src/HOL/Tools/Sledgehammer/sledgehammer_atp_translate.ML
author blanchet
Wed Dec 15 11:26:29 2010 +0100 (2010-12-15)
changeset 41147 0e1903273712
parent 41145 a5ee3b8e5a90
child 41149 d64956b03c70
permissions -rw-r--r--
fix translation of higher-order equality ("fequal") if "precise_overloaded_args" is "true"
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(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_atp_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 for Sledgehammer.
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*)
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signature SLEDGEHAMMER_ATP_TRANSLATE =
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sig
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  type 'a problem = 'a ATP_Problem.problem
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  type translated_formula
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  datatype type_system =
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    Tags of bool |
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    Preds of bool |
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    Const_Args |
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    No_Types
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  val precise_overloaded_args : bool Unsynchronized.ref
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  val fact_prefix : string
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  val conjecture_prefix : string
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  val types_dangerous_types : type_system -> bool
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  val num_atp_type_args : theory -> type_system -> string -> int
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  val translate_atp_fact :
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    Proof.context -> (string * 'a) * thm
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    -> translated_formula option * ((string * 'a) * thm)
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  val prepare_atp_problem :
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    Proof.context -> bool -> bool -> type_system -> bool -> term list -> term
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    -> (translated_formula option * ((string * 'a) * thm)) list
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    -> string problem * string Symtab.table * int * (string * 'a) list vector
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end;
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structure Sledgehammer_ATP_Translate : SLEDGEHAMMER_ATP_TRANSLATE =
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struct
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open ATP_Problem
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open Metis_Translate
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open Sledgehammer_Util
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(* FIXME: Remove references once appropriate defaults have been determined
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   empirically. *)
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val precise_overloaded_args = Unsynchronized.ref false
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val fact_prefix = "fact_"
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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 tfree_prefix = "tfree_"
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(* Freshness almost guaranteed! *)
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val sledgehammer_weak_prefix = "Sledgehammer:"
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type translated_formula =
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  {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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datatype type_system =
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  Tags of bool |
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  Preds of bool |
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  Const_Args |
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  No_Types
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fun types_dangerous_types (Tags _) = true
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  | types_dangerous_types (Preds _) = true
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  | types_dangerous_types _ = false
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(* This is an approximation. If it returns "true" for a constant that isn't
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   overloaded (i.e., that has one uniform definition), needless clutter is
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   generated; if it returns "false" for an overloaded constant, the ATP gets a
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   license to do unsound reasoning if the type system is "overloaded_args". *)
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fun is_overloaded thy s =
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  not (!precise_overloaded_args) orelse
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  (s <> @{const_name HOL.eq} andalso
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   length (Defs.specifications_of (Theory.defs_of thy) s) > 1)
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fun needs_type_args thy type_sys s =
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  case type_sys of
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    Tags full_types => not full_types andalso is_overloaded thy s
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  | Preds _ => is_overloaded thy s (* FIXME: could be more precise *)
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  | Const_Args => is_overloaded thy s
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  | No_Types => false
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fun num_atp_type_args thy type_sys s =
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  if needs_type_args thy type_sys s then num_type_args thy s else 0
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fun atp_type_literals_for_types type_sys Ts =
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  if type_sys = No_Types then [] else type_literals_for_types Ts
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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 close_universally phi =
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  let
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    fun term_vars bounds (ATerm (name as (s, _), tms)) =
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        (is_atp_variable s andalso not (member (op =) bounds name))
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          ? insert (op =) name
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        #> fold (term_vars bounds) tms
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    fun formula_vars bounds (AQuant (_, xs, phi)) =
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        formula_vars (xs @ bounds) phi
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      | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
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      | formula_vars bounds (AAtom tm) = term_vars bounds tm
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  in
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    case formula_vars [] phi [] of [] => phi | xs => AQuant (AForall, xs, phi)
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  end
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fun combformula_for_prop thy eq_as_iff =
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  let
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    fun do_term bs t ts =
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      combterm_from_term thy bs (Envir.eta_contract t)
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      |>> AAtom ||> union (op =) ts
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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 HOL.conj} $ t1 $ t2 => do_conn bs AAnd t1 t2
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      | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr t1 t2
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      | @{const HOL.implies} $ t1 $ t2 => do_conn bs AImplies t1 t2
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      | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
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        if eq_as_iff then do_conn bs AIff t1 t2 else do_term bs t
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      | _ => do_term bs t
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  in do_formula [] end
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val presimplify_term = prop_of o Meson.presimplify oo Skip_Proof.make_thm
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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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(* Removes the lambdas from an equation of the form "t = (%x. u)".
