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

     Author:     Fabian Immler, TU Muenchen

     Author:     Makarius

     Author:     Jasmin Blanchette, TU Muenchen

 Translation of HOL to FOL.

 *)

 signature SLEDGEHAMMER_TRANSLATE =

 sig

   type 'a problem = 'a ATP_Problem.problem

   val axiom_prefix : string

   val conjecture_prefix : string

   val helper_prefix : string

   val class_rel_clause_prefix : string

   val arity_clause_prefix : string

   val tfrees_name : string

   val prepare_problem :

     Proof.context -> bool -> bool -> bool -> bool -> term list -> term

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

     -> string problem * string Symtab.table * int * (string * 'a) vector

 end;

 structure Sledgehammer_Translate : SLEDGEHAMMER_TRANSLATE =

 struct

 open ATP_Problem

 open Metis_Clauses

 open Sledgehammer_Util

 val axiom_prefix = "ax_"

 val conjecture_prefix = "conj_"

 val helper_prefix = "help_"

 val class_rel_clause_prefix = "clrel_";

 val arity_clause_prefix = "arity_"

 val tfrees_name = "tfrees"

 (* Freshness almost guaranteed! *)

 val sledgehammer_weak_prefix = "Sledgehammer:"

 type fol_formula =

   {name: string,

    kind: kind,

    combformula: (name, combterm) formula,

    ctypes_sorts: typ list}

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

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

 fun mk_ahorn [] phi = phi

   | mk_ahorn (phi :: phis) psi =

     AConn (AImplies, [fold (mk_aconn AAnd) phis phi, psi])

 fun combformula_for_prop thy =

   let

     val do_term = combterm_from_term thy

     fun do_quant bs q s T t' =

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

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

         #>> (fn phi => AQuant (q, [`make_bound_var s], phi))

       end

     and do_conn bs c t1 t2 =

       do_formula bs t1 ##>> do_formula bs t2

       #>> (fn (phi1, phi2) => AConn (c, [phi1, phi2]))

     and do_formula bs t =

       case t of

         @{const Not} $ t1 =>

         do_formula bs t1 #>> (fn phi => AConn (ANot, [phi]))

       | 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 "op &"} $ t1 $ t2 => do_conn bs AAnd t1 t2

       | @{const "op |"} $ t1 $ t2 => do_conn bs AOr t1 t2

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

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

         do_conn bs AIff t1 t2

       | _ => (fn ts => do_term bs (Envir.eta_contract t)

                        |>> AAtom ||> union (op =) ts)

   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 ^ Int.toString 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 "Clausifier".) *)

 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 "op ="} 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 = ProofContext.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 "op &"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2

           | (t0 as @{const "op |"}) $ 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 "op ="}, 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

                      |> Clausifier.introduce_combinators_in_cterm

                      |> prop_of |> Logic.dest_equals |> snd

                      |> reveal_bounds Ts

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

       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 unreplable 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

 (* "Object_Logic.atomize_term" isn't as powerful as it could be; for example,

     it leaves metaequalities over "prop"s alone. *)

 val atomize_term =

   let

     fun aux (@{const Trueprop} $ t1) = t1

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

         HOLogic.all_const T $ Abs (s, T, aux t')

       | aux (@{const "==>"} $ t1 $ t2) = HOLogic.mk_imp (pairself aux (t1, t2))

       | aux (Const (@{const_name "=="}, Type (_, [@{typ prop}, _])) $ t1 $ t2) =

         HOLogic.eq_const HOLogic.boolT $ aux t1 $ aux t2

       | aux (Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2) =

         HOLogic.eq_const T $ t1 $ t2

       | aux _ = raise Fail "aux"

   in perhaps (try aux) end

 (* making axiom and conjecture formulas *)

 fun make_formula ctxt presimp name kind t =

   let

     val thy = ProofContext.theory_of ctxt

     val t = t |> Envir.beta_eta_contract

               |> transform_elim_term

               |> atomize_term

     val need_trueprop = (fastype_of t = HOLogic.boolT)

     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, ctypes_sorts) = combformula_for_prop thy t []

   in

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

      ctypes_sorts = ctypes_sorts}

   end

 fun make_axiom ctxt presimp ((name, loc), th) =

   case make_formula ctxt presimp name Axiom (prop_of th) of

     {combformula = AAtom (CombConst (("c_True", _), _, _)), ...} => NONE

   | formula => SOME ((name, loc), formula)

 fun make_conjecture ctxt ts =

   let val last = length ts - 1 in

     map2 (fn j => make_formula ctxt true (Int.toString j)

