src/HOL/Tools/Metis/metis_translate.ML
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properly locate helpers whose constants have several entries in the helper table
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(*  Title:      HOL/Tools/Metis/metis_translate.ML
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    Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA
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    Author:     Kong W. Susanto, Cambridge University Computer Laboratory
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    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
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    Author:     Jasmin Blanchette, TU Muenchen
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Translation of HOL to FOL for Metis.
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*)
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signature METIS_TRANSLATE =
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sig
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  type type_literal = ATP_Translate.type_literal
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  type type_sys = ATP_Translate.type_sys
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  datatype mode = FO | HO | FT | MX
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  datatype isa_thm =
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    Isa_Reflexive_or_Trivial |
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    Isa_Raw of thm
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  type metis_problem =
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    {axioms : (Metis_Thm.thm * isa_thm) list,
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     tfrees : type_literal list,
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     old_skolems : (string * term) list}
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  val metis_equal : string
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  val metis_predicator : string
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  val metis_app_op : string
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  val metis_type_tag : string
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  val metis_generated_var_prefix : string
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  val metis_name_table : ((string * int) * (string * bool)) list
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  val reveal_old_skolem_terms : (string * term) list -> term -> term
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  val string_of_mode : mode -> string
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  val prepare_metis_problem :
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    Proof.context -> mode -> type_sys option -> thm list -> thm list
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    -> mode * int Symtab.table * metis_problem
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end
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structure Metis_Translate : METIS_TRANSLATE =
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struct
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open ATP_Problem
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open ATP_Translate
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val metis_equal = "="
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val metis_predicator = "{}"
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val metis_app_op = "."
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val metis_type_tag = ":"
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val metis_generated_var_prefix = "_"
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val metis_name_table =
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  [((tptp_equal, 2), (metis_equal, false)),
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   ((tptp_old_equal, 2), (metis_equal, false)),
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   ((prefixed_predicator_name, 1), (metis_predicator, false)),
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   ((prefixed_app_op_name, 2), (metis_app_op, false)),
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   ((prefixed_type_tag_name, 2), (metis_type_tag, true))]
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fun predicate_of thy ((@{const Not} $ P), pos) = predicate_of thy (P, not pos)
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  | predicate_of thy (t, pos) =
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    (combterm_from_term thy [] (Envir.eta_contract t), pos)
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fun literals_of_term1 args thy (@{const Trueprop} $ P) =
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    literals_of_term1 args thy P
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  | literals_of_term1 args thy (@{const HOL.disj} $ P $ Q) =
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    literals_of_term1 (literals_of_term1 args thy P) thy Q
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  | literals_of_term1 (lits, ts) thy P =
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    let val ((pred, ts'), pol) = predicate_of thy (P, true) in
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      ((pol, pred) :: lits, union (op =) ts ts')
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    end
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val literals_of_term = literals_of_term1 ([], [])
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fun old_skolem_const_name i j num_T_args =
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  old_skolem_const_prefix ^ Long_Name.separator ^
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  (space_implode Long_Name.separator (map string_of_int [i, j, num_T_args]))
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fun conceal_old_skolem_terms i old_skolems t =
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  if exists_Const (curry (op =) @{const_name Meson.skolem} o fst) t then
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    let
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      fun aux old_skolems
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             (t as (Const (@{const_name Meson.skolem}, Type (_, [_, T])) $ _)) =
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          let
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            val (old_skolems, s) =
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              if i = ~1 then
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                (old_skolems, @{const_name undefined})
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              else case AList.find (op aconv) old_skolems t of
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                s :: _ => (old_skolems, s)
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              | [] =>
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                let
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                  val s = old_skolem_const_name i (length old_skolems)
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                                                (length (Term.add_tvarsT T []))
