diff --git a/src/HOL/Tools/ATP/atp_proof_reconstruct.ML b/src/HOL/Tools/ATP/atp_proof_reconstruct.ML
--- a/src/HOL/Tools/ATP/atp_proof_reconstruct.ML
+++ b/src/HOL/Tools/ATP/atp_proof_reconstruct.ML
@@ -1,798 +1,800 @@
 (*  Title:      HOL/Tools/ATP/atp_proof_reconstruct.ML
     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
     Author:     Claire Quigley, Cambridge University Computer Laboratory
     Author:     Jasmin Blanchette, TU Muenchen
 
 Basic proof reconstruction from ATP proofs.
 *)
 
 signature ATP_PROOF_RECONSTRUCT =
 sig
   type 'a atp_type = 'a ATP_Problem.atp_type
   type ('a, 'b) atp_term = ('a, 'b) ATP_Problem.atp_term
   type ('a, 'b, 'c, 'd) atp_formula = ('a, 'b, 'c, 'd) ATP_Problem.atp_formula
   type stature = ATP_Problem_Generate.stature
   type atp_step_name = ATP_Proof.atp_step_name
   type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
   type 'a atp_proof = 'a ATP_Proof.atp_proof
 
   val metisN : string
   val full_typesN : string
   val partial_typesN : string
   val no_typesN : string
   val really_full_type_enc : string
   val full_type_enc : string
   val partial_type_enc : string
   val no_type_enc : string
   val full_type_encs : string list
   val partial_type_encs : string list
   val default_metis_lam_trans : string
 
   val leo2_extcnf_equal_neg_rule : string
   val satallax_tab_rule_prefix : string
 
   val forall_of : term -> term -> term
   val exists_of : term -> term -> term
   val simplify_bool : term -> term
   val var_name_of_typ : typ -> string
   val rename_bound_vars : term -> term
   val type_enc_aliases : (string * string list) list
   val unalias_type_enc : string -> string list
   val term_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
     bool -> int Symtab.table -> typ option -> (string, string atp_type) atp_term -> term
   val prop_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
     bool -> int Symtab.table ->
     (string, string, (string, string atp_type) atp_term, string) atp_formula -> term
 
   val used_facts_in_atp_proof : Proof.context -> (string * stature) list -> string atp_proof ->
     (string * stature) list
   val used_facts_in_unsound_atp_proof : Proof.context -> (string * stature) list -> 'a atp_proof ->
     string list option
   val atp_proof_prefers_lifting : string atp_proof -> bool
   val is_typed_helper_used_in_atp_proof : string atp_proof -> bool
   val replace_dependencies_in_line : atp_step_name * atp_step_name list -> ('a, 'b) atp_step ->
     ('a, 'b) atp_step
   val termify_atp_proof : Proof.context -> string -> ATP_Problem.atp_format ->
     ATP_Problem_Generate.type_enc -> string Symtab.table -> (string * term) list ->
     int Symtab.table -> string atp_proof -> (term, string) atp_step list
   val repair_waldmeister_endgame : (term, 'a) atp_step list -> (term, 'a) atp_step list
   val infer_formulas_types : Proof.context -> term list -> term list
   val introduce_spass_skolems : (term, string) atp_step list -> (term, string) atp_step list
   val factify_atp_proof : (string * 'a) list -> term list -> term -> (term, string) atp_step list ->
     (term, string) atp_step list
 end;
 
 structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =
 struct
 
 open ATP_Util
 open ATP_Problem
 open ATP_Proof
 open ATP_Problem_Generate
 
 val metisN = "metis"
 
 val full_typesN = "full_types"
 val partial_typesN = "partial_types"
 val no_typesN = "no_types"
 
 val really_full_type_enc = "mono_tags"
 val full_type_enc = "poly_guards_query"
 val partial_type_enc = "poly_args"
 val no_type_enc = "erased"
 
 val full_type_encs = [full_type_enc, really_full_type_enc]
 val partial_type_encs = partial_type_enc :: full_type_encs
 
 val type_enc_aliases =
   [(full_typesN, full_type_encs),
    (partial_typesN, partial_type_encs),
    (no_typesN, [no_type_enc])]
 
 fun unalias_type_enc s =
   AList.lookup (op =) type_enc_aliases s |> the_default [s]
 
 val default_metis_lam_trans = combsN
 
 val leo2_extcnf_equal_neg_rule = "extcnf_equal_neg"
 val satallax_tab_rule_prefix = "tab_"
 
 fun term_name' (Var ((s, _), _)) = perhaps (try Name.dest_skolem) s
   | term_name' _ = ""
 
 fun lambda' v = Term.lambda_name (term_name' v, v)
 
