src/HOL/Tools/Sledgehammer/sledgehammer_atp_translate.ML

tuning

(*  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 '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_system =
    Simple of type_level |
    Preds of polymorphism * type_level |
    Tags of polymorphism * type_level

  type translated_formula

  val readable_names : bool Config.T
  val fact_prefix : string
  val conjecture_prefix : 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 num_atp_type_args : theory -> type_system -> string -> int
  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 -> type_system -> bool -> term list -> term
    -> (translated_formula option * ((string * 'a) * thm)) list
    -> string problem * string Symtab.table * int * int
       * (string * 'a) list vector
  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

(* 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 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

(* official TPTP syntax *)
val tptp_tff_type_of_types = "$tType"
val tptp_tff_bool_type = "$o"
val tptp_false = "$false"
val tptp_true = "$true"

(* 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_system =
  Simple of type_level |
  Preds of polymorphism * type_level |
  Tags of polymorphism * type_level

fun type_sys_from_string s =
  (case try (unprefix "mangled_") s of
     SOME s => (Mangled_Monomorphic, s)
   | NONE =>
     case try (unprefix "mono_") s of
       SOME s => (Monomorphic, s)
     | NONE => (Polymorphic, s))
  ||> (fn s =>
          case try (unsuffix " ?") s of
            SOME s => (Nonmonotonic_Types, s)
          | NONE =>
            case try (unsuffix " !") s of
              SOME s => (Finite_Types, s)
            | NONE => (All_Types, s))
  |> (fn (polymorphism, (level, core)) =>
         case (core, (polymorphism, level)) of
           ("simple", (Polymorphic (* naja *), level)) => Simple level
         | ("preds", extra) => Preds extra
         | ("tags", extra) => Tags extra
         | ("args", (_, All_Types (* naja *))) =>
           Preds (polymorphism, Const_Arg_Types)
         | ("erased", (Polymorphic, All_Types (* naja *))) =>
           Preds (polymorphism, No_Types)
         | _ => error ("Unknown type system: " ^ quote s ^ "."))

fun polymorphism_of_type_sys (Simple _) = Mangled_Monomorphic
  | polymorphism_of_type_sys (Preds (poly, _)) = poly
  | polymorphism_of_type_sys (Tags (poly, _)) = poly

fun level_of_type_sys (Simple level) = level
  | level_of_type_sys (Preds (_, level)) = level
  | level_of_type_sys (Tags (_, level)) = level

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 is_type_level_partial level =
  level = Nonmonotonic_Types orelse level = Finite_Types

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 formula_fold pos f =
  let
    fun aux pos (AQuant (_, _, phi)) = aux pos phi
      | aux pos (AConn (ANot, [phi])) = aux (Option.map not pos) phi
      | aux pos (AConn (AImplies, [phi1, phi2])) =
        aux (Option.map not pos) phi1 #> aux pos phi2
      | aux pos (AConn (AAnd, phis)) = fold (aux pos) phis
      | aux pos (AConn (AOr, phis)) = fold (aux pos) phis
      | aux _ (AConn (_, phis)) = fold (aux NONE) phis
      | aux pos (AAtom tm) = f pos tm
  in aux (SOME 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

val boring_consts = [explicit_app_base, @{const_name Metis.fequal}]

fun should_omit_type_args type_sys s =
  s <> type_pred_base andalso s <> type_tag_name andalso
  (s = @{const_name HOL.eq} orelse level_of_type_sys type_sys = No_Types orelse
   (case type_sys of
      Tags (_, All_Types) => true
    | _ => polymorphism_of_type_sys type_sys <> Mangled_Monomorphic andalso
           member (op =) boring_consts s))

datatype type_arg_policy = No_Type_Args | Explicit_Type_Args | Mangled_Type_Args

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
  else
    Explicit_Type_Args

fun type_arg_policy type_sys s =
  if should_omit_type_args type_sys s then No_Type_Args
  else general_type_arg_policy type_sys

fun num_atp_type_args thy type_sys s =
  if type_arg_policy type_sys s = Explicit_Type_Args then num_type_args thy s
  else 0

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_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 ^ "(" ^ commas (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

val presimplify_term = prop_of o Meson.presimplify oo Skip_Proof.make_thm

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))

