src/HOL/Tools/ATP/atp_translate.ML

declare euclidean_simps [simp] at the point they are proved;
avoid duplicate rule warnings;

(*  Title:      HOL/Tools/ATP/atp_translate.ML
    Author:     Fabian Immler, TU Muenchen
    Author:     Makarius
    Author:     Jasmin Blanchette, TU Muenchen

Translation of HOL to FOL for Metis and Sledgehammer.
*)

signature ATP_TRANSLATE =
sig
  type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
  type connective = ATP_Problem.connective
  type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
  type format = ATP_Problem.format
  type formula_kind = ATP_Problem.formula_kind
  type 'a problem = 'a ATP_Problem.problem

  datatype locality =
    General | Helper | Extensionality | Intro | Elim | Simp | Local | Assum |
    Chained

  datatype order = First_Order | Higher_Order
  datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
  datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
  datatype type_level =
    All_Types |
    Noninf_Nonmono_Types of soundness |
    Fin_Nonmono_Types |
    Const_Arg_Types |
    No_Types
  datatype type_uniformity = Uniform | Nonuniform

  datatype type_enc =
    Simple_Types of order * type_level |
    Guards of polymorphism * type_level * type_uniformity |
    Tags of polymorphism * type_level * type_uniformity

  val no_lambdasN : string
  val concealedN : string
  val liftingN : string
  val combinatorsN : string
  val hybridN : string
  val lambdasN : string
  val smartN : string
  val schematic_var_prefix : string
  val fixed_var_prefix : string
  val tvar_prefix : string
  val tfree_prefix : string
  val const_prefix : string
  val type_const_prefix : string
  val class_prefix : string
  val polymorphic_free_prefix : string
  val skolem_const_prefix : string
  val old_skolem_const_prefix : string
  val new_skolem_const_prefix : string
  val type_decl_prefix : string
  val sym_decl_prefix : string
  val guards_sym_formula_prefix : string
  val tags_sym_formula_prefix : string
  val fact_prefix : string
  val conjecture_prefix : string
  val helper_prefix : string
  val class_rel_clause_prefix : string
  val arity_clause_prefix : string
  val tfree_clause_prefix : string
  val typed_helper_suffix : string
  val untyped_helper_suffix : string
  val type_tag_idempotence_helper_name : string
  val predicator_name : string
  val app_op_name : string
  val type_guard_name : string
  val type_tag_name : string
  val simple_type_prefix : string
  val prefixed_predicator_name : string
  val prefixed_app_op_name : string
  val prefixed_type_tag_name : string
  val ascii_of : string -> string
  val unascii_of : string -> string
  val strip_prefix_and_unascii : string -> string -> string option
  val proxy_table : (string * (string * (thm * (string * string)))) list
  val proxify_const : string -> (string * string) option
  val invert_const : string -> string
  val unproxify_const : string -> string
  val new_skolem_var_name_from_const : string -> string
  val atp_irrelevant_consts : string list
  val atp_schematic_consts_of : term -> typ list Symtab.table
  val type_enc_from_string : soundness -> string -> type_enc
  val is_type_enc_higher_order : type_enc -> bool
  val polymorphism_of_type_enc : type_enc -> polymorphism
  val level_of_type_enc : type_enc -> type_level
  val is_type_enc_quasi_sound : type_enc -> bool
  val is_type_enc_fairly_sound : type_enc -> bool
  val adjust_type_enc : format -> type_enc -> type_enc
  val mk_aconns :
    connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
  val unmangled_const : string -> string * (string, 'b) ho_term list
  val unmangled_const_name : string -> string
  val helper_table : ((string * bool) * thm list) list
  val factsN : string
  val prepare_atp_problem :
    Proof.context -> format -> formula_kind -> formula_kind -> type_enc
    -> bool -> string -> bool -> bool -> term list -> term
    -> ((string * locality) * term) list
    -> string problem * string Symtab.table * int * int
       * (string * locality) list vector * int list * int Symtab.table
  val atp_problem_weights : string problem -> (string * real) list
end;

structure ATP_Translate : ATP_TRANSLATE =
struct

open ATP_Util
open ATP_Problem

type name = string * string

val no_lambdasN = "no_lambdas"
val concealedN = "concealed"
val liftingN = "lifting"
val combinatorsN = "combinators"
val hybridN = "hybrid"
val lambdasN = "lambdas"
val smartN = "smart"

val generate_info = false (* experimental *)

fun isabelle_info s =
  if generate_info then SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
  else NONE

val introN = "intro"
val elimN = "elim"
val simpN = "simp"

val bound_var_prefix = "B_"
val all_bound_var_prefix = "BA_"
val exist_bound_var_prefix = "BE_"
val schematic_var_prefix = "V_"
val fixed_var_prefix = "v_"
val tvar_prefix = "T_"
val tfree_prefix = "t_"
val const_prefix = "c_"
val type_const_prefix = "tc_"
val class_prefix = "cl_"

val polymorphic_free_prefix = "poly_free"

val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"
val old_skolem_const_prefix = skolem_const_prefix ^ "o"
val new_skolem_const_prefix = skolem_const_prefix ^ "n"

val type_decl_prefix = "ty_"
val sym_decl_prefix = "sy_"
val guards_sym_formula_prefix = "gsy_"
val tags_sym_formula_prefix = "tsy_"
val fact_prefix = "fact_"
val conjecture_prefix = "conj_"
val helper_prefix = "help_"
val class_rel_clause_prefix = "clar_"
val arity_clause_prefix = "arity_"
val tfree_clause_prefix = "tfree_"

val lambda_fact_prefix = "ATP.lambda_"
val typed_helper_suffix = "_T"
val untyped_helper_suffix = "_U"
val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"

val predicator_name = "hBOOL"
val app_op_name = "hAPP"
val type_guard_name = "g"
val type_tag_name = "t"
val simple_type_prefix = "ty_"

val prefixed_predicator_name = const_prefix ^ predicator_name
val prefixed_app_op_name = const_prefix ^ app_op_name
val prefixed_type_tag_name = const_prefix ^ type_tag_name

(* Freshness almost guaranteed! *)
val atp_weak_prefix = "ATP:"

(*Escaping of special characters.
  Alphanumeric characters are left unchanged.
  The character _ goes to __
  Characters in the range ASCII space to / go to _A to _P, respectively.
  Other characters go to _nnn where nnn is the decimal ASCII code.*)
val upper_a_minus_space = Char.ord #"A" - Char.ord #" "

fun stringN_of_int 0 _ = ""
  | stringN_of_int k n =
    stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)

fun ascii_of_char c =
  if Char.isAlphaNum c then
    String.str c
  else if c = #"_" then
    "__"
  else if #" " <= c andalso c <= #"/" then
    "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
  else
    (* fixed width, in case more digits follow *)
    "_" ^ stringN_of_int 3 (Char.ord c)

val ascii_of = String.translate ascii_of_char

(** Remove ASCII armoring from names in proof files **)

(* We don't raise error exceptions because this code can run inside a worker
   thread. Also, the errors are impossible. *)
val unascii_of =
  let
    fun un rcs [] = String.implode(rev rcs)
      | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
        (* Three types of _ escapes: __, _A to _P, _nnn *)
      | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
      | un rcs (#"_" :: c :: cs) =
        if #"A" <= c andalso c<= #"P" then
          (* translation of #" " to #"/" *)
          un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
        else
          let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
            case Int.fromString (String.implode digits) of
              SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
            | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
          end
      | un rcs (c :: cs) = un (c :: rcs) cs
  in un [] o String.explode end

(* If string s has the prefix s1, return the result of deleting it,
   un-ASCII'd. *)
fun strip_prefix_and_unascii s1 s =
  if String.isPrefix s1 s then
    SOME (unascii_of (String.extract (s, size s1, NONE)))
  else
    NONE

val proxy_table =
  [("c_False", (@{const_name False}, (@{thm fFalse_def},
       ("fFalse", @{const_name ATP.fFalse})))),
   ("c_True", (@{const_name True}, (@{thm fTrue_def},
       ("fTrue", @{const_name ATP.fTrue})))),
   ("c_Not", (@{const_name Not}, (@{thm fNot_def},
       ("fNot", @{const_name ATP.fNot})))),
   ("c_conj", (@{const_name conj}, (@{thm fconj_def},
       ("fconj", @{const_name ATP.fconj})))),
   ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
       ("fdisj", @{const_name ATP.fdisj})))),
   ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
       ("fimplies", @{const_name ATP.fimplies})))),
   ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
       ("fequal", @{const_name ATP.fequal})))),
   ("c_All", (@{const_name All}, (@{thm fAll_def},
       ("fAll", @{const_name ATP.fAll})))),
   ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
       ("fEx", @{const_name ATP.fEx}))))]

val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)

(* Readable names for the more common symbolic functions. Do not mess with the
   table unless you know what you are doing. *)
val const_trans_table =
  [(@{type_name Product_Type.prod}, "prod"),
   (@{type_name Sum_Type.sum}, "sum"),
   (@{const_name False}, "False"),
   (@{const_name True}, "True"),
   (@{const_name Not}, "Not"),
   (@{const_name conj}, "conj"),
   (@{const_name disj}, "disj"),
   (@{const_name implies}, "implies"),
   (@{const_name HOL.eq}, "equal"),
   (@{const_name All}, "All"),
   (@{const_name Ex}, "Ex"),
   (@{const_name If}, "If"),
   (@{const_name Set.member}, "member"),
   (@{const_name Meson.COMBI}, "COMBI"),
   (@{const_name Meson.COMBK}, "COMBK"),
   (@{const_name Meson.COMBB}, "COMBB"),
   (@{const_name Meson.COMBC}, "COMBC"),
   (@{const_name Meson.COMBS}, "COMBS")]
  |> Symtab.make
  |> fold (Symtab.update o swap o snd o snd o snd) proxy_table