 fun forall_of v t = HOLogic.all_const (fastype_of v) $ lambda' v t
 fun exists_of v t = HOLogic.exists_const (fastype_of v) $ lambda' v t
 
 fun make_tfree ctxt w =
   let val ww = "'" ^ w in
     TFree (ww, the_default \<^sort>\<open>type\<close> (Variable.def_sort ctxt (ww, ~1)))
   end
 
 fun simplify_bool ((all as Const (\<^const_name>\<open>All\<close>, _)) $ Abs (s, T, t)) =
     let val t' = simplify_bool t in
       if loose_bvar1 (t', 0) then all $ Abs (s, T, t') else t'
     end
   | simplify_bool (Const (\<^const_name>\<open>Not\<close>, _) $ t) = s_not (simplify_bool t)
   | simplify_bool (Const (\<^const_name>\<open>conj\<close>, _) $ t $ u) =
     s_conj (simplify_bool t, simplify_bool u)
   | simplify_bool (Const (\<^const_name>\<open>disj\<close>, _) $ t $ u) =
     s_disj (simplify_bool t, simplify_bool u)
   | simplify_bool (Const (\<^const_name>\<open>implies\<close>, _) $ t $ u) =
     s_imp (simplify_bool t, simplify_bool u)
   | simplify_bool ((t as Const (\<^const_name>\<open>HOL.eq\<close>, _)) $ u $ v) =
     (case (u, v) of
       (Const (\<^const_name>\<open>True\<close>, _), _) => v
     | (u, Const (\<^const_name>\<open>True\<close>, _)) => u
     | (Const (\<^const_name>\<open>False\<close>, _), v) => s_not v
     | (u, Const (\<^const_name>\<open>False\<close>, _)) => s_not u
     | _ => if u aconv v then \<^Const>\<open>True\<close> else t $ simplify_bool u $ simplify_bool v)
   | simplify_bool (t $ u) = simplify_bool t $ simplify_bool u
   | simplify_bool (Abs (s, T, t)) = Abs (s, T, simplify_bool t)
   | simplify_bool t = t
 
 fun single_letter upper s =
   let val s' = if String.isPrefix "o" s orelse String.isPrefix "O" s then "z" else s in
     String.extract (Name.desymbolize (SOME upper) (Long_Name.base_name s'), 0, SOME 1)
   end
 
 fun var_name_of_typ (Type (\<^type_name>\<open>fun\<close>, [_, T])) =
     if body_type T = HOLogic.boolT then "p" else "f"
   | var_name_of_typ (Type (\<^type_name>\<open>set\<close>, [T])) =
     let fun default () = single_letter true (var_name_of_typ T) in
       (case T of
         Type (s, [T1, T2]) => if String.isSuffix "prod" s andalso T1 = T2 then "r" else default ()
       | _ => default ())
     end
   | var_name_of_typ (Type (s, Ts)) =
     if String.isSuffix "list" s then var_name_of_typ (the_single Ts) ^ "s"
     else single_letter false (Long_Name.base_name s)
   | var_name_of_typ (TFree (s, _)) = single_letter false (perhaps (try (unprefix "'")) s)
   | var_name_of_typ (TVar ((s, _), T)) = var_name_of_typ (TFree (s, T))
 
 fun rename_bound_vars (t $ u) = rename_bound_vars t $ rename_bound_vars u
   | rename_bound_vars (Abs (_, T, t)) = Abs (var_name_of_typ T, T, rename_bound_vars t)
   | rename_bound_vars t = t
 
 exception ATP_CLASS of string list
 exception ATP_TYPE of string atp_type list
 exception ATP_TERM of (string, string atp_type) atp_term list
 exception ATP_FORMULA of
   (string, string, (string, string atp_type) atp_term, string) atp_formula list
 exception SAME of unit
 
 fun class_of_atp_class cls =
   (case unprefix_and_unascii class_prefix cls of
     SOME s => s
   | NONE => raise ATP_CLASS [cls])
 
 fun atp_type_of_atp_term (ATerm ((s, _), us)) =
   let val tys = map atp_type_of_atp_term us in
     if s = tptp_fun_type then
       (case tys of
         [ty1, ty2] => AFun (ty1, ty2)
       | _ => raise ATP_TYPE tys)
     else
       AType ((s, []), tys)
   end
 