(* Removes the lambdas from an equation of the form "t = (%x. u)".
   (Cf. "extensionalize_theorem" in "Meson_Clausify".) *)
fun extensionalize_term t =
  let
    fun aux j (@{const Trueprop} $ t') = @{const Trueprop} $ aux j t'
      | aux j (t as Const (s, Type (_, [Type (_, [_, T']),
                                        Type (_, [_, res_T])]))
                    $ t2 $ Abs (var_s, var_T, t')) =
        if s = @{const_name HOL.eq} orelse s = @{const_name "=="} then
          let val var_t = Var ((var_s, j), var_T) in
            Const (s, T' --> T' --> res_T)
              $ betapply (t2, var_t) $ subst_bound (var_t, t')
            |> aux (j + 1)
          end
        else
          t
      | aux _ t = t
  in aux (maxidx_of_term t + 1) t 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
              |> extensionalize_term
              |> presimp ? presimplify_term thy
              |> 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 ts =
  let val last = length ts - 1 in
    map2 (fn j => make_formula ctxt true true (string_of_int j) Chained
                               (if j = last then Conjecture else Hypothesis))
         (0 upto last) ts
  end

(** Finite and infinite type inference **)

(* 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 datatype_constrs thy (T as Type (s, Ts)) =
    (case Datatype.get_info thy s of
       SOME {index, descr, ...} =>
       let val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the in
         map (apsnd (fn Us => map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
             constrs
       end
     | NONE => [])
  | datatype_constrs _ _ = []

(* Similar to "Nitpick_HOL.bounded_exact_card_of_type".
   0 means infinite type, 1 means singleton type (e.g., "unit"), and 2 means
   cardinality 2 or more. The specified default cardinality is returned if the
   cardinality of the type can't be determined. *)
fun tiny_card_of_type ctxt default_card T =
  let
    val max = 2 (* 1 would be too small for the "fun" case *)
    fun aux avoid T =
      (if member (op =) avoid T then
         0
       else case T of
         Type (@{type_name fun}, [T1, T2]) =>
         (case (aux avoid T1, aux avoid T2) of
            (_, 1) => 1
          | (0, _) => 0
          | (_, 0) => 0
          | (k1, k2) =>
            if k1 >= max orelse k2 >= max then max
            else Int.min (max, Integer.pow k2 k1))
       | @{typ int} => 0
       | @{typ bool} => 2 (* optimization *)
       | Type _ =>
         let val thy = Proof_Context.theory_of ctxt in
           case datatype_constrs thy T of
             [] => default_card
           | constrs =>
             let
               val constr_cards =
                 map (Integer.prod o map (aux (T :: avoid)) o binder_types
                      o snd) constrs
             in
               if exists (curry (op =) 0) constr_cards then 0
               else Int.min (max, Integer.sum constr_cards)
             end
         end
       | _ => default_card)
  in Int.min (max, aux [] T) end

fun is_type_surely_finite ctxt T = tiny_card_of_type ctxt 0 T <> 0
fun is_type_surely_infinite ctxt T = tiny_card_of_type ctxt 1 T = 0

fun should_encode_type _ _ All_Types _ = true
  | should_encode_type ctxt _ Finite_Types T = is_type_surely_finite ctxt T
  | should_encode_type _ nonmono_Ts Nonmonotonic_Types T =
    exists (curry Type.raw_instance T) nonmono_Ts
  | should_encode_type _ _ _ _ = false

fun should_predicate_on_type ctxt nonmono_Ts (Preds (_, level)) T =
    should_encode_type ctxt nonmono_Ts level T
  | should_predicate_on_type _ _ _ _ = false

fun should_tag_with_type ctxt nonmono_Ts (Tags (_, level)) T =
    should_encode_type ctxt nonmono_Ts level T
  | 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 true (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 impose_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys =
  let
    fun aux (CombApp tmp) = CombApp (pairself aux tmp)
      | aux (CombConst (name as (s, _), T, T_args)) =
        let
          val level = level_of_type_sys type_sys
          val (T, T_args) =
            (* avoid needless identical homogenized versions of "hAPP" *)
            if s = const_prefix ^ explicit_app_base then
              T_args |> map (homogenized_type ctxt nonmono_Ts level)
                     |> (fn Ts => let val T = hd Ts --> nth Ts 1 in
                                    (T --> T, 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'' in
               case type_arg_policy type_sys s'' of
                 No_Type_Args => (name, [])
               | Explicit_Type_Args => (name, T_args)
               | Mangled_Type_Args => (mangled_const_name T_args name, [])
             end)
          |> (fn (name, T_args) => CombConst (name, T, T_args))
        end
      | aux tm = tm
  in aux end

