(*  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 translated_formula

  datatype type_system =

    Many_Typed |

    Mangled of bool |

    Args of bool |

    Tags of bool |

    No_Types

  val readable_names : bool Unsynchronized.ref

  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 is_type_system_sound : type_system -> bool

  val type_system_types_dangerous_types : 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 * 'a) * thm

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

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

   names. For that reason, they are on by default. *)

val readable_names = Unsynchronized.ref 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:"

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 f (AQuant (_, _, phi)) = formula_fold f phi

  | formula_fold f (AConn (_, phis)) = fold (formula_fold f) phis

  | formula_fold f (AAtom tm) = f tm

type translated_formula =

  {name: string,

   kind: formula_kind,

   combformula: (name, typ, combterm) formula,

   atomic_types: typ list}

fun update_combformula f

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

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

   atomic_types = atomic_types} : translated_formula

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

datatype type_system =

  Many_Typed |

  Mangled of bool |

  Args of bool |

  Tags of bool |

  No_Types

fun is_type_system_sound Many_Typed = true

  | is_type_system_sound (Mangled full_types) = full_types

  | is_type_system_sound (Args full_types) = full_types

  | is_type_system_sound (Tags full_types) = full_types

  | is_type_system_sound No_Types = false

fun type_system_types_dangerous_types (Tags _) = true

  | type_system_types_dangerous_types type_sys = is_type_system_sound type_sys

fun type_system_introduces_type_preds (Mangled true) = true

  | type_system_introduces_type_preds (Args true) = true

  | type_system_introduces_type_preds _ = false

fun type_system_declares_sym_types Many_Typed = true

  | type_system_declares_sym_types type_sys =

    type_system_introduces_type_preds type_sys

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

fun should_omit_type_args type_sys s =

  s <> type_pred_base andalso

  (s = @{const_name HOL.eq} orelse

   case type_sys of

     Many_Typed => false

   | Mangled _ => false

   | No_Types => true

   | Tags true => true

   | _ => member (op =) boring_consts s)

datatype type_arg_policy = No_Type_Args | Mangled_Types | Explicit_Type_Args

fun general_type_arg_policy Many_Typed = Mangled_Types

  | general_type_arg_policy (Mangled _) = Mangled_Types

  | general_type_arg_policy _ = 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 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)

val close_combformula_universally = close_universally combterm_vars

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

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

  #> fold term_vars tms

val close_formula_universally = close_universally term_vars

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

val introduce_proxies =

  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 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 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, combformula = combformula, kind = kind,

     atomic_types = atomic_types}

  end

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

  case (keep_trivial, make_formula ctxt eq_as_iff presimp name 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)

                               (if j = last then Conjecture else Hypothesis))

         (0 upto last) ts

  end

(** "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

val add_fact_syms_to_table =

  fact_lift o formula_fold o add_combterm_syms_to_table

val default_sym_table_entries =

  [("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 type_sys =

  let

    fun aux (CombApp tmp) = CombApp (pairself aux tmp)

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

        (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, [])

             | Mangled_Types => (mangled_const_name T_args name, [])

             | Explicit_Type_Args => (name, T_args)

           end)

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

      | aux tm = tm

  in aux end

fun repair_combterm type_sys sym_tab =

  introduce_explicit_apps_in_combterm sym_tab

  #> introduce_predicators_in_combterm sym_tab

  #> impose_type_arg_policy_in_combterm type_sys

val repair_fact = update_combformula o formula_map oo repair_combterm

(** Helper facts **)

fun ti_ti_helper_fact () =

  let

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

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

  in

    Formula (Fof, helper_prefix ^ ascii_of "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_full_types (th, j) =

        ((c ^ "_" ^ string_of_int j ^ (if needs_full_types then "ft" else ""),

          false),

         let val t = th |> prop_of in

           t |> (general_type_arg_policy type_sys = Mangled_Types 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)

    in

      metis_helpers

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

                  if metis_s <> unmangled_s orelse

                     (needs_full_types andalso

                      not (type_system_types_dangerous_types type_sys)) then

                    []

                  else

                    ths ~~ (1 upto length ths)

                    |> map (dub_and_inst mangled_s needs_full_types)

                    |> make_facts (not needs_full_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 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 tag_with_type ty tm = ATerm (`I type_tag_name, [ty, tm])

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

(* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are

   considered dangerous because their "exhaust" properties can easily lead to

   unsound ATP proofs. The checks below are an (unsound) approximation of

   finiteness. *)

fun is_dtyp_dangerous _ (Datatype_Aux.DtTFree _) = true

  | is_dtyp_dangerous ctxt (Datatype_Aux.DtType (s, Us)) =

    is_type_constr_dangerous ctxt s andalso forall (is_dtyp_dangerous ctxt) Us

  | is_dtyp_dangerous _ (Datatype_Aux.DtRec _) = false

and is_type_dangerous ctxt (Type (s, Ts)) =

    is_type_constr_dangerous ctxt s andalso forall (is_type_dangerous ctxt) Ts

  | is_type_dangerous _ _ = false

and is_type_constr_dangerous ctxt s =

  let val thy = Proof_Context.theory_of ctxt in

    case Datatype_Data.get_info thy s of

      SOME {descr, ...} =>

      forall (fn (_, (_, _, constrs)) =>

                 forall (forall (is_dtyp_dangerous ctxt) o snd) constrs) descr

    | NONE =>

      case Typedef.get_info ctxt s of

        ({rep_type, ...}, _) :: _ => is_type_dangerous ctxt rep_type

      | [] => true

  end

fun should_tag_with_type ctxt (Tags full_types) T =

    full_types orelse is_type_dangerous ctxt T

  | should_tag_with_type _ _ _ = false

fun type_pred_combatom type_sys T tm =

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

           tm)