 (* Type variables are given the basic sort "HOL.type". Some will later be constrained by information
    from type literals, or by type inference. *)
 fun typ_of_atp_type ctxt (ty as AType ((a, clss), tys)) =
     let val Ts = map (typ_of_atp_type ctxt) tys in
       (case unprefix_and_unascii native_type_prefix a of
         SOME b => typ_of_atp_type ctxt (atp_type_of_atp_term (unmangled_type b))
       | NONE =>
         (case unprefix_and_unascii type_const_prefix a of
           SOME b => Type (invert_const b, Ts)
         | NONE =>
           if not (null tys) then
             raise ATP_TYPE [ty] (* only "tconst"s have type arguments *)
           else
             (case unprefix_and_unascii tfree_prefix a of
               SOME b => make_tfree ctxt b
             | NONE =>
               (* The term could be an Isabelle variable or a variable from the ATP, say "X1" or "_5018".
                  Sometimes variables from the ATP are indistinguishable from Isabelle variables, which
                  forces us to use a type parameter in all cases. *)
               Type_Infer.param 0 ("'" ^ perhaps (unprefix_and_unascii tvar_prefix) a,
                 (if null clss then \<^sort>\<open>type\<close> else map class_of_atp_class clss)))))
     end
   | typ_of_atp_type ctxt (AFun (ty1, ty2)) = typ_of_atp_type ctxt ty1 --> typ_of_atp_type ctxt ty2
 
 fun typ_of_atp_term ctxt = typ_of_atp_type ctxt o atp_type_of_atp_term
 
 (* Type class literal applied to a type. Returns triple of polarity, class, type. *)
 fun type_constraint_of_term ctxt (u as ATerm ((a, _), us)) =
   (case (unprefix_and_unascii class_prefix a, map (typ_of_atp_term ctxt) us) of
     (SOME b, [T]) => (b, T)
   | _ => raise ATP_TERM [u])
 
 (* Accumulate type constraints in a formula: negative type literals. *)
 fun add_var (key, z) = Vartab.map_default (key, []) (cons z)
 fun add_type_constraint false (cl, TFree (a ,_)) = add_var ((a, ~1), cl)
   | add_type_constraint false (cl, TVar (ix, _)) = add_var (ix, cl)
   | add_type_constraint _ _ = I
 
 fun repair_var_name s =
   (case unprefix_and_unascii schematic_var_prefix s of
     SOME s' => s'
   | NONE => s)
 
 (* The number of type arguments of a constant, zero if it's monomorphic. For (instances of) Skolem
    pseudoconstants, this information is encoded in the constant name. *)
 fun robust_const_num_type_args thy s =
   if String.isPrefix skolem_const_prefix s then
     s |> Long_Name.explode |> List.last |> Int.fromString |> the
   else if String.isPrefix lam_lifted_prefix s then
     if String.isPrefix lam_lifted_poly_prefix s then 2 else 0
   else
     (s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
 
 fun slack_fastype_of t = fastype_of t handle TERM _ => Type_Infer.anyT \<^sort>\<open>type\<close>
 
 val spass_skolem_prefix = "sk" (* "skc" or "skf" *)
 val vampire_skolem_prefix = "sK"
 
 fun var_index_of_textual textual = if textual then 0 else 1
 
 fun quantify_over_var textual quant_of var_s var_T t =
   let
     val vars =
       ((var_s, var_index_of_textual textual), var_T) ::
       filter (fn ((s, _), _) => s = var_s) (Term.add_vars t [])
     val normTs = vars |> AList.group (op =) |> map (apsnd hd)
     fun norm_var_types (Var (x, T)) =
         Var (x, the_default T (AList.lookup (op =) normTs x))
       | norm_var_types t = t
   in t |> map_aterms norm_var_types |> fold_rev quant_of (map Var normTs) end
 
 (* This assumes that distinct names are mapped to distinct names by "Variable.variant_frees". This
    does not hold in general but should hold for ATP-generated Skolem function names, since these end
    with a digit and "variant_frees" appends letters. *)
 fun fresh_up ctxt s = fst (hd (Variable.variant_frees ctxt [] [(s, ())]))
 
 (* Higher-order translation. Variables are typed (although we don't use that information). Lambdas
    are typed. The code is similar to "term_of_atp_fo". *)
 fun term_of_atp_ho ctxt sym_tab =
   let
     val thy = Proof_Context.theory_of ctxt
     val var_index = var_index_of_textual true
 