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                in ((s, t) :: old_skolems, s) end
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          in (old_skolems, Const (s, T)) end
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        | aux old_skolems (t1 $ t2) =
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          let
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            val (old_skolems, t1) = aux old_skolems t1
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            val (old_skolems, t2) = aux old_skolems t2
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          in (old_skolems, t1 $ t2) end
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        | aux old_skolems (Abs (s, T, t')) =
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          let val (old_skolems, t') = aux old_skolems t' in
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            (old_skolems, Abs (s, T, t'))
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          end
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        | aux old_skolems t = (old_skolems, t)
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    in aux old_skolems t end
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  else
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    (old_skolems, t)
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fun reveal_old_skolem_terms old_skolems =
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  map_aterms (fn t as Const (s, _) =>
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                 if String.isPrefix old_skolem_const_prefix s then
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                   AList.lookup (op =) old_skolems s |> the
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                   |> map_types Type_Infer.paramify_vars
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                 else
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                   t
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               | t => t)
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(* ------------------------------------------------------------------------- *)
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(* HOL to FOL  (Isabelle to Metis)                                           *)
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(* ------------------------------------------------------------------------- *)
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(* first-order, higher-order, fully-typed, mode X (fleXible) *)
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datatype mode = FO | HO | FT | MX
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fun string_of_mode FO = "FO"
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  | string_of_mode HO = "HO"
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  | string_of_mode FT = "FT"
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  | string_of_mode MX = "MX"
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fun fn_isa_to_met_sublevel "equal" = "c_fequal"
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  | fn_isa_to_met_sublevel "c_False" = "c_fFalse"
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  | fn_isa_to_met_sublevel "c_True" = "c_fTrue"
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  | fn_isa_to_met_sublevel "c_Not" = "c_fNot"
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  | fn_isa_to_met_sublevel "c_conj" = "c_fconj"
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  | fn_isa_to_met_sublevel "c_disj" = "c_fdisj"
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  | fn_isa_to_met_sublevel "c_implies" = "c_fimplies"
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  | fn_isa_to_met_sublevel x = x
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fun fn_isa_to_met_toplevel "equal" = metis_equal
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  | fn_isa_to_met_toplevel x = x
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fun metis_lit b c args = (b, (c, args));
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fun metis_term_from_typ (Type (s, Ts)) =
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    Metis_Term.Fn (make_fixed_type_const s, map metis_term_from_typ Ts)
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  | metis_term_from_typ (TFree (s, _)) =
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    Metis_Term.Fn (make_fixed_type_var s, [])
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  | metis_term_from_typ (TVar (x, _)) =
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    Metis_Term.Var (make_schematic_type_var x)
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(*These two functions insert type literals before the real literals. That is the
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  opposite order from TPTP linkup, but maybe OK.*)
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fun hol_term_to_fol_FO tm =
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  case strip_combterm_comb tm of
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      (CombConst ((c, _), _, Ts), tms) =>
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        let val tyargs = map metis_term_from_typ Ts
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            val args = map hol_term_to_fol_FO tms
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        in Metis_Term.Fn (c, tyargs @ args) end
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    | (CombVar ((v, _), _), []) => Metis_Term.Var v
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    | _ => raise Fail "non-first-order combterm"
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fun hol_term_to_fol_HO (CombConst ((a, _), _, Ts)) =
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    Metis_Term.Fn (fn_isa_to_met_sublevel a, map metis_term_from_typ Ts)
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  | hol_term_to_fol_HO (CombVar ((s, _), _)) = Metis_Term.Var s
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  | hol_term_to_fol_HO (CombApp (tm1, tm2)) =
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    Metis_Term.Fn (metis_app_op, map hol_term_to_fol_HO [tm1, tm2])
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(*The fully-typed translation, to avoid type errors*)
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fun tag_with_type tm T =
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  Metis_Term.Fn (metis_type_tag, [tm, metis_term_from_typ T])
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fun hol_term_to_fol_FT (CombVar ((s, _), ty)) =
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    tag_with_type (Metis_Term.Var s) ty
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  | hol_term_to_fol_FT (CombConst ((a, _), ty, _)) =
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    tag_with_type (Metis_Term.Fn (fn_isa_to_met_sublevel a, [])) ty
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  | hol_term_to_fol_FT (tm as CombApp (tm1,tm2)) =
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    tag_with_type
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        (Metis_Term.Fn (metis_app_op, map hol_term_to_fol_FT [tm1, tm2]))
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        (combtyp_of tm)
