(*  Title:      HOL/Tools/ATP/atp_problem_generate.ML

     Author:     Fabian Immler, TU Muenchen

     Author:     Makarius

     Author:     Jasmin Blanchette, TU Muenchen

 Translation of HOL to FOL for Metis and Sledgehammer.

 *)

 signature ATP_PROBLEM_GENERATE =

 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 atp_format = ATP_Problem.atp_format

   type formula_kind = ATP_Problem.formula_kind

   type 'a problem = 'a ATP_Problem.problem

   datatype mode = Metis | Sledgehammer | Sledgehammer_Aggressive | Exporter

   datatype scope = Global | Local | Assum | Chained

   datatype status =

     General | Induction | Intro | Inductive | Elim | Simp | Def

   type stature = scope * status

   datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic

   datatype strictness = Strict | Non_Strict

   datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars

   datatype type_level =

     All_Types |

     Noninf_Nonmono_Types of strictness * granularity |

     Fin_Nonmono_Types of granularity |

     Const_Arg_Types |

     No_Types

   type type_enc

   val no_lamsN : string

   val hide_lamsN : string

   val liftingN : string

   val combsN : string

   val combs_and_liftingN : string

   val combs_or_liftingN : string

   val lam_liftingN : string

   val keep_lamsN : 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 lam_lifted_prefix : string

   val lam_lifted_mono_prefix : string

   val lam_lifted_poly_prefix : string

   val skolem_const_prefix : string

   val old_skolem_const_prefix : string

   val new_skolem_const_prefix : string

   val combinator_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 lam_fact_prefix : string

   val typed_helper_suffix : string

   val untyped_helper_suffix : string

   val predicator_name : string

   val app_op_name : string

   val type_guard_name : string

   val type_tag_name : string

   val native_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 unprefix_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 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 type_enc_from_string : strictness -> string -> type_enc

   val adjust_type_enc : atp_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 list

   val helper_table : ((string * bool) * (status * thm) list) list

   val trans_lams_from_string :

     Proof.context -> type_enc -> string -> term list -> term list * term list

   val factsN : string

   val prepare_atp_problem :

     Proof.context -> atp_format -> formula_kind -> type_enc -> mode -> string

     -> bool -> bool -> bool -> term list -> term

     -> ((string * stature) * term) list

     -> string problem * string Symtab.table * (string * stature) list vector

        * (string * term) list * int Symtab.table

   val atp_problem_selection_weights : string problem -> (string * real) list

   val atp_problem_term_order_info : string problem -> (string * int) list

 end;

 structure ATP_Problem_Generate : ATP_PROBLEM_GENERATE =

 struct

 open ATP_Util

 open ATP_Problem

 type name = string * string

 datatype mode = Metis | Sledgehammer | Sledgehammer_Aggressive | Exporter

 datatype scope = Global | Local | Assum | Chained

 datatype status = General | Induction | Intro | Inductive | Elim | Simp | Def

 type stature = scope * status

 datatype order =

   First_Order |

   Higher_Order of thf_flavor

 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic

 datatype strictness = Strict | Non_Strict

 datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars

 datatype type_level =

   All_Types |

   Noninf_Nonmono_Types of strictness * granularity |

   Fin_Nonmono_Types of granularity |

   Const_Arg_Types |

   No_Types

 datatype type_enc =

   Native of order * polymorphism * type_level |

   Guards of polymorphism * type_level |

   Tags of polymorphism * type_level

 fun is_type_enc_native (Native _) = true

   | is_type_enc_native _ = false

 fun is_type_enc_higher_order (Native (Higher_Order _, _, _)) = true

   | is_type_enc_higher_order _ = false

 fun polymorphism_of_type_enc (Native (_, poly, _)) = poly

   | polymorphism_of_type_enc (Guards (poly, _)) = poly

   | polymorphism_of_type_enc (Tags (poly, _)) = poly

 fun level_of_type_enc (Native (_, _, level)) = level

   | level_of_type_enc (Guards (_, level)) = level

   | level_of_type_enc (Tags (_, level)) = level

 fun granularity_of_type_level (Noninf_Nonmono_Types (_, grain)) = grain

   | granularity_of_type_level (Fin_Nonmono_Types grain) = grain

   | granularity_of_type_level _ = All_Vars

 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 (Fin_Nonmono_Types _) = true

   | is_type_level_fairly_sound level = is_type_level_quasi_sound level

 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

 val no_lamsN = "no_lams" (* used internally; undocumented *)

 val hide_lamsN = "hide_lams"

 val liftingN = "lifting"

 val combsN = "combs"

 val combs_and_liftingN = "combs_and_lifting"

 val combs_or_liftingN = "combs_or_lifting"

 val keep_lamsN = "keep_lams"

 val lam_liftingN = "lam_lifting" (* legacy *)

 (* It's still unclear whether all TFF1 implementations will support type

    signatures such as "!>[A : $tType] : $o", with ghost type variables. *)

 val avoid_first_order_ghost_type_vars = false

 val bound_var_prefix = "B_"

 val all_bound_var_prefix = "A_"

 val exist_bound_var_prefix = "E_"

 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 native_type_prefix = "n_"

 val class_prefix = "cl_"

 (* Freshness almost guaranteed! *)

 val atp_prefix = "ATP" ^ Long_Name.separator

 val atp_weak_prefix = "ATP:"

 val lam_lifted_prefix = atp_weak_prefix ^ "Lam"

 val lam_lifted_mono_prefix = lam_lifted_prefix ^ "m"

 val lam_lifted_poly_prefix = lam_lifted_prefix ^ "p"

 val skolem_const_prefix = atp_prefix ^ "Sko"

 val old_skolem_const_prefix = skolem_const_prefix ^ "o"

 val new_skolem_const_prefix = skolem_const_prefix ^ "n"

 val combinator_prefix = "COMB"

 val type_decl_prefix = "ty_"

 val sym_decl_prefix = "sy_"

 val guards_sym_formula_prefix = "gsy_"

 val tags_sym_formula_prefix = "tsy_"

 val uncurried_alias_eq_prefix = "unc_"

 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 lam_fact_prefix = "ATP.lambda_"

 val typed_helper_suffix = "_T"

 val untyped_helper_suffix = "_U"

 val predicator_name = "pp"

 val app_op_name = "aa"

 val type_guard_name = "gg"

 val type_tag_name = "tt"

 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

 (*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 unprefix_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}, combinator_prefix ^ "I"),

    (@{const_name Meson.COMBK}, combinator_prefix ^ "K"),

    (@{const_name Meson.COMBB}, combinator_prefix ^ "B"),

    (@{const_name Meson.COMBC}, combinator_prefix ^ "C"),

    (@{const_name Meson.COMBS}, combinator_prefix ^ "S")]

   |> 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" and choice are mapped to the ATP's equivalents *)

 local

   val choice_const = (fst o dest_Const o HOLogic.choice_const) Term.dummyT

   fun default c = const_prefix ^ lookup_const c

 in

   fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal

     | make_fixed_const (SOME (Native (Higher_Order THF_With_Choice, _, _))) c =

       if c = choice_const then tptp_choice else default c

     | make_fixed_const _ c = default c

 end

 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 **)

 (** Isabelle arities **)

 type arity_atom = name * name * name list

 val type_class = the_single @{sort type}

 type arity_clause =

   {name : string,

    prem_atoms : arity_atom list,

    concl_atom : arity_atom}

 fun add_prem_atom tvar =

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

 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)

 fun make_axiom_arity_clause (tcons, name, (cls, args)) =

   let

     val tvars = map (prefix tvar_prefix o string_of_int) (1 upto length args)

     val tvars_srts = ListPair.zip (tvars, args)

   in

     {name = name,

      prem_atoms = [] |> fold (uncurry add_prem_atom) tvars_srts,

      concl_atom = (`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 atomic_types_of T = fold_atyps (insert (op =)) T []

 val tvar_a_str = "'a"

 val tvar_a = TVar ((tvar_a_str, 0), HOLogic.typeS)

 val tvar_a_name = (make_schematic_type_var (tvar_a_str, 0), tvar_a_str)

 val itself_name = `make_fixed_type_const @{type_name itself}

 val TYPE_name = `(make_fixed_const NONE) @{const_name TYPE}

 val tvar_a_atype = AType (tvar_a_name, [])

 val a_itself_atype = AType (itself_name, [tvar_a_atype])

 fun new_skolem_const_name s num_T_args =

   [new_skolem_const_prefix, s, string_of_int num_T_args]

   |> Long_Name.implode

 val alpha_to_beta = Logic.varifyT_global @{typ "'a => 'b"}

 val alpha_to_beta_to_alpha_to_beta = alpha_to_beta --> alpha_to_beta

 fun robust_const_type thy s =

   if s = app_op_name then

     alpha_to_beta_to_alpha_to_beta

   else if String.isPrefix lam_lifted_prefix s then

     alpha_to_beta

   else

     (* Old Skolems throw a "TYPE" exception here, which will be caught. *)

     s |> Sign.the_const_type thy

 val robust_const_ary =

   let

     fun ary (Type (@{type_name fun}, [_, T])) = 1 + ary T

       | ary _ = 0

   in ary oo robust_const_type end

 (* This function only makes sense if "T" is as general as possible. *)

 fun robust_const_typargs thy (s, T) =

   if s = app_op_name then

     let val (T1, T2) = T |> domain_type |> dest_funT in [T1, T2] end

   else if String.isPrefix old_skolem_const_prefix s then

     [] |> Term.add_tvarsT T |> rev |> map TVar

   else if String.isPrefix lam_lifted_prefix s then

     if String.isPrefix lam_lifted_poly_prefix s then

       let val (T1, T2) = T |> dest_funT in [T1, T2] end

     else

       []

   else

     (s, T) |> Sign.const_typargs thy

 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.