                                (if j = last then Conjecture else Hypothesis))

          (0 upto last) ts

   end

 (** Helper facts **)

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

     Symtab.map_entry s (Integer.add 1)

   | count_combterm (CombVar _) = I

   | count_combterm (CombApp (t1, t2)) = fold count_combterm [t1, t2]

 fun count_combformula (AQuant (_, _, phi)) = count_combformula phi

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

   | count_combformula (AAtom tm) = count_combterm tm

 fun count_fol_formula ({combformula, ...} : fol_formula) =

   count_combformula combformula

 val optional_helpers =

   [(["c_COMBI", "c_COMBK"], @{thms COMBI_def COMBK_def}),

    (["c_COMBB", "c_COMBC"], @{thms COMBB_def COMBC_def}),

    (["c_COMBS"], @{thms COMBS_def})]

 val optional_typed_helpers =

   [(["c_True", "c_False", "c_If"], @{thms True_or_False}),

    (["c_If"], @{thms if_True if_False})]

 val mandatory_helpers = @{thms fequal_def}

 val init_counters =

   [optional_helpers, optional_typed_helpers] |> maps (maps fst)

   |> sort_distinct string_ord |> map (rpair 0) |> Symtab.make

 fun get_helper_facts ctxt is_FO full_types conjectures axioms =

   let

     val ct = fold (fold count_fol_formula) [conjectures, axioms] init_counters

     fun is_needed c = the (Symtab.lookup ct c) > 0

     fun baptize th = ((Thm.get_name_hint th, false), th)

   in

     (optional_helpers

      |> full_types ? append optional_typed_helpers

      |> maps (fn (ss, ths) =>

                  if exists is_needed ss then map baptize ths else [])) @

     (if is_FO then [] else map baptize mandatory_helpers)

     |> map_filter (Option.map snd o make_axiom ctxt false)

   end

 fun prepare_formulas ctxt full_types hyp_ts concl_t axiom_pairs =

   let

     val thy = ProofContext.theory_of ctxt

     val axiom_ts = map (prop_of o snd) axiom_pairs

     val hyp_ts =

       if null hyp_ts then

         []

       else

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

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

         let

           val axiom_table = fold (Termtab.update o rpair ()) axiom_ts

                                  Termtab.empty

         in hyp_ts |> filter_out (Termtab.defined axiom_table) end

     val goal_t = Logic.list_implies (hyp_ts, concl_t)

     val is_FO = Meson.is_fol_term thy goal_t

     val subs = tfree_classes_of_terms [goal_t]

     val supers = tvar_classes_of_terms axiom_ts

     val tycons = type_consts_of_terms thy (goal_t :: axiom_ts)

     (* TFrees in the conjecture; TVars in the axioms *)

     val conjectures = make_conjecture ctxt (hyp_ts @ [concl_t])

     val (axiom_names, axioms) =

       ListPair.unzip (map_filter (make_axiom ctxt true) axiom_pairs)

     val helper_facts = get_helper_facts ctxt is_FO full_types conjectures axioms

     val (supers', arity_clauses) = make_arity_clauses thy tycons supers

     val class_rel_clauses = make_class_rel_clauses thy subs supers'

   in

     (Vector.fromList axiom_names,

      (conjectures, axioms, helper_facts, class_rel_clauses, arity_clauses))

   end

 fun wrap_type ty t = ATerm ((type_wrapper_name, type_wrapper_name), [ty, t])

 fun fo_term_for_combtyp (CombTVar name) = ATerm (name, [])

   | fo_term_for_combtyp (CombTFree name) = ATerm (name, [])

   | fo_term_for_combtyp (CombType (name, tys)) =

     ATerm (name, map fo_term_for_combtyp tys)

 fun fo_literal_for_type_literal (TyLitVar (class, name)) =

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

   | fo_literal_for_type_literal (TyLitFree (class, name)) =

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

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

 fun fo_term_for_combterm full_types =

   let

     fun aux top_level u =

       let

         val (head, args) = strip_combterm_comb u

         val (x, ty_args) =

           case head of

             CombConst (name as (s, s'), _, ty_args) =>

             let val ty_args = if full_types then [] else ty_args in

               if s = "equal" then

                 if top_level andalso length args = 2 then (name, [])

                 else (("c_fequal", @{const_name fequal}), ty_args)

               else if top_level then

                 case s of

                   "c_False" => (("$false", s'), [])

                 | "c_True" => (("$true", s'), [])

                 | _ => (name, ty_args)

               else

                 (name, ty_args)