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fun hol_literal_to_fol FO (pos, tm) =
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      let
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        val (CombConst((p, _), _, Ts), tms) = strip_combterm_comb tm
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        val tylits = if p = "equal" then [] else map metis_term_from_typ Ts
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        val lits = map hol_term_to_fol_FO tms
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      in metis_lit pos (fn_isa_to_met_toplevel p) (tylits @ lits) end
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  | hol_literal_to_fol HO (pos, tm) =
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     (case strip_combterm_comb tm of
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          (CombConst(("equal", _), _, _), tms) =>
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            metis_lit pos metis_equal (map hol_term_to_fol_HO tms)
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        | _ => metis_lit pos metis_predicator [hol_term_to_fol_HO tm])
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  | hol_literal_to_fol FT (pos, tm) =
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     (case strip_combterm_comb tm of
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          (CombConst(("equal", _), _, _), tms) =>
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            metis_lit pos metis_equal (map hol_term_to_fol_FT tms)
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        | _ => metis_lit pos metis_predicator [hol_term_to_fol_FT tm])
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fun literals_of_hol_term thy mode t =
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  let val (lits, types_sorts) = literals_of_term thy t in
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    (map (hol_literal_to_fol mode) lits, types_sorts)
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  end
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(*Sign should be "true" for conjecture type constraints, "false" for type lits in clauses.*)
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fun metis_of_type_literals pos (TyLitVar ((s, _), (s', _))) =
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    metis_lit pos s [Metis_Term.Var s']
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  | metis_of_type_literals pos (TyLitFree ((s, _), (s', _))) =
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    metis_lit pos s [Metis_Term.Fn (s',[])]
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fun has_default_sort _ (TVar _) = false
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  | has_default_sort ctxt (TFree (x, s)) =
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    (s = the_default [] (Variable.def_sort ctxt (x, ~1)));
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fun metis_of_tfree tf =
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  Metis_Thm.axiom (Metis_LiteralSet.singleton (metis_of_type_literals true tf));
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fun hol_thm_to_fol is_conjecture ctxt mode j old_skolems th =
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  let
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    val thy = Proof_Context.theory_of ctxt
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    val (old_skolems, (mlits, types_sorts)) =
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     th |> prop_of |> Logic.strip_imp_concl
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        |> conceal_old_skolem_terms j old_skolems
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        ||> (HOLogic.dest_Trueprop #> literals_of_hol_term thy mode)
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  in
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    if is_conjecture then
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      (Metis_Thm.axiom (Metis_LiteralSet.fromList mlits),
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       raw_type_literals_for_types types_sorts, old_skolems)
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    else
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      let
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        val tylits = types_sorts |> filter_out (has_default_sort ctxt)
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                                 |> raw_type_literals_for_types
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        val mtylits = map (metis_of_type_literals false) tylits
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      in
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        (Metis_Thm.axiom (Metis_LiteralSet.fromList(mtylits @ mlits)), [],
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         old_skolems)
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      end
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  end;
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(* ------------------------------------------------------------------------- *)
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(* Logic maps manage the interface between HOL and first-order logic.        *)
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(* ------------------------------------------------------------------------- *)
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datatype isa_thm =
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  Isa_Reflexive_or_Trivial |
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  Isa_Raw of thm
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type metis_problem =
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  {axioms : (Metis_Thm.thm * isa_thm) list,
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   tfrees : type_literal list,
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   old_skolems : (string * term) list}
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fun is_quasi_fol_clause thy =
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  Meson.is_fol_term thy o snd o conceal_old_skolem_terms ~1 [] o prop_of
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(*Extract TFree constraints from context to include as conjecture clauses*)
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fun init_tfrees ctxt =
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  let fun add ((a,i),s) Ts = if i = ~1 then TFree(a,s) :: Ts else Ts in
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    Vartab.fold add (#2 (Variable.constraints_of ctxt)) []
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    |> raw_type_literals_for_types
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  end;
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fun const_in_metis c (pred, tm_list) =
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  let
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    fun in_mterm (Metis_Term.Var _) = false
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      | in_mterm (Metis_Term.Fn (nm, tm_list)) =
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        c = nm orelse exists in_mterm tm_list
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  in c = pred orelse exists in_mterm tm_list end