(* Invert the table of translations between Isabelle and ATPs. *)
val const_trans_table_inv =
  const_trans_table |> Symtab.dest |> map swap |> Symtab.make
val const_trans_table_unprox =
  Symtab.empty
  |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table

val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)

fun lookup_const c =
  case Symtab.lookup const_trans_table c of
    SOME c' => c'
  | NONE => ascii_of c

fun ascii_of_indexname (v, 0) = ascii_of v
  | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i

fun make_bound_var x = bound_var_prefix ^ ascii_of x
fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
fun make_fixed_var x = fixed_var_prefix ^ ascii_of x

fun make_schematic_type_var (x, i) =
      tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))

(* "HOL.eq" is mapped to the ATP's equality. *)
fun make_fixed_const @{const_name HOL.eq} = tptp_old_equal
  | make_fixed_const c = const_prefix ^ lookup_const c

fun make_fixed_type_const c = type_const_prefix ^ lookup_const c

fun make_type_class clas = class_prefix ^ ascii_of clas

fun new_skolem_var_name_from_const s =
  let val ss = s |> space_explode Long_Name.separator in
    nth ss (length ss - 2)
  end

(* These are either simplified away by "Meson.presimplify" (most of the time) or
   handled specially via "fFalse", "fTrue", ..., "fequal". *)
val atp_irrelevant_consts =
  [@{const_name False}, @{const_name True}, @{const_name Not},
   @{const_name conj}, @{const_name disj}, @{const_name implies},
   @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]

val atp_monomorph_bad_consts =
  atp_irrelevant_consts @
  (* These are ignored anyway by the relevance filter (unless they appear in
     higher-order places) but not by the monomorphizer. *)
  [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
   @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
   @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]

fun add_schematic_const (x as (_, T)) =
  Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
val add_schematic_consts_of =
  Term.fold_aterms (fn Const (x as (s, _)) =>
                       not (member (op =) atp_monomorph_bad_consts s)
                       ? add_schematic_const x
                      | _ => I)
fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty

(** Definitions and functions for FOL clauses and formulas for TPTP **)

(* The first component is the type class; the second is a "TVar" or "TFree". *)
datatype type_literal =
  TyLitVar of name * name |
  TyLitFree of name * name


(** Isabelle arities **)

datatype arity_literal =
  TConsLit of name * name * name list |
  TVarLit of name * name

fun gen_TVars 0 = []
  | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)

val type_class = the_single @{sort type}

fun add_packed_sort tvar =
  fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))

type arity_clause =
  {name : string,
   prem_lits : arity_literal list,
   concl_lits : arity_literal}

(* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
fun make_axiom_arity_clause (tcons, name, (cls, args)) =
  let
    val tvars = gen_TVars (length args)
    val tvars_srts = ListPair.zip (tvars, args)
  in
    {name = name,
     prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,
     concl_lits = TConsLit (`make_type_class cls,
                            `make_fixed_type_const tcons,
                            tvars ~~ tvars)}
  end

fun arity_clause _ _ (_, []) = []
  | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)
    arity_clause seen n (tcons, ars)
  | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
    if member (op =) seen class then
      (* multiple arities for the same (tycon, class) pair *)
      make_axiom_arity_clause (tcons,
          lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
          ar) ::
      arity_clause seen (n + 1) (tcons, ars)
    else
      make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
                               ascii_of class, ar) ::
      arity_clause (class :: seen) n (tcons, ars)

fun multi_arity_clause [] = []
  | multi_arity_clause ((tcons, ars) :: tc_arlists) =
      arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists

(* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
   theory thy provided its arguments have the corresponding sorts. *)
fun type_class_pairs thy tycons classes =
  let
    val alg = Sign.classes_of thy
    fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
    fun add_class tycon class =
      cons (class, domain_sorts tycon class)
      handle Sorts.CLASS_ERROR _ => I
    fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
  in map try_classes tycons end

(*Proving one (tycon, class) membership may require proving others, so iterate.*)
fun iter_type_class_pairs _ _ [] = ([], [])
  | iter_type_class_pairs thy tycons classes =
      let
        fun maybe_insert_class s =
          (s <> type_class andalso not (member (op =) classes s))
          ? insert (op =) s
        val cpairs = type_class_pairs thy tycons classes
        val newclasses =
          [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
        val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
      in (classes' @ classes, union (op =) cpairs' cpairs) end

fun make_arity_clauses thy tycons =
  iter_type_class_pairs thy tycons ##> multi_arity_clause


(** Isabelle class relations **)

type class_rel_clause =
  {name : string,
   subclass : name,
   superclass : name}

(* Generate all pairs (sub, super) such that sub is a proper subclass of super
   in theory "thy". *)
fun class_pairs _ [] _ = []
  | class_pairs thy subs supers =
      let
        val class_less = Sorts.class_less (Sign.classes_of thy)
        fun add_super sub super = class_less (sub, super) ? cons (sub, super)
        fun add_supers sub = fold (add_super sub) supers
      in fold add_supers subs [] end

fun make_class_rel_clause (sub, super) =
  {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
   superclass = `make_type_class super}

fun make_class_rel_clauses thy subs supers =
  map make_class_rel_clause (class_pairs thy subs supers)

(* intermediate terms *)
datatype iterm =
  IConst of name * typ * typ list |
  IVar of name * typ |
  IApp of iterm * iterm |
  IAbs of (name * typ) * iterm

fun ityp_of (IConst (_, T, _)) = T
  | ityp_of (IVar (_, T)) = T
  | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
  | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm

(*gets the head of a combinator application, along with the list of arguments*)
fun strip_iterm_comb u =
  let
    fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
      | stripc x = x
  in stripc (u, []) end

fun atyps_of T = fold_atyps (insert (op =)) T []

fun new_skolem_const_name s num_T_args =
  [new_skolem_const_prefix, s, string_of_int num_T_args]
  |> space_implode Long_Name.separator

(* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
   Also accumulates sort infomation. *)
fun iterm_from_term thy bs (P $ Q) =
    let
      val (P', P_atomics_Ts) = iterm_from_term thy bs P
      val (Q', Q_atomics_Ts) = iterm_from_term thy bs Q
    in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
  | iterm_from_term thy _ (Const (c, T)) =
    (IConst (`make_fixed_const c, T,
             if String.isPrefix old_skolem_const_prefix c then
               [] |> Term.add_tvarsT T |> map TVar
             else
               (c, T) |> Sign.const_typargs thy),
     atyps_of T)
  | iterm_from_term _ _ (Free (s, T)) =
    (IConst (`make_fixed_var s, T,
             if String.isPrefix polymorphic_free_prefix s then [T] else []),
     atyps_of T)
  | iterm_from_term _ _ (Var (v as (s, _), T)) =
    (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
       let
         val Ts = T |> strip_type |> swap |> op ::
         val s' = new_skolem_const_name s (length Ts)
       in IConst (`make_fixed_const s', T, Ts) end
     else
       IVar ((make_schematic_var v, s), T), atyps_of T)
  | iterm_from_term _ bs (Bound j) =
    nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))
  | iterm_from_term thy bs (Abs (s, T, t)) =
    let
      fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
      val s = vary s
      val name = `make_bound_var s
      val (tm, atomic_Ts) = iterm_from_term thy ((s, (name, T)) :: bs) t
    in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end

datatype locality =
  General | Helper | Extensionality | Intro | Elim | Simp | Local | Assum |
  Chained

datatype order = First_Order | Higher_Order
datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
datatype type_level =
  All_Types |
  Noninf_Nonmono_Types of soundness |
  Fin_Nonmono_Types |
  Const_Arg_Types |
  No_Types
datatype type_uniformity = Uniform | Nonuniform

datatype type_enc =
  Simple_Types of order * type_level |
  Guards of polymorphism * type_level * type_uniformity |
  Tags of polymorphism * type_level * type_uniformity

fun try_unsuffixes ss s =
  fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE

fun type_enc_from_string soundness s =
  (case try (unprefix "poly_") s of
     SOME s => (SOME Polymorphic, s)
   | NONE =>
     case try (unprefix "mono_") s of
       SOME s => (SOME Monomorphic, s)
     | NONE =>
       case try (unprefix "mangled_") s of
         SOME s => (SOME Mangled_Monomorphic, s)
       | NONE => (NONE, s))
  ||> (fn s =>
          (* "_query" and "_bang" are for the ASCII-challenged Metis and
             Mirabelle. *)
          case try_unsuffixes ["?", "_query"] s of
            SOME s => (Noninf_Nonmono_Types soundness, s)
          | NONE =>
            case try_unsuffixes ["!", "_bang"] s of
              SOME s => (Fin_Nonmono_Types, s)
            | NONE => (All_Types, s))
  ||> apsnd (fn s =>
                case try (unsuffix "_uniform") s of
                  SOME s => (Uniform, s)
                | NONE => (Nonuniform, s))
  |> (fn (poly, (level, (uniformity, core))) =>
         case (core, (poly, level, uniformity)) of
           ("simple", (NONE, _, Nonuniform)) =>
           Simple_Types (First_Order, level)
         | ("simple_higher", (NONE, _, Nonuniform)) =>
           (case level of
              Noninf_Nonmono_Types _ => raise Same.SAME
            | _ => Simple_Types (Higher_Order, level))
         | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)
         | ("tags", (SOME Polymorphic, _, _)) =>
           Tags (Polymorphic, level, uniformity)
         | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)
         | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>
           Guards (poly, Const_Arg_Types, Nonuniform)
         | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>
           Guards (Polymorphic, No_Types, Nonuniform)
         | _ => raise Same.SAME)
  handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")