     fun do_term opt_T u =
       (case u of
         AAbs (((var, ty), term), []) =>
         let
           val typ = typ_of_atp_type ctxt ty
           val var_name = repair_var_name var
           val tm = do_term NONE term
         in quantify_over_var true lambda' var_name typ tm end
       | ATerm ((s, tys), us) =>
         if s = "" (* special marker generated on parse error *) then
           error "Isar proof reconstruction failed because the ATP proof contains unparsable \
             \material"
         else if s = tptp_equal then
           list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
             map (do_term NONE) us)
+        else if s = tptp_not_equal then
+          \<^const>\<open>HOL.Not\<close> $ list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
+            map (do_term NONE) us)
         else if not (null us) then
           let
             val args = map (do_term NONE) us
             val opt_T' = SOME (map slack_fastype_of args ---> the_default dummyT opt_T)
             val func = do_term opt_T' (ATerm ((s, tys), []))
           in foldl1 (op $) (func :: args) end
         else if s = tptp_or then HOLogic.disj
         else if s = tptp_and then HOLogic.conj
         else if s = tptp_implies then HOLogic.imp
         else if s = tptp_iff orelse s = tptp_equal then HOLogic.eq_const dummyT
-        else if s = tptp_not_iff orelse s = tptp_not_equal then \<^term>\<open>\<lambda>P Q. Q \<noteq> P\<close>
+        else if s = tptp_not_iff then \<^term>\<open>\<lambda>x y. x \<noteq> y\<close>
         else if s = tptp_if then \<^term>\<open>\<lambda>P Q. Q \<longrightarrow> P\<close>
         else if s = tptp_not_and then \<^term>\<open>\<lambda>P Q. \<not> (P \<and> Q)\<close>
         else if s = tptp_not_or then \<^term>\<open>\<lambda>P Q. \<not> (P \<or> Q)\<close>
         else if s = tptp_not then HOLogic.Not
         else if s = tptp_ho_forall then HOLogic.all_const dummyT
         else if s = tptp_ho_exists then HOLogic.exists_const dummyT
         else if s = tptp_hilbert_choice then HOLogic.choice_const dummyT
         else if s = tptp_hilbert_the then \<^term>\<open>The\<close>
         else
           (case unprefix_and_unascii const_prefix s of
             SOME s' =>
             let
               val ((s', _), mangled_us) = s' |> unmangled_const |>> `invert_const
               val num_ty_args = length us - the_default 0 (Symtab.lookup sym_tab s)
               val (type_us, term_us) = chop num_ty_args us |>> append mangled_us
               val term_ts = map (do_term NONE) term_us
               val Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
               val T =
                 (if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
                    try (Sign.const_instance thy) (s', Ts)
                  else
                    NONE)
                 |> (fn SOME T => T
                      | NONE =>
                        map slack_fastype_of term_ts --->
                        the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
               val t = Const (unproxify_const s', T)
             in list_comb (t, term_ts) end
           | NONE => (* a free or schematic variable *)
             let
               val ts = map (do_term NONE) us
               val T =
                 (case tys of
                   [ty] => typ_of_atp_type ctxt ty
                 | _ =>
                   map slack_fastype_of ts --->
                   (case opt_T of
                     SOME T => T
                   | NONE => Type_Infer.anyT \<^sort>\<open>type\<close>))
               val t =
                 (case unprefix_and_unascii fixed_var_prefix s of
                   SOME s => Free (s, T)
                 | NONE =>
                   if not (is_tptp_variable s) then Free (fresh_up ctxt s, T)
                   else Var ((repair_var_name s, var_index), T))
             in list_comb (t, ts) end))
   in do_term end
 
 (* First-order translation. No types are known for variables. "Type_Infer.anyT @{sort type}"
    should allow them to be inferred. *)
 fun term_of_atp_fo ctxt textual sym_tab =
   let
     val thy = Proof_Context.theory_of ctxt
     (* For Metis, we use 1 rather than 0 because variable references in clauses may otherwise
        conflict with variable constraints in the goal. At least, type inference often fails
        otherwise. See also "axiom_inference" in "Metis_Reconstruct". *)
     val var_index = var_index_of_textual textual
 
     fun do_term extra_ts opt_T u =
       (case u of
         ATerm ((s, tys), us) =>
         if s = "" (* special marker generated on parse error *) then
           error "Isar proof reconstruction failed because the ATP proof contains unparsable \
             \material"
         else if String.isPrefix native_type_prefix s then
           \<^Const>\<open>True\<close> (* ignore TPTP type information (needed?) *)
         else if s = tptp_equal then
           list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
             map (do_term [] NONE) us)
         else
           (case unprefix_and_unascii const_prefix s of
             SOME s' =>
             let val ((s', s''), mangled_us) = s' |> unmangled_const |>> `invert_const in
               if s' = type_tag_name then
                 (case mangled_us @ us of
                   [typ_u, term_u] => do_term extra_ts (SOME (typ_of_atp_term ctxt typ_u)) term_u
                 | _ => raise ATP_TERM us)
               else if s' = predicator_name then
                 do_term [] (SOME \<^typ>\<open>bool\<close>) (hd us)
               else if s' = app_op_name then
                 let val extra_t = do_term [] NONE (List.last us) in
                   do_term (extra_t :: extra_ts)
                     (case opt_T of SOME T => SOME (slack_fastype_of extra_t --> T) | NONE => NONE)
                     (nth us (length us - 2))
                 end
               else if s' = type_guard_name then
                 \<^Const>\<open>True\<close> (* ignore type predicates *)
               else
                 let
                   val new_skolem = String.isPrefix new_skolem_const_prefix s''
                   val num_ty_args = length us - the_default 0 (Symtab.lookup sym_tab s)
                   val (type_us, term_us) = chop num_ty_args us |>> append mangled_us
                   val term_ts = map (do_term [] NONE) term_us
 