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(* ARITY CLAUSE *)
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fun m_arity_cls (TConsLit ((c, _), (t, _), args)) =
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    metis_lit true c [Metis_Term.Fn(t, map (Metis_Term.Var o fst) args)]
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  | m_arity_cls (TVarLit ((c, _), (s, _))) =
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    metis_lit false c [Metis_Term.Var s]
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(* TrueI is returned as the Isabelle counterpart because there isn't any. *)
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fun arity_cls ({prem_lits, concl_lits, ...} : arity_clause) =
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  (TrueI,
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   Metis_Thm.axiom (Metis_LiteralSet.fromList
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                        (map m_arity_cls (concl_lits :: prem_lits))));
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(* CLASSREL CLAUSE *)
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fun m_class_rel_cls (subclass, _) (superclass, _) =
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  [metis_lit false subclass [Metis_Term.Var "T"],
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   metis_lit true superclass [Metis_Term.Var "T"]]
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fun class_rel_cls ({subclass, superclass, ...} : class_rel_clause) =
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  (TrueI, m_class_rel_cls subclass superclass
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          |> Metis_LiteralSet.fromList |> Metis_Thm.axiom)
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fun type_ext thy tms =
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  let
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    val subs = tfree_classes_of_terms tms
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    val supers = tvar_classes_of_terms tms
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    val tycons = type_consts_of_terms thy tms
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    val (supers', arity_clauses) = make_arity_clauses thy tycons supers
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    val class_rel_clauses = make_class_rel_clauses thy subs supers'
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  in map class_rel_cls class_rel_clauses @ map arity_cls arity_clauses end
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val proxy_defs = map (fst o snd o snd) proxy_table
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val prepare_helper =
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  Meson.make_meta_clause #> rewrite_rule (map safe_mk_meta_eq proxy_defs)
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fun metis_name_from_atp s ary =
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  AList.lookup (op =) metis_name_table (s, ary) |> the_default (s, false)
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fun metis_term_from_atp (ATerm (s, tms)) =
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  if is_tptp_variable s then
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    Metis_Term.Var s
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  else
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    let val (s, swap) = metis_name_from_atp s (length tms) in
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      Metis_Term.Fn (s, tms |> map metis_term_from_atp |> swap ? rev)
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    end
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fun metis_atom_from_atp (AAtom tm) =
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    (case metis_term_from_atp tm of
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       Metis_Term.Fn x => x
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     | _ => raise Fail "non CNF -- expected function")
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  | metis_atom_from_atp _ = raise Fail "not CNF -- expected atom"
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fun metis_literal_from_atp (AConn (ANot, [phi])) =
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    (false, metis_atom_from_atp phi)
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  | metis_literal_from_atp phi = (true, metis_atom_from_atp phi)
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fun metis_literals_from_atp (AConn (AOr, [phi1, phi2])) =
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    uncurry (union (op =)) (pairself metis_literals_from_atp (phi1, phi2))
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  | metis_literals_from_atp phi = [metis_literal_from_atp phi]
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fun metis_axiom_from_atp clauses (Formula (ident, _, phi, _, _)) =
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    let
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      fun some isa =
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        SOME (phi |> metis_literals_from_atp |> Metis_LiteralSet.fromList
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                  |> Metis_Thm.axiom, isa)
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    in
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      if ident = type_tag_idempotence_helper_name orelse
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         String.isPrefix lightweight_tags_sym_formula_prefix ident then
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        Isa_Reflexive_or_Trivial |> some
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      else if String.isPrefix helper_prefix ident then
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        case space_explode "_" ident  of
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          _ :: const :: j :: _ =>
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          nth (helper_table |> filter (curry (op =) const o fst)
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                            |> maps (snd o snd))
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              (the (Int.fromString j) - 1)
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          |> prepare_helper |> Isa_Raw |> some
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        | _ => raise Fail ("malformed helper identifier " ^ quote ident)
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      else case try (unprefix conjecture_prefix) ident of
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        SOME s =>
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        let val j = the (Int.fromString s) in
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          if j = length clauses then NONE
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          else Meson.make_meta_clause (nth clauses j) |> Isa_Raw |> some
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        end
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      | NONE => TrueI |> Isa_Raw |> some
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    end
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  | metis_axiom_from_atp _ _ = raise Fail "not CNF -- expected formula"