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   (Cf. "extensionalize_theorem" in "Meson_Clausify".) *)
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fun extensionalize_term t =
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  let
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    fun aux j (@{const Trueprop} $ t') = @{const Trueprop} $ aux j t'
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      | aux j (t as Const (s, Type (_, [Type (_, [_, T']),
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                                        Type (_, [_, res_T])]))
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                    $ t2 $ Abs (var_s, var_T, t')) =
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        if s = @{const_name HOL.eq} orelse s = @{const_name "=="} then
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          let val var_t = Var ((var_s, j), var_T) in
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            Const (s, T' --> T' --> res_T)
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              $ betapply (t2, var_t) $ subst_bound (var_t, t')
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            |> aux (j + 1)
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          end
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        else
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          t
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      | aux _ t = t
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  in aux (maxidx_of_term t + 1) t end
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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 All}, _)) $ t1 =>
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            aux Ts (t0 $ eta_expand Ts t1 1)
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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 (@{const_name Ex}, _)) $ t1 =>
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            aux Ts (t0 $ eta_expand Ts t1 1)
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          | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
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          | (t0 as Const (@{const_name HOL.eq}, 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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                     |> Meson_Clausify.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 |>> the_single
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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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             if kind = Conjecture then HOLogic.false_const
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             else HOLogic.true_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 fact and conjecture formulas *)
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fun make_formula ctxt eq_as_iff 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 |> Envir.beta_eta_contract
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              |> transform_elim_term
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              |> atomize_term
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    val need_trueprop = (fastype_of t = HOLogic.boolT)
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    val t = t |> need_trueprop ? 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 <> Axiom ? freeze_term
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    val (combformula, ctypes_sorts) = combformula_for_prop thy eq_as_iff t []
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  in
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    {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_fact ctxt eq_as_iff presimp ((name, _), th) =
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  case make_formula ctxt eq_as_iff presimp name Axiom (prop_of th) of
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    {combformula = AAtom (CombConst (("c_True", _), _, _)), ...} => NONE
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  | formula => SOME formula
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fun make_conjecture ctxt ts =
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  let val last = length ts - 1 in
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    map2 (fn j => make_formula ctxt true true (Int.toString j)
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                               (if j = last then Conjecture else Hypothesis))
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         (0 upto last) ts
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  end
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(** Helper facts **)
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fun count_term (ATerm ((s, _), tms)) =
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  (if is_atp_variable s then I
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   else Symtab.map_entry s (Integer.add 1))
