src/HOL/Tools/Sledgehammer/sledgehammer_translate.ML
author blanchet
Thu Aug 19 14:39:31 2010 +0200 (2010-08-19)
changeset 38605 970ee38495e4
parent 38604 cda5f2000427
child 38606 3003ddbd46d9
permissions -rw-r--r--
fix atomize
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(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_translate.ML
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    Author:     Fabian Immler, TU Muenchen
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    Author:     Makarius
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    Author:     Jasmin Blanchette, TU Muenchen
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Translation of HOL to FOL.
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*)
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signature SLEDGEHAMMER_TRANSLATE =
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sig
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  type 'a problem = 'a ATP_Problem.problem
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  val axiom_prefix : string
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  val conjecture_prefix : string
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  val helper_prefix : string
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  val class_rel_clause_prefix : string
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  val arity_clause_prefix : string
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  val tfrees_name : string
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  val prepare_problem :
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    Proof.context -> bool -> bool -> bool -> bool -> term list -> term
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    -> (string * thm) list
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    -> string problem * string Symtab.table * int * string Vector.vector
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end;
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structure Sledgehammer_Translate : SLEDGEHAMMER_TRANSLATE =
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struct
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open ATP_Problem
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open Metis_Clauses
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open Sledgehammer_Util
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val axiom_prefix = "ax_"
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val conjecture_prefix = "conj_"
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val helper_prefix = "help_"
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val class_rel_clause_prefix = "clrel_";
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val arity_clause_prefix = "arity_"
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val tfrees_name = "tfrees"
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(* Freshness almost guaranteed! *)
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val sledgehammer_weak_prefix = "Sledgehammer:"
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datatype fol_formula =
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  FOLFormula of {name: string,
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                 kind: kind,
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                 combformula: (name, combterm) formula,
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                 ctypes_sorts: typ list}
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fun mk_anot phi = AConn (ANot, [phi])
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fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])
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fun mk_ahorn [] phi = phi
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  | mk_ahorn (phi :: phis) psi =
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    AConn (AImplies, [fold (mk_aconn AAnd) phis phi, psi])
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fun combformula_for_prop thy =
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  let
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    val do_term = combterm_from_term thy
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    fun do_quant bs q s T t' =
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      let val s = Name.variant (map fst bs) s in
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        do_formula ((s, T) :: bs) t'
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        #>> (fn phi => AQuant (q, [`make_bound_var s], phi))
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      end
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    and do_conn bs c t1 t2 =
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      do_formula bs t1 ##>> do_formula bs t2
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      #>> (fn (phi1, phi2) => AConn (c, [phi1, phi2]))
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    and do_formula bs t =
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      case t of
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        @{const Not} $ t1 =>
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        do_formula bs t1 #>> (fn phi => AConn (ANot, [phi]))
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      | Const (@{const_name All}, _) $ Abs (s, T, t') =>
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        do_quant bs AForall s T t'
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      | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
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        do_quant bs AExists s T t'
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      | @{const "op &"} $ t1 $ t2 => do_conn bs AAnd t1 t2
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      | @{const "op |"} $ t1 $ t2 => do_conn bs AOr t1 t2
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      | @{const "op -->"} $ t1 $ t2 => do_conn bs AImplies t1 t2
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      | Const (@{const_name "op ="}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
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        do_conn bs AIff t1 t2
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      | _ => (fn ts => do_term bs (Envir.eta_contract t)
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                       |>> AAtom ||> union (op =) ts)
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  in do_formula [] end
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(* Converts an elim-rule into an equivalent theorem that does not have the
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   predicate variable. Leaves other theorems unchanged. We simply instantiate
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   the conclusion variable to False. (Cf. "transform_elim_term" in
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   "ATP_Systems".) *)
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fun transform_elim_term t =
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  case Logic.strip_imp_concl t of
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    @{const Trueprop} $ Var (z, @{typ bool}) =>
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    subst_Vars [(z, @{const False})] t
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  | Var (z, @{typ prop}) => subst_Vars [(z, @{prop False})] t
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  | _ => t
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fun presimplify_term thy =
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  Skip_Proof.make_thm thy