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val default_type_sys = Preds (Polymorphic, Nonmonotonic_Types, Lightweight)
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(* Function to generate metis clauses, including comb and type clauses *)
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fun prepare_metis_problem ctxt MX type_sys conj_clauses fact_clauses =
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    let
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      val type_sys = type_sys |> the_default default_type_sys
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      val explicit_apply = NONE
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      val clauses =
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        conj_clauses @ fact_clauses
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        |> (if polymorphism_of_type_sys type_sys = Polymorphic then
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              I
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            else
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              map (pair 0)
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              #> rpair ctxt
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              #-> Monomorph.monomorph Monomorph.all_schematic_consts_of
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              #> fst #> maps (map (zero_var_indexes o snd)))
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      val (atp_problem, _, _, _, _, _, sym_tab) =
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        prepare_atp_problem ctxt CNF Hypothesis Axiom type_sys explicit_apply
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                            false false (map prop_of clauses) @{prop False} []
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      val axioms =
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        atp_problem |> maps (map_filter (metis_axiom_from_atp clauses) o snd)
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    in
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      (MX, sym_tab,
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       {axioms = axioms, tfrees = [], old_skolems = [] (* FIXME ### *)})
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    end
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  | prepare_metis_problem ctxt mode _ conj_clauses fact_clauses =
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    let
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      val thy = Proof_Context.theory_of ctxt
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      (* The modes FO and FT are sticky. HO can be downgraded to FO. *)
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      val mode =
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        if mode = HO andalso
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           forall (forall (is_quasi_fol_clause thy))
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                  [conj_clauses, fact_clauses] then
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          FO
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        else
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          mode
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      fun add_thm is_conjecture (isa_ith, metis_ith)
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                  {axioms, tfrees, old_skolems} : metis_problem =
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        let
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          val (mth, tfree_lits, old_skolems) =
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            hol_thm_to_fol is_conjecture ctxt mode (length axioms) old_skolems
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                           metis_ith
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        in
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          {axioms = (mth, Isa_Raw isa_ith) :: axioms,
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           tfrees = union (op =) tfree_lits tfrees, old_skolems = old_skolems}
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        end;
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      fun add_type_thm (ith, mth) {axioms, tfrees, old_skolems} =
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        {axioms = (mth, Isa_Raw ith) :: axioms, tfrees = tfrees,
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         old_skolems = old_skolems}
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      fun add_tfrees {axioms, tfrees, old_skolems} =
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        {axioms = map (rpair (Isa_Raw TrueI) o metis_of_tfree)
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                      (distinct (op =) tfrees) @ axioms,
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         tfrees = tfrees, old_skolems = old_skolems}
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      val problem =
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        {axioms = [], tfrees = init_tfrees ctxt, old_skolems = []}
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        |> fold (add_thm true o `Meson.make_meta_clause) conj_clauses
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        |> add_tfrees
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        |> fold (add_thm false o `Meson.make_meta_clause) fact_clauses
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      val clause_lists = map (Metis_Thm.clause o #1) (#axioms problem)
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      fun is_used c =
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        exists (Metis_LiteralSet.exists (const_in_metis c o #2)) clause_lists
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      val problem =
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        if mode = FO then
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          problem
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        else
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          let
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            val helper_ths =
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              helper_table
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              |> filter (is_used o prefix const_prefix o fst)
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              |> maps (fn (_, (needs_full_types, thms)) =>
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                          if needs_full_types andalso mode <> FT then []
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                          else map (`prepare_helper) thms)
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          in problem |> fold (add_thm false) helper_ths end
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      val type_ths = type_ext thy (map prop_of (conj_clauses @ fact_clauses))
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    in (mode, Symtab.empty, fold add_type_thm type_ths problem) end
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end;