(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_atp_translate.ML

    Author:     Fabian Immler, TU Muenchen

    Author:     Makarius

    Author:     Jasmin Blanchette, TU Muenchen

Translation of HOL to FOL for Sledgehammer.

*)

signature SLEDGEHAMMER_ATP_TRANSLATE =

sig

  type 'a fo_term = 'a ATP_Problem.fo_term

  type format = ATP_Problem.format

  type formula_kind = ATP_Problem.formula_kind

  type 'a problem = 'a ATP_Problem.problem

  type locality = Sledgehammer_Filter.locality

  datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

  datatype type_level =

    All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types

  datatype type_heaviness = Heavy | Light

  datatype type_system =

    Simple_Types of type_level |

    Preds of polymorphism * type_level * type_heaviness |

    Tags of polymorphism * type_level * type_heaviness

  type translated_formula

  val readable_names : bool Config.T

  val fact_prefix : string

  val conjecture_prefix : string

  val helper_prefix : string

  val typed_helper_suffix : string

  val predicator_base : string

  val explicit_app_base : string

  val type_pred_base : string

  val tff_type_prefix : string

  val type_sys_from_string : string -> type_system

  val polymorphism_of_type_sys : type_system -> polymorphism

  val level_of_type_sys : type_system -> type_level

  val is_type_sys_virtually_sound : type_system -> bool

  val is_type_sys_fairly_sound : type_system -> bool

  val unmangled_const : string -> string * string fo_term list

  val translate_atp_fact :

    Proof.context -> bool -> (string * locality) * thm

    -> translated_formula option * ((string * locality) * thm)

  val prepare_atp_problem :

    Proof.context -> format -> formula_kind -> formula_kind -> type_system

    -> bool -> term list -> term

    -> (translated_formula option * ((string * 'a) * thm)) list

    -> string problem * string Symtab.table * int * int

       * (string * 'a) list vector * int list * int Symtab.table

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

end;

structure Sledgehammer_ATP_Translate : SLEDGEHAMMER_ATP_TRANSLATE =

struct

open ATP_Problem

open Metis_Translate

open Sledgehammer_Util

open Sledgehammer_Filter

(* experimental *)

val generate_useful_info = false

fun useful_isabelle_info s =

  if generate_useful_info then

    SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))

  else

    NONE

val intro_info = useful_isabelle_info "intro"

val elim_info = useful_isabelle_info "elim"

val simp_info = useful_isabelle_info "simp"

(* Readable names are often much shorter, especially if types are mangled in

   names. Also, the logic for generating legal SNARK sort names is only

   implemented for readable names. Finally, readable names are, well, more

   readable. For these reason, they are enabled by default. *)

val readable_names =

  Attrib.setup_config_bool @{binding sledgehammer_atp_readable_names} (K true)

val type_decl_prefix = "type_"

val sym_decl_prefix = "sym_"

val fact_prefix = "fact_"

val conjecture_prefix = "conj_"

val helper_prefix = "help_"

val class_rel_clause_prefix = "crel_";

val arity_clause_prefix = "arity_"

val tfree_prefix = "tfree_"

val typed_helper_suffix = "_T"

val untyped_helper_suffix = "_U"

val predicator_base = "hBOOL"

val explicit_app_base = "hAPP"

val type_pred_base = "is"

val tff_type_prefix = "ty_"

fun make_tff_type s = tff_type_prefix ^ ascii_of s

(* Freshness almost guaranteed! *)

val sledgehammer_weak_prefix = "Sledgehammer:"

datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

datatype type_level =

  All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types

datatype type_heaviness = Heavy | Light

datatype type_system =

  Simple_Types of type_level |

  Preds of polymorphism * type_level * type_heaviness |

  Tags of polymorphism * type_level * type_heaviness

fun try_unsuffixes ss s =

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

fun type_sys_from_string s =

  (case try (unprefix "poly_") s of

     SOME s => (SOME Polymorphic, s)

   | NONE =>

     case try (unprefix "mono_") s of

       SOME s => (SOME Monomorphic, s)

     | NONE =>

       case try (unprefix "mangled_") s of

         SOME s => (SOME Mangled_Monomorphic, s)

       | NONE => (NONE, s))

  ||> (fn s =>

          (* "_query" and "_bang" are for the ASCII-challenged Mirabelle. *)

          case try_unsuffixes ["?", "_query"] s of

            SOME s => (Nonmonotonic_Types, s)

          | NONE =>

            case try_unsuffixes ["!", "_bang"] s of

              SOME s => (Finite_Types, s)

            | NONE => (All_Types, s))

  ||> apsnd (fn s =>

                case try (unsuffix "_heavy") s of

                  SOME s => (Heavy, s)

                | NONE => (Light, s))