                   val Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
                   val T =
                     (if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
                        if new_skolem then SOME (Type_Infer.paramify_vars (tl Ts ---> hd Ts))
                        else if textual then try (Sign.const_instance thy) (s', Ts)
                        else NONE
                      else
                        NONE)
                     |> (fn SOME T => T
                          | NONE =>
                            map slack_fastype_of term_ts --->
                            the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
                   val t =
                     if new_skolem then Var ((new_skolem_var_name_of_const s'', var_index), T)
                     else Const (unproxify_const s', T)
                 in
                   list_comb (t, term_ts @ extra_ts)
                 end
             end
           | NONE => (* a free or schematic variable *)
             let
               val term_ts =
                 map (do_term [] NONE) us
                 (* SPASS (3.8ds) and Vampire (2.6) pass arguments to Skolem functions in reverse
                    order, which is incompatible with "metis"'s new skolemizer. *)
                 |> exists (fn pre => String.isPrefix pre s)
                   [spass_skolem_prefix, vampire_skolem_prefix] ? rev
               val ts = term_ts @ extra_ts
               val T =
                 (case tys of
                   [ty] => typ_of_atp_type ctxt ty
                 | _ =>
                   (case opt_T of
                     SOME T => map slack_fastype_of term_ts ---> T
                   | NONE => map slack_fastype_of ts ---> Type_Infer.anyT \<^sort>\<open>type\<close>))
               val t =
                 (case unprefix_and_unascii fixed_var_prefix s of
                   SOME s => Free (s, T)
                 | NONE =>
                   if textual andalso not (is_tptp_variable s) then
                     Free (s |> textual ? fresh_up ctxt, T)
                   else
                     Var ((repair_var_name s, var_index), T))
             in list_comb (t, ts) end))
   in do_term [] end
 
 fun term_of_atp ctxt (ATP_Problem.THF _) type_enc =
     if ATP_Problem_Generate.is_type_enc_higher_order type_enc then K (term_of_atp_ho ctxt)
     else error "Unsupported Isar reconstruction"
   | term_of_atp ctxt _ type_enc =
     if not (ATP_Problem_Generate.is_type_enc_higher_order type_enc) then term_of_atp_fo ctxt
     else error "Unsupported Isar reconstruction"
 
 fun term_of_atom ctxt format type_enc textual sym_tab pos (u as ATerm ((s, _), _)) =
   if String.isPrefix class_prefix s then
     add_type_constraint pos (type_constraint_of_term ctxt u)
     #> pair \<^Const>\<open>True\<close>
   else
     pair (term_of_atp ctxt format type_enc textual sym_tab (SOME \<^typ>\<open>bool\<close>) u)
 
 (* Update schematic type variables with detected sort constraints. It's not
    totally clear whether this code is necessary. *)
 fun repair_tvar_sorts (t, tvar_tab) =
   let
     fun do_type (Type (a, Ts)) = Type (a, map do_type Ts)
       | do_type (TVar (xi, s)) =
         TVar (xi, the_default s (Vartab.lookup tvar_tab xi))
       | do_type (TFree z) = TFree z
     fun do_term (Const (a, T)) = Const (a, do_type T)
       | do_term (Free (a, T)) = Free (a, do_type T)
       | do_term (Var (xi, T)) = Var (xi, do_type T)
       | do_term (t as Bound _) = t
       | do_term (Abs (a, T, t)) = Abs (a, do_type T, do_term t)
       | do_term (t1 $ t2) = do_term t1 $ do_term t2
   in t |> not (Vartab.is_empty tvar_tab) ? do_term end
 
 (* Interpret an ATP formula as a HOL term, extracting sort constraints as they appear in the
    formula. *)
 fun prop_of_atp ctxt format type_enc textual sym_tab phi =
   let
     fun do_formula pos phi =
       (case phi of
         AQuant (_, [], phi) => do_formula pos phi
       | AQuant (q, (s, _) :: xs, phi') =>
         do_formula pos (AQuant (q, xs, phi'))
         (* FIXME: TFF *)
         #>> quantify_over_var textual (case q of AForall => forall_of | AExists => exists_of)
           (repair_var_name s) dummyT
       | AConn (ANot, [phi']) => do_formula (not pos) phi' #>> s_not
       | AConn (c, [phi1, phi2]) =>
         do_formula (pos |> c = AImplies ? not) phi1
         ##>> do_formula pos phi2
         #>> (case c of
               AAnd => s_conj
             | AOr => s_disj
             | AImplies => s_imp
             | AIff => s_iff
             | ANot => raise Fail "impossible connective")
       | AAtom tm => term_of_atom ctxt format type_enc textual sym_tab pos tm
       | _ => raise ATP_FORMULA [phi])
   in
     repair_tvar_sorts (do_formula true phi Vartab.empty)
   end
 