fun is_type_enc_higher_order (Simple_Types (Higher_Order, _)) = true
  | is_type_enc_higher_order _ = false

fun polymorphism_of_type_enc (Simple_Types _) = Mangled_Monomorphic
  | polymorphism_of_type_enc (Guards (poly, _, _)) = poly
  | polymorphism_of_type_enc (Tags (poly, _, _)) = poly

fun level_of_type_enc (Simple_Types (_, level)) = level
  | level_of_type_enc (Guards (_, level, _)) = level
  | level_of_type_enc (Tags (_, level, _)) = level

fun uniformity_of_type_enc (Simple_Types _) = Uniform
  | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity
  | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity

fun is_type_level_quasi_sound All_Types = true
  | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
  | is_type_level_quasi_sound _ = false
val is_type_enc_quasi_sound =
  is_type_level_quasi_sound o level_of_type_enc

fun is_type_level_fairly_sound level =
  is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types
val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc

fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
  | is_type_level_monotonicity_based Fin_Nonmono_Types = true
  | is_type_level_monotonicity_based _ = false

fun adjust_type_enc (THF _) type_enc = type_enc
  | adjust_type_enc TFF (Simple_Types (_, level)) =
    Simple_Types (First_Order, level)
  | adjust_type_enc format (Simple_Types (_, level)) =
    adjust_type_enc format (Guards (Mangled_Monomorphic, level, Uniform))
  | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
    (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
  | adjust_type_enc _ type_enc = type_enc

fun lift_lambdas ctxt type_enc =
  map (close_form o Envir.eta_contract) #> rpair ctxt
  #-> Lambda_Lifting.lift_lambdas
          (if polymorphism_of_type_enc type_enc = Polymorphic then
             SOME polymorphic_free_prefix
           else
             NONE)
          Lambda_Lifting.is_quantifier
  #> fst

fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
    intentionalize_def t
  | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
    let
      fun lam T t = Abs (Name.uu, T, t)
      val (head, args) = strip_comb t ||> rev
      val head_T = fastype_of head
      val n = length args
      val arg_Ts = head_T |> binder_types |> take n |> rev
      val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
    in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
  | intentionalize_def t = t

type translated_formula =
  {name : string,
   locality : locality,
   kind : formula_kind,
   iformula : (name, typ, iterm) formula,
   atomic_types : typ list}

fun update_iformula f ({name, locality, kind, iformula, atomic_types}
                       : translated_formula) =
  {name = name, locality = locality, kind = kind, iformula = f iformula,
   atomic_types = atomic_types} : translated_formula

fun fact_lift f ({iformula, ...} : translated_formula) = f iformula

fun insert_type ctxt get_T x xs =
  let val T = get_T x in
    if exists (type_instance ctxt T o get_T) xs then xs
    else x :: filter_out (type_generalization ctxt T o get_T) xs
  end

(* The Booleans indicate whether all type arguments should be kept. *)
datatype type_arg_policy =
  Explicit_Type_Args of bool |
  Mangled_Type_Args of bool |
  No_Type_Args

fun should_drop_arg_type_args (Simple_Types _) =
    false (* since TFF doesn't support overloading *)
  | should_drop_arg_type_args type_enc =
    level_of_type_enc type_enc = All_Types andalso
    uniformity_of_type_enc type_enc = Uniform

fun type_arg_policy type_enc s =
  if s = type_tag_name then
    (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
       Mangled_Type_Args
     else
       Explicit_Type_Args) false
  else case type_enc of
    Tags (_, All_Types, Uniform) => No_Type_Args
  | _ =>
    let val level = level_of_type_enc type_enc in
      if level = No_Types orelse s = @{const_name HOL.eq} orelse
         (s = app_op_name andalso level = Const_Arg_Types) then
        No_Type_Args
      else
        should_drop_arg_type_args type_enc
        |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
              Mangled_Type_Args
            else
              Explicit_Type_Args)
    end

(* Make literals for sorted type variables. *)
fun generic_add_sorts_on_type (_, []) = I
  | generic_add_sorts_on_type ((x, i), s :: ss) =
    generic_add_sorts_on_type ((x, i), ss)
    #> (if s = the_single @{sort HOL.type} then
          I
        else if i = ~1 then
          insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))
        else
          insert (op =) (TyLitVar (`make_type_class s,
                                   (make_schematic_type_var (x, i), x))))
fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
  | add_sorts_on_tfree _ = I
fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
  | add_sorts_on_tvar _ = I

fun type_literals_for_types type_enc add_sorts_on_typ Ts =
  [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts

fun mk_aconns c phis =
  let val (phis', phi') = split_last phis in
    fold_rev (mk_aconn c) phis' phi'
  end
fun mk_ahorn [] phi = phi
  | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
fun mk_aquant _ [] phi = phi
  | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
    if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
  | mk_aquant q xs phi = AQuant (q, xs, phi)

fun close_universally atom_vars phi =
  let
    fun formula_vars bounds (AQuant (_, xs, phi)) =
        formula_vars (map fst xs @ bounds) phi
      | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
      | formula_vars bounds (AAtom tm) =
        union (op =) (atom_vars tm []
                      |> filter_out (member (op =) bounds o fst))
  in mk_aquant AForall (formula_vars [] phi []) phi end

fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]
  | iterm_vars (IConst _) = I
  | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)
  | iterm_vars (IAbs (_, tm)) = iterm_vars tm
fun close_iformula_universally phi = close_universally iterm_vars phi

fun term_vars bounds (ATerm (name as (s, _), tms)) =
    (is_tptp_variable s andalso not (member (op =) bounds name))
    ? insert (op =) (name, NONE) #> fold (term_vars bounds) tms
  | term_vars bounds (AAbs ((name, _), tm)) = term_vars (name :: bounds) tm
fun close_formula_universally phi = close_universally (term_vars []) phi

val homo_infinite_type_name = @{type_name ind} (* any infinite type *)
val homo_infinite_type = Type (homo_infinite_type_name, [])

fun ho_term_from_typ format type_enc =
  let
    fun term (Type (s, Ts)) =
      ATerm (case (is_type_enc_higher_order type_enc, s) of
               (true, @{type_name bool}) => `I tptp_bool_type
             | (true, @{type_name fun}) => `I tptp_fun_type
             | _ => if s = homo_infinite_type_name andalso
                       is_format_typed format then
                      `I tptp_individual_type
                    else
                      `make_fixed_type_const s,
             map term Ts)
    | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
    | term (TVar ((x as (s, _)), _)) =
      ATerm ((make_schematic_type_var x, s), [])
  in term end

fun ho_term_for_type_arg format type_enc T =
  if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)

(* This shouldn't clash with anything else. *)
val mangled_type_sep = "\000"

fun generic_mangled_type_name f (ATerm (name, [])) = f name
  | generic_mangled_type_name f (ATerm (name, tys)) =
    f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
    ^ ")"
  | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"

fun mangled_type format type_enc =
  generic_mangled_type_name fst o ho_term_from_typ format type_enc

val bool_atype = AType (`I tptp_bool_type)

fun make_simple_type s =
  if s = tptp_bool_type orelse s = tptp_fun_type orelse
     s = tptp_individual_type then
    s
  else
    simple_type_prefix ^ ascii_of s

fun ho_type_from_ho_term type_enc pred_sym ary =
  let
    fun to_atype ty =
      AType ((make_simple_type (generic_mangled_type_name fst ty),
              generic_mangled_type_name snd ty))
    fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
    fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
      | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
      | to_fo _ _ = raise Fail "unexpected type abstraction"
    fun to_ho (ty as ATerm ((s, _), tys)) =
        if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
      | to_ho _ = raise Fail "unexpected type abstraction"
  in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end

fun ho_type_from_typ format type_enc pred_sym ary =
  ho_type_from_ho_term type_enc pred_sym ary
  o ho_term_from_typ format type_enc

fun mangled_const_name format type_enc T_args (s, s') =
  let
    val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)
    fun type_suffix f g =
      fold_rev (curry (op ^) o g o prefix mangled_type_sep
                o generic_mangled_type_name f) ty_args ""
  in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end

val parse_mangled_ident =
  Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode

fun parse_mangled_type x =
  (parse_mangled_ident
   -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
                    [] >> ATerm) x
and parse_mangled_types x =
  (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x

fun unmangled_type s =
  s |> suffix ")" |> raw_explode
    |> Scan.finite Symbol.stopper
           (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
                                                quote s)) parse_mangled_type))
    |> fst