    Also accumulates sort infomation. *)

 fun iterm_from_term thy type_enc bs (P $ Q) =

     let

       val (P', P_atomics_Ts) = iterm_from_term thy type_enc bs P

       val (Q', Q_atomics_Ts) = iterm_from_term thy type_enc bs Q

     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end

   | iterm_from_term thy type_enc _ (Const (c, T)) =

     (IConst (`(make_fixed_const (SOME type_enc)) c, T,

              robust_const_typargs thy (c, T)),

      atomic_types_of T)

   | iterm_from_term _ _ _ (Free (s, T)) =

     (IConst (`make_fixed_var s, T, []), atomic_types_of T)

   | iterm_from_term _ type_enc _ (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 (SOME type_enc)) s', T, Ts) end

      else

        IVar ((make_schematic_var v, s), T), atomic_types_of T)

   | iterm_from_term _ _ bs (Bound j) =

     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atomic_types_of T))

   | iterm_from_term thy type_enc 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 type_enc ((s, (name, T)) :: bs) t

     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atomic_types_of T)) end

 (* "_query", "_bang", and "_at" are for the ASCII-challenged Metis and

    Mirabelle. *)

 val queries = ["?", "_query"]

 val bangs = ["!", "_bang"]

 val ats = ["@", "_at"]

 fun try_unsuffixes ss s =

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

 fun try_nonmono constr suffixes fallback s =

   case try_unsuffixes suffixes s of

     SOME s =>

     (case try_unsuffixes suffixes s of

        SOME s => (constr Positively_Naked_Vars, s)

      | NONE =>

        case try_unsuffixes ats s of

          SOME s => (constr Ghost_Type_Arg_Vars, s)

        | NONE => (constr All_Vars, s))

   | NONE => fallback s

 fun type_enc_from_string strictness s =

   (case try (unprefix "poly_") s of

      SOME s => (SOME Polymorphic, s)

    | NONE =>

      case try (unprefix "raw_mono_") s of

        SOME s => (SOME Raw_Monomorphic, s)

      | NONE =>

        case try (unprefix "mono_") s of

          SOME s => (SOME Mangled_Monomorphic, s)

        | NONE => (NONE, s))

   ||> (pair All_Types

        |> try_nonmono Fin_Nonmono_Types bangs

        |> try_nonmono (curry Noninf_Nonmono_Types strictness) queries)

   |> (fn (poly, (level, core)) =>

          case (core, (poly, level)) of

            ("native", (SOME poly, _)) =>

            (case (poly, level) of

               (Polymorphic, All_Types) =>

               Native (First_Order, Polymorphic, All_Types)

             | (Mangled_Monomorphic, _) =>

               if granularity_of_type_level level = All_Vars then

                 Native (First_Order, Mangled_Monomorphic, level)

               else

                 raise Same.SAME

             | _ => raise Same.SAME)

          | ("native_higher", (SOME poly, _)) =>

            (case (poly, level) of

               (Polymorphic, All_Types) =>

               Native (Higher_Order THF_With_Choice, Polymorphic, All_Types)

             | (_, Noninf_Nonmono_Types _) => raise Same.SAME

             | (Mangled_Monomorphic, _) =>

               if granularity_of_type_level level = All_Vars then

                 Native (Higher_Order THF_With_Choice, Mangled_Monomorphic,

                         level)

               else

                 raise Same.SAME

             | _ => raise Same.SAME)

          | ("guards", (SOME poly, _)) =>

            if poly = Mangled_Monomorphic andalso

               granularity_of_type_level level = Ghost_Type_Arg_Vars then

              raise Same.SAME

            else

              Guards (poly, level)

          | ("tags", (SOME poly, _)) =>

            if granularity_of_type_level level = Ghost_Type_Arg_Vars then

              raise Same.SAME

            else

              Tags (poly, level)

          | ("args", (SOME poly, All_Types (* naja *))) =>

            Guards (poly, Const_Arg_Types)

          | ("erased", (NONE, All_Types (* naja *))) =>

            Guards (Polymorphic, No_Types)

          | _ => raise Same.SAME)

   handle Same.SAME => error ("Unknown type encoding: " ^ quote s ^ ".")

 fun adjust_order THF_Without_Choice (Higher_Order _) =

     Higher_Order THF_Without_Choice

   | adjust_order _ type_enc = type_enc

 fun adjust_type_enc (THF (TPTP_Polymorphic, _, flavor))

                     (Native (order, poly, level)) =

     Native (adjust_order flavor order, poly, level)

   | adjust_type_enc (THF (TPTP_Monomorphic, _, flavor))

                          (Native (order, _, level)) =

     Native (adjust_order flavor order, Mangled_Monomorphic, level)

   | adjust_type_enc (TFF (TPTP_Monomorphic, _)) (Native (_, _, level)) =

     Native (First_Order, Mangled_Monomorphic, level)

   | adjust_type_enc (DFG DFG_Sorted) (Native (_, _, level)) =

     Native (First_Order, Mangled_Monomorphic, level)

   | adjust_type_enc (TFF _) (Native (_, poly, level)) =

     Native (First_Order, poly, level)

   | adjust_type_enc format (Native (_, poly, level)) =

     adjust_type_enc format (Guards (poly, level))

   | 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 is_fol_term t =

   case t of

     @{const Not} $ t1 => is_fol_term t1

   | Const (@{const_name All}, _) $ Abs (_, _, t') => is_fol_term t'

   | Const (@{const_name All}, _) $ t1 => is_fol_term t1

   | Const (@{const_name Ex}, _) $ Abs (_, _, t') => is_fol_term t'

   | Const (@{const_name Ex}, _) $ t1 => is_fol_term t1

   | @{const HOL.conj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2

   | @{const HOL.disj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2

   | @{const HOL.implies} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2

   | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>

     is_fol_term t1 andalso is_fol_term t2

   | _ => not (exists_subterm (fn Abs _ => true | _ => false) t)

 fun simple_translate_lambdas do_lambdas ctxt t =

   if is_fol_term t then

     t

   else

     let

       fun trans Ts t =

         case t of

           @{const Not} $ t1 => @{const Not} $ trans Ts t1

         | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>

           t0 $ Abs (s, T, trans (T :: Ts) t')

         | (t0 as Const (@{const_name All}, _)) $ t1 =>

           trans Ts (t0 $ eta_expand Ts t1 1)

         | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>

           t0 $ Abs (s, T, trans (T :: Ts) t')

         | (t0 as Const (@{const_name Ex}, _)) $ t1 =>

           trans Ts (t0 $ eta_expand Ts t1 1)

         | (t0 as @{const HOL.conj}) $ t1 $ t2 =>

           t0 $ trans Ts t1 $ trans Ts t2

         | (t0 as @{const HOL.disj}) $ t1 $ t2 =>

           t0 $ trans Ts t1 $ trans Ts t2

         | (t0 as @{const HOL.implies}) $ t1 $ t2 =>

           t0 $ trans Ts t1 $ trans Ts t2

         | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))

             $ t1 $ t2 =>

           t0 $ trans Ts t1 $ trans 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 |> trans [] |> singleton (Variable.export_terms ctxt' ctxt) 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)) =

     Const (lam_lifted_poly_prefix ^ serial_string (),

            T --> fastype_of1 (T :: Ts, t))

   | do_cheaply_conceal_lambdas _ t = t

 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 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 constify_lifted (t $ u) = constify_lifted t $ constify_lifted u

   | constify_lifted (Abs (s, T, t)) = Abs (s, T, constify_lifted t)

   | constify_lifted (Free (x as (s, _))) =

     (if String.isPrefix lam_lifted_prefix s then Const else Free) x

   | constify_lifted t = t

 fun lift_lams_part_1 ctxt type_enc =

   map close_form #> rpair ctxt

   #-> Lambda_Lifting.lift_lambdas

           (SOME ((if polymorphism_of_type_enc type_enc = Polymorphic then

                     lam_lifted_poly_prefix

                   else

                     lam_lifted_mono_prefix) ^ "_a"))

           Lambda_Lifting.is_quantifier

   #> fst

 fun lift_lams_part_2 ctxt (facts, lifted) =

   (facts, lifted)

   (* Lambda-lifting sometimes leaves some lambdas around; we need some way to get rid

      of them *)

   |> pairself (map (introduce_combinators ctxt))

   |> pairself (map constify_lifted)

   (* Requires bound variables not to clash with any schematic variables (as

      should be the case right after lambda-lifting). *)

   |>> map (open_form (unprefix close_form_prefix))

   ||> map (open_form I)

 fun lift_lams ctxt = lift_lams_part_2 ctxt oo lift_lams_part_1 ctxt

 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,

    stature : stature,

    kind : formula_kind,

    iformula : (name, typ, iterm) formula,

    atomic_types : typ list}

 fun update_iformula f ({name, stature, kind, iformula, atomic_types}

                        : translated_formula) =

   {name = name, stature = stature, kind = kind, iformula = f iformula,

    atomic_types = atomic_types} : translated_formula

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

 fun insert_type thy get_T x xs =

   let val T = get_T x in

     if exists (type_instance thy T o get_T) xs then xs

     else x :: filter_out (type_generalization thy 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 (* infer_from_term_args *) |