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  #> Meson.presimplify
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  #> prop_of
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fun concealed_bound_name j = sledgehammer_weak_prefix ^ Int.toString j
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fun conceal_bounds Ts t =
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  subst_bounds (map (Free o apfst concealed_bound_name)
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                    (0 upto length Ts - 1 ~~ Ts), t)
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fun reveal_bounds Ts =
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  subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
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                    (0 upto length Ts - 1 ~~ Ts))
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fun introduce_combinators_in_term ctxt kind t =
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  let val thy = ProofContext.theory_of ctxt in
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    if Meson.is_fol_term thy t then
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      t
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    else
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      let
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        fun aux Ts t =
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          case t of
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            @{const Not} $ t1 => @{const Not} $ aux Ts t1
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          | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
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            t0 $ Abs (s, T, aux (T :: Ts) t')
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          | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
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            t0 $ Abs (s, T, aux (T :: Ts) t')
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          | (t0 as @{const "op &"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as @{const "op |"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as @{const "op -->"}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as Const (@{const_name "op ="}, Type (_, [@{typ bool}, _])))
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              $ t1 $ t2 =>
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            t0 $ aux Ts t1 $ aux Ts t2
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          | _ => if not (exists_subterm (fn Abs _ => true | _ => false) t) then
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                   t
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                 else
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                   t |> conceal_bounds Ts
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                     |> Envir.eta_contract
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                     |> cterm_of thy
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                     |> Clausifier.introduce_combinators_in_cterm
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                     |> prop_of |> Logic.dest_equals |> snd
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                     |> reveal_bounds Ts
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        val ([t], ctxt') = Variable.import_terms true [t] ctxt
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      in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
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      handle THM _ =>
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             (* A type variable of sort "{}" will make abstraction fail. *)
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             case kind of
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               Axiom => HOLogic.true_const
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             | Conjecture => HOLogic.false_const
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  end
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(* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
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   same in Sledgehammer to prevent the discovery of unreplable proofs. *)
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fun freeze_term t =
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  let
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    fun aux (t $ u) = aux t $ aux u
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      | aux (Abs (s, T, t)) = Abs (s, T, aux t)
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      | aux (Var ((s, i), T)) =
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        Free (sledgehammer_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
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      | aux t = t
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  in t |> exists_subterm is_Var t ? aux end
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(* "Object_Logic.atomize_term" isn't as powerful as it could be; for example,
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    it leaves metaequalities over "prop"s alone. *)
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val atomize_term =
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  let
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    fun aux (@{const Trueprop} $ t1) = t1
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      | aux (Const (@{const_name all}, _) $ Abs (s, T, t')) =
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        HOLogic.all_const T $ Abs (s, T, aux t')
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      | aux (@{const "==>"} $ t1 $ t2) = HOLogic.mk_imp (pairself aux (t1, t2))
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      | aux (Const (@{const_name "=="}, Type (_, [@{typ prop}, _])) $ t1 $ t2) =
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        HOLogic.eq_const HOLogic.boolT $ aux t1 $ aux t2
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      | aux (Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2) =
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        HOLogic.eq_const T $ t1 $ t2
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      | aux _ = raise Fail "aux"
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  in perhaps (try aux) end
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(* making axiom and conjecture formulas *)
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fun make_formula ctxt presimp (name, kind, t) =
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  let
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    val thy = ProofContext.theory_of ctxt
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    val t = t |> transform_elim_term
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              |> atomize_term
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    val t = t |> fastype_of t = HOLogic.boolT ? HOLogic.mk_Trueprop
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              |> extensionalize_term
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              |> presimp ? presimplify_term thy
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              |> perhaps (try (HOLogic.dest_Trueprop))
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              |> introduce_combinators_in_term ctxt kind