val unmangled_const_name = space_explode mangled_type_sep #> hd
fun unmangled_const s =
  let val ss = space_explode mangled_type_sep s in
    (hd ss, map unmangled_type (tl ss))
  end

fun introduce_proxies type_enc =
  let
    fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
      | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
                       _ =
        (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
           limitation. This works in conjuction with special code in
           "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
           possible. *)
        IAbs ((`I "P", p_T),
              IApp (IConst (`I ho_quant, T, []),
                    IAbs ((`I "X", x_T),
                          IApp (IConst (`I "P", p_T, []),
                                IConst (`I "X", x_T, [])))))
      | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
    fun intro top_level args (IApp (tm1, tm2)) =
        IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
      | intro top_level args (IConst (name as (s, _), T, T_args)) =
        (case proxify_const s of
           SOME proxy_base =>
           if top_level orelse is_type_enc_higher_order type_enc then
             case (top_level, s) of
               (_, "c_False") => IConst (`I tptp_false, T, [])
             | (_, "c_True") => IConst (`I tptp_true, T, [])
             | (false, "c_Not") => IConst (`I tptp_not, T, [])
             | (false, "c_conj") => IConst (`I tptp_and, T, [])
             | (false, "c_disj") => IConst (`I tptp_or, T, [])
             | (false, "c_implies") => IConst (`I tptp_implies, T, [])
             | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
             | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
             | (false, s) =>
               if is_tptp_equal s andalso length args = 2 then
                 IConst (`I tptp_equal, T, [])
               else
                 (* Use a proxy even for partially applied THF equality, because
                    the LEO-II and Satallax parsers complain about not being
                    able to infer the type of "=". *)
                 IConst (proxy_base |>> prefix const_prefix, T, T_args)
             | _ => IConst (name, T, [])
           else
             IConst (proxy_base |>> prefix const_prefix, T, T_args)
          | NONE => IConst (name, T, T_args))
      | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
      | intro _ _ tm = tm
  in intro true [] end

fun iformula_from_prop thy type_enc eq_as_iff =
  let
    fun do_term bs t atomic_types =
      iterm_from_term thy bs (Envir.eta_contract t)
      |>> (introduce_proxies type_enc #> AAtom)
      ||> union (op =) atomic_types
    fun do_quant bs q pos s T t' =
      let
        val s = singleton (Name.variant_list (map fst bs)) s
        val universal = Option.map (q = AExists ? not) pos
        val name =
          s |> `(case universal of
                   SOME true => make_all_bound_var
                 | SOME false => make_exist_bound_var
                 | NONE => make_bound_var)
      in
        do_formula ((s, (name, T)) :: bs) pos t'
        #>> mk_aquant q [(name, SOME T)]
      end
    and do_conn bs c pos1 t1 pos2 t2 =
      do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
    and do_formula bs pos t =
      case t of
        @{const Trueprop} $ t1 => do_formula bs pos t1
      | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
      | Const (@{const_name All}, _) $ Abs (s, T, t') =>
        do_quant bs AForall pos s T t'
      | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
        do_quant bs AExists pos s T t'
      | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
      | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
      | @{const HOL.implies} $ t1 $ t2 =>
        do_conn bs AImplies (Option.map not pos) t1 pos t2
      | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
        if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
      | _ => do_term bs t
  in do_formula [] end

fun presimplify_term _ [] t = t
  | presimplify_term ctxt presimp_consts t =
    t |> exists_Const (member (op =) presimp_consts o fst) t
         ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
            #> Meson.presimplify ctxt
            #> prop_of)

fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
fun conceal_bounds Ts t =
  subst_bounds (map (Free o apfst concealed_bound_name)
                    (0 upto length Ts - 1 ~~ Ts), t)
fun reveal_bounds Ts =
  subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
                    (0 upto length Ts - 1 ~~ Ts))

fun is_fun_equality (@{const_name HOL.eq},
                     Type (_, [Type (@{type_name fun}, _), _])) = true
  | is_fun_equality _ = false

fun extensionalize_term ctxt t =
  if exists_Const is_fun_equality t then
    let val thy = Proof_Context.theory_of ctxt in
      t |> cterm_of thy |> Meson.extensionalize_conv ctxt
        |> prop_of |> Logic.dest_equals |> snd
    end
  else
    t

fun simple_translate_lambdas do_lambdas ctxt t =
  let val thy = Proof_Context.theory_of ctxt in
    if Meson.is_fol_term thy t then
      t
    else
      let
        fun aux Ts t =
          case t of
            @{const Not} $ t1 => @{const Not} $ aux Ts t1
          | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
            t0 $ Abs (s, T, aux (T :: Ts) t')
          | (t0 as Const (@{const_name All}, _)) $ t1 =>
            aux Ts (t0 $ eta_expand Ts t1 1)
          | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
            t0 $ Abs (s, T, aux (T :: Ts) t')
          | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
            aux Ts (t0 $ eta_expand Ts t1 1)
          | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
          | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
          | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
          | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
              $ t1 $ t2 =>
            t0 $ aux Ts t1 $ aux Ts t2
          | _ =>
            if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
            else t |> Envir.eta_contract |> do_lambdas ctxt Ts
        val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
      in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
  end

fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
    do_cheaply_conceal_lambdas Ts t1
    $ do_cheaply_conceal_lambdas Ts t2
  | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
    Free (polymorphic_free_prefix ^ serial_string (),
          T --> fastype_of1 (T :: Ts, t))
  | do_cheaply_conceal_lambdas _ t = t

fun do_introduce_combinators ctxt Ts t =
  let val thy = Proof_Context.theory_of ctxt in
    t |> conceal_bounds Ts
      |> cterm_of thy
      |> Meson_Clausify.introduce_combinators_in_cterm
      |> prop_of |> Logic.dest_equals |> snd
      |> reveal_bounds Ts
  end
  (* A type variable of sort "{}" will make abstraction fail. *)
  handle THM _ => t |> do_cheaply_conceal_lambdas Ts
val introduce_combinators = simple_translate_lambdas do_introduce_combinators

fun preprocess_abstractions_in_terms trans_lambdas facts =
  let
    val (facts, lambda_ts) =
      facts |> map (snd o snd) |> trans_lambdas 
            |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
    val lambda_facts =
      map2 (fn t => fn j =>
               ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
           lambda_ts (1 upto length lambda_ts)
  in (facts, lambda_facts) end

(* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
   same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
fun freeze_term t =
  let
    fun aux (t $ u) = aux t $ aux u
      | aux (Abs (s, T, t)) = Abs (s, T, aux t)
      | aux (Var ((s, i), T)) =
        Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
      | aux t = t
  in t |> exists_subterm is_Var t ? aux end

fun presimp_prop ctxt presimp_consts t =
  let
    val thy = Proof_Context.theory_of ctxt
    val t = t |> Envir.beta_eta_contract
              |> transform_elim_prop
              |> Object_Logic.atomize_term thy
    val need_trueprop = (fastype_of t = @{typ bool})
  in
    t |> need_trueprop ? HOLogic.mk_Trueprop
      |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
      |> extensionalize_term ctxt
      |> presimplify_term ctxt presimp_consts
      |> perhaps (try (HOLogic.dest_Trueprop))
  end

(* making fact and conjecture formulas *)
fun make_formula thy type_enc eq_as_iff name loc kind t =
  let
    val (iformula, atomic_types) =
      iformula_from_prop thy type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
  in
    {name = name, locality = loc, kind = kind, iformula = iformula,
     atomic_types = atomic_types}
  end

fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  let val thy = Proof_Context.theory_of ctxt in
    case t |> make_formula thy type_enc (eq_as_iff andalso format <> CNF) name
                           loc Axiom of
      formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
      if s = tptp_true then NONE else SOME formula
    | formula => SOME formula
  end

fun make_conjecture ctxt format type_enc ps =
  let
    val thy = Proof_Context.theory_of ctxt
    val last = length ps - 1
  in
    map2 (fn j => fn ((name, loc), (kind, t)) =>
             t |> make_formula thy type_enc (format <> CNF) name loc kind
               |> (j <> last) = (kind = Conjecture) ? update_iformula mk_anot)
         (0 upto last) ps
  end

(** Finite and infinite type inference **)

type monotonicity_info =
  {maybe_finite_Ts : typ list,
   surely_finite_Ts : typ list,
   maybe_infinite_Ts : typ list,
   surely_infinite_Ts : typ list,
   maybe_nonmono_Ts : typ list}

(* These types witness that the type classes they belong to allow infinite
   models and hence that any types with these type classes is monotonic. *)
val known_infinite_types =
  [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]

fun is_type_surely_infinite' ctxt soundness cached_Ts T =
  (* Unlike virtually any other polymorphic fact whose type variables can be
     instantiated by a known infinite type, extensionality actually encodes a
     cardinality constraints. *)
  soundness <> Sound andalso
  is_type_surely_infinite ctxt (soundness = Unsound) cached_Ts T

(* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
   dangerous because their "exhaust" properties can easily lead to unsound ATP
   proofs. On the other hand, all HOL infinite types can be given the same
   models in first-order logic (via Löwenheim-Skolem). *)

fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
                             maybe_nonmono_Ts, ...}
                       (Noninf_Nonmono_Types soundness) T =
    exists (type_instance ctxt T orf type_generalization ctxt T)
           maybe_nonmono_Ts andalso
    not (exists (type_instance ctxt T) surely_infinite_Ts orelse
         (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
          is_type_surely_infinite' ctxt soundness surely_infinite_Ts T))
  | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
                             maybe_nonmono_Ts, ...}
                       Fin_Nonmono_Types T =
    exists (type_instance ctxt T orf type_generalization ctxt T)
           maybe_nonmono_Ts andalso
    (exists (type_instance ctxt T) surely_finite_Ts orelse
     (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
      is_type_surely_finite ctxt T))
  | should_encode_type _ _ _ _ = false

fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var
                      T =
    (uniformity = Uniform orelse should_guard_var ()) andalso
    should_encode_type ctxt mono level T
  | should_guard_type _ _ _ _ _ = false

fun is_maybe_universal_var (IConst ((s, _), _, _)) =
    String.isPrefix bound_var_prefix s orelse
    String.isPrefix all_bound_var_prefix s
  | is_maybe_universal_var (IVar _) = true
  | is_maybe_universal_var _ = false

datatype tag_site =
  Top_Level of bool option |
  Eq_Arg of bool option |
  Elsewhere

fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =
    (case uniformity of
       Uniform => should_encode_type ctxt mono level T
     | Nonuniform =>
       case (site, is_maybe_universal_var u) of
         (Eq_Arg _, true) => should_encode_type ctxt mono level T
       | _ => false)
  | should_tag_with_type _ _ _ _ _ _ = false

fun homogenized_type ctxt mono level =
  let
    val should_encode = should_encode_type ctxt mono level
    fun homo 0 T = if should_encode T then T else homo_infinite_type
      | homo ary (Type (@{type_name fun}, [T1, T2])) =
        homo 0 T1 --> homo (ary - 1) T2
      | homo _ _ = raise Fail "expected function type"
  in homo end

(** "hBOOL" and "hAPP" **)

type sym_info =
  {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}

fun add_iterm_syms_to_table ctxt explicit_apply =
  let
    fun consider_var_arity const_T var_T max_ary =
      let
        fun iter ary T =
          if ary = max_ary orelse type_instance ctxt var_T T orelse
             type_instance ctxt T var_T then
            ary
          else
            iter (ary + 1) (range_type T)
      in iter 0 const_T end
    fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
      if explicit_apply = NONE andalso
         (can dest_funT T orelse T = @{typ bool}) then
        let
          val bool_vars' = bool_vars orelse body_type T = @{typ bool}
          fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
            {pred_sym = pred_sym andalso not bool_vars',
             min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
             max_ary = max_ary, types = types}
          val fun_var_Ts' =
            fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
        in
          if bool_vars' = bool_vars andalso
             pointer_eq (fun_var_Ts', fun_var_Ts) then
            accum
          else
            ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
        end
      else
        accum
    fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
      let val (head, args) = strip_iterm_comb tm in
        (case head of
           IConst ((s, _), T, _) =>
           if String.isPrefix bound_var_prefix s orelse
              String.isPrefix all_bound_var_prefix s then
             add_universal_var T accum
           else
             let val ary = length args in
               ((bool_vars, fun_var_Ts),
                case Symtab.lookup sym_tab s of
                  SOME {pred_sym, min_ary, max_ary, types} =>
                  let
                    val pred_sym =
                      pred_sym andalso top_level andalso not bool_vars
                    val types' = types |> insert_type ctxt I T
                    val min_ary =
                      if is_some explicit_apply orelse
                         pointer_eq (types', types) then
                        min_ary
                      else
                        fold (consider_var_arity T) fun_var_Ts min_ary
                  in
                    Symtab.update (s, {pred_sym = pred_sym,
                                       min_ary = Int.min (ary, min_ary),
                                       max_ary = Int.max (ary, max_ary),
                                       types = types'})
                                  sym_tab
                  end
                | NONE =>
                  let
                    val pred_sym = top_level andalso not bool_vars
                    val min_ary =
                      case explicit_apply of
                        SOME true => 0
                      | SOME false => ary
                      | NONE => fold (consider_var_arity T) fun_var_Ts ary
                  in
                    Symtab.update_new (s, {pred_sym = pred_sym,
                                           min_ary = min_ary, max_ary = ary,
                                           types = [T]})
                                      sym_tab
                  end)
             end
         | IVar (_, T) => add_universal_var T accum
         | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
         | _ => accum)
        |> fold (add false) args
      end
  in add true end
fun add_fact_syms_to_table ctxt explicit_apply =
  fact_lift (formula_fold NONE
                          (K (add_iterm_syms_to_table ctxt explicit_apply)))

val tvar_a = TVar (("'a", 0), HOLogic.typeS)

val default_sym_tab_entries : (string * sym_info) list =
  (prefixed_predicator_name,
   {pred_sym = true, min_ary = 1, max_ary = 1, types = []}) ::
  (make_fixed_const @{const_name undefined},
   {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  ([tptp_false, tptp_true]
   |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  ([tptp_equal, tptp_old_equal]
   |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))

fun sym_table_for_facts ctxt explicit_apply facts =
  ((false, []), Symtab.empty)
  |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  |> fold Symtab.update default_sym_tab_entries

fun min_arity_of sym_tab s =
  case Symtab.lookup sym_tab s of
    SOME ({min_ary, ...} : sym_info) => min_ary
  | NONE =>
    case strip_prefix_and_unascii const_prefix s of
      SOME s =>
      let val s = s |> unmangled_const_name |> invert_const in
        if s = predicator_name then 1
        else if s = app_op_name then 2
        else if s = type_guard_name then 1
        else 0
      end
    | NONE => 0

(* True if the constant ever appears outside of the top-level position in
   literals, or if it appears with different arities (e.g., because of different
   type instantiations). If false, the constant always receives all of its
   arguments and is used as a predicate. *)
fun is_pred_sym sym_tab s =
  case Symtab.lookup sym_tab s of
    SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
    pred_sym andalso min_ary = max_ary
  | NONE => false

val predicator_combconst =
  IConst (`make_fixed_const predicator_name, @{typ "bool => bool"}, [])
fun predicator tm = IApp (predicator_combconst, tm)

fun introduce_predicators_in_iterm sym_tab tm =
  case strip_iterm_comb tm of
    (IConst ((s, _), _, _), _) =>
    if is_pred_sym sym_tab s then tm else predicator tm
  | _ => predicator tm

fun list_app head args = fold (curry (IApp o swap)) args head

val app_op = `make_fixed_const app_op_name

fun explicit_app arg head =
  let
    val head_T = ityp_of head
    val (arg_T, res_T) = dest_funT head_T
    val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
  in list_app explicit_app [head, arg] end
fun list_explicit_app head args = fold explicit_app args head

fun introduce_explicit_apps_in_iterm sym_tab =
  let
    fun aux tm =
      case strip_iterm_comb tm of
        (head as IConst ((s, _), _, _), args) =>
        args |> map aux
             |> chop (min_arity_of sym_tab s)
             |>> list_app head
             |-> list_explicit_app
      | (head, args) => list_explicit_app head (map aux args)
  in aux end

fun chop_fun 0 T = ([], T)
  | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
    chop_fun (n - 1) ran_T |>> cons dom_T
  | chop_fun _ _ = raise Fail "unexpected non-function"

fun filter_type_args _ _ _ [] = []
  | filter_type_args thy s arity T_args =
    let
      (* will throw "TYPE" for pseudo-constants *)
      val U = if s = app_op_name then
                @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global
              else
                s |> Sign.the_const_type thy
    in
      case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of
        [] => []
      | res_U_vars =>
        let val U_args = (s, U) |> Sign.const_typargs thy in
          U_args ~~ T_args
          |> map (fn (U, T) =>
                     if member (op =) res_U_vars (dest_TVar U) then T
                     else dummyT)
        end
    end
    handle TYPE _ => T_args

fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
  let
    val thy = Proof_Context.theory_of ctxt
    fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
      | aux arity (IConst (name as (s, _), T, T_args)) =
        (case strip_prefix_and_unascii const_prefix s of
           NONE =>
           (name, if level_of_type_enc type_enc = No_Types then [] else T_args)
         | SOME s'' =>
           let
             val s'' = invert_const s''
             fun filtered_T_args false = T_args
               | filtered_T_args true = filter_type_args thy s'' arity T_args
           in
             case type_arg_policy type_enc s'' of
               Explicit_Type_Args drop_args =>
               (name, filtered_T_args drop_args)
             | Mangled_Type_Args drop_args =>
               (mangled_const_name format type_enc (filtered_T_args drop_args)
                                   name, [])
             | No_Type_Args => (name, [])
           end)
        |> (fn (name, T_args) => IConst (name, T, T_args))
      | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
      | aux _ tm = tm
  in aux 0 end

fun repair_iterm ctxt format type_enc sym_tab =
  not (is_type_enc_higher_order type_enc)
  ? (introduce_explicit_apps_in_iterm sym_tab
     #> introduce_predicators_in_iterm sym_tab)
  #> enforce_type_arg_policy_in_iterm ctxt format type_enc
fun repair_fact ctxt format type_enc sym_tab =
  update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))

(** Helper facts **)