   Mangled_Type_Args |

   No_Type_Args

 fun type_arg_policy monom_constrs type_enc s =

   let val poly = polymorphism_of_type_enc type_enc in

     if s = type_tag_name then

       if poly = Mangled_Monomorphic then Mangled_Type_Args

       else Explicit_Type_Args false

     else case type_enc of

       Native (_, Polymorphic, _) => Explicit_Type_Args false

     | Tags (_, All_Types) => 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 if poly = Mangled_Monomorphic then

           Mangled_Type_Args

         else if member (op =) monom_constrs s andalso

                 granularity_of_type_level level = Positively_Naked_Vars then

           No_Type_Args

         else

           Explicit_Type_Args

               (level = All_Types orelse

                granularity_of_type_level level = Ghost_Type_Arg_Vars)

       end

   end

 (* Make atoms 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 =) (`make_type_class s, `make_fixed_type_var x)

         else

           insert (op =) (`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_class_formula type_enc class arg =

   AAtom (ATerm (class, arg ::

       (case type_enc of

          Native (First_Order, Polymorphic, _) =>

          if avoid_first_order_ghost_type_vars then [ATerm (TYPE_name, [arg])]

          else []

        | _ => [])))

 fun formulas_for_types type_enc add_sorts_on_typ Ts =

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

      |> map (fn (class, name) =>

                 type_class_formula type_enc class (ATerm (name, [])))

 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 add_term_vars phi =

   let

     fun add_formula_vars bounds (AQuant (_, xs, phi)) =

         add_formula_vars (map fst xs @ bounds) phi

       | add_formula_vars bounds (AConn (_, phis)) =

         fold (add_formula_vars bounds) phis

       | add_formula_vars bounds (AAtom tm) = add_term_vars bounds tm

   in mk_aquant AForall (add_formula_vars [] phi []) phi end

 fun add_term_vars bounds (ATerm (name as (s, _), tms)) =

     (if is_tptp_variable s andalso

         not (String.isPrefix tvar_prefix s) andalso

         not (member (op =) bounds name) then

        insert (op =) (name, NONE)

      else

        I)

     #> fold (add_term_vars bounds) tms

   | add_term_vars bounds (AAbs (((name, _), tm), args)) =

     add_term_vars (name :: bounds) tm #> fold (add_term_vars bounds) args

 fun close_formula_universally phi = close_universally add_term_vars phi

 fun add_iterm_vars bounds (IApp (tm1, tm2)) =

     fold (add_iterm_vars bounds) [tm1, tm2]

   | add_iterm_vars _ (IConst _) = I

   | add_iterm_vars bounds (IVar (name, T)) =

     not (member (op =) bounds name) ? insert (op =) (name, SOME T)

   | add_iterm_vars bounds (IAbs (_, tm)) = add_iterm_vars bounds tm

 fun close_iformula_universally phi = close_universally add_iterm_vars phi

 val fused_infinite_type_name = "ATP.fused_inf" (* shouldn't clash *)

 val fused_infinite_type = Type (fused_infinite_type_name, [])

 fun tvar_name (x as (s, _)) = (make_schematic_type_var x, s)

 fun ho_term_from_typ 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 = fused_infinite_type_name andalso

                        is_type_enc_native type_enc 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, _)) = ATerm (tvar_name x, [])

   in term end

 fun ho_term_for_type_arg type_enc T =

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

 (* This shouldn't clash with anything else. *)

 val uncurried_alias_sep = "\000"

 val mangled_type_sep = "\001"

 val ascii_of_uncurried_alias_sep = ascii_of uncurried_alias_sep

 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 type_enc =

   generic_mangled_type_name fst o ho_term_from_typ type_enc

 fun make_native_type s =

   if s = tptp_bool_type orelse s = tptp_fun_type orelse

      s = tptp_individual_type then

     s

   else

     native_type_prefix ^ ascii_of s

 fun ho_type_from_ho_term type_enc pred_sym ary =

   let

     fun to_mangled_atype ty =

       AType ((make_native_type (generic_mangled_type_name fst ty),

               generic_mangled_type_name snd ty), [])

     fun to_poly_atype (ATerm (name, tys)) = AType (name, map to_poly_atype tys)

       | to_poly_atype _ = raise Fail "unexpected type abstraction"

     val to_atype =

       if polymorphism_of_type_enc type_enc = Polymorphic then to_poly_atype

       else to_mangled_atype

     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 type_enc pred_sym ary =

   ho_type_from_ho_term type_enc pred_sym ary

   o ho_term_from_typ type_enc

 fun aliased_uncurried ary (s, s') =

   (s ^ ascii_of_uncurried_alias_sep ^ string_of_int ary, s' ^ string_of_int ary)

 fun unaliased_uncurried (s, s') =

   case space_explode uncurried_alias_sep s of

     [_] => (s, s')

   | [s1, s2] => (s1, unsuffix s2 s')

   | _ => raise Fail "ill-formed explicit application alias"

 fun raw_mangled_const_name type_name ty_args (s, s') =

   let

     fun type_suffix f g =

       fold_rev (curry (op ^) o g o prefix mangled_type_sep o type_name f)

                ty_args ""

   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end

 fun mangled_const_name type_enc =

   map_filter (ho_term_for_type_arg type_enc)

   #> raw_mangled_const_name generic_mangled_type_name

 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

 fun unmangled_const_name s =

   (s, s) |> unaliased_uncurried |> fst |> space_explode mangled_type_sep

 fun unmangled_const s =

   let val ss = unmangled_const_name s in

     (hd ss, map unmangled_type (tl ss))

   end

 fun introduce_proxies_in_iterm 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 then

                  if length args = 2 then

                    IConst (`I tptp_equal, T, [])

                  else

                    (* Eta-expand partially applied THF equality, because the

                       LEO-II and Satallax parsers complain about not being able to

                       infer the type of "=". *)

                    let val i_T = domain_type T in

                      IAbs ((`I "Y", i_T),

                            IAbs ((`I "Z", i_T),

                                  IApp (IApp (IConst (`I tptp_equal, T, []),

                                              IConst (`I "Y", i_T, [])),

                                        IConst (`I "Z", i_T, []))))

                    end

                else

                  IConst (name, T, [])

              | _ => IConst (name, T, [])

            else

              IConst (proxy_base |>> prefix const_prefix, T, T_args)

           | NONE => if s = tptp_choice then tweak_ho_quant tptp_choice T args

                     else IConst (name, T, T_args))

       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)

       | intro _ _ tm = tm

   in intro true [] end

 fun mangle_type_args_in_const type_enc (name as (s, _)) T_args =

   case unprefix_and_unascii const_prefix s of

     NONE => (name, T_args)

   | SOME s'' =>

     case type_arg_policy [] type_enc (invert_const s'') of

       Mangled_Type_Args => (mangled_const_name type_enc T_args name, [])

     | _ => (name, T_args)

 fun mangle_type_args_in_iterm type_enc =

   if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then

     let

       fun mangle (IApp (tm1, tm2)) = IApp (mangle tm1, mangle tm2)

         | mangle (tm as IConst (_, _, [])) = tm

         | mangle (IConst (name, T, T_args)) =

           mangle_type_args_in_const type_enc name T_args

           |> (fn (name, T_args) => IConst (name, T, T_args))

         | mangle (IAbs (bound, tm)) = IAbs (bound, mangle tm)

         | mangle tm = tm

     in mangle end

   else

     I

 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 _ T = ([], T)

 fun filter_const_type_args _ _ _ [] = []

   | filter_const_type_args thy s ary T_args =

     let

       val U = robust_const_type thy s

       val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun ary |> fst) []

       val U_args = (s, U) |> robust_const_typargs thy

     in

       U_args ~~ T_args

       |> map (fn (U, T) =>

                  if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)

     end

     handle TYPE _ => T_args

 fun filter_type_args_in_const _ _ _ _ _ [] = []

   | filter_type_args_in_const thy monom_constrs type_enc ary s T_args =

     case unprefix_and_unascii const_prefix s of

       NONE =>

       if level_of_type_enc type_enc = No_Types orelse s = tptp_choice then []

       else T_args

     | SOME s'' =>

       let

         val s'' = invert_const s''

         fun filter_T_args false = T_args

           | filter_T_args true = filter_const_type_args thy s'' ary T_args

       in

         case type_arg_policy monom_constrs type_enc s'' of

           Explicit_Type_Args infer_from_term_args =>

           filter_T_args infer_from_term_args

         | No_Type_Args => []

         | Mangled_Type_Args => raise Fail "unexpected (un)mangled symbol"

       end

 fun filter_type_args_in_iterm thy monom_constrs type_enc =

   let

     fun filt ary (IApp (tm1, tm2)) = IApp (filt (ary + 1) tm1, filt 0 tm2)

       | filt ary (IConst (name as (s, _), T, T_args)) =

         filter_type_args_in_const thy monom_constrs type_enc ary s T_args

         |> (fn T_args => IConst (name, T, T_args))

       | filt _ (IAbs (bound, tm)) = IAbs (bound, filt 0 tm)

       | filt _ tm = tm

   in filt 0 end

 fun iformula_from_prop ctxt type_enc iff_for_eq =

   let

     val thy = Proof_Context.theory_of ctxt

     fun do_term bs t atomic_Ts =

       iterm_from_term thy type_enc bs (Envir.eta_contract t)

       |>> (introduce_proxies_in_iterm type_enc

            #> mangle_type_args_in_iterm type_enc #> AAtom)

       ||> union (op =) atomic_Ts

     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)]