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              |> kind = Conjecture ? freeze_term
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    val (combformula, ctypes_sorts) = combformula_for_prop thy t []
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  in
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    FOLFormula {name = name, combformula = combformula, kind = kind,
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                ctypes_sorts = ctypes_sorts}
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  end
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fun make_axiom ctxt presimp (name, th) =
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  (name, make_formula ctxt presimp (name, Axiom, prop_of th))
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fun make_conjectures ctxt ts =
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  map2 (fn j => fn t => make_formula ctxt true (Int.toString j, Conjecture, t))
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       (0 upto length ts - 1) ts
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(** Helper facts **)
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fun count_combterm (CombConst ((s, _), _, _)) =
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    Symtab.map_entry s (Integer.add 1)
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  | count_combterm (CombVar _) = I
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  | count_combterm (CombApp (t1, t2)) = fold count_combterm [t1, t2]
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fun count_combformula (AQuant (_, _, phi)) = count_combformula phi
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  | count_combformula (AConn (_, phis)) = fold count_combformula phis
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  | count_combformula (AAtom tm) = count_combterm tm
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fun count_fol_formula (FOLFormula {combformula, ...}) =
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  count_combformula combformula
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val optional_helpers =
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  [(["c_COMBI", "c_COMBK"], @{thms COMBI_def COMBK_def}),
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   (["c_COMBB", "c_COMBC"], @{thms COMBB_def COMBC_def}),
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   (["c_COMBS"], @{thms COMBS_def})]
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val optional_typed_helpers =
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  [(["c_True", "c_False"], @{thms True_or_False}),
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   (["c_If"], @{thms if_True if_False True_or_False})]
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val mandatory_helpers = @{thms fequal_imp_equal equal_imp_fequal}
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val init_counters =
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  Symtab.make (maps (maps (map (rpair 0) o fst))
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                    [optional_helpers, optional_typed_helpers])
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fun get_helper_facts ctxt is_FO full_types conjectures axioms =
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  let
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    val ct = fold (fold count_fol_formula) [conjectures, axioms] init_counters
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    fun is_needed c = the (Symtab.lookup ct c) > 0
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  in
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    (optional_helpers
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     |> full_types ? append optional_typed_helpers
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     |> maps (fn (ss, ths) =>
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                 if exists is_needed ss then map (`Thm.get_name_hint) ths
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                 else [])) @
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    (if is_FO then [] else map (`Thm.get_name_hint) mandatory_helpers)
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    |> map (snd o make_axiom ctxt false)
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  end
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fun meta_not t = @{const "==>"} $ t $ @{prop False}
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fun prepare_formulas ctxt full_types hyp_ts concl_t axioms =
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  let
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    val thy = ProofContext.theory_of ctxt
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    val axiom_ts = map (prop_of o snd) axioms
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    val hyp_ts =
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      if null hyp_ts then
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        []
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      else
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        (* Remove existing axioms from the conjecture, as this can dramatically
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           boost an ATP's performance (for some reason). *)
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        let
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          val axiom_table = fold (Termtab.update o rpair ()) axiom_ts
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                                 Termtab.empty
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        in hyp_ts |> filter_out (Termtab.defined axiom_table) end
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    val goal_t = Logic.list_implies (hyp_ts, concl_t)
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    val is_FO = Meson.is_fol_term thy goal_t
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    val subs = tfree_classes_of_terms [goal_t]
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    val supers = tvar_classes_of_terms axiom_ts
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    val tycons = type_consts_of_terms thy (goal_t :: axiom_ts)
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    (* TFrees in the conjecture; TVars in the axioms *)
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    val conjectures = map meta_not hyp_ts @ [concl_t] |> make_conjectures ctxt
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    val (axiom_names, axioms) =
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      ListPair.unzip (map (make_axiom ctxt true) axioms)
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    val helper_facts = get_helper_facts ctxt is_FO full_types conjectures axioms
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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
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    (Vector.fromList axiom_names,
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     (conjectures, axioms, helper_facts, class_rel_clauses, arity_clauses))
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  end
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fun wrap_type ty t = ATerm ((type_wrapper_name, type_wrapper_name), [ty, t])
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fun fo_term_for_combtyp (CombTVar name) = ATerm (name, [])
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  | fo_term_for_combtyp (CombTFree name) = ATerm (name, [])
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  | fo_term_for_combtyp (CombType (name, tys)) =