 val unprefix_fact_number = space_implode "_" o tl o space_explode "_"
 
 fun resolve_fact facts s =
   (case try (unprefix fact_prefix) s of
     SOME s' =>
     let val s' = s' |> unprefix_fact_number |> unascii_of in
       AList.lookup (op =) facts s' |> Option.map (pair s')
     end
   | NONE => NONE)
 
 fun resolve_conjecture s =
   (case try (unprefix conjecture_prefix) s of
     SOME s' => Int.fromString s'
   | NONE => NONE)
 
 fun resolve_facts facts = map_filter (resolve_fact facts)
 val resolve_conjectures = map_filter resolve_conjecture
 
 fun is_axiom_used_in_proof pred =
   exists (fn ((_, ss), _, _, _, []) => exists pred ss | _ => false)
 
 fun add_fact ctxt facts ((num, ss), _, _, rule, deps) =
   (if member (op =) [agsyhol_core_rule, leo2_extcnf_equal_neg_rule] rule orelse
       String.isPrefix satallax_tab_rule_prefix rule then
      insert (op =) (short_thm_name ctxt ext, (Global, General))
    else
      I)
   #> (if null deps then union (op =) (resolve_facts facts (num :: ss)) else I)
 
 fun used_facts_in_atp_proof ctxt facts atp_proof =
   if null atp_proof then facts else fold (add_fact ctxt facts) atp_proof []
 
 fun used_facts_in_unsound_atp_proof _ _ [] = NONE
   | used_facts_in_unsound_atp_proof ctxt facts atp_proof =
     let
       val used_facts = used_facts_in_atp_proof ctxt facts atp_proof
       val all_global_facts = forall (fn (_, (sc, _)) => sc = Global) used_facts
       val axiom_used = is_axiom_used_in_proof (is_some o resolve_conjecture) atp_proof
     in
       if all_global_facts andalso not axiom_used then
         SOME (map fst used_facts)
       else
         NONE
     end
 
 val ascii_of_lam_fact_prefix = ascii_of lam_fact_prefix
 
 (* overapproximation (good enough) *)
 fun is_lam_lifted s =
   String.isPrefix fact_prefix s andalso
   String.isSubstring ascii_of_lam_fact_prefix s
 
 val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)
 
 fun atp_proof_prefers_lifting atp_proof =
   (is_axiom_used_in_proof is_combinator_def atp_proof,
    is_axiom_used_in_proof is_lam_lifted atp_proof) = (false, true)
 
 val is_typed_helper_name =
   String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix
 
 fun is_typed_helper_used_in_atp_proof atp_proof =
   is_axiom_used_in_proof is_typed_helper_name atp_proof
 
 fun replace_one_dependency (old, new) dep = if is_same_atp_step dep old then new else [dep]
 fun replace_dependencies_in_line old_new (name, role, t, rule, deps) =
   (name, role, t, rule, fold (union (op =) o replace_one_dependency old_new) deps [])
 
 fun repair_name "$true" = "c_True"
   | repair_name "$false" = "c_False"
   | repair_name "$$e" = tptp_equal (* seen in Vampire proofs *)
   | repair_name s =
     if is_tptp_equal s orelse
        (* seen in Vampire proofs *)
        (String.isPrefix "sQ" s andalso String.isSuffix "_eqProxy" s) then
       tptp_equal
     else
       s
 
 fun set_var_index j = map_aterms (fn Var ((s, 0), T) => Var ((s, j), T) | t => t)
 
 fun infer_formulas_types ctxt =
   map_index (uncurry (fn j => set_var_index j #> Type.constraint HOLogic.boolT))
   #> Syntax.check_terms (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
-  #> map (set_var_index 0)
 
 val combinator_table =
   [(\<^const_name>\<open>Meson.COMBI\<close>, @{thm Meson.COMBI_def [abs_def]}),
    (\<^const_name>\<open>Meson.COMBK\<close>, @{thm Meson.COMBK_def [abs_def]}),
    (\<^const_name>\<open>Meson.COMBB\<close>, @{thm Meson.COMBB_def [abs_def]}),
    (\<^const_name>\<open>Meson.COMBC\<close>, @{thm Meson.COMBC_def [abs_def]}),
    (\<^const_name>\<open>Meson.COMBS\<close>, @{thm Meson.COMBS_def [abs_def]})]
 