         ##> union (op =) (atomic_types_of 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'

       | (t0 as Const (@{const_name All}, _)) $ t1 =>

         do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)

       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>

         do_quant bs AExists pos s T t'

       | (t0 as Const (@{const_name Ex}, _)) $ t1 =>

         do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)

       | @{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 iff_for_eq 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 thy t =

   if exists_Const (member (op =) Meson.presimplified_consts o fst) t then

     t |> Skip_Proof.make_thm thy

       |> Meson.presimplify

       |> prop_of

   else

     t

 fun preprocess_abstractions_in_terms trans_lams facts =

   let

     val (facts, lambda_ts) =

       facts |> map (snd o snd) |> trans_lams

             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts

     val lam_facts =

       map2 (fn t => fn j =>

                ((lam_fact_prefix ^ Int.toString j, (Global, Def)), (Axiom, t)))

            lambda_ts (1 upto length lambda_ts)

   in (facts, lam_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 freeze (t $ u) = freeze t $ freeze u

       | freeze (Abs (s, T, t)) = Abs (s, T, freeze t)

       | freeze (Var ((s, i), T)) =

         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)

       | freeze t = t

   in t |> exists_subterm is_Var t ? freeze end

 fun unextensionalize_def t =

   case t of

     @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ lhs $ rhs) =>

     (case strip_comb lhs of

        (c as Const (_, T), args) =>

        if forall is_Var args andalso not (has_duplicates (op =) args) then

          @{const Trueprop}

          $ (Const (@{const_name HOL.eq}, T --> T --> @{typ bool})

             $ c $ fold_rev lambda args rhs)

        else

          t

      | _ => t)

   | _ => t

 fun presimp_prop ctxt type_enc 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})

     val is_ho = is_type_enc_higher_order type_enc

   in

     t |> need_trueprop ? HOLogic.mk_Trueprop

       |> (if is_ho then unextensionalize_def

           else cong_extensionalize_term thy #> abs_extensionalize_term ctxt)

       |> presimplify_term thy

       |> HOLogic.dest_Trueprop

   end

   handle TERM _ => @{const True}

 (* Satallax prefers "=" to "<=>" (for definitions) and Metis (CNF) requires "="

    for obscure technical reasons. *)

 fun should_use_iff_for_eq CNF _ = false

   | should_use_iff_for_eq (THF _) format = not (is_type_enc_higher_order format)

   | should_use_iff_for_eq _ _ = true

 fun make_formula ctxt format type_enc iff_for_eq name stature kind t =

   let

     val iff_for_eq = iff_for_eq andalso should_use_iff_for_eq format type_enc

     val (iformula, atomic_Ts) =

       iformula_from_prop ctxt type_enc iff_for_eq (SOME (kind <> Conjecture)) t

                          []

       |>> close_iformula_universally

   in

     {name = name, stature = stature, kind = kind, iformula = iformula,

      atomic_types = atomic_Ts}

   end

 fun make_fact ctxt format type_enc iff_for_eq ((name, stature), t) =

   case t |> make_formula ctxt format type_enc iff_for_eq name stature Axiom of

     formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>

     if s = tptp_true then NONE else SOME formula

   | formula => SOME formula

 fun s_not_prop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t

   | s_not_prop _ = @{prop True} (* "t" is too meta for "metis" *)

 (*

   | s_not_prop (@{const "==>"} $ t $ @{prop False}) = t

   | s_not_prop t = @{const "==>"} $ t $ @{prop False}

 *)

 fun make_conjecture ctxt format type_enc =

   map (fn ((name, stature), (kind, t)) =>

           t |> kind = Conjecture ? s_not

             |> make_formula ctxt format type_enc true name stature kind)

 (** Finite and infinite type inference **)

 fun tvar_footprint thy s ary =

   (case unprefix_and_unascii const_prefix s of

      SOME s =>

      s |> invert_const |> robust_const_type thy |> chop_fun ary |> fst

        |> map (fn T => Term.add_tvarsT T [] |> map fst)

    | NONE => [])

   handle TYPE _ => []

 fun ghost_type_args thy s ary =

   if is_tptp_equal s then

     0 upto ary - 1

   else

     let

       val footprint = tvar_footprint thy s ary

       val eq = (s = @{const_name HOL.eq})

       fun ghosts _ [] = []

         | ghosts seen ((i, tvars) :: args) =

           ghosts (union (op =) seen tvars) args

           |> (eq orelse exists (fn tvar => not (member (op =) seen tvar)) tvars)

              ? cons i

     in

       if forall null footprint then

         []

       else

         0 upto length footprint - 1 ~~ footprint

         |> sort (rev_order o list_ord Term_Ord.indexname_ord o pairself snd)

         |> ghosts []

     end

 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}, HOLogic.intT, HOLogic.realT, @{typ "nat => bool"}]

 fun is_type_kind_of_surely_infinite ctxt strictness cached_Ts T =

   strictness <> Strict andalso is_type_surely_infinite ctxt true 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 (strictness, grain)) T =

     let val thy = Proof_Context.theory_of ctxt in

       grain = Ghost_Type_Arg_Vars orelse

       (exists (type_intersect thy T) maybe_nonmono_Ts andalso

        not (exists (type_instance thy T) surely_infinite_Ts orelse

             (not (member (type_equiv thy) maybe_finite_Ts T) andalso

              is_type_kind_of_surely_infinite ctxt strictness surely_infinite_Ts

                                              T)))

     end

   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,

                              maybe_nonmono_Ts, ...}

                        (Fin_Nonmono_Types grain) T =

     let val thy = Proof_Context.theory_of ctxt in

       grain = Ghost_Type_Arg_Vars orelse

       (exists (type_intersect thy T) maybe_nonmono_Ts andalso

        (exists (type_generalization thy T) surely_finite_Ts orelse

         (not (member (type_equiv thy) maybe_infinite_Ts T) andalso

          is_type_surely_finite ctxt T)))

     end

   | should_encode_type _ _ _ _ = false

 fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =

     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 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)) site u T =

     if granularity_of_type_level level = All_Vars then

       should_encode_type ctxt mono level T

     else

       (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 fused_type ctxt mono level =

   let

     val should_encode = should_encode_type ctxt mono level

     fun fuse 0 T = if should_encode T then T else fused_infinite_type

       | fuse ary (Type (@{type_name fun}, [T1, T2])) =

         fuse 0 T1 --> fuse (ary - 1) T2

       | fuse _ _ = raise Fail "expected function type"

   in fuse end

 (** predicators and application operators **)

 type sym_info =

   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list,

    in_conj : bool}

 fun default_sym_tab_entries type_enc =

   (make_fixed_const NONE @{const_name undefined},

        {pred_sym = false, min_ary = 0, max_ary = 0, types = [],

         in_conj = false}) ::

   ([tptp_false, tptp_true]

    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = [],

                   in_conj = false})) @

   ([tptp_equal, tptp_old_equal]

    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = [],

                   in_conj = false}))

   |> not (is_type_enc_higher_order type_enc)

      ? cons (prefixed_predicator_name,

              {pred_sym = true, min_ary = 1, max_ary = 1, types = [],

               in_conj = false})

 datatype app_op_level =

   Min_App_Op |

   Sufficient_App_Op |

   Sufficient_App_Op_And_Predicator |

   Full_App_Op_And_Predicator

 fun add_iterm_syms_to_sym_table ctxt app_op_level conj_fact =

   let

     val thy = Proof_Context.theory_of ctxt

     fun consider_var_ary const_T var_T max_ary =

       let

         fun iter ary T =

           if ary = max_ary orelse type_instance thy var_T T orelse

              type_instance thy 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 (app_op_level = Sufficient_App_Op andalso can dest_funT T) orelse

          (app_op_level = Sufficient_App_Op_And_Predicator andalso

           (can dest_funT T orelse T = @{typ bool})) then

         let

           val bool_vars' =

             bool_vars orelse

             (app_op_level = Sufficient_App_Op_And_Predicator andalso

              body_type T = @{typ bool})

           fun repair_min_ary {pred_sym, min_ary, max_ary, types, in_conj} =

             {pred_sym = pred_sym andalso not bool_vars',

              min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,

              max_ary = max_ary, types = types, in_conj = in_conj}

           val fun_var_Ts' =

             fun_var_Ts |> can dest_funT T ? insert_type thy 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_ary) sym_tab)

         end

       else

         accum

     fun add_iterm_syms 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 if String.isPrefix exist_bound_var_prefix s then

              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, in_conj} =>

                   let

                     val pred_sym =

                       pred_sym andalso top_level andalso not bool_vars

                     val types' = types |> insert_type thy I T

                     val in_conj = in_conj orelse conj_fact

                     val min_ary =

                       if (app_op_level = Sufficient_App_Op orelse

                           app_op_level = Sufficient_App_Op_And_Predicator)

                          andalso not (pointer_eq (types', types)) then

                         fold (consider_var_ary T) fun_var_Ts min_ary

                       else

                         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', in_conj = in_conj})

                                   sym_tab

                   end

                 | NONE =>

                   let

                     val pred_sym = top_level andalso not bool_vars

                     val ary =

                       case unprefix_and_unascii const_prefix s of

                         SOME s =>

                         (if String.isSubstring uncurried_alias_sep s then

                            ary

                          else case try (robust_const_ary thy

                                         o invert_const o hd

                                         o unmangled_const_name) s of

                            SOME ary0 => Int.min (ary0, ary)