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    ATerm (name, map fo_term_for_combtyp tys)
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fun fo_literal_for_type_literal (TyLitVar (class, name)) =
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    (true, ATerm (class, [ATerm (name, [])]))
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  | fo_literal_for_type_literal (TyLitFree (class, name)) =
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    (true, ATerm (class, [ATerm (name, [])]))
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fun formula_for_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
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fun fo_term_for_combterm full_types =
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  let
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    fun aux top_level u =
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      let
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        val (head, args) = strip_combterm_comb u
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        val (x, ty_args) =
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          case head of
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            CombConst (name as (s, s'), _, ty_args) =>
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            let val ty_args = if full_types then [] else ty_args in
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              if s = "equal" then
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                if top_level andalso length args = 2 then (name, [])
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                else (("c_fequal", @{const_name fequal}), ty_args)
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              else if top_level then
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                case s of
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                  "c_False" => (("$false", s'), [])
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                | "c_True" => (("$true", s'), [])
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                | _ => (name, ty_args)
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              else
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                (name, ty_args)
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            end
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          | CombVar (name, _) => (name, [])
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          | CombApp _ => raise Fail "impossible \"CombApp\""
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        val t = ATerm (x, map fo_term_for_combtyp ty_args @
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                          map (aux false) args)
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    in
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      if full_types then wrap_type (fo_term_for_combtyp (combtyp_of u)) t else t
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    end
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  in aux true end
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fun formula_for_combformula full_types =
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  let
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    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
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      | aux (AConn (c, phis)) = AConn (c, map aux phis)
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      | aux (AAtom tm) = AAtom (fo_term_for_combterm full_types tm)
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  in aux end
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fun formula_for_axiom full_types (FOLFormula {combformula, ctypes_sorts, ...}) =
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  mk_ahorn (map (formula_for_fo_literal o fo_literal_for_type_literal)
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                (type_literals_for_types ctypes_sorts))
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           (formula_for_combformula full_types combformula)
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fun problem_line_for_fact prefix full_types
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                          (formula as FOLFormula {name, kind, ...}) =
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  Fof (prefix ^ ascii_of name, kind, formula_for_axiom full_types formula)
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fun problem_line_for_class_rel_clause (ClassRelClause {name, subclass,
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                                                       superclass, ...}) =
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  let val ty_arg = ATerm (("T", "T"), []) in
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    Fof (class_rel_clause_prefix ^ ascii_of name, Axiom,
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         AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
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                           AAtom (ATerm (superclass, [ty_arg]))]))
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  end
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fun fo_literal_for_arity_literal (TConsLit (c, t, args)) =
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    (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
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  | fo_literal_for_arity_literal (TVarLit (c, sort)) =
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    (false, ATerm (c, [ATerm (sort, [])]))
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fun problem_line_for_arity_clause (ArityClause {name, conclLit, premLits,
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                                                ...}) =
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  Fof (arity_clause_prefix ^ ascii_of name, Axiom,
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       mk_ahorn (map (formula_for_fo_literal o apfst not
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                      o fo_literal_for_arity_literal) premLits)
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                (formula_for_fo_literal
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                     (fo_literal_for_arity_literal conclLit)))
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fun problem_line_for_conjecture full_types
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                                (FOLFormula {name, kind, combformula, ...}) =
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  Fof (conjecture_prefix ^ name, kind,
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       formula_for_combformula full_types combformula)
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fun free_type_literals_for_conjecture (FOLFormula {ctypes_sorts, ...}) =
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  map fo_literal_for_type_literal (type_literals_for_types ctypes_sorts)
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fun problem_line_for_free_type lit =
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  Fof (tfrees_name, Conjecture, mk_anot (formula_for_fo_literal lit))
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fun problem_lines_for_free_types conjectures =
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  let