(* The Boolean indicates that a fairly sound type encoding is needed. *)
val helper_table =
  [(("COMBI", false), @{thms Meson.COMBI_def}),
   (("COMBK", false), @{thms Meson.COMBK_def}),
   (("COMBB", false), @{thms Meson.COMBB_def}),
   (("COMBC", false), @{thms Meson.COMBC_def}),
   (("COMBS", false), @{thms Meson.COMBS_def}),
   (("fFalse", false), [@{lemma "~ fFalse" by (unfold fFalse_def) fast}]),
   (("fFalse", true), @{thms True_or_False}),
   (("fTrue", false), [@{lemma "fTrue" by (unfold fTrue_def) fast}]),
   (("fTrue", true), @{thms True_or_False}),
   (("fNot", false),
    @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
           fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
   (("fconj", false),
    @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
        by (unfold fconj_def) fast+}),
   (("fdisj", false),
    @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
        by (unfold fdisj_def) fast+}),
   (("fimplies", false),
    @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
        by (unfold fimplies_def) fast+}),
   (("fequal", true),
    (* This is a lie: Higher-order equality doesn't need a sound type encoding.
       However, this is done so for backward compatibility: Including the
       equality helpers by default in Metis breaks a few existing proofs. *)
    @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
           fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
   (* Partial characterization of "fAll" and "fEx". A complete characterization
      would require the axiom of choice for replay with Metis. *)
   (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
   (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
   (("If", true), @{thms if_True if_False True_or_False})]
  |> map (apsnd (map zero_var_indexes))

fun fo_literal_from_type_literal (TyLitVar (class, name)) =
    (true, ATerm (class, [ATerm (name, [])]))
  | fo_literal_from_type_literal (TyLitFree (class, name)) =
    (true, ATerm (class, [ATerm (name, [])]))

fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot

fun bound_tvars type_enc Ts =
  mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
                (type_literals_for_types type_enc add_sorts_on_tvar Ts))

fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
   else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  |> bound_tvars type_enc atomic_Ts
  |> close_formula_universally

val type_tag = `make_fixed_const type_tag_name

fun type_tag_idempotence_fact type_enc =
  let
    fun var s = ATerm (`I s, [])
    fun tag tm = ATerm (type_tag, [var "A", tm])
    val tagged_var = tag (var "X")
  in
    Formula (type_tag_idempotence_helper_name, Axiom,
             eq_formula type_enc [] false (tag tagged_var) tagged_var,
             isabelle_info simpN, NONE)
  end

fun should_specialize_helper type_enc t =
  polymorphism_of_type_enc type_enc = Mangled_Monomorphic andalso
  level_of_type_enc type_enc <> No_Types andalso
  not (null (Term.hidden_polymorphism t))

fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  case strip_prefix_and_unascii const_prefix s of
    SOME mangled_s =>
    let
      val thy = Proof_Context.theory_of ctxt
      val unmangled_s = mangled_s |> unmangled_const_name
      fun dub needs_fairly_sound j k =
        (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
         (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
         (if needs_fairly_sound then typed_helper_suffix
          else untyped_helper_suffix),
         Helper)
      fun dub_and_inst needs_fairly_sound (th, j) =
        let val t = prop_of th in
          if should_specialize_helper type_enc t then
            map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
                types
          else
            [t]
        end
        |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
      val make_facts = map_filter (make_fact ctxt format type_enc false)
      val fairly_sound = is_type_enc_fairly_sound type_enc
    in
      helper_table
      |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
                  if helper_s <> unmangled_s orelse
                     (needs_fairly_sound andalso not fairly_sound) then
                    []
                  else
                    ths ~~ (1 upto length ths)
                    |> maps (dub_and_inst needs_fairly_sound)
                    |> make_facts)
    end
  | NONE => []
fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
                  []

(***************************************************************)
(* Type Classes Present in the Axiom or Conjecture Clauses     *)
(***************************************************************)

fun set_insert (x, s) = Symtab.update (x, ()) s

fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)

(* Remove this trivial type class (FIXME: similar code elsewhere) *)
fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset

fun classes_of_terms get_Ts =
  map (map snd o get_Ts)
  #> List.foldl add_classes Symtab.empty
  #> delete_type #> Symtab.keys

val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars

fun fold_type_constrs f (Type (s, Ts)) x =
    fold (fold_type_constrs f) Ts (f (s, x))
  | fold_type_constrs _ _ x = x

(* Type constructors used to instantiate overloaded constants are the only ones
   needed. *)
fun add_type_constrs_in_term thy =
  let
    fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
      | add (t $ u) = add t #> add u
      | add (Const (x as (s, _))) =
        if String.isPrefix skolem_const_prefix s then I
        else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)
      | add (Free (s, T)) =
        if String.isPrefix polymorphic_free_prefix s then
          T |> fold_type_constrs set_insert
        else
          I
      | add (Abs (_, _, u)) = add u
      | add _ = I
  in add end

fun type_constrs_of_terms thy ts =
  Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)

fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
                       hyp_ts concl_t facts =
  let
    val thy = Proof_Context.theory_of ctxt
    val presimp_consts = Meson.presimplified_consts ctxt
    val fact_ts = facts |> map snd
    (* Remove existing facts from the conjecture, as this can dramatically
       boost an ATP's performance (for some reason). *)
    val hyp_ts =
      hyp_ts
      |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
    val facts = facts |> map (apsnd (pair Axiom))
    val conjs =
      map (pair prem_kind) hyp_ts @ [(Conjecture, concl_t)]
      |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
    val ((conjs, facts), lambdas) =
      if preproc then
        conjs @ facts
        |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
        |> preprocess_abstractions_in_terms trans_lambdas
        |>> chop (length conjs)
        |>> apfst (map (apsnd (apsnd freeze_term)))
      else
        ((conjs, facts), [])
    val conjs = conjs |> make_conjecture ctxt format type_enc
    val (fact_names, facts) =
      facts
      |> map_filter (fn (name, (_, t)) =>
                        make_fact ctxt format type_enc true (name, t)
                        |> Option.map (pair name))
      |> ListPair.unzip
    val lambdas =
      lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
    val all_ts = concl_t :: hyp_ts @ fact_ts
    val subs = tfree_classes_of_terms all_ts
    val supers = tvar_classes_of_terms all_ts
    val tycons = type_constrs_of_terms thy all_ts
    val (supers, arity_clauses) =
      if level_of_type_enc type_enc = No_Types then ([], [])
      else make_arity_clauses thy tycons supers
    val class_rel_clauses = make_class_rel_clauses thy subs supers
  in
    (fact_names |> map single,
     (conjs, facts @ lambdas, class_rel_clauses, arity_clauses))
  end

val type_guard = `make_fixed_const type_guard_name

fun type_guard_iterm ctxt format type_enc T tm =
  IApp (IConst (type_guard, T --> @{typ bool}, [T])
        |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)

fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
    accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  | is_var_positively_naked_in_term _ _ _ _ = true

fun should_guard_var_in_formula pos phi (SOME true) name =
    formula_fold pos (is_var_positively_naked_in_term name) phi false
  | should_guard_var_in_formula _ _ _ _ = true

fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =
    should_encode_type ctxt mono level T
  | should_generate_tag_bound_decl _ _ _ _ _ = false

fun mk_aterm format type_enc name T_args args =
  ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)

fun tag_with_type ctxt format mono type_enc pos T tm =
  IConst (type_tag, T --> T, [T])
  |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
       | _ => raise Fail "unexpected lambda-abstraction")
and ho_term_from_iterm ctxt format mono type_enc =
  let
    fun aux site u =
      let
        val (head, args) = strip_iterm_comb u
        val pos =
          case site of
            Top_Level pos => pos
          | Eq_Arg pos => pos
          | Elsewhere => NONE
        val t =
          case head of
            IConst (name as (s, _), _, T_args) =>
            let
              val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
            in
              mk_aterm format type_enc name T_args (map (aux arg_site) args)
            end
          | IVar (name, _) =>
            mk_aterm format type_enc name [] (map (aux Elsewhere) args)
          | IAbs ((name, T), tm) =>
            AAbs ((name, ho_type_from_typ format type_enc true 0 T),
                  aux Elsewhere tm)
          | IApp _ => raise Fail "impossible \"IApp\""
        val T = ityp_of u
      in
        t |> (if should_tag_with_type ctxt mono type_enc site u T then
                tag_with_type ctxt format mono type_enc pos T
              else
                I)
      end
  in aux end
and formula_from_iformula ctxt format mono type_enc should_guard_var =
  let
    val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
    val do_bound_type =
      case type_enc of
        Simple_Types (_, level) =>
        homogenized_type ctxt mono level 0
        #> ho_type_from_typ format type_enc false 0 #> SOME
      | _ => K NONE
    fun do_out_of_bound_type pos phi universal (name, T) =
      if should_guard_type ctxt mono type_enc
             (fn () => should_guard_var pos phi universal name) T then
        IVar (name, T)
        |> type_guard_iterm ctxt format type_enc T
        |> do_term pos |> AAtom |> SOME
      else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
        let
          val var = ATerm (name, [])
          val tagged_var =
            ATerm (type_tag, [ho_term_from_typ format type_enc T, var])
        in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
      else
        NONE
    fun do_formula pos (AQuant (q, xs, phi)) =
        let
          val phi = phi |> do_formula pos
          val universal = Option.map (q = AExists ? not) pos
        in
          AQuant (q, xs |> map (apsnd (fn NONE => NONE
                                        | SOME T => do_bound_type T)),
                  (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
                      (map_filter
                           (fn (_, NONE) => NONE
                             | (s, SOME T) =>
                               do_out_of_bound_type pos phi universal (s, T))
                           xs)
                      phi)
        end
      | do_formula pos (AConn conn) = aconn_map pos do_formula conn
      | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  in do_formula end