                          | NONE => ary)

                       | NONE => ary

                     val min_ary =

                       case app_op_level of

                         Min_App_Op => ary

                       | Full_App_Op_And_Predicator => 0

                       | _ => fold (consider_var_ary T) fun_var_Ts ary

                   in

                     Symtab.update_new (s,

                         {pred_sym = pred_sym, min_ary = min_ary,

                          max_ary = ary, types = [T], in_conj = conj_fact})

                         sym_tab

                   end)

              end

          | IVar (_, T) => add_universal_var T accum

          | IAbs ((_, T), tm) =>

            accum |> add_universal_var T |> add_iterm_syms false tm

          | _ => accum)

         |> fold (add_iterm_syms false) args

       end

   in add_iterm_syms end

 fun sym_table_for_facts ctxt type_enc app_op_level conjs facts =

   let

     fun add_iterm_syms conj_fact =

       add_iterm_syms_to_sym_table ctxt app_op_level conj_fact true

     fun add_fact_syms conj_fact =

       K (add_iterm_syms conj_fact) |> formula_fold NONE |> fact_lift

   in

     ((false, []), Symtab.empty)

     |> fold (add_fact_syms true) conjs

     |> fold (add_fact_syms false) facts

     ||> fold Symtab.update (default_sym_tab_entries type_enc)

   end

 fun min_ary_of sym_tab s =

   case Symtab.lookup sym_tab s of

     SOME ({min_ary, ...} : sym_info) => min_ary

   | NONE =>

     case unprefix_and_unascii const_prefix s of

       SOME s =>

       let val s = s |> unmangled_const_name |> hd |> 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 fTrue_iconst =

   IConst ((const_prefix ^ "fTrue", @{const_name ATP.fTrue}), @{typ bool}, [])

 val predicator_iconst =

   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])

 fun predicatify aggressive tm =

   if aggressive then

     IApp (IApp (IConst (`I tptp_equal, @{typ "bool => bool => bool"}, []), tm),

           fTrue_iconst)

   else

     IApp (predicator_iconst, tm)

 val app_op = `(make_fixed_const NONE) app_op_name

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

 fun mk_app_op type_enc head arg =

   let

     val head_T = ityp_of head

     val (arg_T, res_T) = dest_funT head_T

     val app =

       IConst (app_op, head_T --> head_T, [arg_T, res_T])

       |> mangle_type_args_in_iterm type_enc

   in list_app app [head, arg] end

 fun firstorderize_fact thy monom_constrs type_enc sym_tab uncurried_aliases

                        aggressive =

   let

     fun do_app arg head = mk_app_op type_enc head arg

     fun list_app_ops head args = fold do_app args head

     fun introduce_app_ops tm =

       let val (head, args) = tm |> strip_iterm_comb ||> map introduce_app_ops in

         case head of

           IConst (name as (s, _), T, T_args) =>

           if uncurried_aliases andalso String.isPrefix const_prefix s then

             let

               val ary = length args

               val name =

                 name |> ary > min_ary_of sym_tab s ? aliased_uncurried ary

             in list_app (IConst (name, T, T_args)) args end

           else

             args |> chop (min_ary_of sym_tab s)

                  |>> list_app head |-> list_app_ops

         | _ => list_app_ops head args

       end

     fun introduce_predicators tm =

       case strip_iterm_comb tm of

         (IConst ((s, _), _, _), _) =>

         if is_pred_sym sym_tab s then tm else predicatify aggressive tm

       | _ => predicatify aggressive tm

     val do_iterm =

       not (is_type_enc_higher_order type_enc)

       ? (introduce_app_ops #> introduce_predicators)

       #> filter_type_args_in_iterm thy monom_constrs type_enc

   in update_iformula (formula_map do_iterm) end

 (** Helper facts **)

 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}

 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}

 (* The Boolean indicates that a fairly sound type encoding is needed. *)

 val base_helper_table =

   [(("COMBI", false), [(Def, @{thm Meson.COMBI_def})]),

    (("COMBK", false), [(Def, @{thm Meson.COMBK_def})]),

    (("COMBB", false), [(Def, @{thm Meson.COMBB_def})]),

    (("COMBC", false), [(Def, @{thm Meson.COMBC_def})]),

    (("COMBS", false), [(Def, @{thm Meson.COMBS_def})]),

    ((predicator_name, false), [(General, not_ffalse), (General, ftrue)]),

    (("fFalse", false), [(General, not_ffalse)]),

    (("fFalse", true), [(General, @{thm True_or_False})]),

    (("fTrue", false), [(General, ftrue)]),

    (("fTrue", true), [(General, @{thm True_or_False})]),

    (("If", true),

     [(Def, @{thm if_True}), (Def, @{thm if_False}),

      (General, @{thm True_or_False})])]

 val helper_table =

   base_helper_table @

   [(("fNot", false),

     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]

            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}

     |> map (pair Def)),

    (("fconj", false),

     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"

         by (unfold fconj_def) fast+}

     |> map (pair General)),

    (("fdisj", false),

     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"

         by (unfold fdisj_def) fast+}

     |> map (pair General)),

    (("fimplies", false),

     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"

         by (unfold fimplies_def) fast+}

     |> map (pair General)),

    (("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]}

     |> map (pair General)),

    (* Partial characterization of "fAll" and "fEx". A complete characterization

       would require the axiom of choice for replay with Metis. *)

    (("fAll", false),

     [(General, @{lemma "~ fAll P | P x" by (auto simp: fAll_def)})]),

    (("fEx", false),

     [(General, @{lemma "~ P x | fEx P" by (auto simp: fEx_def)})])]

   |> map (apsnd (map (apsnd zero_var_indexes)))

 val aggressive_helper_table =

   base_helper_table @

   [((predicator_name, true),

     @{thms True_or_False fTrue_ne_fFalse} |> map (pair General)),

    ((app_op_name, true),

     [(General, @{lemma "EX x. ~ f x = g x | f = g" by blast})]),

    (("fconj", false),

     @{thms fconj_table fconj_laws fdisj_laws} |> map (pair Def)),

    (("fdisj", false),

     @{thms fdisj_table fconj_laws fdisj_laws} |> map (pair Def)),

    (("fimplies", false),

     @{thms fimplies_table fconj_laws fdisj_laws fimplies_laws}

     |> map (pair Def)),

    (("fequal", false),

     (@{thms fequal_table} |> map (pair Def)) @

     (@{thms fequal_laws} |> map (pair General))),

    (("fAll", false),

     @{thms fAll_table fComp_law fAll_law fEx_law} |> map (pair Def)),

    (("fEx", false),

     @{thms fEx_table fComp_law fAll_law fEx_law} |> map (pair Def))]

   |> map (apsnd (map (apsnd zero_var_indexes)))

 fun atype_of_type_vars (Native (_, Polymorphic, _)) = SOME atype_of_types

   | atype_of_type_vars _ = NONE

 fun bound_tvars type_enc sorts Ts =

   (sorts ? mk_ahorn (formulas_for_types type_enc add_sorts_on_tvar Ts))

   #> mk_aquant AForall

         (map_filter (fn TVar (x as (s, _), _) =>

                         SOME ((make_schematic_type_var x, s),

                               atype_of_type_vars type_enc)

                       | _ => NONE) Ts)

 fun eq_formula type_enc atomic_Ts bounds pred_sym tm1 tm2 =

   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])

    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))

   |> mk_aquant AForall bounds

   |> close_formula_universally

   |> bound_tvars type_enc true atomic_Ts

 val helper_rank = default_rank

 val min_rank = 9 * helper_rank div 10

 val max_rank = 4 * min_rank

 fun rank_of_fact_num n j = min_rank + (max_rank - min_rank) * j div n

 val type_tag = `(make_fixed_const NONE) type_tag_name

 fun could_specialize_helpers type_enc =

   polymorphism_of_type_enc type_enc <> Polymorphic andalso

   level_of_type_enc type_enc <> No_Types

 fun should_specialize_helper type_enc t =

   could_specialize_helpers type_enc andalso

   not (null (Term.hidden_polymorphism t))

 fun add_helper_facts_for_sym ctxt format type_enc aggressive

                              (s, {types, ...} : sym_info) =

   case unprefix_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 |> hd

       fun dub needs_fairly_sound j k =

         ascii_of 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)

       fun specialize_helper t T =

         if unmangled_s = app_op_name then

           let

             val tyenv =

               Sign.typ_match thy (alpha_to_beta, domain_type T) Vartab.empty

           in monomorphic_term tyenv t end

         else

           specialize_type thy (invert_const unmangled_s, T) t

       fun dub_and_inst needs_fairly_sound ((status, t), j) =

         (if should_specialize_helper type_enc t then

            map_filter (try (specialize_helper t)) types

          else

            [t])

         |> tag_list 1

         |> map (fn (k, t) =>

                    ((dub needs_fairly_sound j k, (Global, status)), t))

       val make_facts = map_filter (make_fact ctxt format type_enc false)

       val fairly_sound = is_type_enc_fairly_sound type_enc

       val could_specialize = could_specialize_helpers type_enc

     in

       fold (fn ((helper_s, needs_fairly_sound), ths) =>

                if (needs_fairly_sound andalso not fairly_sound) orelse

                   (helper_s <> unmangled_s andalso

                    (not aggressive orelse could_specialize)) then

                  I

                else

                  ths ~~ (1 upto length ths)

                  |> maps (dub_and_inst needs_fairly_sound

                           o apfst (apsnd prop_of))

                  |> make_facts

                  |> union (op = o pairself #iformula))

            (if aggressive then aggressive_helper_table else helper_table)

     end

   | NONE => I

 fun helper_facts_for_sym_table ctxt format type_enc aggressive sym_tab =

   Symtab.fold_rev (add_helper_facts_for_sym ctxt format type_enc aggressive)