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

         case (core, (poly, level, heaviness)) of

           ("simple", (NONE, _, Light)) => Simple_Types level

         | ("preds", (SOME poly, _, _)) => Preds (poly, level, heaviness)

         | ("tags", (SOME Polymorphic, All_Types, _)) =>

           Tags (Polymorphic, All_Types, heaviness)

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

           (* The actual light encoding is very unsound. *)

           Tags (Polymorphic, level, Heavy)

         | ("tags", (SOME poly, _, _)) => Tags (poly, level, heaviness)

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

           Preds (poly, Const_Arg_Types, Light)

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

           Preds (Polymorphic, No_Types, Light)

         | _ => raise Same.SAME)

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

fun polymorphism_of_type_sys (Simple_Types _) = Mangled_Monomorphic

  | polymorphism_of_type_sys (Preds (poly, _, _)) = poly

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

fun level_of_type_sys (Simple_Types level) = level

  | level_of_type_sys (Preds (_, level, _)) = level

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

fun heaviness_of_type_sys (Simple_Types _) = Heavy

  | heaviness_of_type_sys (Preds (_, _, heaviness)) = heaviness

  | heaviness_of_type_sys (Tags (_, _, heaviness)) = heaviness

fun is_type_level_virtually_sound level =

  level = All_Types orelse level = Nonmonotonic_Types

val is_type_sys_virtually_sound =

  is_type_level_virtually_sound o level_of_type_sys

fun is_type_level_fairly_sound level =

  is_type_level_virtually_sound level orelse level = Finite_Types

val is_type_sys_fairly_sound = is_type_level_fairly_sound o level_of_type_sys

fun formula_map f (AQuant (q, xs, phi)) = AQuant (q, xs, formula_map f phi)

  | formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)

  | formula_map f (AAtom tm) = AAtom (f tm)

fun aconn_fold pos f (ANot, [phi]) = f (Option.map not pos) phi

  | aconn_fold pos f (AImplies, [phi1, phi2]) =

    f (Option.map not pos) phi1 #> f pos phi2

  | aconn_fold pos f (AAnd, phis) = fold (f pos) phis

  | aconn_fold pos f (AOr, phis) = fold (f pos) phis

  | aconn_fold _ f (_, phis) = fold (f NONE) phis

fun aconn_map pos f (ANot, [phi]) = AConn (ANot, [f (Option.map not pos) phi])

  | aconn_map pos f (AImplies, [phi1, phi2]) =

    AConn (AImplies, [f (Option.map not pos) phi1, f pos phi2])

  | aconn_map pos f (AAnd, phis) = AConn (AAnd, map (f pos) phis)

  | aconn_map pos f (AOr, phis) = AConn (AOr, map (f pos) phis)

  | aconn_map _ f (c, phis) = AConn (c, map (f NONE) phis)

fun formula_fold pos f =

  let

    fun aux pos (AQuant (_, _, phi)) = aux pos phi

      | aux pos (AConn conn) = aconn_fold pos aux conn

      | aux pos (AAtom tm) = f pos tm

  in aux pos end

type translated_formula =

  {name: string,

   locality: locality,

   kind: formula_kind,

   combformula: (name, typ, combterm) formula,

   atomic_types: typ list}

fun update_combformula f ({name, locality, kind, combformula, atomic_types}

                          : translated_formula) =

  {name = name, locality = locality, kind = kind, combformula = f combformula,

   atomic_types = atomic_types} : translated_formula

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

(* The Booleans indicate whether all type arguments should be kept. *)

datatype type_arg_policy =

  Explicit_Type_Args of bool |

  Mangled_Type_Args of bool |

  No_Type_Args

fun should_drop_arg_type_args (Simple_Types _) =

    false (* since TFF doesn't support overloading *)

  | should_drop_arg_type_args type_sys =

    level_of_type_sys type_sys = All_Types andalso

    heaviness_of_type_sys type_sys = Heavy

fun general_type_arg_policy type_sys =

  if level_of_type_sys type_sys = No_Types then

    No_Type_Args

  else if polymorphism_of_type_sys type_sys = Mangled_Monomorphic then

    Mangled_Type_Args (should_drop_arg_type_args type_sys)

  else

    Explicit_Type_Args (should_drop_arg_type_args type_sys)

fun type_arg_policy _ @{const_name HOL.eq} = No_Type_Args

  | type_arg_policy type_sys _ = general_type_arg_policy type_sys

fun atp_type_literals_for_types type_sys kind Ts =

  if level_of_type_sys type_sys = No_Types then

    []

