src/HOL/Tools/ATP/atp_proof_reconstruct.ML

         end
       | aux subst depth nextp (step :: proof) =
         step :: aux subst depth nextp proof
   in aux [] 0 (1, 1) end
 
-fun string_for_proof ctxt0 type_enc lam_trans i n =
-  let
-    val ctxt = ctxt0
-(* FIXME: Implement proper handling of type constraints:
-      |> Config.put show_free_types false
-      |> Config.put show_types false
-      |> Config.put show_sorts false
-*)
+
+(** Type annotations **)
+
+fun post_traverse_term_type' f _ (t as Const (_, T)) s = f t T s
+  | post_traverse_term_type' f _ (t as Free (_, T)) s = f t T s
+  | post_traverse_term_type' f _ (t as Var (_, T)) s = f t T s
+  | post_traverse_term_type' f env (t as Bound i) s = f t (nth env i) s
+  | post_traverse_term_type' f env (Abs (x, T1, b)) s =
+    let
+      val ((b', s'), T2) = post_traverse_term_type' f (T1 :: env) b s
+    in f (Abs (x, T1, b')) (T1 --> T2) s' end
+  | post_traverse_term_type' f env (u $ v) s =
+    let
+      val ((u', s'), Type (_, [_, T])) = post_traverse_term_type' f env u s
+      val ((v', s''), _) = post_traverse_term_type' f env v s'
+    in f (u' $ v') T s'' end
+
+fun post_traverse_term_type f s t =
+  post_traverse_term_type' (fn t => fn T => fn s => (f t T s, T)) [] t s |> fst
+fun post_fold_term_type f s t =
+  post_traverse_term_type (fn t => fn T => fn s => (t, f t T s)) s t |> snd
+
+(* Data structures, orders *)
+val cost_ord = prod_ord int_ord (prod_ord int_ord int_ord)
+
+structure Var_Set_Tab = Table(
+  type key = indexname list
+  val ord = list_ord Term_Ord.fast_indexname_ord)
+
+(* (1) Generalize Types *)
+fun generalize_types ctxt t =
+  t |> map_types (fn _ => dummyT)
+    |> Syntax.check_term
+         (Proof_Context.set_mode Proof_Context.mode_pattern ctxt)
+
+(* (2) Typing-spot Table *)
+local
+fun key_of_atype (TVar (idxn, _)) =
+    Ord_List.insert Term_Ord.fast_indexname_ord idxn
+  | key_of_atype _ = I
+fun key_of_type T = fold_atyps key_of_atype T []
+fun update_tab t T (tab, pos) =
+  (case key_of_type T of
+     [] => tab
+   | key =>
+     let val cost = (size_of_typ T, (size_of_term t, pos)) in
+       case Var_Set_Tab.lookup tab key of
+         NONE => Var_Set_Tab.update_new (key, cost) tab
+       | SOME old_cost =>
+         (case cost_ord (cost, old_cost) of
+            LESS => Var_Set_Tab.update (key, cost) tab
+          | _ => tab)
+     end,
+   pos + 1)
+in
+val typing_spot_table =
+  post_fold_term_type update_tab (Var_Set_Tab.empty, 0) #> fst
+end
+
+(* (3) Reverse-Greedy *)
+fun reverse_greedy typing_spot_tab =
+  let
+    fun update_count z =
+      fold (fn tvar => fn tab =>
+        let val c = Vartab.lookup tab tvar |> the_default 0 in
+          Vartab.update (tvar, c + z) tab
+        end)
+    fun superfluous tcount =
+      forall (fn tvar => the (Vartab.lookup tcount tvar) > 1)
+    fun drop_superfluous (tvars, (_, (_, spot))) (spots, tcount) =
+      if superfluous tcount tvars then (spots, update_count ~1 tvars tcount)
+      else (spot :: spots, tcount)
+    val (typing_spots, tvar_count_tab) =
+      Var_Set_Tab.fold
+        (fn kv as (k, _) => apfst (cons kv) #> apsnd (update_count 1 k))
+        typing_spot_tab ([], Vartab.empty)
+      |>> sort_distinct (rev_order o cost_ord o pairself snd)
+  in fold drop_superfluous typing_spots ([], tvar_count_tab) |> fst end
+
+(* (4) Introduce Annotations *)
+fun introduce_annotations thy spots t t' =
+  let
+    val get_types = post_fold_term_type (K cons) []