                   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) =

         x |> robust_const_typargs thy |> fold (fold_type_constrs set_insert)

       | 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 extract_lambda_def (Const (@{const_name HOL.eq}, _) $ t $ u) =

     let val (head, args) = strip_comb t in

       (head |> dest_Const |> fst,

        fold_rev (fn t as Var ((s, _), T) =>

                     (fn u => Abs (s, T, abstract_over (t, u)))

                   | _ => raise Fail "expected \"Var\"") args u)

     end

   | extract_lambda_def _ = raise Fail "malformed lifted lambda"

 fun trans_lams_from_string ctxt type_enc lam_trans =

   if lam_trans = no_lamsN then

     rpair []

   else if lam_trans = hide_lamsN then

     lift_lams ctxt type_enc ##> K []

   else if lam_trans = liftingN orelse lam_trans = lam_liftingN then

     lift_lams ctxt type_enc

   else if lam_trans = combsN then

     map (introduce_combinators ctxt) #> rpair []

   else if lam_trans = combs_and_liftingN then

     lift_lams_part_1 ctxt type_enc

     ##> maps (fn t => [t, introduce_combinators ctxt (intentionalize_def t)])

     #> lift_lams_part_2 ctxt

   else if lam_trans = combs_or_liftingN then

     lift_lams_part_1 ctxt type_enc

     ##> map (fn t => case head_of (strip_qnt_body @{const_name All} t) of

                        @{term "op =::bool => bool => bool"} => t

                      | _ => introduce_combinators ctxt (intentionalize_def t))

     #> lift_lams_part_2 ctxt

   else if lam_trans = keep_lamsN then

     map (Envir.eta_contract) #> rpair []

   else

     error ("Unknown lambda translation scheme: " ^ quote lam_trans ^ ".")

 fun translate_formulas ctxt prem_kind format type_enc lam_trans presimp hyp_ts

                        concl_t facts =

   let

     val thy = Proof_Context.theory_of ctxt

     val trans_lams = trans_lams_from_string ctxt type_enc lam_trans

     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, s_not_prop concl_t)]

       |> map (apsnd freeze_term)

       |> map2 (pair o rpair (Local, General) o string_of_int)

               (0 upto length hyp_ts)

     val ((conjs, facts), lam_facts) =

       (conjs, facts)

       |> presimp ? pairself (map (apsnd (apsnd (presimp_prop ctxt type_enc))))

       |> (if lam_trans = no_lamsN then

             rpair []

           else

             op @

             #> preprocess_abstractions_in_terms trans_lams

             #>> chop (length conjs))

     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 lifted = lam_facts |> map (extract_lambda_def o snd o snd)

     val lam_facts =

       lam_facts |> 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, union (op =) subs supers, conjs,

      facts @ lam_facts, class_rel_clauses, arity_clauses, lifted)

   end

 val type_guard = `(make_fixed_const NONE) type_guard_name

 fun type_guard_iterm type_enc T tm =

   IApp (IConst (type_guard, T --> @{typ bool}, [T])

         |> mangle_type_args_in_iterm 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 is_var_ghost_type_arg_in_term thy polym_constrs name pos tm accum =

   is_var_positively_naked_in_term name pos tm accum orelse

   let

     val var = ATerm (name, [])

     fun is_nasty_in_term (ATerm (_, [])) = false

       | is_nasty_in_term (ATerm ((s, _), tms)) =

         let

           val ary = length tms

           val polym_constr = member (op =) polym_constrs s

           val ghosts = ghost_type_args thy s ary

         in

           exists (fn (j, tm) =>

                      if polym_constr then

                        member (op =) ghosts j andalso

                        (tm = var orelse is_nasty_in_term tm)

                      else

                        tm = var andalso member (op =) ghosts j)

                  (0 upto ary - 1 ~~ tms)

           orelse (not polym_constr andalso exists is_nasty_in_term tms)

         end

       | is_nasty_in_term _ = true

   in is_nasty_in_term tm end

 fun should_guard_var_in_formula thy polym_constrs level pos phi (SOME true)

                                 name =

     (case granularity_of_type_level level of

        All_Vars => true

      | Positively_Naked_Vars =>

        formula_fold pos (is_var_positively_naked_in_term name) phi false

      | Ghost_Type_Arg_Vars =>

        formula_fold pos (is_var_ghost_type_arg_in_term thy polym_constrs name) phi

                     false)

   | should_guard_var_in_formula _ _ _ _ _ _ _ = true

 fun always_guard_var_in_formula _ _ _ _ _ _ _ = true

 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false

   | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =

     granularity_of_type_level level <> All_Vars andalso

     should_encode_type ctxt mono level T

   | should_generate_tag_bound_decl _ _ _ _ _ = false

 fun mk_aterm type_enc name T_args args =

   ATerm (name, map_filter (ho_term_for_type_arg type_enc) T_args @ args)

 fun do_bound_type ctxt mono type_enc =

   case type_enc of

     Native (_, _, level) =>

     fused_type ctxt mono level 0 #> ho_type_from_typ type_enc false 0 #> SOME

   | _ => K NONE

 fun tag_with_type ctxt mono type_enc pos T tm =

   IConst (type_tag, T --> T, [T])

   |> mangle_type_args_in_iterm type_enc

   |> ho_term_from_iterm ctxt mono type_enc pos

   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])

        | _ => raise Fail "unexpected lambda-abstraction")

 and ho_term_from_iterm ctxt mono type_enc pos =

   let

     fun term site u =

       let

         val (head, args) = strip_iterm_comb u

         val pos =

           case site of

             Top_Level pos => pos

           | Eq_Arg pos => pos

           | _ => 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 map (term arg_site) args |> mk_aterm type_enc name T_args end

           | IVar (name, _) =>

             map (term Elsewhere) args |> mk_aterm type_enc name []

           | IAbs ((name, T), tm) =>

             if is_type_enc_higher_order type_enc then

               AAbs (((name, ho_type_from_typ type_enc true 0 T),

                      term Elsewhere tm), map (term Elsewhere) args)

             else

               raise Fail "unexpected lambda-abstraction"

           | IApp _ => raise Fail "impossible \"IApp\""

         val T = ityp_of u

       in

         if should_tag_with_type ctxt mono type_enc site u T then

           tag_with_type ctxt mono type_enc pos T t

         else

           t

       end

   in term (Top_Level pos) end

 and formula_from_iformula ctxt polym_constrs mono type_enc should_guard_var =

   let

     val thy = Proof_Context.theory_of ctxt

     val level = level_of_type_enc type_enc

     val do_term = ho_term_from_iterm ctxt mono type_enc

     fun do_out_of_bound_type pos phi universal (name, T) =

       if should_guard_type ctxt mono type_enc

              (fn () => should_guard_var thy polym_constrs level pos phi

                                         universal name) T then

         IVar (name, T)

         |> type_guard_iterm 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 = tag_with_type ctxt mono type_enc pos 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

           val do_bound_type = do_bound_type ctxt mono type_enc

         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 polym_constrs prefix encode freshen pos

         mono type_enc rank (j, {name, stature, kind, iformula, atomic_types}) =

   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,

    iformula

    |> formula_from_iformula ctxt polym_constrs mono type_enc

           should_guard_var_in_formula (if pos then SOME true else NONE)

    |> close_formula_universally

    |> bound_tvars type_enc true atomic_types,

    NONE,

    let val rank = rank j in

      case snd stature of

        Intro => isabelle_info introN rank

      | Inductive => isabelle_info inductiveN rank

      | Elim => isabelle_info elimN rank

      | Simp => isabelle_info simpN rank

      | Def => isabelle_info defN rank

      | _ => isabelle_info "" rank

    end)

   |> Formula

 fun formula_line_for_class_rel_clause type_enc

         ({name, subclass, superclass, ...} : class_rel_clause) =

   let val ty_arg = ATerm (tvar_a_name, []) in

     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,

              AConn (AImplies,

                     [type_class_formula type_enc subclass ty_arg,

                      type_class_formula type_enc superclass ty_arg])

              |> mk_aquant AForall

                           [(tvar_a_name, atype_of_type_vars type_enc)],

              NONE, isabelle_info inductiveN helper_rank)

   end

 fun formula_from_arity_atom type_enc (class, t, args) =

   ATerm (t, map (fn arg => ATerm (arg, [])) args)

   |> type_class_formula type_enc class

 fun formula_line_for_arity_clause type_enc

         ({name, prem_atoms, concl_atom} : arity_clause) =

   Formula (arity_clause_prefix ^ name, Axiom,

            mk_ahorn (map (formula_from_arity_atom type_enc) prem_atoms)

                     (formula_from_arity_atom type_enc concl_atom)

            |> mk_aquant AForall

                   (map (rpair (atype_of_type_vars type_enc)) (#3 concl_atom)),

            NONE, isabelle_info inductiveN helper_rank)

 fun formula_line_for_conjecture ctxt polym_constrs mono type_enc

         ({name, kind, iformula, atomic_types, ...} : translated_formula) =

   Formula (conjecture_prefix ^ name, kind,

            iformula

            |> formula_from_iformula ctxt polym_constrs mono type_enc

                   should_guard_var_in_formula (SOME false)

            |> close_formula_universally

            |> bound_tvars type_enc true atomic_types, NONE, [])

 fun type_enc_needs_free_types (Native (_, Polymorphic, _)) = true

   | type_enc_needs_free_types (Native _) = false

   | type_enc_needs_free_types _ = true

 fun formula_line_for_free_type j phi =

   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis, phi, NONE, [])

 fun formula_lines_for_free_types type_enc (facts : translated_formula list) =

   if type_enc_needs_free_types type_enc then

     let

       val phis =

         fold (union (op =)) (map #atomic_types facts) []

         |> formulas_for_types type_enc add_sorts_on_tfree

     in map2 formula_line_for_free_type (0 upto length phis - 1) phis end

   else

     []