  else

    Ts |> type_literals_for_types

       |> filter (fn TyLitVar _ => kind <> Conjecture

                   | TyLitFree _ => kind = Conjecture)

fun mk_anot phi = AConn (ANot, [phi])

fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])

fun mk_aconns c phis =

  let val (phis', phi') = split_last phis in

    fold_rev (mk_aconn c) phis' phi'

  end

fun mk_ahorn [] phi = phi

  | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])

fun mk_aquant _ [] phi = phi

  | mk_aquant q xs (phi as AQuant (q', xs', phi')) =

    if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)

  | mk_aquant q xs phi = AQuant (q, xs, phi)

fun close_universally atom_vars phi =

  let

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

        formula_vars (map fst xs @ bounds) phi

      | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis

      | formula_vars bounds (AAtom tm) =

        union (op =) (atom_vars tm []

                      |> filter_out (member (op =) bounds o fst))

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

fun combterm_vars (CombApp (tm1, tm2)) = fold combterm_vars [tm1, tm2]

  | combterm_vars (CombConst _) = I

  | combterm_vars (CombVar (name, T)) = insert (op =) (name, SOME T)

fun close_combformula_universally phi = close_universally combterm_vars phi

fun term_vars (ATerm (name as (s, _), tms)) =

  is_atp_variable s ? insert (op =) (name, NONE)

  #> fold term_vars tms

fun close_formula_universally phi = close_universally term_vars phi

fun fo_term_from_typ (Type (s, Ts)) =

    ATerm (`make_fixed_type_const s, map fo_term_from_typ Ts)

  | fo_term_from_typ (TFree (s, _)) =

    ATerm (`make_fixed_type_var s, [])

  | fo_term_from_typ (TVar ((x as (s, _)), _)) =

    ATerm ((make_schematic_type_var x, s), [])

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

val mangled_type_sep = "\000"

fun generic_mangled_type_name f (ATerm (name, [])) = f name

  | generic_mangled_type_name f (ATerm (name, tys)) =

    f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)

    ^ ")"

val mangled_type_name =

  fo_term_from_typ

  #> (fn ty => (make_tff_type (generic_mangled_type_name fst ty),

                generic_mangled_type_name snd ty))

fun generic_mangled_type_suffix f g Ts =

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

            o generic_mangled_type_name f) Ts ""

fun mangled_const_name T_args (s, s') =

  let val ty_args = map fo_term_from_typ T_args in

    (s ^ generic_mangled_type_suffix fst ascii_of ty_args,

     s' ^ generic_mangled_type_suffix snd I ty_args)

  end

val parse_mangled_ident =

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

fun parse_mangled_type x =

  (parse_mangled_ident

   -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")

                    [] >> ATerm) x

and parse_mangled_types x =

  (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x

fun unmangled_type s =

  s |> suffix ")" |> raw_explode

    |> Scan.finite Symbol.stopper

           (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^

                                                quote s)) parse_mangled_type))

    |> fst

val unmangled_const_name = space_explode mangled_type_sep #> hd

fun unmangled_const s =

  let val ss = space_explode mangled_type_sep s in

    (hd ss, map unmangled_type (tl ss))

  end

fun introduce_proxies tm =

  let

    fun aux top_level (CombApp (tm1, tm2)) =

        CombApp (aux top_level tm1, aux false tm2)

      | aux top_level (CombConst (name as (s, s'), T, T_args)) =

        (case proxify_const s of

           SOME proxy_base =>

           if top_level then

             (case s of

                "c_False" => (tptp_false, s')

              | "c_True" => (tptp_true, s')

              | _ => name, [])

           else

             (proxy_base |>> prefix const_prefix, T_args)

          | NONE => (name, T_args))

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

      | aux _ tm = tm

  in aux true tm end

fun combformula_from_prop thy eq_as_iff =

  let

    fun do_term bs t atomic_types =

      combterm_from_term thy bs (Envir.eta_contract t)

      |>> (introduce_proxies #> AAtom)

      ||> union (op =) atomic_types

    fun do_quant bs q s T t' =

      let val s = Name.variant (map fst bs) s in

        do_formula ((s, T) :: bs) t'

        #>> mk_aquant q [(`make_bound_var s, SOME T)]

      end

    and do_conn bs c t1 t2 =

      do_formula bs t1 ##>> do_formula bs t2

      #>> uncurry (mk_aconn c)

    and do_formula bs t =

      case t of

        @{const Not} $ t1 => do_formula bs t1 #>> mk_anot

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

        do_quant bs AForall s T t'