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  #> fold count_term tms
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fun count_formula (AQuant (_, _, phi)) = count_formula phi
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  | count_formula (AConn (_, phis)) = fold count_formula phis
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  | count_formula (AAtom tm) = count_term tm
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val init_counters =
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  metis_helpers |> map fst |> sort_distinct string_ord |> map (rpair 0)
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  |> Symtab.make
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fun get_helper_facts ctxt explicit_forall type_sys formulas =
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  let
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    val no_dangerous_types = types_dangerous_types type_sys
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    val ct = init_counters |> fold count_formula formulas
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    fun is_used s = the (Symtab.lookup ct s) > 0
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    fun dub c needs_full_types (th, j) =
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      ((c ^ "_" ^ string_of_int j ^ (if needs_full_types then "ft" else ""),
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        false), th)
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    fun make_facts eq_as_iff = map_filter (make_fact ctxt eq_as_iff false)
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  in
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    (metis_helpers
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     |> filter (is_used o fst)
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     |> maps (fn (c, (needs_full_types, ths)) =>
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                 if needs_full_types andalso not no_dangerous_types then
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                   []
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                 else
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                   ths ~~ (1 upto length ths)
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                   |> map (dub c needs_full_types)
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                   |> make_facts (not needs_full_types)),
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     if type_sys = Tags false then
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       let
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         fun var s = ATerm (`I s, [])
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         fun tag tm = ATerm (`I type_tag_name, [var "X", tm])
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       in
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         [Fof (helper_prefix ^ ascii_of "ti_ti", Axiom,
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               AAtom (ATerm (`I "equal", [tag (tag (var "Y")), tag (var "Y")]))
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   308
               |> explicit_forall ? close_universally)]
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   309
       end
blanchet@41145
   310
     else
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   311
       [])
blanchet@38282
   312
  end
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   313
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   314
fun translate_atp_fact ctxt = `(make_fact ctxt true true)
blanchet@39004
   315
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   316
fun translate_formulas ctxt type_sys hyp_ts concl_t rich_facts =
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   317
  let
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   318
    val thy = ProofContext.theory_of ctxt
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   319
    val fact_ts = map (prop_of o snd o snd) rich_facts
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   320
    val (facts, fact_names) =
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   321
      rich_facts
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   322
      |> map_filter (fn (NONE, _) => NONE
blanchet@41091
   323
                      | (SOME fact, (name, _)) => SOME (fact, name))
blanchet@41091
   324
      |> ListPair.unzip
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   325
    (* Remove existing facts from the conjecture, as this can dramatically
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   326
       boost an ATP's performance (for some reason). *)
blanchet@40204
   327
    val hyp_ts = hyp_ts |> filter_out (member (op aconv) fact_ts)
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   328
    val goal_t = Logic.list_implies (hyp_ts, concl_t)
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   329
    val subs = tfree_classes_of_terms [goal_t]