 (** Symbol declarations **)

 fun decl_line_for_class order s =

   let val name as (s, _) = `make_type_class s in

     Decl (sym_decl_prefix ^ s, name,

           if order = First_Order then

             ATyAbs ([tvar_a_name],

                     if avoid_first_order_ghost_type_vars then

                       AFun (a_itself_atype, bool_atype)

                     else

                       bool_atype)

           else

             AFun (atype_of_types, bool_atype))

   end

 fun decl_lines_for_classes type_enc classes =

   case type_enc of

     Native (order, Polymorphic, _) => map (decl_line_for_class order) classes

   | _ => []

 fun sym_decl_table_for_facts thy type_enc sym_tab (conjs, facts, extra_tms) =

   let

     fun add_iterm_syms tm =

       let val (head, args) = strip_iterm_comb tm in

         (case head of

            IConst ((s, s'), T, T_args) =>

            let

              val (pred_sym, in_conj) =

                case Symtab.lookup sym_tab s of

                  SOME ({pred_sym, in_conj, ...} : sym_info) =>

                  (pred_sym, in_conj)

                | NONE => (false, false)

              val decl_sym =

                (case type_enc of

                   Guards _ => not pred_sym

                 | _ => true) andalso

                is_tptp_user_symbol s

            in

              if decl_sym then

                Symtab.map_default (s, [])

                    (insert_type thy #3 (s', T_args, T, pred_sym, length args,

                                         in_conj))

              else

                I

            end

          | IAbs (_, tm) => add_iterm_syms tm

          | _ => I)

         #> fold add_iterm_syms args

       end

     val add_fact_syms = K add_iterm_syms |> formula_fold NONE |> fact_lift

     fun add_formula_var_types (AQuant (_, xs, phi)) =

         fold (fn (_, SOME T) => insert_type thy 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 NONE) @{const_name undefined}

           |> (case type_arg_policy [] type_enc @{const_name undefined} of

                 Mangled_Type_Args => mangled_const_name type_enc [T]

               | _ => I)

       in

         Symtab.map_default (s, [])

                            (insert_type thy #3 (s', [T], T, false, 0, false))

       end

     fun add_TYPE_const () =

       let val (s, s') = TYPE_name in

         Symtab.map_default (s, [])

             (insert_type thy #3

                          (s', [tvar_a], @{typ "'a itself"}, false, 0, false))

       end

   in

     Symtab.empty

     |> is_type_enc_fairly_sound type_enc

        ? (fold (fold add_fact_syms) [conjs, facts]

           #> fold add_iterm_syms extra_tms

           #> (case type_enc of

                 Native (First_Order, Polymorphic, _) =>

                 if avoid_first_order_ghost_type_vars then add_TYPE_const ()

                 else I

               | Native _ => I

               | _ => fold add_undefined_const (var_types ())))

   end

 (* We add "bool" in case the helper "True_or_False" is included later. *)

 fun default_mono level =

   {maybe_finite_Ts = [@{typ bool}],

    surely_finite_Ts = [@{typ bool}],

    maybe_infinite_Ts = known_infinite_types,

    surely_infinite_Ts =

      case level of

        Noninf_Nonmono_Types (Strict, _) => []

      | _ => 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}) =

     let val thy = Proof_Context.theory_of ctxt in

       if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then

         case level of

           Noninf_Nonmono_Types (strictness, _) =>

           if exists (type_instance thy T) surely_infinite_Ts orelse

              member (type_equiv thy) maybe_finite_Ts T then

             mono

           else if is_type_kind_of_surely_infinite ctxt strictness

                                                   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 thy I T,

              maybe_nonmono_Ts = maybe_nonmono_Ts}

           else

             {maybe_finite_Ts = maybe_finite_Ts |> insert (type_equiv thy) 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 thy I T}

         | Fin_Nonmono_Types _ =>

           if exists (type_instance thy T) surely_finite_Ts orelse

              member (type_equiv thy) 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 thy I T,

              maybe_infinite_Ts = maybe_infinite_Ts,

              surely_infinite_Ts = surely_infinite_Ts,

              maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}

           else

             {maybe_finite_Ts = maybe_finite_Ts,

              surely_finite_Ts = surely_finite_Ts,

              maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_equiv thy) T,

              surely_infinite_Ts = surely_infinite_Ts,

              maybe_nonmono_Ts = maybe_nonmono_Ts}

         | _ => mono

       else

         mono

     end

   | 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 level

     |> is_type_level_monotonicity_based level

        ? fold (add_fact_mononotonicity_info ctxt level) facts

   end

 fun add_iformula_monotonic_types ctxt mono type_enc =

   let

     val thy = Proof_Context.theory_of ctxt

     val level = level_of_type_enc type_enc

     val should_encode = should_encode_type ctxt mono level

     fun add_type T = not (should_encode T) ? insert_type thy I T

     fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2

       | add_args _ = I

     and add_term tm = add_type (ityp_of tm) #> add_args tm

   in formula_fold NONE (K add_term) end

 fun add_fact_monotonic_types ctxt mono type_enc =

   add_iformula_monotonic_types ctxt mono type_enc |> fact_lift

 fun monotonic_types_for_facts ctxt mono type_enc facts =

   let val level = level_of_type_enc type_enc in

     [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso

            is_type_level_monotonicity_based level andalso

            granularity_of_type_level level <> Ghost_Type_Arg_Vars)

           ? fold (add_fact_monotonic_types ctxt mono type_enc) facts

   end

 fun formula_line_for_guards_mono_type ctxt mono type_enc T =

   Formula (guards_sym_formula_prefix ^

            ascii_of (mangled_type type_enc T),

            Axiom,

            IConst (`make_bound_var "X", T, [])

            |> type_guard_iterm type_enc T

            |> AAtom

            |> formula_from_iformula ctxt [] mono type_enc

                                     always_guard_var_in_formula (SOME true)

            |> close_formula_universally

            |> bound_tvars type_enc true (atomic_types_of T),

            NONE, isabelle_info inductiveN helper_rank)

 fun formula_line_for_tags_mono_type ctxt mono type_enc T =

   let val x_var = ATerm (`make_bound_var "X", []) in

     Formula (tags_sym_formula_prefix ^

              ascii_of (mangled_type type_enc T),

              Axiom,

              eq_formula type_enc (atomic_types_of T) [] false

                   (tag_with_type ctxt mono type_enc NONE T x_var) x_var,

              NONE, isabelle_info defN helper_rank)

   end

 fun problem_lines_for_mono_types ctxt mono type_enc Ts =

   case type_enc of

     Native _ => []

   | Guards _ => map (formula_line_for_guards_mono_type ctxt mono type_enc) Ts

   | Tags _ => map (formula_line_for_tags_mono_type ctxt mono type_enc) Ts

 fun decl_line_for_sym ctxt mono type_enc s

                       (s', T_args, T, pred_sym, ary, _) =

   let

     val thy = Proof_Context.theory_of ctxt

     val (T, T_args) =

       if null T_args then

         (T, [])

       else case unprefix_and_unascii const_prefix s of

         SOME s' =>

         let

           val s' = s' |> invert_const

           val T = s' |> robust_const_type thy

         in (T, robust_const_typargs thy (s', T)) end

       | NONE => raise Fail "unexpected type arguments"

   in

     Decl (sym_decl_prefix ^ s, (s, s'),

           T |> fused_type ctxt mono (level_of_type_enc type_enc) ary

             |> ho_type_from_typ type_enc pred_sym ary

             |> not (null T_args)

                ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))

   end

 fun honor_conj_sym_kind in_conj =

   if in_conj then (Hypothesis, I) else (Axiom, I)

 fun formula_line_for_guards_sym_decl ctxt mono type_enc n s j

                                      (s', T_args, T, _, ary, in_conj) =

   let

     val thy = Proof_Context.theory_of ctxt

     val (kind, maybe_negate) = honor_conj_sym_kind in_conj

     val (arg_Ts, res_T) = chop_fun ary T

     val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)

     val bounds =

       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))

     val bound_Ts =

       if exists (curry (op =) dummyT) T_args then

         case level_of_type_enc type_enc of

           All_Types => map SOME arg_Ts

         | level =>

           if granularity_of_type_level level = Ghost_Type_Arg_Vars then

             let val ghosts = ghost_type_args thy s ary in

               map2 (fn j => if member (op =) ghosts j then SOME else K NONE)

                    (0 upto ary - 1) arg_Ts

             end

           else

             replicate ary NONE

       else

         replicate ary 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 type_enc res_T