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

        do_quant bs AExists s T t'

      | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd t1 t2

      | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr t1 t2

      | @{const HOL.implies} $ t1 $ t2 => do_conn bs AImplies t1 t2

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

        if eq_as_iff then do_conn bs AIff t1 t2 else do_term bs t

      | _ => do_term bs t

  in do_formula [] end

fun presimplify_term ctxt =

  Skip_Proof.make_thm (Proof_Context.theory_of ctxt)

  #> Meson.presimplify ctxt

  #> prop_of

fun concealed_bound_name j = sledgehammer_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 extensionalize_term ctxt t =

  let val thy = Proof_Context.theory_of ctxt in

    t |> cterm_of thy |> Meson.extensionalize_conv ctxt

      |> prop_of |> Logic.dest_equals |> snd

  end

fun introduce_combinators_in_term ctxt kind t =

  let val thy = Proof_Context.theory_of ctxt in

    if Meson.is_fol_term thy t then

      t

    else

      let

        fun aux Ts t =

          case t of

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

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

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

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

            aux Ts (t0 $ eta_expand Ts t1 1)

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

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

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

            aux Ts (t0 $ eta_expand Ts t1 1)

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

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

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

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

              $ t1 $ t2 =>

            t0 $ aux Ts t1 $ aux Ts t2

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

                   t

                 else

                   t |> conceal_bounds Ts

                     |> Envir.eta_contract

                     |> cterm_of thy

                     |> Meson_Clausify.introduce_combinators_in_cterm

                     |> prop_of |> Logic.dest_equals |> snd

                     |> reveal_bounds Ts

        val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single

      in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end

      handle THM _ =>

             (* A type variable of sort "{}" will make abstraction fail. *)

             if kind = Conjecture then HOLogic.false_const

             else HOLogic.true_const

  end

(* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the

   same in Sledgehammer to prevent the discovery of unreplayable proofs. *)

fun freeze_term t =

  let

    fun aux (t $ u) = aux t $ aux u

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

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

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

      | aux t = t

  in t |> exists_subterm is_Var t ? aux end

(* making fact and conjecture formulas *)

fun make_formula ctxt eq_as_iff presimp name loc kind t =

  let

    val thy = Proof_Context.theory_of ctxt

    val t = t |> Envir.beta_eta_contract

              |> transform_elim_term

              |> Object_Logic.atomize_term thy

    val need_trueprop = (fastype_of t = @{typ bool})

    val t = t |> need_trueprop ? HOLogic.mk_Trueprop

              |> Raw_Simplifier.rewrite_term thy

                     (Meson.unfold_set_const_simps ctxt) []

              |> extensionalize_term ctxt

              |> presimp ? presimplify_term ctxt

              |> perhaps (try (HOLogic.dest_Trueprop))

              |> introduce_combinators_in_term ctxt kind

              |> kind <> Axiom ? freeze_term

    val (combformula, atomic_types) = combformula_from_prop thy eq_as_iff t []

  in

    {name = name, locality = loc, kind = kind, combformula = combformula,

     atomic_types = atomic_types}

  end

fun make_fact ctxt keep_trivial eq_as_iff presimp ((name, loc), t) =

  case (keep_trivial, make_formula ctxt eq_as_iff presimp name loc Axiom t) of

    (false, {combformula = AAtom (CombConst (("c_True", _), _, _)), ...}) =>

    NONE

  | (_, formula) => SOME formula

fun make_conjecture ctxt prem_kind ts =

  let val last = length ts - 1 in

    map2 (fn j => fn t =>

             let

               val (kind, maybe_negate) =

                 if j = last then

                   (Conjecture, I)

                 else

                   (prem_kind,

                    if prem_kind = Conjecture then update_combformula mk_anot

                    else I)

              in

                make_formula ctxt true true (string_of_int j) General kind t

                |> maybe_negate

              end)

         (0 upto last) ts

  end

(** Finite and infinite type inference **)

fun deep_freeze_atyp (TVar (_, S)) = TFree ("v", S)

  | deep_freeze_atyp T = T

val deep_freeze_type = map_atyps deep_freeze_atyp

val type_instance = Sign.typ_instance o Proof_Context.theory_of

(* 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 ctxt (nonmono_Ts as _ :: _) _ T =

    exists (curry (type_instance ctxt) (deep_freeze_type T)) nonmono_Ts

  | should_encode_type _ _ All_Types _ = true

  | should_encode_type ctxt _ Finite_Types T = is_type_surely_finite ctxt T

  | should_encode_type _ _ _ _ = false

fun should_predicate_on_type ctxt nonmono_Ts (Preds (_, level, heaviness))