blanchet@40204
   330
    val supers = tvar_classes_of_terms fact_ts
blanchet@40204
   331
    val tycons = type_consts_of_terms thy (goal_t :: fact_ts)
blanchet@40204
   332
    (* TFrees in the conjecture; TVars in the facts *)
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   333
    val conjectures = make_conjecture ctxt (hyp_ts @ [concl_t])
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   334
    val (supers', arity_clauses) =
blanchet@41137
   335
      if type_sys = No_Types then ([], [])
blanchet@41137
   336
      else make_arity_clauses thy tycons supers
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   337
    val class_rel_clauses = make_class_rel_clauses thy subs supers'
blanchet@38282
   338
  in
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   339
    (fact_names |> map single |> Vector.fromList,
blanchet@41140
   340
     (conjectures, facts, class_rel_clauses, arity_clauses))
blanchet@38282
   341
  end
blanchet@38282
   342
blanchet@41138
   343
fun tag_with_type ty t = ATerm (`I type_tag_name, [ty, t])
blanchet@38282
   344
blanchet@38282
   345
fun fo_term_for_combtyp (CombTVar name) = ATerm (name, [])
blanchet@38282
   346
  | fo_term_for_combtyp (CombTFree name) = ATerm (name, [])
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   347
  | fo_term_for_combtyp (CombType (name, tys)) =
blanchet@38282
   348
    ATerm (name, map fo_term_for_combtyp tys)
blanchet@38282
   349
blanchet@38282
   350
fun fo_literal_for_type_literal (TyLitVar (class, name)) =
blanchet@38282
   351
    (true, ATerm (class, [ATerm (name, [])]))
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   352
  | fo_literal_for_type_literal (TyLitFree (class, name)) =
blanchet@38282
   353
    (true, ATerm (class, [ATerm (name, [])]))
blanchet@38282
   354
blanchet@38282
   355
fun formula_for_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
blanchet@38282
   356
blanchet@41138
   357
(* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
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   358
   considered dangerous because their "exhaust" properties can easily lead to
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   359
   unsound ATP proofs. The checks below are an (unsound) approximation of
blanchet@41138
   360
   finiteness. *)
blanchet@41138
   361
blanchet@41138
   362
fun is_dtyp_dangerous _ (Datatype_Aux.DtTFree _) = true
blanchet@41138
   363
  | is_dtyp_dangerous ctxt (Datatype_Aux.DtType (s, Us)) =
blanchet@41138
   364
    is_type_constr_dangerous ctxt s andalso forall (is_dtyp_dangerous ctxt) Us
blanchet@41138
   365
  | is_dtyp_dangerous _ (Datatype_Aux.DtRec _) = false
blanchet@41138
   366
and is_type_dangerous ctxt (Type (s, Ts)) =
blanchet@41138
   367
    is_type_constr_dangerous ctxt s andalso forall (is_type_dangerous ctxt) Ts
blanchet@41140
   368
  | is_type_dangerous _ _ = false
blanchet@41138
   369
and is_type_constr_dangerous ctxt s =
blanchet@41138
   370
  let val thy = ProofContext.theory_of ctxt in
blanchet@41138
   371
    case Datatype_Data.get_info thy s of
blanchet@41138
   372
      SOME {descr, ...} =>
blanchet@41138
   373
      forall (fn (_, (_, _, constrs)) =>
blanchet@41138
   374
                 forall (forall (is_dtyp_dangerous ctxt) o snd) constrs) descr
blanchet@41138
   375
    | NONE =>
blanchet@41138
   376
      case Typedef.get_info ctxt s of
blanchet@41138
   377
        ({rep_type, ...}, _) :: _ => is_type_dangerous ctxt rep_type
blanchet@41138
   378
      | [] => true
blanchet@41138
   379
  end
blanchet@41138
   380
blanchet@41138
   381
fun is_combtyp_dangerous ctxt (CombType ((s, _), tys)) =
blanchet@41138
   382
    (case strip_prefix_and_unascii type_const_prefix s of
blanchet@41138
   383
       SOME s' => forall (is_combtyp_dangerous ctxt) tys andalso
blanchet@41138
   384
                  is_type_constr_dangerous ctxt (invert_const s')
blanchet@41138
   385
     | NONE => false)
blanchet@41138
   386
  | is_combtyp_dangerous _ _ = false
blanchet@41138
   387
blanchet@41138
   388
fun should_tag_with_type ctxt (Tags full_types) ty =
blanchet@41138
   389
    full_types orelse is_combtyp_dangerous ctxt ty
blanchet@41138
   390
  | should_tag_with_type _ _ _ = false
blanchet@41138
   391
blanchet@41140
   392
val fname_table =
blanchet@41140
   393
  [("c_False", (0, ("c_fFalse", @{const_name Metis.fFalse}))),
blanchet@41140
   394
   ("c_True", (0, ("c_fTrue", @{const_name Metis.fTrue}))),
blanchet@41140
   395
   ("c_Not", (1, ("c_fNot", @{const_name Metis.fNot}))),
blanchet@41140
   396
   ("c_conj", (2, ("c_fconj", @{const_name Metis.fconj}))),
blanchet@41140
   397
   ("c_disj", (2, ("c_fdisj", @{const_name Metis.fdisj}))),
blanchet@41140
   398
   ("c_implies", (2, ("c_fimplies", @{const_name Metis.fimplies}))),
blanchet@41140
   399
   ("equal", (2, ("c_fequal", @{const_name Metis.fequal})))]
blanchet@41140
   400
blanchet@41138
   401
fun fo_term_for_combterm ctxt type_sys =
blanchet@38282
   402
  let
blanchet@41138
   403