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    val litss = map free_type_literals_for_conjecture conjectures
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    val lits = fold (union (op =)) litss []
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  in map problem_line_for_free_type lits end
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(** "hBOOL" and "hAPP" **)
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type const_info = {min_arity: int, max_arity: int, sub_level: bool}
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fun consider_term top_level (ATerm ((s, _), ts)) =
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  (if is_tptp_variable s then
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     I
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   else
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     let val n = length ts in
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       Symtab.map_default
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           (s, {min_arity = n, max_arity = 0, sub_level = false})
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           (fn {min_arity, max_arity, sub_level} =>
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               {min_arity = Int.min (n, min_arity),
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                max_arity = Int.max (n, max_arity),
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                sub_level = sub_level orelse not top_level})
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     end)
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  #> fold (consider_term (top_level andalso s = type_wrapper_name)) ts
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fun consider_formula (AQuant (_, _, phi)) = consider_formula phi
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  | consider_formula (AConn (_, phis)) = fold consider_formula phis
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  | consider_formula (AAtom tm) = consider_term true tm
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fun consider_problem_line (Fof (_, _, phi)) = consider_formula phi
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fun consider_problem problem = fold (fold consider_problem_line o snd) problem
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fun const_table_for_problem explicit_apply problem =
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  if explicit_apply then NONE
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  else SOME (Symtab.empty |> consider_problem problem)
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fun min_arity_of thy full_types NONE s =
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    (if s = "equal" orelse s = type_wrapper_name orelse
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        String.isPrefix type_const_prefix s orelse
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        String.isPrefix class_prefix s then
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       16383 (* large number *)
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   394
     else if full_types then
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       0
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     else case strip_prefix_and_undo_ascii const_prefix s of
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       SOME s' => num_type_args thy (invert_const s')
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     | NONE => 0)
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  | min_arity_of _ _ (SOME the_const_tab) s =
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   400
    case Symtab.lookup the_const_tab s of
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      SOME ({min_arity, ...} : const_info) => min_arity
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    | NONE => 0
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   403
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   404
fun full_type_of (ATerm ((s, _), [ty, _])) =
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    if s = type_wrapper_name then ty else raise Fail "expected type wrapper"
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  | full_type_of _ = raise Fail "expected type wrapper"
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   408
fun list_hAPP_rev _ t1 [] = t1
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  | list_hAPP_rev NONE t1 (t2 :: ts2) =
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    ATerm (`I "hAPP", [list_hAPP_rev NONE t1 ts2, t2])
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  | list_hAPP_rev (SOME ty) t1 (t2 :: ts2) =
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    let val ty' = ATerm (`make_fixed_type_const @{type_name fun},
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   413
                         [full_type_of t2, ty]) in
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      ATerm (`I "hAPP", [wrap_type ty' (list_hAPP_rev (SOME ty') t1 ts2), t2])
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   415
    end
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fun repair_applications_in_term thy full_types const_tab =
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  let
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    fun aux opt_ty (ATerm (name as (s, _), ts)) =
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   420
      if s = type_wrapper_name then
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   421
        case ts of
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   422
          [t1, t2] => ATerm (name, [aux NONE t1, aux (SOME t1) t2])
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   423
        | _ => raise Fail "malformed type wrapper"
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   424
      else
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        let
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          val ts = map (aux NONE) ts
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   427
          val (ts1, ts2) = chop (min_arity_of thy full_types const_tab s) ts
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   428
        in list_hAPP_rev opt_ty (ATerm (name, ts1)) (rev ts2) end
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  in aux NONE end
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   430
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   431
fun boolify t = ATerm (`I "hBOOL", [t])
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   432
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   433
(* True if the constant ever appears outside of the top-level position in
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   434
   literals, or if it appears with different arities (e.g., because of different
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   435
   type instantiations). If false, the constant always receives all of its
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   436
   arguments and is used as a predicate. *)
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   437
fun is_predicate NONE s =
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   438
    s = "equal" orelse s = "$false" orelse s = "$true" orelse
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   439
    String.isPrefix type_const_prefix s orelse String.isPrefix class_prefix s
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   440
  | is_predicate (SOME the_const_tab) s =
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   441
    case Symtab.lookup the_const_tab s of
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   442