              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)

              |> formula_from_iformula ctxt [] mono type_enc

                                       always_guard_var_in_formula (SOME true)

              |> close_formula_universally

              |> bound_tvars type_enc (n > 1) (atomic_types_of T)

              |> maybe_negate,

              NONE, isabelle_info inductiveN helper_rank)

   end

 fun formula_lines_for_tags_sym_decl ctxt mono type_enc n s

         (j, (s', T_args, T, pred_sym, ary, in_conj)) =

   let

     val thy = Proof_Context.theory_of ctxt

     val level = level_of_type_enc type_enc

     val grain = granularity_of_type_level level

     val ident_base =

       tags_sym_formula_prefix ^ s ^

       (if n > 1 then "_" ^ string_of_int j else "")

     val (kind, maybe_negate) = honor_conj_sym_kind in_conj

     val (arg_Ts, res_T) = chop_fun ary T

     val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)

     val bounds = bound_names |> map (fn name => ATerm (name, []))

     val cst = mk_aterm type_enc (s, s') T_args

     val eq = maybe_negate oo eq_formula type_enc (atomic_types_of T) [] pred_sym

     val should_encode = should_encode_type ctxt mono level

     val tag_with = tag_with_type ctxt mono type_enc NONE

     val add_formula_for_res =

       if should_encode res_T then

         let

           val tagged_bounds =

             if grain = Ghost_Type_Arg_Vars then

               let val ghosts = ghost_type_args thy s ary in

                 map2 (fn (j, arg_T) => member (op =) ghosts j ? tag_with arg_T)

                      (0 upto ary - 1 ~~ arg_Ts) bounds

               end

             else

               bounds

         in

           cons (Formula (ident_base ^ "_res", kind,

                          eq (tag_with res_T (cst bounds)) (cst tagged_bounds),

                          NONE, isabelle_info defN helper_rank))

         end

       else

         I

   in [] |> not pred_sym ? add_formula_for_res end

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

 fun rationalize_decls thy (decls as decl :: (decls' as _ :: _)) =

     let

       val T = result_type_of_decl decl

               |> map_type_tvar (fn (z, _) => TVar (z, HOLogic.typeS))

     in

       if forall (type_generalization thy T o result_type_of_decl) decls' then

         [decl]

       else

         decls

     end

   | rationalize_decls _ decls = decls

 fun problem_lines_for_sym_decls ctxt mono type_enc (s, decls) =

   case type_enc of

     Native _ => [decl_line_for_sym ctxt mono type_enc s (hd decls)]

   | Guards (_, level) =>

     let

       val thy = Proof_Context.theory_of ctxt

       val decls = decls |> rationalize_decls thy

       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 mono type_enc n s)

     end

   | Tags (_, level) =>

     if granularity_of_type_level level = All_Vars then

       []

     else

       let val n = length decls in

         (0 upto n - 1 ~~ decls)

         |> maps (formula_lines_for_tags_sym_decl ctxt mono type_enc n s)

       end

 fun problem_lines_for_sym_decl_table ctxt mono type_enc mono_Ts sym_decl_tab =

   let

     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst

     val mono_lines = problem_lines_for_mono_types ctxt mono type_enc mono_Ts

     val decl_lines =

       fold_rev (append o problem_lines_for_sym_decls ctxt mono type_enc) syms []

   in mono_lines @ decl_lines end

 fun pair_append (xs1, xs2) (ys1, ys2) = (xs1 @ ys1, xs2 @ ys2)

 fun do_uncurried_alias_lines_for_sym ctxt monom_constrs mono type_enc sym_tab0

                                      sym_tab base_s0 types in_conj =

   let

     fun do_alias ary =

       let

         val thy = Proof_Context.theory_of ctxt

         val (kind, maybe_negate) = honor_conj_sym_kind in_conj

         val base_name = base_s0 |> `(make_fixed_const (SOME type_enc))

         val T = case types of [T] => T | _ => robust_const_type thy base_s0

         val T_args = robust_const_typargs thy (base_s0, T)

         val (base_name as (base_s, _), T_args) =

           mangle_type_args_in_const type_enc base_name T_args

         val base_ary = min_ary_of sym_tab0 base_s

         fun do_const name = IConst (name, T, T_args)

         val filter_ty_args =

           filter_type_args_in_iterm thy monom_constrs type_enc

         val ho_term_of = ho_term_from_iterm ctxt mono type_enc (SOME true)

         val name1 as (s1, _) =

           base_name |> ary - 1 > base_ary ? aliased_uncurried (ary - 1)

         val name2 as (s2, _) = base_name |> aliased_uncurried ary

         val (arg_Ts, _) = chop_fun ary T

         val bound_names =

           1 upto ary |> map (`I o make_bound_var o string_of_int)

         val bounds = bound_names ~~ arg_Ts

         val (first_bounds, last_bound) =

           bounds |> map (fn (name, T) => IConst (name, T, [])) |> split_last

         val tm1 =

           mk_app_op type_enc (list_app (do_const name1) first_bounds) last_bound

           |> filter_ty_args

         val tm2 =

           list_app (do_const name2) (first_bounds @ [last_bound])

           |> filter_ty_args

         val do_bound_type = do_bound_type ctxt mono type_enc

         val eq =

           eq_formula type_enc (atomic_types_of T)

                      (map (apsnd do_bound_type) bounds) false

                      (ho_term_of tm1) (ho_term_of tm2)

       in

         ([tm1, tm2],

          [Formula (uncurried_alias_eq_prefix ^ s2, kind, eq |> maybe_negate,

                    NONE, isabelle_info defN helper_rank)])

         |> (if ary - 1 = base_ary orelse Symtab.defined sym_tab s1 then I

             else pair_append (do_alias (ary - 1)))

       end

   in do_alias end

 fun uncurried_alias_lines_for_sym ctxt monom_constrs mono type_enc sym_tab0

         sym_tab (s, {min_ary, types, in_conj, ...} : sym_info) =

   case unprefix_and_unascii const_prefix s of

     SOME mangled_s =>

     if String.isSubstring uncurried_alias_sep mangled_s then

       let

         val base_s0 = mangled_s |> unmangled_const_name |> hd |> invert_const

       in

         do_uncurried_alias_lines_for_sym ctxt monom_constrs mono type_enc

             sym_tab0 sym_tab base_s0 types in_conj min_ary

       end

     else

       ([], [])

   | NONE => ([], [])

 fun uncurried_alias_lines_for_sym_table ctxt monom_constrs mono type_enc

                                         uncurried_aliases sym_tab0 sym_tab =

   ([], [])

   |> uncurried_aliases

      ? Symtab.fold_rev

            (pair_append

             o uncurried_alias_lines_for_sym ctxt monom_constrs mono type_enc

                                             sym_tab0 sym_tab) sym_tab

 val implicit_declsN = "Should-be-implicit typings"

 val explicit_declsN = "Explicit typings"

 val uncurried_alias_eqsN = "Uncurried aliases"

 val factsN = "Relevant facts"

 val class_relsN = "Class relationships"

 val aritiesN = "Arities"

 val helpersN = "Helper facts"

 val conjsN = "Conjectures"

 val free_typesN = "Type variables"

 (* TFF allows implicit declarations of types, function symbols, and predicate

    symbols (with "$i" as the type of individuals), but some provers (e.g.,

    SNARK) require explicit declarations. The situation is similar for THF. *)

 fun default_type type_enc pred_sym s =

   let

     val ind =

       case type_enc of

         Native _ =>

         if String.isPrefix type_const_prefix s orelse

            String.isPrefix tfree_prefix s then

           atype_of_types

         else

           individual_atype

       | _ => individual_atype

     fun typ 0 = if pred_sym then bool_atype else ind

       | typ ary = AFun (ind, typ (ary - 1))

   in typ end

 fun nary_type_constr_type n =

   funpow n (curry AFun atype_of_types) atype_of_types

 fun undeclared_syms_in_problem type_enc problem =

   let

     fun do_sym (name as (s, _)) ty =

       if is_tptp_user_symbol s then

         Symtab.default (s, (name, ty))

       else

         I

     fun do_type (AType (name, tys)) =

         do_sym name (fn () => nary_type_constr_type (length tys))

         #> fold do_type tys

       | do_type (AFun (ty1, ty2)) = do_type ty1 #> do_type ty2

       | do_type (ATyAbs (_, ty)) = do_type ty

     fun do_term pred_sym (ATerm (name as (s, _), tms)) =

         do_sym name (fn _ => default_type type_enc pred_sym s (length tms))

         #> fold (do_term false) tms

       | do_term _ (AAbs (((_, ty), tm), args)) =

         do_type ty #> do_term false tm #> fold (do_term false) args

     fun do_formula (AQuant (_, xs, phi)) =

         fold do_type (map_filter snd xs) #> do_formula phi

       | do_formula (AConn (_, phis)) = fold do_formula phis

       | do_formula (AAtom tm) = do_term true tm

     fun do_problem_line (Decl (_, _, ty)) = do_type ty

       | do_problem_line (Formula (_, _, phi, _, _)) = do_formula phi

   in

     Symtab.empty

     |> fold (fn (s, _) => Symtab.default (s, (("", ""), K tvar_a_atype)))

             (declared_syms_in_problem problem)

     |> fold (fold do_problem_line o snd) problem

   end

 fun declare_undeclared_syms_in_atp_problem type_enc problem =

   let

     val decls =

       Symtab.fold (fn (_, (("", ""), _)) => I (* already declared *)

                     | (s, (sym, ty)) =>

                       cons (Decl (type_decl_prefix ^ s, sym, ty ())))

                   (undeclared_syms_in_problem type_enc problem) []

   in (implicit_declsN, decls) :: problem end

 fun exists_subdtype P =

   let

     fun ex U = P U orelse

       (case U of Datatype.DtType (_, Us) => exists ex Us | _ => false)

   in ex end

 fun is_poly_constr (_, Us) =

   exists (exists_subdtype (fn Datatype.DtTFree _ => true | _ => false)) Us

 fun all_constrs_of_polymorphic_datatypes thy =

   Symtab.fold (snd

                #> #descr

                #> maps (snd #> #3)