fun repair_combterm ctxt nonmono_Ts type_sys sym_tab =
  introduce_explicit_apps_in_combterm sym_tab
  #> introduce_predicators_in_combterm sym_tab
  #> impose_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, NONE, 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_some_types (th, j) =
        ((c ^ "_" ^ string_of_int j ^ (if needs_some_types then "T" else ""),
          Chained),
         let val t = th |> prop_of in
           t |> (general_type_arg_policy type_sys = Mangled_Type_Args 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 has_some_types = is_type_sys_fairly_sound type_sys
    in
      metis_helpers
      |> maps (fn (metis_s, (needs_some_types, ths)) =>
                  if metis_s <> unmangled_s orelse
                     (needs_some_types andalso not has_some_types) then
                    []
                  else
                    ths ~~ (1 upto length ths)
                    |> map (dub_and_inst mangled_s needs_some_types)
                    |> make_facts (not needs_some_types))
    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 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 (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_combatom ctxt nonmono_Ts type_sys T tm =
  CombApp (CombConst (`make_fixed_const type_pred_base, T --> @{typ bool}, [T]),
           tm)
  |> impose_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys
  |> AAtom

fun formula_from_combformula ctxt nonmono_Ts type_sys =
  let
    fun tag_with_type type_sys T tm =
      CombConst (`make_fixed_const type_tag_name, T --> T, [T])
      |> impose_type_arg_policy_in_combterm ctxt nonmono_Ts type_sys
      |> do_term true
      |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm]))
    and do_term top_level u =
      let
        val (head, args) = strip_combterm_comb u
        val (x, T_args) =
          case head of
            CombConst (name, _, T_args) => (name, T_args)
          | CombVar (name, _) => (name, [])
          | CombApp _ => raise Fail "impossible \"CombApp\""
        val t = ATerm (x, map fo_term_from_typ T_args @
                          map (do_term false) args)
        val T = combtyp_of u
      in
        t |> (if not top_level andalso
                should_tag_with_type ctxt nonmono_Ts type_sys T then
                tag_with_type type_sys T
              else
                I)
      end
    val do_bound_type =
      case type_sys of
        Simple level =>
        SOME o mangled_type_name o homogenized_type ctxt nonmono_Ts level
      | _ => K NONE
    fun do_out_of_bound_type (s, T) =
      if should_predicate_on_type ctxt nonmono_Ts type_sys T then
        type_pred_combatom ctxt nonmono_Ts type_sys T (CombVar (s, T))
        |> do_formula |> SOME
      else
        NONE
    and do_formula (AQuant (q, xs, phi)) =
        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 (s, T)) xs)
                    (do_formula phi))
      | do_formula (AConn (c, phis)) = AConn (c, map do_formula phis)
      | do_formula (AAtom tm) = AAtom (do_term true tm)
  in do_formula end

fun formula_for_fact ctxt nonmono_Ts type_sys
                     ({combformula, atomic_types, ...} : translated_formula) =
  mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
                (atp_type_literals_for_types type_sys Axiom atomic_types))
           (formula_from_combformula ctxt nonmono_Ts type_sys
                (close_combformula_universally combformula))
  |> close_formula_universally

fun useful_isabelle_info s = SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))

(* 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,
           if generate_useful_info then
             case locality of
               Intro => useful_isabelle_info "intro"
             | Elim => useful_isabelle_info "elim"
             | Simp => useful_isabelle_info "simp"
             | _ => NONE
           else
             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, NONE, 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, conclLit, premLits,
                                                ...}) =
  Formula (arity_clause_prefix ^ ascii_of name, Axiom,
           mk_ahorn (map (formula_from_fo_literal o apfst not
                          o fo_literal_from_arity_literal) premLits)
                    (formula_from_fo_literal
                         (fo_literal_from_arity_literal conclLit))
           |> close_formula_universally, NONE, 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
                                    (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 get_T x xs =
  let val T = get_T x in
    if exists (curry Type.raw_instance T o get_T) xs then xs
    else x :: filter_out ((fn T' => Type.raw_instance (T', 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 "$" s) andalso
  ((case type_sys of Simple _ => true | _ => false) orelse not pred_sym)