                             should_predicate_on_var T =

    (heaviness = Heavy orelse should_predicate_on_var ()) andalso

    should_encode_type ctxt nonmono_Ts level T

  | should_predicate_on_type _ _ _ _ _ = false

fun is_var_or_bound_var (CombConst ((s, _), _, _)) =

    String.isPrefix bound_var_prefix s

  | is_var_or_bound_var (CombVar _) = true

  | is_var_or_bound_var _ = false

datatype tag_site = Top_Level | Eq_Arg | Elsewhere

fun should_tag_with_type _ _ _ Top_Level _ _ = false

  | should_tag_with_type ctxt nonmono_Ts (Tags (_, level, heaviness)) site u T =

    (case heaviness of

       Heavy => should_encode_type ctxt nonmono_Ts level T

     | Light =>

       case (site, is_var_or_bound_var u) of

         (Eq_Arg, true) => should_encode_type ctxt nonmono_Ts level T

       | _ => false)

  | should_tag_with_type _ _ _ _ _ _ = false

val homo_infinite_T = @{typ ind} (* any infinite type *)

fun homogenized_type ctxt nonmono_Ts level T =

  if should_encode_type ctxt nonmono_Ts level T then T else homo_infinite_T

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

type sym_info =

  {pred_sym : bool, min_ary : int, max_ary : int, typ : typ option}

fun add_combterm_syms_to_table explicit_apply =

  let

    fun aux top_level tm =

      let val (head, args) = strip_combterm_comb tm in

        (case head of

           CombConst ((s, _), T, _) =>

           if String.isPrefix bound_var_prefix s then

             I

           else

             let val ary = length args in

               Symtab.map_default

                   (s, {pred_sym = true,

                        min_ary = if explicit_apply then 0 else ary,

                        max_ary = 0, typ = SOME T})

                   (fn {pred_sym, min_ary, max_ary, typ} =>

                       {pred_sym = pred_sym andalso top_level,

                        min_ary = Int.min (ary, min_ary),

                        max_ary = Int.max (ary, max_ary),

                        typ = if typ = SOME T then typ else NONE})

            end

         | _ => I)

        #> fold (aux false) args

      end

  in aux true end

fun add_fact_syms_to_table explicit_apply =

  fact_lift (formula_fold NONE (K (add_combterm_syms_to_table explicit_apply)))

val default_sym_table_entries : (string * sym_info) list =

  [("equal", {pred_sym = true, min_ary = 2, max_ary = 2, typ = NONE}),

   (make_fixed_const predicator_base,

    {pred_sym = true, min_ary = 1, max_ary = 1, typ = NONE})] @

  ([tptp_false, tptp_true]

   |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, typ = NONE}))

fun sym_table_for_facts explicit_apply facts =

  Symtab.empty |> fold Symtab.default default_sym_table_entries

               |> fold (add_fact_syms_to_table explicit_apply) facts

fun min_arity_of sym_tab s =

  case Symtab.lookup sym_tab s of

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

  | NONE =>

    case strip_prefix_and_unascii const_prefix s of

      SOME s =>

      let val s = s |> unmangled_const_name |> invert_const in

        if s = predicator_base then 1

        else if s = explicit_app_base then 2

        else if s = type_pred_base then 1

        else 0

      end

    | NONE => 0

(* True if the constant ever appears outside of the top-level position in

   literals, or if it appears with different arities (e.g., because of different

   type instantiations). If false, the constant always receives all of its

   arguments and is used as a predicate. *)

fun is_pred_sym sym_tab s =

  case Symtab.lookup sym_tab s of

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

    pred_sym andalso min_ary = max_ary

  | NONE => false

val predicator_combconst =

  CombConst (`make_fixed_const predicator_base, @{typ "bool => bool"}, [])

fun predicator tm = CombApp (predicator_combconst, tm)

fun introduce_predicators_in_combterm sym_tab tm =

  case strip_combterm_comb tm of

    (CombConst ((s, _), _, _), _) =>

    if is_pred_sym sym_tab s then tm else predicator tm

  | _ => predicator tm

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

fun explicit_app arg head =

  let

    val head_T = combtyp_of head

    val (arg_T, res_T) = dest_funT head_T

    val explicit_app =

      CombConst (`make_fixed_const explicit_app_base, head_T --> head_T,

                 [arg_T, res_T])

  in list_app explicit_app [head, arg] end

fun list_explicit_app head args = fold explicit_app args head

fun introduce_explicit_apps_in_combterm sym_tab =

  let

    fun aux tm =

      case strip_combterm_comb tm of

        (head as CombConst ((s, _), _, _), args) =>

        args |> map aux

             |> chop (min_arity_of sym_tab s)

             |>> list_app head

             |-> list_explicit_app

      | (head, args) => list_explicit_app head (map aux args)

