

   in (Proofterm.instantiate insts prf, ctxt') end;

 fun import is_open ths ctxt =

   let

     val thy = Proof_Context.theory_of ctxt;

     val (insts, ctxt') = import_inst is_open (map Thm.full_prop_of ths) ctxt;

     val insts' = Thm.certify_inst thy insts;

     val ths' = map (Thm.instantiate insts') ths;

   in ((insts', ths'), ctxt') end;

 (* import/export *)

 fun gen_trade imp exp f ctxt ths =

   let val ((_, ths'), ctxt') = imp ths ctxt

   in exp ctxt' ctxt (f ctxt' ths') end;

 val tradeT = gen_trade importT exportT;

 val trade = gen_trade (import true) export;

 (* focus on outermost parameters: !!x y z. B *)

 fun focus_params t ctxt =

   let

     val (xs, Ts) =

       split_list (Term.variant_frees t (Term.strip_all_vars t));  (*as they are printed :-*)

     val (xs', ctxt') = variant_fixes xs ctxt;

     val ps = xs' ~~ Ts;

     val ctxt'' = ctxt' |> fold (declare_constraints o Free) ps;

   in ((xs, ps), ctxt'') end;

 fun focus t ctxt =

   let

     val ((xs, ps), ctxt') = focus_params t ctxt;

     val t' = Term.subst_bounds (rev (map Free ps), Term.strip_all_body t);

   in (((xs ~~ ps), t'), ctxt') end;

 fun forall_elim_prop t prop =

   Thm.beta_conversion false (Thm.apply (Thm.dest_arg prop) t)

   |> Thm.cprop_of |> Thm.dest_arg;

 fun focus_cterm goal ctxt =

   let

     val cert = Thm.cterm_of (Thm.theory_of_cterm goal);

     val ((xs, ps), ctxt') = focus_params (Thm.term_of goal) ctxt;

     val ps' = map (cert o Free) ps;

     val goal' = fold forall_elim_prop ps' goal;

   in ((xs ~~ ps', goal'), ctxt') end;

 fun focus_subgoal i st =

   let

     val all_vars = Thm.fold_terms Term.add_vars st [];

     val no_binds = map (fn (xi, _) => (xi, NONE)) all_vars;

   in

     fold bind_term no_binds #>

     fold (declare_constraints o Var) all_vars #>

     focus_cterm (Thm.cprem_of st i)

   end;

 (** implicit polymorphism **)

 (* warn_extra_tfrees *)

 fun warn_extra_tfrees ctxt1 ctxt2 =

   let

     fun occs_typ a = Term.exists_subtype (fn TFree (b, _) => a = b | _ => false);

     fun occs_free a x =

       (case def_type ctxt1 false (x, ~1) of

         SOME T => if occs_typ a T then I else cons (a, x)

       | NONE => cons (a, x));

     val occs1 = type_occs_of ctxt1;

     val occs2 = type_occs_of ctxt2;

     val extras = Symtab.fold (fn (a, xs) =>

       if Symtab.defined occs1 a then I else fold (occs_free a) xs) occs2 [];

     val tfrees = map #1 extras |> sort_distinct string_ord;

     val frees = map #2 extras |> sort_distinct string_ord;

   in

     if null extras orelse not (Context_Position.is_visible ctxt2) then ()

     else warning ("Introduced fixed type variable(s): " ^ commas tfrees ^ " in " ^

       space_implode " or " (map quote frees))

   end;

 (* polymorphic terms *)

 fun polymorphic_types ctxt ts =

   let

     val ctxt' = fold declare_term ts ctxt;

     val occs = type_occs_of ctxt;

     val occs' = type_occs_of ctxt';

     val types = Symtab.fold (fn (a, _) => if Symtab.defined occs a then I else cons a) occs' [];

     val idx = maxidx_of ctxt' + 1;

     val Ts' = (fold o fold_types o fold_atyps)

       (fn T as TFree _ =>

           (case Term_Subst.generalizeT types idx T of TVar v => insert (op =) v | _ => I)

         | _ => I) ts [];

     val ts' = map (Term_Subst.generalize (types, []) idx) ts;

   in (rev Ts', ts') end;

 fun polymorphic ctxt ts = snd (polymorphic_types ctxt ts);

 end;