fun add_combterm_syms_to_decl_table type_sys repaired_sym_tab =
  let
    fun declare_sym decl decls =
      case type_sys of
        Preds (Polymorphic, All_Types) => insert_type #3 decl decls
      | _ => insert (op =) decl decls
    fun do_term 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, [])
                   (declare_sym (s', T_args, T, pred_sym, length args))
             else
               I
           end
         | _ => I)
        #> fold do_term args
      end
  in do_term end
fun add_fact_syms_to_decl_table type_sys repaired_sym_tab =
  fact_lift (formula_fold true
      (K (add_combterm_syms_to_decl_table type_sys repaired_sym_tab)))
fun sym_decl_table_for_facts type_sys repaired_sym_tab facts =
  Symtab.empty |> is_type_sys_fairly_sound type_sys
                  ? fold (add_fact_syms_to_decl_table type_sys repaired_sym_tab)
                         facts

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

(* 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_combterm_nonmonotonic_types _ (SOME false) _ = I
  | add_combterm_nonmonotonic_types ctxt _
        (CombApp (CombApp (CombConst (("equal", _), Type (_, [T, _]), _), tm1),
                  tm2)) =
    (exists is_var_or_bound_var [tm1, tm2] andalso
     not (is_type_surely_infinite ctxt T)) ? insert_type I T
  | add_combterm_nonmonotonic_types _ _ _ = I
fun add_fact_nonmonotonic_types ctxt ({kind, combformula, ...}
                                      : translated_formula) =
  formula_fold (kind <> Conjecture) (add_combterm_nonmonotonic_types ctxt)
               combformula
fun add_nonmonotonic_types_for_facts ctxt type_sys facts =
  level_of_type_sys type_sys = Nonmonotonic_Types
  ? (insert_type I @{typ bool} (* in case helper "True_or_False" is included *)
     #> fold (add_fact_nonmonotonic_types ctxt) facts)

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

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

fun decl_line_for_sym s (s', _, T, pred_sym, ary) =
  let val (arg_Ts, res_T) = n_ary_strip_type ary T in
    Decl (sym_decl_prefix ^ s, (s, s'), map mangled_type_name arg_Ts,
          if pred_sym then `I tptp_tff_bool_type else mangled_type_name res_T)
  end

fun is_polymorphic_type T = fold_atyps (fn TVar _ => K true | _ => I) T false

fun formula_line_for_sym_decl ctxt nonmono_Ts type_sys n s j
                              (s', T_args, T, _, ary) =
  let
    val (arg_Ts, res_T) = n_ary_strip_type ary T
    val bound_names =
      1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
    val bound_tms =
      bound_names ~~ arg_Ts |> map (fn (name, T) => CombConst (name, T, []))
    val bound_Ts =
      arg_Ts |> map (fn T => if n > 1 orelse is_polymorphic_type T then SOME T
                             else NONE)
  in
    Formula (sym_decl_prefix ^ s ^
             (if n > 1 then "_" ^ string_of_int j else ""), Axiom,
             CombConst ((s, s'), T, T_args)
             |> fold (curry (CombApp o swap)) bound_tms
             |> type_pred_combatom ctxt nonmono_Ts type_sys res_T
             |> mk_aquant AForall (bound_names ~~ bound_Ts)
             |> formula_from_combformula ctxt nonmono_Ts type_sys
             |> close_formula_universally,
             NONE, NONE)
  end

fun problem_lines_for_sym_decls ctxt nonmono_Ts type_sys (s, decls) =
  case type_sys of
    Simple _ => map (decl_line_for_sym s) decls
  | _ =>
    let
      val decls =
        case decls of
          decl :: (decls' as _ :: _) =>
          let val T = result_type_of_decl decl in
            if forall ((fn T' => Type.raw_instance (T', T))
                       o result_type_of_decl) decls' then
              [decl]
            else
              decls
          end
        | _ => decls
      val n = length decls
      val decls =
        decls |> filter (should_predicate_on_type ctxt nonmono_Ts type_sys
                         o result_type_of_decl)
    in
      map2 (formula_line_for_sym_decl ctxt nonmono_Ts type_sys n s)
           (0 upto length decls - 1) decls
    end