  in aux end

fun chop_fun 0 T = ([], T)

  | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =

    chop_fun (n - 1) ran_T |>> cons dom_T

  | chop_fun _ _ = raise Fail "unexpected non-function"

fun filter_type_args _ _ _ [] = []

  | filter_type_args thy s arity T_args =

    let

      (* will throw "TYPE" for pseudo-constants *)

      val U = if s = explicit_app_base then

                @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global

              else

                s |> Sign.the_const_type thy

    in

      case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of

        [] => []

      | res_U_vars =>

        let val U_args = (s, U) |> Sign.const_typargs thy in

          U_args ~~ T_args

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

                            if member (op =) res_U_vars (dest_TVar U) then

                              SOME T

                            else

                              NONE)

        end

    end

    handle TYPE _ => T_args

fun enforce_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys =

  let

    val thy = Proof_Context.theory_of ctxt

    fun aux arity (CombApp (tm1, tm2)) =

        CombApp (aux (arity + 1) tm1, aux 0 tm2)

      | aux arity (CombConst (name as (s, _), T, T_args)) =

        let

          val level = level_of_type_sys type_sys

          val (T, T_args) =

            (* Aggressively merge most "hAPPs" if the type system is unsound

               anyway, by distinguishing overloads only on the homogenized

               result type. Don't do it for lightweight type systems, though,

               since it leads to too many unsound proofs. *)

            if s = const_prefix ^ explicit_app_base andalso

               length T_args = 2 andalso

               not (is_type_sys_virtually_sound type_sys) andalso

               heaviness_of_type_sys type_sys = Heavy then

              T_args |> map (homogenized_type ctxt nonmono_Ts level)

                     |> (fn Ts => let val T = hd Ts --> nth Ts 1 in

                                    (T --> T, tl Ts)

                                  end)

            else

              (T, T_args)

        in

          (case strip_prefix_and_unascii const_prefix s of

             NONE => (name, T_args)

           | SOME s'' =>

             let

               val s'' = invert_const s''

               fun filtered_T_args false = T_args

                 | filtered_T_args true = filter_type_args thy s'' arity T_args

             in

               case type_arg_policy type_sys s'' of

                 Explicit_Type_Args drop_args =>

                 (name, filtered_T_args drop_args)

               | Mangled_Type_Args drop_args =>

                 (mangled_const_name (filtered_T_args drop_args) name, [])

               | No_Type_Args => (name, [])

             end)

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

        end

      | aux _ tm = tm

  in aux 0 end

fun repair_combterm ctxt nonmono_Ts type_sys sym_tab =

  introduce_explicit_apps_in_combterm sym_tab

  #> introduce_predicators_in_combterm sym_tab

  #> enforce_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys

fun repair_fact ctxt nonmono_Ts type_sys sym_tab =

  update_combformula (formula_map

      (repair_combterm ctxt nonmono_Ts type_sys sym_tab))

(** Helper facts **)

fun ti_ti_helper_fact () =

  let

    fun var s = ATerm (`I s, [])

    fun tag tm = ATerm (`make_fixed_const type_tag_name, [var "X", tm])

  in

    Formula (helper_prefix ^ "ti_ti", Axiom,

             AAtom (ATerm (`I "equal", [tag (tag (var "Y")), tag (var "Y")]))

             |> close_formula_universally, simp_info, NONE)

  end

fun helper_facts_for_sym ctxt type_sys (s, {typ, ...} : sym_info) =

  case strip_prefix_and_unascii const_prefix s of

    SOME mangled_s =>

    let

      val thy = Proof_Context.theory_of ctxt

      val unmangled_s = mangled_s |> unmangled_const_name

      fun dub_and_inst c needs_fairly_sound (th, j) =

        ((c ^ "_" ^ string_of_int j ^

          (if needs_fairly_sound then typed_helper_suffix

           else untyped_helper_suffix),

          General),

         let val t = th |> prop_of in

           t |> ((case general_type_arg_policy type_sys of

                    Mangled_Type_Args _ => true

                  | _ => false) andalso

                 not (null (Term.hidden_polymorphism t)))