+    fun match_types tp =
+      fold (Sign.typ_match thy) (op ~~ (pairself get_types tp)) Vartab.empty
+    fun unica' b x [] = if b then [x] else []
+      | unica' b x (y :: ys) =
+        if x = y then unica' false x ys
+        else unica' true y ys |> b ? cons x
+    fun unica ord xs =
+      case sort ord xs of x :: ys => unica' true x ys | [] => []
+    val add_all_tfree_namesT = fold_atyps (fn TFree (x, _) => cons x | _ => I)
+    fun erase_unica_tfrees env =
+      let
+        val unica =
+          Vartab.fold (add_all_tfree_namesT o snd o snd) env []
+          |> unica fast_string_ord
+        val erase_unica = map_atyps
+          (fn T as TFree (s, _) =>
+              if Ord_List.member fast_string_ord unica s then dummyT else T
+            | T => T)
+      in Vartab.map (K (apsnd erase_unica)) env end
+    val env = match_types (t', t) |> erase_unica_tfrees
+    fun get_annot env (TFree _) = (false, (env, dummyT))
+      | get_annot env (T as TVar (v, S)) =
+        let val T' = Envir.subst_type env T in
+          if T' = dummyT then (false, (env, dummyT))
+          else (true, (Vartab.update (v, (S, dummyT)) env, T'))
+        end
+      | get_annot env (Type (S, Ts)) =
+        (case fold_rev (fn T => fn (b, (env, Ts)) =>
+                  let
+                    val (b', (env', T)) = get_annot env T
+                  in (b orelse b', (env', T :: Ts)) end)
+                Ts (false, (env, [])) of
+           (true, (env', Ts)) => (true, (env', Type (S, Ts)))
+         | (false, (env', _)) => (false, (env', dummyT)))
+    fun post1 _ T (env, cp, ps as p :: ps', annots) =
+        if p <> cp then
+          (env, cp + 1, ps, annots)
+        else
+          let val (_, (env', T')) = get_annot env T in
+            (env', cp + 1, ps', (p, T') :: annots)
+          end
+      | post1 _ _ accum = accum
+    val (_, _, _, annots) = post_fold_term_type post1 (env, 0, spots, []) t'
+    fun post2 t _ (cp, annots as (p, T) :: annots') =
+        if p <> cp then (t, (cp + 1, annots))
+        else (Type.constraint T t, (cp + 1, annots'))
+      | post2 t _ x = (t, x)
+  in post_traverse_term_type post2 (0, rev annots) t |> fst end
+
+(* (5) Annotate *)
+fun annotate_types ctxt t =
+  let
+    val thy = Proof_Context.theory_of ctxt
+    val t' = generalize_types ctxt t
+    val typing_spots =
+      t' |> typing_spot_table
+         |> reverse_greedy
+         |> sort int_ord
+  in introduce_annotations thy typing_spots t t' end
+
+fun string_for_proof ctxt type_enc lam_trans i n =
+  let
     fun fix_print_mode f x =
       Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)
                                (print_mode_value ())) f x
     fun do_indent ind = replicate_string (ind * indent_size) " "
     fun do_free (s, T) =

       (if member (op =) qs Ultimately then "ultimately " else "") ^
       (if member (op =) qs Then then
          if member (op =) qs Show then "thus" else "hence"
        else
          if member (op =) qs Show then "show" else "have")
-    val do_term = maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)
+    val do_term =
+      maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)
+      o annotate_types ctxt
     val reconstr = Metis (type_enc, lam_trans)
     fun do_facts (ls, ss) =
       reconstructor_command reconstr 1 1 [] 0
           (ls |> sort_distinct (prod_ord string_ord int_ord),
            ss |> sort_distinct string_ord)