  |> impose_type_arg_policy_in_combterm type_sys

  |> AAtom

fun formula_from_combformula ctxt type_sys =

  let

    fun 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 type_sys T then

                tag_with_type (fo_term_from_typ T)

              else

                I)

      end

    val do_bound_type =

      if type_sys = Many_Typed then SOME o mangled_type_name else K NONE

    fun do_out_of_bound_type (s, T) =

      if type_system_introduces_type_preds type_sys then

        type_pred_combatom 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 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 type_sys

                (close_combformula_universally combformula))

  |> close_formula_universally

fun logic_for_type_sys Many_Typed = Tff

  | logic_for_type_sys _ = Fof

(* 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 type_sys

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

  Formula (logic_for_type_sys type_sys,

           prefix ^ string_of_int j ^ "_" ^ ascii_of name, kind,

           formula_for_fact ctxt type_sys formula, NONE, NONE)

fun formula_line_for_class_rel_clause (ClassRelClause {name, subclass,

                                                       superclass, ...}) =

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

    Formula (Fof, 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 (Fof, 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 type_sys

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

  Formula (logic_for_type_sys type_sys, conjecture_prefix ^ name, kind,

           formula_from_combformula ctxt 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 (Fof, 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 should_declare_sym type_sys pred_sym s =

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

  (type_sys = Many_Typed orelse not pred_sym)

fun add_combterm_syms_to_decl_table type_sys repaired_sym_tab =

  let

    fun declare_sym (decl as (_, _, T, _, _)) decls =

      if exists (curry Type.raw_instance T o #3) decls then decls

      else decl :: filter_out ((fn T' => Type.raw_instance (T', T)) o #3) 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

      (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 |> type_system_declares_sym_types type_sys

                  ? fold (add_fact_syms_to_decl_table type_sys repaired_sym_tab)

                         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 problem_line_for_sym_decl ctxt type_sys need_bound_types s j

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

  if type_sys = Many_Typed then

    let val (arg_Ts, res_T) = n_ary_strip_type ary T in

      Decl (sym_decl_prefix ^ ascii_of 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

  else

    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 (if need_bound_types then SOME else K NONE)

      val freshener =

        case type_sys of Args _ => string_of_int j ^ "_" | _ => ""

    in

      Formula (Fof, sym_decl_prefix ^ freshener ^ "_" ^ ascii_of s, Axiom,

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

               |> fold (curry (CombApp o swap)) bound_tms

               |> type_pred_combatom type_sys res_T

               |> mk_aquant AForall (bound_names ~~ bound_Ts)

               |> formula_from_combformula ctxt type_sys,

               NONE, NONE)

    end

fun problem_lines_for_sym_decls ctxt type_sys (s, decls) =

  let

    val n = length decls

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

    val need_bound_types =

      case decls of

        decl :: (decls as _ :: _) =>

        exists (curry (op <>) (result_type_of_decl decl)

                o result_type_of_decl) decls

      | _ => false

  in

    map2 (problem_line_for_sym_decl ctxt type_sys need_bound_types s)

         (0 upto n - 1) decls

  end

fun problem_lines_for_sym_decl_table ctxt type_sys sym_decl_tab =

  Symtab.fold_rev (append o problem_lines_for_sym_decls ctxt 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 (conjs, facts) =

      (conjs, facts) |> pairself (map (repair_fact type_sys sym_tab))

    val repaired_sym_tab = conjs @ facts |> sym_table_for_facts false

    val sym_decl_lines =

      conjs @ facts

      |> sym_decl_table_for_facts type_sys repaired_sym_tab

      |> problem_lines_for_sym_decl_table ctxt type_sys

    val helpers =

      helper_facts_for_sym_table ctxt type_sys repaired_sym_tab

      |> map (repair_fact type_sys sym_tab)

    (* 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 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 type_sys)

                      (0 upto length helpers - 1 ~~ helpers)

                  |> (if type_sys = Tags false then cons (ti_ti_helper_fact ())

                      else I)),

       (conjsN, map (formula_line_for_conjecture ctxt type_sys) conjs),

       (free_typesN, formula_lines_for_free_types type_sys (facts @ conjs))]

    val problem =

      problem

      |> (if type_sys = Many_Typed then

            cons (type_declsN,

                  map decl_line_for_tff_type (tff_types_in_problem problem))

          else

            I)

    val (problem, pool) = problem |> nice_atp_problem (!readable_names)

  in

    (problem,

     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,

     offset_of_heading_in_problem factsN problem 0,

     offset_of_heading_in_problem conjsN 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

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 (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 =

  Symtab.empty

  |> add_conjectures_weights (these (AList.lookup (op =) problem conjsN))

  |> add_facts_weights (these (AList.lookup (op =) problem factsN))

  |> Symtab.dest

  |> sort (prod_ord Real.compare string_ord o pairself swap)

end;