 fun uncombine_term thy =
   let
     fun uncomb (t1 $ t2) = betapply (uncomb t1, uncomb t2)
       | uncomb (Abs (s, T, t)) = Abs (s, T, uncomb t)
       | uncomb (t as Const (x as (s, _))) =
         (case AList.lookup (op =) combinator_table s of
           SOME thm => thm |> Thm.prop_of |> specialize_type thy x |> Logic.dest_equals |> snd
         | NONE => t)
       | uncomb t = t
   in uncomb end
 
 fun unlift_aterm lifted (t as Const (s, _)) =
     if String.isPrefix lam_lifted_prefix s then
       (* FIXME: do something about the types *)
       (case AList.lookup (op =) lifted s of
         SOME t' => unlift_term lifted t'
       | NONE => t)
     else
       t
   | unlift_aterm _ t = t
 and unlift_term lifted =
   map_aterms (unlift_aterm lifted)
 
 fun termify_line _ _ _ _ _ (_, Type_Role, _, _, _) = NONE
   | termify_line ctxt format type_enc lifted sym_tab (name, role, u, rule, deps) =
     let
       val thy = Proof_Context.theory_of ctxt
       val t = u
         |> prop_of_atp ctxt format type_enc true sym_tab
         |> unlift_term lifted
         |> uncombine_term thy
         |> simplify_bool
     in
       SOME (name, role, t, rule, deps)
     end
 
 val waldmeister_conjecture_num = "1.0.0.0"
 
 fun repair_waldmeister_endgame proof =
   let
     fun repair_tail (name, _, \<^Const>\<open>Trueprop for t\<close>, rule, deps) =
       (name, Negated_Conjecture, \<^Const>\<open>Trueprop for \<open>s_not t\<close>\<close>, rule, deps)
     fun repair_body [] = []
       | repair_body ((line as ((num, _), _, _, _, _)) :: lines) =
         if num = waldmeister_conjecture_num then map repair_tail (line :: lines)
         else line :: repair_body lines
   in
     repair_body proof
   end
 
 fun map_proof_terms f (lines : ('a * 'b * 'c * 'd * 'e) list) =
   map2 (fn c => fn (a, b, _, d, e) => (a, b, c, d, e)) (f (map #3 lines)) lines
 
 fun termify_atp_proof ctxt local_prover format type_enc pool lifted sym_tab =
   nasty_atp_proof pool
   #> map_term_names_in_atp_proof repair_name
   #> map_filter (termify_line ctxt format type_enc lifted sym_tab)
   #> map_proof_terms (infer_formulas_types ctxt #> map HOLogic.mk_Trueprop)
   #> local_prover = waldmeisterN ? repair_waldmeister_endgame
 
 fun unskolemize_term skos =
   let
     val is_skolem_name = member (op =) skos
 
     fun find_argless_skolem (Free _ $ Var _) = NONE
       | find_argless_skolem (Free (x as (s, _))) = if is_skolem_name s then SOME x else NONE
       | find_argless_skolem (t $ u) =
         (case find_argless_skolem t of NONE => find_argless_skolem u | sk => sk)
       | find_argless_skolem (Abs (_, _, t)) = find_argless_skolem t
       | find_argless_skolem _ = NONE
 
     fun find_skolem_arg (Free (s, _) $ Var v) = if is_skolem_name s then SOME v else NONE
       | find_skolem_arg (t $ u) = (case find_skolem_arg t of NONE => find_skolem_arg u | v => v)
       | find_skolem_arg (Abs (_, _, t)) = find_skolem_arg t
       | find_skolem_arg _ = NONE
 
     fun kill_skolem_arg (t as Free (s, T) $ Var _) =
         if is_skolem_name s then Free (s, range_type T) else t
       | kill_skolem_arg (t $ u) = kill_skolem_arg t $ kill_skolem_arg u
       | kill_skolem_arg (Abs (s, T, t)) = Abs (s, T, kill_skolem_arg t)
       | kill_skolem_arg t = t
 
     fun find_var (Var v) = SOME v
       | find_var (t $ u) = (case find_var t of NONE => find_var u | v => v)
       | find_var (Abs (_, _, t)) = find_var t
       | find_var _ = NONE
 
     val safe_abstract_over = abstract_over o apsnd (incr_boundvars 1)
 
     fun unskolem t =
       (case find_argless_skolem t of
         SOME (x as (s, T)) =>
         HOLogic.exists_const T $ Abs (s, T, unskolem (safe_abstract_over (Free x, t)))
       | NONE =>
         (case find_skolem_arg t of
           SOME (v as ((s, _), T)) =>
           let
             val (haves, have_nots) =
               HOLogic.disjuncts t
               |> List.partition (exists_subterm (curry (op =) (Var v)))
               |> apply2 (fn lits => fold (curry s_disj) lits \<^term>\<open>False\<close>)
           in
             s_disj (HOLogic.all_const T
                 $ Abs (s, T, unskolem (safe_abstract_over (Var v, kill_skolem_arg haves))),
               unskolem have_nots)
           end
         | NONE =>
           (case find_var t of
             SOME (v as ((s, _), T)) =>
             HOLogic.all_const T $ Abs (s, T, unskolem (safe_abstract_over (Var v, t)))
           | NONE => t)))
   in
     HOLogic.mk_Trueprop o unskolem o HOLogic.dest_Trueprop
   end
 