      SOME {min_arity, max_arity, sub_level} =>
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   443
      not sub_level andalso min_arity = max_arity
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   444
    | NONE => false
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   445
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   446
fun repair_predicates_in_term const_tab (t as ATerm ((s, _), ts)) =
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   447
  if s = type_wrapper_name then
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   448
    case ts of
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   449
      [_, t' as ATerm ((s', _), _)] =>
blanchet@38282
   450
      if is_predicate const_tab s' then t' else boolify t
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   451
    | _ => raise Fail "malformed type wrapper"
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   452
  else
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   453
    t |> not (is_predicate const_tab s) ? boolify
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   454
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   455
fun close_universally phi =
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   456
  let
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   457
    fun term_vars bounds (ATerm (name as (s, _), tms)) =
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   458
        (is_tptp_variable s andalso not (member (op =) bounds name))
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   459
          ? insert (op =) name
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   460
        #> fold (term_vars bounds) tms
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   461
    fun formula_vars bounds (AQuant (q, xs, phi)) =
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   462
        formula_vars (xs @ bounds) phi
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   463
      | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
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   464
      | formula_vars bounds (AAtom tm) = term_vars bounds tm
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   465
  in
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   466
    case formula_vars [] phi [] of [] => phi | xs => AQuant (AForall, xs, phi)
blanchet@38282
   467
  end
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   468
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   469
fun repair_formula thy explicit_forall full_types const_tab =
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   470
  let
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   471
    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
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   472
      | aux (AConn (c, phis)) = AConn (c, map aux phis)
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   473
      | aux (AAtom tm) =
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   474
        AAtom (tm |> repair_applications_in_term thy full_types const_tab
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   475
                  |> repair_predicates_in_term const_tab)
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   476
  in aux #> explicit_forall ? close_universally end
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   477
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   478
fun repair_problem_line thy explicit_forall full_types const_tab
blanchet@38282
   479
                        (Fof (ident, kind, phi)) =
blanchet@38282
   480
  Fof (ident, kind, repair_formula thy explicit_forall full_types const_tab phi)
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   481
fun repair_problem_with_const_table thy =
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   482
  map o apsnd o map ooo repair_problem_line thy
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   483
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   484
fun repair_problem thy explicit_forall full_types explicit_apply problem =
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   485
  repair_problem_with_const_table thy explicit_forall full_types
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   486
      (const_table_for_problem explicit_apply problem) problem
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   487
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   488
fun prepare_problem ctxt readable_names explicit_forall full_types
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   489
                    explicit_apply hyp_ts concl_t axiom_ts =
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   490
  let
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   491
    val thy = ProofContext.theory_of ctxt
blanchet@38282
   492
    val (axiom_names, (conjectures, axioms, helper_facts, class_rel_clauses,
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   493
                       arity_clauses)) =
blanchet@38282
   494
      prepare_formulas ctxt full_types hyp_ts concl_t axiom_ts
blanchet@38282
   495
    val axiom_lines = map (problem_line_for_fact axiom_prefix full_types) axioms
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   496
    val helper_lines =
blanchet@38282
   497
      map (problem_line_for_fact helper_prefix full_types) helper_facts
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   498
    val conjecture_lines =
blanchet@38282
   499
      map (problem_line_for_conjecture full_types) conjectures
blanchet@38282
   500
    val tfree_lines = problem_lines_for_free_types conjectures
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   501
    val class_rel_lines =
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   502
      map problem_line_for_class_rel_clause class_rel_clauses
blanchet@38282
   503
    val arity_lines = map problem_line_for_arity_clause arity_clauses
blanchet@38282
   504
    (* Reordering these might or might not confuse the proof reconstruction
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   505
       code or the SPASS Flotter hack. *)
blanchet@38282
   506
    val problem =
blanchet@38282
   507
      [("Relevant facts", axiom_lines),
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   508
       ("Class relationships", class_rel_lines),
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   509
       ("Arity declarations", arity_lines),
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   510
       ("Helper facts", helper_lines),
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   511
       ("Conjectures", conjecture_lines),
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   512
       ("Type variables", tfree_lines)]
blanchet@38282
   513
      |> repair_problem thy explicit_forall full_types explicit_apply
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   514
    val (problem, pool) = nice_tptp_problem readable_names problem
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   515
    val conjecture_offset =
blanchet@38282
   516
      length axiom_lines + length class_rel_lines + length arity_lines
blanchet@38282
   517
      + length helper_lines
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   518
  in
blanchet@38282
   519
    (problem,
blanchet@38282
   520
     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
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   521
     conjecture_offset, axiom_names)
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   522
  end
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   523
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   524
end;