                ? (case typ of

                     SOME T => specialize_type thy (invert_const unmangled_s, T)

                   | NONE => I)

         end)

      fun make_facts eq_as_iff =

        map_filter (make_fact ctxt false eq_as_iff false)

      val fairly_sound = is_type_sys_fairly_sound type_sys

    in

      metis_helpers

      |> maps (fn (metis_s, (needs_fairly_sound, ths)) =>

                  if metis_s <> unmangled_s orelse

                     (needs_fairly_sound andalso not fairly_sound) then

                    []

                  else

                    ths ~~ (1 upto length ths)

                    |> map (dub_and_inst mangled_s needs_fairly_sound)

                    |> make_facts (not needs_fairly_sound))

    end

  | NONE => []

fun helper_facts_for_sym_table ctxt type_sys sym_tab =

  Symtab.fold_rev (append o helper_facts_for_sym ctxt type_sys) sym_tab []

fun translate_atp_fact ctxt keep_trivial =

  `(make_fact ctxt keep_trivial true true o apsnd prop_of)

fun translate_formulas ctxt prem_kind type_sys hyp_ts concl_t rich_facts =

  let

    val thy = Proof_Context.theory_of ctxt

    val fact_ts = map (prop_of o snd o snd) rich_facts

    val (facts, fact_names) =

      rich_facts

      |> map_filter (fn (NONE, _) => NONE

                      | (SOME fact, (name, _)) => SOME (fact, name))

      |> ListPair.unzip

    (* Remove existing facts from the conjecture, as this can dramatically

       boost an ATP's performance (for some reason). *)

    val hyp_ts = hyp_ts |> filter_out (member (op aconv) fact_ts)

    val goal_t = Logic.list_implies (hyp_ts, concl_t)

    val all_ts = goal_t :: fact_ts

    val subs = tfree_classes_of_terms all_ts

    val supers = tvar_classes_of_terms all_ts

    val tycons = type_consts_of_terms thy all_ts

    val conjs = make_conjecture ctxt prem_kind (hyp_ts @ [concl_t])

    val (supers', arity_clauses) =

      if level_of_type_sys type_sys = No_Types then ([], [])

      else make_arity_clauses thy tycons supers

    val class_rel_clauses = make_class_rel_clauses thy subs supers'

  in

    (fact_names |> map single, (conjs, facts, class_rel_clauses, arity_clauses))

  end

fun fo_literal_from_type_literal (TyLitVar (class, name)) =

    (true, ATerm (class, [ATerm (name, [])]))

  | fo_literal_from_type_literal (TyLitFree (class, name)) =

    (true, ATerm (class, [ATerm (name, [])]))

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

fun type_pred_combterm ctxt nonmono_Ts type_sys T tm =

  CombApp (CombConst (`make_fixed_const type_pred_base, T --> @{typ bool}, [T])

           |> enforce_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys,

           tm)

fun var_occurs_positively_naked_in_term _ (SOME false) _ accum = accum

  | var_occurs_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =

    accum orelse (s = "equal" andalso member (op =) tms (ATerm (name, [])))

fun is_var_nonmonotonic_in_formula _ _ (SOME false) _ = false

  | is_var_nonmonotonic_in_formula pos phi _ name =

    formula_fold pos (var_occurs_positively_naked_in_term name) phi false

fun tag_with_type ctxt nonmono_Ts type_sys T tm =

  CombConst (`make_fixed_const type_tag_name, T --> T, [T])

  |> enforce_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys

  |> term_from_combterm ctxt nonmono_Ts type_sys Top_Level

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

and term_from_combterm ctxt nonmono_Ts type_sys site u =

  let

    val (head, args) = strip_combterm_comb u

    val (x as (s, _), T_args) =

      case head of

        CombConst (name, _, T_args) => (name, T_args)

      | CombVar (name, _) => (name, [])

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

    val arg_site = if site = Top_Level andalso s = "equal" then Eq_Arg

                   else Elsewhere

    val t = ATerm (x, map fo_term_from_typ T_args @

                      map (term_from_combterm ctxt nonmono_Ts type_sys arg_site)

                          args)

    val T = combtyp_of u

  in

    t |> (if should_tag_with_type ctxt nonmono_Ts type_sys site u T then

            tag_with_type ctxt nonmono_Ts type_sys T

          else

            I)

  end

and formula_from_combformula ctxt nonmono_Ts type_sys should_predicate_on_var =

  let

    val do_term = term_from_combterm ctxt nonmono_Ts type_sys Top_Level

    val do_bound_type =

      case type_sys of

        Simple_Types level =>

        SOME o mangled_type_name o homogenized_type ctxt nonmono_Ts level

      | _ => K NONE

    fun do_out_of_bound_type pos phi universal (name, T) =

      if should_predicate_on_type ctxt nonmono_Ts type_sys

             (fn () => should_predicate_on_var pos phi universal name) T then

        CombVar (name, T)

        |> type_pred_combterm ctxt nonmono_Ts type_sys T

        |> do_term |> AAtom |> SOME

      else

        NONE

    fun do_formula pos (AQuant (q, xs, phi)) =

        let

          val phi = phi |> do_formula pos

          val universal = Option.map (q = AExists ? not) pos

        in

          AQuant (q, xs |> map (apsnd (fn NONE => NONE

                                        | SOME T => do_bound_type T)),

                  (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))