 fun rename_skolem_args t =
   let
     fun add_skolem_args (Abs (_, _, t)) = add_skolem_args t
       | add_skolem_args t =
         (case strip_comb t of
           (Free (s, _), args as _ :: _) =>
           if String.isPrefix spass_skolem_prefix s then
             insert (op =) (s, take_prefix is_Var args)
           else
             fold add_skolem_args args
         | (u, args as _ :: _) => fold add_skolem_args (u :: args)
         | _ => I)
 
     fun subst_of_skolem (sk, args) =
       map_index (fn (j, Var (z, T)) => (z, Var ((sk ^ "_" ^ string_of_int j, 0), T))) args
 
     val subst = maps subst_of_skolem (add_skolem_args t [])
   in
     subst_vars ([], subst) t
   end
 
 fun introduce_spass_skolems proof =
   let
     fun add_skolem (Free (s, _)) = String.isPrefix spass_skolem_prefix s ? insert (op =) s
       | add_skolem _ = I
 
     (* union-find would be faster *)
     fun add_cycle [] = I
       | add_cycle ss =
         fold (fn s => Graph.default_node (s, ())) ss
         #> fold Graph.add_edge (ss ~~ tl ss @ [hd ss])
 
     val (input_steps, other_steps) = List.partition (null o #5) proof
 
     (* The names of the formulas are added to the Skolem constants, to ensure that formulas giving
        rise to several clauses are skolemized together. *)
     val skoXss = map (fn ((_, ss), _, t, _, _) => Term.fold_aterms add_skolem t ss) input_steps
     val groups0 = Graph.strong_conn (fold add_cycle skoXss Graph.empty)
     val groups = filter (exists (String.isPrefix spass_skolem_prefix)) groups0
 
     val skoXss_input_steps = skoXss ~~ input_steps
 
     fun step_name_of_group skoXs = (implode skoXs, [])
     fun in_group group = member (op =) group o hd
     fun group_of skoX = find_first (fn group => in_group group skoX) groups
 
     fun new_steps (skoXss_steps : (string list * (term, 'a) atp_step) list) group =
       let
         val name = step_name_of_group group
         val name0 = name |>> prefix "0"
         val t =
           (case map (snd #> #3) skoXss_steps of
             [t] => t
           | ts => ts
             |> map (HOLogic.dest_Trueprop #> rename_skolem_args)
             |> Library.foldr1 s_conj
             |> HOLogic.mk_Trueprop)
         val skos =
           fold (union (op =) o filter (String.isPrefix spass_skolem_prefix) o fst) skoXss_steps []
         val deps = map (snd #> #1) skoXss_steps
       in
         [(name0, Unknown, unskolemize_term skos t, spass_pre_skolemize_rule, deps),
          (name, Unknown, t, spass_skolemize_rule, [name0])]
       end
 
     val sko_steps =
       maps (fn group => new_steps (filter (in_group group o fst) skoXss_input_steps) group) groups
 
     val old_news =
       map_filter (fn (skoXs, (name, _, _, _, _)) =>
           Option.map (pair name o single o step_name_of_group) (group_of skoXs))
         skoXss_input_steps
     val repair_deps = fold replace_dependencies_in_line old_news
   in
     input_steps @ sko_steps @ map repair_deps other_steps
   end
 
 fun factify_atp_proof facts hyp_ts concl_t atp_proof =
   let
     fun factify_step ((num, ss), role, t, rule, deps) =
       let
         val (ss', role', t') =
           (case resolve_conjectures ss of
             [j] =>
             if j = length hyp_ts then ([], Conjecture, concl_t)
             else ([], Hypothesis, close_form (nth hyp_ts j))
           | _ =>
             (case resolve_facts facts (num :: ss) of
               [] => (ss, if role = Lemma then Lemma else Plain, t)
             | facts => (map fst facts, Axiom, t)))
       in
         ((num, ss'), role', t', rule, deps)
       end
 
     val atp_proof = map factify_step atp_proof
     val names = map #1 atp_proof
 
     fun repair_dep (num, ss) = (num, the_default ss (AList.lookup (op =) names num))
     fun repair_deps (name, role, t, rule, deps) = (name, role, t, rule, map repair_dep deps)
   in
     map repair_deps atp_proof
   end
 
 end;