    val thy = ProofContext.theory_of ctxt
blanchet@38282
   404
    fun aux top_level u =
blanchet@38282
   405
      let
blanchet@38282
   406
        val (head, args) = strip_combterm_comb u
blanchet@38282
   407
        val (x, ty_args) =
blanchet@38282
   408
          case head of
blanchet@38282
   409
            CombConst (name as (s, s'), _, ty_args) =>
blanchet@41140
   410
            (case AList.lookup (op =) fname_table s of
blanchet@41140
   411
               SOME (n, fname) =>
blanchet@41147
   412
               (if top_level andalso length args = n then name else fname, [])
blanchet@41140
   413
             | NONE =>
blanchet@41140
   414
               case strip_prefix_and_unascii const_prefix s of
blanchet@41140
   415
                 NONE => (name, ty_args)
blanchet@41140
   416
               | SOME s'' =>
blanchet@41140
   417
                 let
blanchet@41140
   418
                   val s'' = invert_const s''
blanchet@41140
   419
                   val ty_args =
blanchet@41140
   420
                     if needs_type_args thy type_sys s'' then ty_args else []
blanchet@41140
   421
                  in
blanchet@41140
   422
                    if top_level then
blanchet@41140
   423
                      case s of
blanchet@41140
   424
                        "c_False" => (("$false", s'), [])
blanchet@41140
   425
                      | "c_True" => (("$true", s'), [])
blanchet@41140
   426
                      | _ => (name, ty_args)
blanchet@41140
   427
                    else
blanchet@41140
   428
                      (name, ty_args)
blanchet@41140
   429
                  end)
blanchet@38282
   430
          | CombVar (name, _) => (name, [])
blanchet@38282
   431
          | CombApp _ => raise Fail "impossible \"CombApp\""
blanchet@41138
   432
        val t =
blanchet@41138
   433
          ATerm (x, map fo_term_for_combtyp ty_args @ map (aux false) args)
blanchet@41138
   434
        val ty = combtyp_of u
blanchet@38282
   435
    in
blanchet@41138
   436
      t |> (if should_tag_with_type ctxt type_sys ty then
blanchet@41138
   437
              tag_with_type (fo_term_for_combtyp ty)
blanchet@41134
   438
            else
blanchet@41134
   439
              I)
blanchet@38282
   440
    end
blanchet@38282
   441
  in aux true end
blanchet@38282
   442
blanchet@41138
   443
fun formula_for_combformula ctxt type_sys =
blanchet@38282
   444
  let
blanchet@38282
   445
    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
blanchet@38282
   446
      | aux (AConn (c, phis)) = AConn (c, map aux phis)
blanchet@41138
   447
      | aux (AAtom tm) = AAtom (fo_term_for_combterm ctxt type_sys tm)
blanchet@38282
   448
  in aux end
blanchet@38282
   449
blanchet@41138
   450
fun formula_for_fact ctxt type_sys
blanchet@40204
   451
                     ({combformula, ctypes_sorts, ...} : translated_formula) =
blanchet@38282
   452
  mk_ahorn (map (formula_for_fo_literal o fo_literal_for_type_literal)
blanchet@41137
   453
                (atp_type_literals_for_types type_sys ctypes_sorts))
blanchet@41138
   454
           (formula_for_combformula ctxt type_sys combformula)
blanchet@38282
   455
blanchet@41138
   456
fun problem_line_for_fact ctxt prefix type_sys (formula as {name, kind, ...}) =
blanchet@41138
   457
  Fof (prefix ^ ascii_of name, kind, formula_for_fact ctxt type_sys formula)
blanchet@38282
   458
blanchet@38282
   459
fun problem_line_for_class_rel_clause (ClassRelClause {name, subclass,
blanchet@38282
   460
                                                       superclass, ...}) =
blanchet@38282
   461
  let val ty_arg = ATerm (("T", "T"), []) in
blanchet@38282
   462
    Fof (class_rel_clause_prefix ^ ascii_of name, Axiom,
blanchet@38282
   463
         AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
blanchet@38282
   464
                           AAtom (ATerm (superclass, [ty_arg]))]))
blanchet@38282
   465
  end
blanchet@38282
   466
blanchet@38282
   467
fun fo_literal_for_arity_literal (TConsLit (c, t, args)) =
blanchet@38282
   468
    (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
blanchet@38282
   469
  | fo_literal_for_arity_literal (TVarLit (c, sort)) =
blanchet@38282
   470
    (false, ATerm (c, [ATerm (sort, [])]))
blanchet@38282
   471
blanchet@38282
   472
fun problem_line_for_arity_clause (ArityClause {name, conclLit, premLits,
blanchet@38282
   473
                                                ...}) =
blanchet@38282
   474
  Fof (arity_clause_prefix ^ ascii_of name, Axiom,
blanchet@38282
   475
       mk_ahorn (map (formula_for_fo_literal o apfst not
blanchet@38282
   476
                      o fo_literal_for_arity_literal) premLits)
blanchet@38282
   477
                (formula_for_fo_literal
blanchet@38282
   478
                     (fo_literal_for_arity_literal conclLit)))
blanchet@38282
   479
blanchet@41138
   480
fun problem_line_for_conjecture ctxt type_sys
blanchet@40114
   481
        ({name, kind, combformula, ...} : translated_formula) =
blanchet@38282
   482
  Fof (conjecture_prefix ^ name, kind,