(* Each fact is given a unique fact number to avoid name clashes (e.g., because
   of monomorphization). The TPTP explicitly forbids name clashes, and some of
   the remote provers might care. *)
fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
                          (j, {name, locality, kind, iformula, atomic_types}) =
  (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
   iformula
   |> close_iformula_universally
   |> formula_from_iformula ctxt format mono type_enc
                            should_guard_var_in_formula
                            (if pos then SOME true else NONE)
   |> bound_tvars type_enc atomic_types
   |> close_formula_universally,
   NONE,
   case locality of
     Intro => isabelle_info introN
   | Elim => isabelle_info elimN
   | Simp => isabelle_info simpN
   | _ => NONE)
  |> Formula

fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}
                                       : class_rel_clause) =
  let val ty_arg = ATerm (`I "T", []) in
    Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
             AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
                               AAtom (ATerm (superclass, [ty_arg]))])
             |> close_formula_universally, isabelle_info introN, NONE)
  end

fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
    (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
  | fo_literal_from_arity_literal (TVarLit (c, sort)) =
    (false, ATerm (c, [ATerm (sort, [])]))

fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}
                                   : arity_clause) =
  Formula (arity_clause_prefix ^ name, Axiom,
           mk_ahorn (map (formula_from_fo_literal o apfst not
                          o fo_literal_from_arity_literal) prem_lits)
                    (formula_from_fo_literal
                         (fo_literal_from_arity_literal concl_lits))
           |> close_formula_universally, isabelle_info introN, NONE)

fun formula_line_for_conjecture ctxt format mono type_enc
        ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  Formula (conjecture_prefix ^ name, kind,
           formula_from_iformula ctxt format mono type_enc
               should_guard_var_in_formula (SOME false)
               (close_iformula_universally iformula)
           |> bound_tvars type_enc atomic_types
           |> close_formula_universally, NONE, NONE)

fun free_type_literals type_enc ({atomic_types, ...} : translated_formula) =
  atomic_types |> type_literals_for_types type_enc add_sorts_on_tfree
               |> map fo_literal_from_type_literal

fun formula_line_for_free_type j lit =
  Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,
           formula_from_fo_literal lit, NONE, NONE)
fun formula_lines_for_free_types type_enc facts =
  let
    val litss = map (free_type_literals type_enc) facts
    val lits = fold (union (op =)) litss []
  in map2 formula_line_for_free_type (0 upto length lits - 1) lits end

(** Symbol declarations **)

fun should_declare_sym type_enc pred_sym s =
  (case type_enc of
     Guards _ => not pred_sym
   | _ => true) andalso
  is_tptp_user_symbol s andalso
  forall (fn prefix => not (String.isPrefix prefix s))
         [bound_var_prefix, all_bound_var_prefix, exist_bound_var_prefix]

fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
                             (conjs, facts) =
  let
    fun add_iterm_syms in_conj tm =
      let val (head, args) = strip_iterm_comb tm in
        (case head of
           IConst ((s, s'), T, T_args) =>
           let val pred_sym = is_pred_sym repaired_sym_tab s in
             if should_declare_sym type_enc pred_sym s then
               Symtab.map_default (s, [])
                   (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
                                         in_conj))
             else
               I
           end
         | IAbs (_, tm) => add_iterm_syms in_conj tm
         | _ => I)
        #> fold (add_iterm_syms in_conj) args
      end
    fun add_fact_syms in_conj =
      fact_lift (formula_fold NONE (K (add_iterm_syms in_conj)))
    fun add_formula_var_types (AQuant (_, xs, phi)) =
        fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
        #> add_formula_var_types phi
      | add_formula_var_types (AConn (_, phis)) =
        fold add_formula_var_types phis
      | add_formula_var_types _ = I
    fun var_types () =
      if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
      else fold (fact_lift add_formula_var_types) (conjs @ facts) []
    fun add_undefined_const T =
      let
        val (s, s') =
          `make_fixed_const @{const_name undefined}
          |> (case type_arg_policy type_enc @{const_name undefined} of
                Mangled_Type_Args _ => mangled_const_name format type_enc [T]
              | _ => I)
      in
        Symtab.map_default (s, [])
                           (insert_type ctxt #3 (s', [T], T, false, 0, false))
      end
  in
    Symtab.empty
    |> is_type_enc_fairly_sound type_enc
       ? (fold (add_fact_syms true) conjs
          #> fold (add_fact_syms false) facts
          #> (case type_enc of
                Simple_Types _ => I
              | _ => fold add_undefined_const (var_types ())))
  end

(* We add "bool" in case the helper "True_or_False" is included later. *)
val default_mono =
  {maybe_finite_Ts = [@{typ bool}],
   surely_finite_Ts = [@{typ bool}],
   maybe_infinite_Ts = known_infinite_types,
   surely_infinite_Ts = known_infinite_types,
   maybe_nonmono_Ts = [@{typ bool}]}

(* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
   out with monotonicity" paper presented at CADE 2011. *)
fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  | add_iterm_mononotonicity_info ctxt level _
        (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
        (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
                  surely_infinite_Ts, maybe_nonmono_Ts}) =
    if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
      case level of
        Noninf_Nonmono_Types soundness =>
        if exists (type_instance ctxt T) surely_infinite_Ts orelse
           member (type_aconv ctxt) maybe_finite_Ts T then
          mono
        else if is_type_surely_infinite' ctxt soundness surely_infinite_Ts
                                         T then
          {maybe_finite_Ts = maybe_finite_Ts,
           surely_finite_Ts = surely_finite_Ts,
           maybe_infinite_Ts = maybe_infinite_Ts,
           surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
           maybe_nonmono_Ts = maybe_nonmono_Ts}
        else
          {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
           surely_finite_Ts = surely_finite_Ts,
           maybe_infinite_Ts = maybe_infinite_Ts,
           surely_infinite_Ts = surely_infinite_Ts,
           maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
      | Fin_Nonmono_Types =>
        if exists (type_instance ctxt T) surely_finite_Ts orelse
           member (type_aconv ctxt) maybe_infinite_Ts T then
          mono
        else if is_type_surely_finite ctxt T then
          {maybe_finite_Ts = maybe_finite_Ts,
           surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
           maybe_infinite_Ts = maybe_infinite_Ts,
           surely_infinite_Ts = surely_infinite_Ts,
           maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
        else
          {maybe_finite_Ts = maybe_finite_Ts,
           surely_finite_Ts = surely_finite_Ts,
           maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
           surely_infinite_Ts = surely_infinite_Ts,
           maybe_nonmono_Ts = maybe_nonmono_Ts}
      | _ => mono
    else
      mono
  | add_iterm_mononotonicity_info _ _ _ _ mono = mono
fun add_fact_mononotonicity_info ctxt level
        ({kind, iformula, ...} : translated_formula) =
  formula_fold (SOME (kind <> Conjecture))
               (add_iterm_mononotonicity_info ctxt level) iformula
fun mononotonicity_info_for_facts ctxt type_enc facts =
  let val level = level_of_type_enc type_enc in
    default_mono
    |> is_type_level_monotonicity_based level
       ? fold (add_fact_mononotonicity_info ctxt level) facts
  end

fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  Formula (guards_sym_formula_prefix ^
           ascii_of (mangled_type format type_enc T),
           Axiom,
           IConst (`make_bound_var "X", T, [])
           |> type_guard_iterm ctxt format type_enc T
           |> AAtom
           |> formula_from_iformula ctxt format mono type_enc
                                    (K (K (K (K true)))) (SOME true)
           |> bound_tvars type_enc (atyps_of T)
           |> close_formula_universally,
           isabelle_info introN, NONE)

fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  let val x_var = ATerm (`make_bound_var "X", []) in
    Formula (tags_sym_formula_prefix ^
             ascii_of (mangled_type format type_enc T),
             Axiom,
             eq_formula type_enc (atyps_of T) false
                        (tag_with_type ctxt format mono type_enc NONE T x_var)
                        x_var,
             isabelle_info simpN, NONE)
  end

fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  case type_enc of
    Simple_Types _ => []
  | Guards _ =>
    map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts

fun decl_line_for_sym ctxt format mono type_enc s
                      (s', T_args, T, pred_sym, ary, _) =
  let
    val (T_arg_Ts, level) =
      case type_enc of
        Simple_Types (_, level) => ([], level)
      | _ => (replicate (length T_args) homo_infinite_type, No_Types)
  in
    Decl (sym_decl_prefix ^ s, (s, s'),
          (T_arg_Ts ---> (T |> homogenized_type ctxt mono level ary))
          |> ho_type_from_typ format type_enc pred_sym (length T_arg_Ts + ary))
  end

fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
                                     j (s', T_args, T, _, ary, in_conj) =
  let
    val (kind, maybe_negate) =
      if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
      else (Axiom, I)
    val (arg_Ts, res_T) = chop_fun ary T
    val num_args = length arg_Ts
    val bound_names =
      1 upto num_args |> map (`I o make_bound_var o string_of_int)
    val bounds =
      bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
    val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
    fun should_keep_arg_type T =
      sym_needs_arg_types andalso
      should_guard_type ctxt mono type_enc (K true) T
    val bound_Ts =
      arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  in
    Formula (guards_sym_formula_prefix ^ s ^
             (if n > 1 then "_" ^ string_of_int j else ""), kind,
             IConst ((s, s'), T, T_args)
             |> fold (curry (IApp o swap)) bounds
             |> type_guard_iterm ctxt format type_enc res_T
             |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
             |> formula_from_iformula ctxt format mono type_enc
                                      (K (K (K (K true)))) (SOME true)
             |> n > 1 ? bound_tvars type_enc (atyps_of T)
             |> close_formula_universally
             |> maybe_negate,
             isabelle_info introN, NONE)
  end

fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
        type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  let
    val ident_base =
      tags_sym_formula_prefix ^ s ^
      (if n > 1 then "_" ^ string_of_int j else "")
    val (kind, maybe_negate) =
      if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
      else (Axiom, I)
    val (arg_Ts, res_T) = chop_fun ary T
    val bound_names =
      1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
    val bounds = bound_names |> map (fn name => ATerm (name, []))
    val cst = mk_aterm format type_enc (s, s') T_args
    val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
    val should_encode =
      should_encode_type ctxt mono (level_of_type_enc type_enc)
    val tag_with = tag_with_type ctxt format mono type_enc NONE
    val add_formula_for_res =
      if should_encode res_T then
        cons (Formula (ident_base ^ "_res", kind,
                       eq (tag_with res_T (cst bounds)) (cst bounds),
                       isabelle_info simpN, NONE))
      else
        I
    fun add_formula_for_arg k =
      let val arg_T = nth arg_Ts k in
        if should_encode arg_T then
          case chop k bounds of
            (bounds1, bound :: bounds2) =>
            cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
                           eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
                              (cst bounds),
                           isabelle_info simpN, NONE))
          | _ => raise Fail "expected nonempty tail"
        else
          I
      end
  in
    [] |> not pred_sym ? add_formula_for_res
       |> fold add_formula_for_arg (ary - 1 downto 0)
  end

fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd

fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
                                (s, decls) =
  case type_enc of
    Simple_Types _ =>
    decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  | Guards (_, level, _) =>
    let
      val decls =
        case decls of
          decl :: (decls' as _ :: _) =>
          let val T = result_type_of_decl decl in
            if forall (type_generalization ctxt T o result_type_of_decl)
                      decls' then
              [decl]
            else
              decls
          end
        | _ => decls
      val n = length decls
      val decls =
        decls |> filter (should_encode_type ctxt mono level
                         o result_type_of_decl)
    in
      (0 upto length decls - 1, decls)
      |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
                                                 type_enc n s)
    end
  | Tags (_, _, uniformity) =>
    (case uniformity of
       Uniform => []
     | Nonuniform =>
       let val n = length decls in
         (0 upto n - 1 ~~ decls)
         |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
                      conj_sym_kind mono type_enc n s)
       end)

fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
                                     sym_decl_tab =
  let
    val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
    val mono_Ts =
      if polymorphism_of_type_enc type_enc = Polymorphic then
        syms |> maps (map result_type_of_decl o snd)
             |> filter_out (should_encode_type ctxt mono
                                (level_of_type_enc type_enc))
             |> rpair [] |-> fold (insert_type ctxt I)
      else
        []
    val mono_lines =
      problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
    val decl_lines =
      fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
                                                     mono type_enc)
               syms []
  in mono_lines @ decl_lines end

fun needs_type_tag_idempotence (Tags (poly, level, uniformity)) =
    poly <> Mangled_Monomorphic andalso
    ((level = All_Types andalso uniformity = Nonuniform) orelse
     is_type_level_monotonicity_based level)
  | needs_type_tag_idempotence _ = false

fun offset_of_heading_in_problem _ [] j = j
  | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
    if heading = needle then j
    else offset_of_heading_in_problem needle problem (j + length lines)

val implicit_declsN = "Should-be-implicit typings"
val explicit_declsN = "Explicit typings"
val factsN = "Relevant facts"
val class_relsN = "Class relationships"
val aritiesN = "Arities"
val helpersN = "Helper facts"
val conjsN = "Conjectures"
val free_typesN = "Type variables"

val explicit_apply = NONE (* for experiments *)

fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
        lambda_trans readable_names preproc hyp_ts concl_t facts =
  let
    val type_enc = type_enc |> adjust_type_enc format
    val lambda_trans =
      if lambda_trans = smartN then
        if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
      else if lambda_trans = lambdasN andalso
              not (is_type_enc_higher_order type_enc) then
        error ("Lambda translation method incompatible with first-order \
               \encoding.")
      else
        lambda_trans
    val trans_lambdas =
      if lambda_trans = no_lambdasN then
        rpair []
      else if lambda_trans = concealedN then
        lift_lambdas ctxt type_enc ##> K []
      else if lambda_trans = liftingN then
        lift_lambdas ctxt type_enc
      else if lambda_trans = combinatorsN then
        map (introduce_combinators ctxt) #> rpair []
      else if lambda_trans = hybridN then
        lift_lambdas ctxt type_enc
        ##> maps (fn t => [t, introduce_combinators ctxt
                                  (intentionalize_def t)])
      else if lambda_trans = lambdasN then
        map (Envir.eta_contract) #> rpair []
      else
        error ("Unknown lambda translation method: " ^
               quote lambda_trans ^ ".")
    val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
      translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
                         hyp_ts concl_t facts
    val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
    val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
    val repair = repair_fact ctxt format type_enc sym_tab
    val (conjs, facts) = (conjs, facts) |> pairself (map repair)
    val repaired_sym_tab =
      conjs @ facts |> sym_table_for_facts ctxt (SOME false)
    val helpers =
      repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
                       |> map repair
    val sym_decl_lines =
      (conjs, helpers @ facts)
      |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
      |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
                                               type_enc
    val helper_lines =
      0 upto length helpers - 1 ~~ helpers
      |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
                                    type_enc)
      |> (if needs_type_tag_idempotence type_enc then
            cons (type_tag_idempotence_fact type_enc)
          else
            I)
    (* Reordering these might confuse the proof reconstruction code or the SPASS
       FLOTTER hack. *)
    val problem =
      [(explicit_declsN, sym_decl_lines),
       (factsN,
        map (formula_line_for_fact ctxt format fact_prefix ascii_of
                                   (not exporter) (not exporter) mono
                                   type_enc)
            (0 upto length facts - 1 ~~ facts)),
       (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
       (aritiesN, map formula_line_for_arity_clause arity_clauses),
       (helpersN, helper_lines),
       (conjsN,
        map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
       (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
    val problem =
      problem
      |> (case format of
            CNF => ensure_cnf_problem
          | CNF_UEQ => filter_cnf_ueq_problem
          | _ => I)
      |> (if is_format_typed format then
            declare_undeclared_syms_in_atp_problem type_decl_prefix
                                                   implicit_declsN
          else
            I)
    val (problem, pool) = problem |> nice_atp_problem readable_names
    val helpers_offset = offset_of_heading_in_problem helpersN problem 0
    val typed_helpers =
      map_filter (fn (j, {name, ...}) =>
                     if String.isSuffix typed_helper_suffix name then SOME j
                     else NONE)
                 ((helpers_offset + 1 upto helpers_offset + length helpers)
                  ~~ helpers)
    fun add_sym_arity (s, {min_ary, ...} : sym_info) =
      if min_ary > 0 then
        case strip_prefix_and_unascii const_prefix s of
          SOME s => Symtab.insert (op =) (s, min_ary)
        | NONE => I
      else
        I
  in
    (problem,
     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
     offset_of_heading_in_problem conjsN problem 0,
     offset_of_heading_in_problem factsN problem 0,
     fact_names |> Vector.fromList,
     typed_helpers,
     Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
  end

(* FUDGE *)
val conj_weight = 0.0
val hyp_weight = 0.1
val fact_min_weight = 0.2
val fact_max_weight = 1.0
val type_info_default_weight = 0.8

fun add_term_weights weight (ATerm (s, tms)) =
    is_tptp_user_symbol s ? Symtab.default (s, weight)
    #> fold (add_term_weights weight) tms
  | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
    formula_fold NONE (K (add_term_weights weight)) phi
  | add_problem_line_weights _ _ = I

fun add_conjectures_weights [] = I
  | add_conjectures_weights conjs =
    let val (hyps, conj) = split_last conjs in
      add_problem_line_weights conj_weight conj
      #> fold (add_problem_line_weights hyp_weight) hyps
    end

fun add_facts_weights facts =
  let
    val num_facts = length facts
    fun weight_of j =
      fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
                        / Real.fromInt num_facts
  in
    map weight_of (0 upto num_facts - 1) ~~ facts
    |> fold (uncurry add_problem_line_weights)
  end

(* Weights are from 0.0 (most important) to 1.0 (least important). *)
fun atp_problem_weights problem =
  let val get = these o AList.lookup (op =) problem in
    Symtab.empty
    |> add_conjectures_weights (get free_typesN @ get conjsN)
    |> add_facts_weights (get factsN)
    |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
            [explicit_declsN, class_relsN, aritiesN]
    |> Symtab.dest
    |> sort (prod_ord Real.compare string_ord o pairself swap)
  end

end;