                      (map_filter

                           (fn (_, NONE) => NONE

                             | (s, SOME T) =>

                               do_out_of_bound_type pos phi universal (s, T))

                           xs)

                      phi)

        end

      | do_formula pos (AConn conn) = aconn_map pos do_formula conn

      | do_formula _ (AAtom tm) = AAtom (do_term tm)

  in do_formula o SOME end

fun bound_atomic_types type_sys Ts =

  mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)

                (atp_type_literals_for_types type_sys Axiom Ts))

fun formula_for_fact ctxt nonmono_Ts type_sys

                     ({combformula, atomic_types, ...} : translated_formula) =

  combformula

  |> close_combformula_universally

  |> formula_from_combformula ctxt nonmono_Ts type_sys

                              is_var_nonmonotonic_in_formula true

  |> bound_atomic_types type_sys atomic_types

  |> close_formula_universally

(* 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 prefix nonmono_Ts type_sys

                          (j, formula as {name, locality, kind, ...}) =

  Formula (prefix ^ (if polymorphism_of_type_sys type_sys = Polymorphic then ""

                     else string_of_int j ^ "_") ^

           ascii_of name,

           kind, formula_for_fact ctxt nonmono_Ts type_sys formula, NONE,

           case locality of

             Intro => intro_info

           | Elim => elim_info

           | Simp => simp_info

           | _ => NONE)

fun formula_line_for_class_rel_clause

        (ClassRelClause {name, subclass, superclass, ...}) =

  let val ty_arg = ATerm (`I "T", []) in

    Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,

             AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),

                               AAtom (ATerm (superclass, [ty_arg]))])

             |> close_formula_universally, intro_info, NONE)

  end

fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =

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

  | fo_literal_from_arity_literal (TVarLit (c, sort)) =

    (false, ATerm (c, [ATerm (sort, [])]))

fun formula_line_for_arity_clause

        (ArityClause {name, prem_lits, concl_lits, ...}) =

  Formula (arity_clause_prefix ^ ascii_of name, Axiom,

           mk_ahorn (map (formula_from_fo_literal o apfst not

                          o fo_literal_from_arity_literal) prem_lits)

                    (formula_from_fo_literal

                         (fo_literal_from_arity_literal concl_lits))

           |> close_formula_universally, intro_info, NONE)

fun formula_line_for_conjecture ctxt nonmono_Ts type_sys

        ({name, kind, combformula, ...} : translated_formula) =

  Formula (conjecture_prefix ^ name, kind,

           formula_from_combformula ctxt nonmono_Ts type_sys

               is_var_nonmonotonic_in_formula false

               (close_combformula_universally combformula)

           |> close_formula_universally, NONE, NONE)

fun free_type_literals type_sys ({atomic_types, ...} : translated_formula) =

  atomic_types |> atp_type_literals_for_types type_sys Conjecture

               |> map fo_literal_from_type_literal

fun formula_line_for_free_type j lit =

  Formula (tfree_prefix ^ string_of_int j, Hypothesis,

           formula_from_fo_literal lit, NONE, NONE)

fun formula_lines_for_free_types type_sys facts =

  let

    val litss = map (free_type_literals type_sys) facts

    val lits = fold (union (op =)) litss []

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

(** Symbol declarations **)

fun insert_type ctxt get_T x xs =

  let val T = get_T x in

    if exists (curry (type_instance ctxt) T o get_T) xs then xs

    else x :: filter_out (curry (type_instance ctxt o swap) T o get_T) xs

  end

fun should_declare_sym type_sys pred_sym s =

  not (String.isPrefix bound_var_prefix s) andalso s <> "equal" andalso

  not (String.isPrefix tptp_special_prefix s) andalso

  (case type_sys of

     Simple_Types _ => true

   | Tags (_, _, Light) => true

   | _ => not pred_sym)

fun sym_decl_table_for_facts ctxt type_sys repaired_sym_tab (conjs, facts) =

  let

    fun add_combterm in_conj tm =

      let val (head, args) = strip_combterm_comb tm in

        (case head of

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

           let val pred_sym = is_pred_sym repaired_sym_tab s in

             if should_declare_sym type_sys pred_sym s then

               Symtab.map_default (s, [])

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

                                         in_conj))

             else

               I

           end

         | _ => I)

        #> fold (add_combterm in_conj) args

      end

    fun add_fact in_conj =

      fact_lift (formula_fold NONE (K (add_combterm in_conj)))

  in

    Symtab.empty

    |> is_type_sys_fairly_sound type_sys