fun problem_lines_for_sym_decl_table ctxt nonmono_Ts type_sys sym_decl_tab =
  Symtab.fold_rev (append o problem_lines_for_sym_decls ctxt nonmono_Ts
                                                        type_sys)
                  sym_decl_tab []

fun add_tff_types_in_formula (AQuant (_, xs, phi)) =
    union (op =) (map_filter snd xs) #> add_tff_types_in_formula phi
  | add_tff_types_in_formula (AConn (_, phis)) =
    fold add_tff_types_in_formula phis
  | add_tff_types_in_formula (AAtom _) = I

fun add_tff_types_in_problem_line (Decl (_, _, arg_Ts, res_T)) =
    union (op =) (res_T :: arg_Ts)
  | add_tff_types_in_problem_line (Formula (_, _, phi, _, _)) =
    add_tff_types_in_formula phi

fun tff_types_in_problem problem =
  fold (fold add_tff_types_in_problem_line o snd) problem []

fun decl_line_for_tff_type (s, s') =
  Decl (type_decl_prefix ^ ascii_of s, (s, s'), [], `I tptp_tff_type_of_types)

val type_declsN = "Types"
val sym_declsN = "Symbol types"
val factsN = "Relevant facts"
val class_relsN = "Class relationships"
val aritiesN = "Arities"
val helpersN = "Helper facts"
val conjsN = "Conjectures"
val free_typesN = "Type variables"

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

fun prepare_atp_problem ctxt type_sys explicit_apply hyp_ts concl_t facts =
  let
    val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
      translate_formulas ctxt type_sys hyp_ts concl_t facts
    val sym_tab = conjs @ facts |> sym_table_for_facts explicit_apply
    val nonmono_Ts =
      [] |> fold (add_nonmonotonic_types_for_facts ctxt type_sys) [facts, conjs]
    val repair = repair_fact ctxt nonmono_Ts type_sys sym_tab
    val (conjs, facts) = (conjs, facts) |> pairself (map repair)
    val repaired_sym_tab = conjs @ facts |> sym_table_for_facts false
    val helpers =
      repaired_sym_tab |> helper_facts_for_sym_table ctxt type_sys |> map repair
    val sym_decl_lines =
      conjs @ facts
      |> sym_decl_table_for_facts type_sys repaired_sym_tab
      |> problem_lines_for_sym_decl_table ctxt nonmono_Ts type_sys
    (* Reordering these might confuse the proof reconstruction code or the SPASS
       Flotter hack. *)
    val problem =
      [(sym_declsN, sym_decl_lines),
       (factsN, map (formula_line_for_fact ctxt fact_prefix nonmono_Ts type_sys)
                    (0 upto length facts - 1 ~~ facts)),
       (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
       (aritiesN, map formula_line_for_arity_clause arity_clauses),
       (helpersN, map (formula_line_for_fact ctxt helper_prefix nonmono_Ts
                                             type_sys)
                      (0 upto length helpers - 1 ~~ helpers)
                  |> (case type_sys of
                        Tags (Polymorphic, level) =>
                        is_type_level_partial level
                        ? cons (ti_ti_helper_fact ())
                      | _ => I)),
       (conjsN, map (formula_line_for_conjecture ctxt nonmono_Ts type_sys)
                    conjs),
       (free_typesN, formula_lines_for_free_types type_sys (facts @ conjs))]
    val problem =
      problem
      |> (case type_sys of
            Simple _ =>
            cons (type_declsN,
                  map decl_line_for_tff_type (tff_types_in_problem problem))
          | _ => I)
    val (problem, pool) =
      problem |> nice_atp_problem (Config.get ctxt readable_names)
  in
    (problem,
     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
     offset_of_heading_in_problem conjsN problem 0,
     offset_of_heading_in_problem factsN problem 0,
     fact_names |> Vector.fromList)
  end

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

fun add_term_weights weight (ATerm (s, tms)) =
  (not (is_atp_variable s) andalso s <> "equal") ? Symtab.default (s, weight)
  #> fold (add_term_weights weight) tms
fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
    formula_fold true (K (add_term_weights weight)) phi
  | add_problem_line_weights _ _ = I

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

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

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

end;