             end

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

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

         val t = ATerm (x, map fo_term_for_combtyp ty_args @

                           map (aux false) args)

     in

       if full_types then wrap_type (fo_term_for_combtyp (combtyp_of u)) t else t

     end

   in aux true end

 fun formula_for_combformula full_types =

   let

     fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)

       | aux (AConn (c, phis)) = AConn (c, map aux phis)

       | aux (AAtom tm) = AAtom (fo_term_for_combterm full_types tm)

   in aux end

 fun formula_for_axiom full_types

                       ({combformula, ctypes_sorts, ...} : fol_formula) =

   mk_ahorn (map (formula_for_fo_literal o fo_literal_for_type_literal)

                 (type_literals_for_types ctypes_sorts))

            (formula_for_combformula full_types combformula)

 fun problem_line_for_fact prefix full_types (formula as {name, kind, ...}) =

   Fof (prefix ^ ascii_of name, kind, formula_for_axiom full_types formula)

 fun problem_line_for_class_rel_clause (ClassRelClause {name, subclass,

                                                        superclass, ...}) =

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

     Fof (class_rel_clause_prefix ^ ascii_of name, Axiom,

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

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

   end

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

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

   | fo_literal_for_arity_literal (TVarLit (c, sort)) =

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

 fun problem_line_for_arity_clause (ArityClause {name, conclLit, premLits,

                                                 ...}) =

   Fof (arity_clause_prefix ^ ascii_of name, Axiom,

        mk_ahorn (map (formula_for_fo_literal o apfst not

                       o fo_literal_for_arity_literal) premLits)

                 (formula_for_fo_literal

                      (fo_literal_for_arity_literal conclLit)))

 fun problem_line_for_conjecture full_types

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

   Fof (conjecture_prefix ^ name, kind,

        formula_for_combformula full_types combformula)

 fun free_type_literals_for_conjecture ({ctypes_sorts, ...} : fol_formula) =

   map fo_literal_for_type_literal (type_literals_for_types ctypes_sorts)

 fun problem_line_for_free_type lit =

   Fof (tfrees_name, Hypothesis, formula_for_fo_literal lit)

 fun problem_lines_for_free_types conjectures =

   let

     val litss = map free_type_literals_for_conjecture conjectures

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

   in map problem_line_for_free_type lits end

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

 type const_info = {min_arity: int, max_arity: int, sub_level: bool}

 fun consider_term top_level (ATerm ((s, _), ts)) =

   (if is_tptp_variable s then

      I

    else

      let val n = length ts in

        Symtab.map_default

            (s, {min_arity = n, max_arity = 0, sub_level = false})

            (fn {min_arity, max_arity, sub_level} =>

                {min_arity = Int.min (n, min_arity),

                 max_arity = Int.max (n, max_arity),

                 sub_level = sub_level orelse not top_level})

      end)

   #> fold (consider_term (top_level andalso s = type_wrapper_name)) ts

 fun consider_formula (AQuant (_, _, phi)) = consider_formula phi

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

   | consider_formula (AAtom tm) = consider_term true tm

 fun consider_problem_line (Fof (_, _, phi)) = consider_formula phi

 fun consider_problem problem = fold (fold consider_problem_line o snd) problem

 fun const_table_for_problem explicit_apply problem =

   if explicit_apply then NONE

   else SOME (Symtab.empty |> consider_problem problem)

 fun min_arity_of thy full_types NONE s =

     (if s = "equal" orelse s = type_wrapper_name orelse

         String.isPrefix type_const_prefix s orelse

         String.isPrefix class_prefix s then

        16383 (* large number *)

      else if full_types then

        0

      else case strip_prefix_and_unascii const_prefix s of

        SOME s' => num_type_args thy (invert_const s')

      | NONE => 0)

   | min_arity_of _ _ (SOME the_const_tab) s =

     case Symtab.lookup the_const_tab s of

       SOME ({min_arity, ...} : const_info) => min_arity

     | NONE => 0

 fun full_type_of (ATerm ((s, _), [ty, _])) =

     if s = type_wrapper_name then ty else raise Fail "expected type wrapper"

   | full_type_of _ = raise Fail "expected type wrapper"

 fun list_hAPP_rev _ t1 [] = t1

   | list_hAPP_rev NONE t1 (t2 :: ts2) =

     ATerm (`I "hAPP", [list_hAPP_rev NONE t1 ts2, t2])

   | list_hAPP_rev (SOME ty) t1 (t2 :: ts2) =

     let val ty' = ATerm (`make_fixed_type_const @{type_name fun},

                          [full_type_of t2, ty]) in

       ATerm (`I "hAPP", [wrap_type ty' (list_hAPP_rev (SOME ty') t1 ts2), t2])

     end

 fun repair_applications_in_term thy full_types const_tab =

   let

     fun aux opt_ty (ATerm (name as (s, _), ts)) =

       if s = type_wrapper_name then

         case ts of

           [t1, t2] => ATerm (name, [aux NONE t1, aux (SOME t1) t2])

         | _ => raise Fail "malformed type wrapper"

       else

         let

           val ts = map (aux NONE) ts

           val (ts1, ts2) = chop (min_arity_of thy full_types const_tab s) ts

         in list_hAPP_rev opt_ty (ATerm (name, ts1)) (rev ts2) end

   in aux NONE end

 fun boolify t = ATerm (`I "hBOOL", [t])

 (* 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_predicate NONE s =

     s = "equal" orelse s = "$false" orelse s = "$true" orelse

     String.isPrefix type_const_prefix s orelse String.isPrefix class_prefix s

   | is_predicate (SOME the_const_tab) s =

     case Symtab.lookup the_const_tab s of

       SOME {min_arity, max_arity, sub_level} =>

       not sub_level andalso min_arity = max_arity

     | NONE => false

 fun repair_predicates_in_term const_tab (t as ATerm ((s, _), ts)) =

   if s = type_wrapper_name then

     case ts of

       [_, t' as ATerm ((s', _), _)] =>

       if is_predicate const_tab s' then t' else boolify t

     | _ => raise Fail "malformed type wrapper"

   else

     t |> not (is_predicate const_tab s) ? boolify

 fun close_universally phi =

   let

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

         (is_tptp_variable s andalso not (member (op =) bounds name))

           ? insert (op =) name

         #> fold (term_vars bounds) tms

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

         formula_vars (xs @ bounds) phi

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

       | formula_vars bounds (AAtom tm) = term_vars bounds tm

   in

     case formula_vars [] phi [] of [] => phi | xs => AQuant (AForall, xs, phi)

   end

 fun repair_formula thy explicit_forall full_types const_tab =

   let

     fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)

       | aux (AConn (c, phis)) = AConn (c, map aux phis)

       | aux (AAtom tm) =

         AAtom (tm |> repair_applications_in_term thy full_types const_tab

                   |> repair_predicates_in_term const_tab)

   in aux #> explicit_forall ? close_universally end

 fun repair_problem_line thy explicit_forall full_types const_tab

                         (Fof (ident, kind, phi)) =

   Fof (ident, kind, repair_formula thy explicit_forall full_types const_tab phi)

 fun repair_problem_with_const_table thy =

   map o apsnd o map ooo repair_problem_line thy

 fun repair_problem thy explicit_forall full_types explicit_apply problem =

   repair_problem_with_const_table thy explicit_forall full_types

       (const_table_for_problem explicit_apply problem) problem

 fun prepare_problem ctxt readable_names explicit_forall full_types

                     explicit_apply hyp_ts concl_t axiom_pairs =

   let

     val thy = ProofContext.theory_of ctxt

     val (axiom_names, (conjectures, axioms, helper_facts, class_rel_clauses,

                        arity_clauses)) =

       prepare_formulas ctxt full_types hyp_ts concl_t axiom_pairs

     val axiom_lines = map (problem_line_for_fact axiom_prefix full_types) axioms

     val helper_lines =

       map (problem_line_for_fact helper_prefix full_types) helper_facts

     val conjecture_lines =

       map (problem_line_for_conjecture full_types) conjectures

     val tfree_lines = problem_lines_for_free_types conjectures

     val class_rel_lines =

       map problem_line_for_class_rel_clause class_rel_clauses

     val arity_lines = map problem_line_for_arity_clause arity_clauses

     (* Reordering these might or might not confuse the proof reconstruction

        code or the SPASS Flotter hack. *)

     val problem =

       [("Relevant facts", axiom_lines),

        ("Class relationships", class_rel_lines),

        ("Arity declarations", arity_lines),

        ("Helper facts", helper_lines),

        ("Conjectures", conjecture_lines),

        ("Type variables", tfree_lines)]

       |> repair_problem thy explicit_forall full_types explicit_apply

     val (problem, pool) = nice_tptp_problem readable_names problem

     val conjecture_offset =

       length axiom_lines + length class_rel_lines + length arity_lines

       + length helper_lines

   in

     (problem,

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

      conjecture_offset, axiom_names)

   end

 end;