blanchet@41138
   483
       formula_for_combformula ctxt type_sys combformula)
blanchet@38282
   484
blanchet@41137
   485
fun free_type_literals_for_conjecture type_sys
blanchet@40114
   486
        ({ctypes_sorts, ...} : translated_formula) =
blanchet@41137
   487
  ctypes_sorts |> atp_type_literals_for_types type_sys
blanchet@41137
   488
               |> map fo_literal_for_type_literal
blanchet@38282
   489
blanchet@39975
   490
fun problem_line_for_free_type j lit =
blanchet@39975
   491
  Fof (tfree_prefix ^ string_of_int j, Hypothesis, formula_for_fo_literal lit)
blanchet@41137
   492
fun problem_lines_for_free_types type_sys conjectures =
blanchet@38282
   493
  let
blanchet@41137
   494
    val litss = map (free_type_literals_for_conjecture type_sys) conjectures
blanchet@38282
   495
    val lits = fold (union (op =)) litss []
blanchet@39975
   496
  in map2 problem_line_for_free_type (0 upto length lits - 1) lits end
blanchet@38282
   497
blanchet@38282
   498
(** "hBOOL" and "hAPP" **)
blanchet@38282
   499
blanchet@38282
   500
type const_info = {min_arity: int, max_arity: int, sub_level: bool}
blanchet@38282
   501
blanchet@38282
   502
fun consider_term top_level (ATerm ((s, _), ts)) =
blanchet@39452
   503
  (if is_atp_variable s then
blanchet@38282
   504
     I
blanchet@38282
   505
   else
blanchet@38282
   506
     let val n = length ts in
blanchet@38282
   507
       Symtab.map_default
blanchet@38282
   508
           (s, {min_arity = n, max_arity = 0, sub_level = false})
blanchet@38282
   509
           (fn {min_arity, max_arity, sub_level} =>
blanchet@38282
   510
               {min_arity = Int.min (n, min_arity),
blanchet@38282
   511
                max_arity = Int.max (n, max_arity),
blanchet@38282
   512
                sub_level = sub_level orelse not top_level})
blanchet@38282
   513
     end)
blanchet@41138
   514
  #> fold (consider_term (top_level andalso s = type_tag_name)) ts
blanchet@38282
   515
fun consider_formula (AQuant (_, _, phi)) = consider_formula phi
blanchet@38282
   516
  | consider_formula (AConn (_, phis)) = fold consider_formula phis
blanchet@38282
   517
  | consider_formula (AAtom tm) = consider_term true tm
blanchet@38282
   518
blanchet@38282
   519
fun consider_problem_line (Fof (_, _, phi)) = consider_formula phi
blanchet@38282
   520
fun consider_problem problem = fold (fold consider_problem_line o snd) problem
blanchet@38282
   521
blanchet@41140
   522
(* needed for helper facts if the problem otherwise does not involve equality *)
blanchet@41140
   523
val equal_entry = ("equal", {min_arity = 2, max_arity = 2, sub_level = false})
blanchet@41140
   524
blanchet@38282
   525
fun const_table_for_problem explicit_apply problem =
blanchet@41140
   526
  if explicit_apply then
blanchet@41140
   527
    NONE
blanchet@41140
   528
  else
blanchet@41147
   529
    SOME (Symtab.empty |> Symtab.default equal_entry |> consider_problem problem)
blanchet@38282
   530
blanchet@41134
   531
fun min_arity_of thy type_sys NONE s =
blanchet@41138
   532
    (if s = "equal" orelse s = type_tag_name orelse
blanchet@38282
   533
        String.isPrefix type_const_prefix s orelse
blanchet@38282
   534
        String.isPrefix class_prefix s then
blanchet@38282
   535
       16383 (* large number *)
blanchet@38748
   536
     else case strip_prefix_and_unascii const_prefix s of
blanchet@41136
   537
       SOME s' => num_atp_type_args thy type_sys (invert_const s')
blanchet@38282
   538
     | NONE => 0)
blanchet@38282
   539
  | min_arity_of _ _ (SOME the_const_tab) s =
blanchet@38282
   540
    case Symtab.lookup the_const_tab s of
blanchet@38282
   541
      SOME ({min_arity, ...} : const_info) => min_arity
blanchet@38282
   542
    | NONE => 0
blanchet@38282
   543
blanchet@38282
   544
fun full_type_of (ATerm ((s, _), [ty, _])) =
blanchet@41138
   545
    if s = type_tag_name then SOME ty else NONE
blanchet@41138
   546
  | full_type_of _ = NONE
blanchet@38282
   547
blanchet@38282
   548
fun list_hAPP_rev _ t1 [] = t1
blanchet@38282
   549
  | list_hAPP_rev NONE t1 (t2 :: ts2) =
blanchet@38282
   550
    ATerm (`I "hAPP", [list_hAPP_rev NONE t1 ts2, t2])
blanchet@38282
   551
  | list_hAPP_rev (SOME ty) t1 (t2 :: ts2) =
blanchet@41138
   552
    case full_type_of t2 of
blanchet@41138
   553
      SOME ty2 =>
blanchet@41138
   554
      let val ty' = ATerm (`make_fixed_type_const @{type_name fun},
blanchet@41138
   555
                           [ty2, ty]) in
blanchet@41138
   556
        ATerm (`I "hAPP",
blanchet@41138
   557
               [tag_with_type ty' (list_hAPP_rev (SOME ty') t1 ts2), t2])
blanchet@41138
   558
      end
blanchet@41138
   559
    | NONE => list_hAPP_rev NONE t1 (t2 :: ts2)
blanchet@38282
   560
blanchet@41134
   561
fun repair_applications_in_term thy type_sys const_tab =
blanchet@38282
   562
  let
blanchet@38282
   563
    fun aux opt_ty (ATerm (name as (s, _), ts)) =
blanchet@41138
   564
      if s = type_tag_name then
blanchet@38282
   565
        case ts of
blanchet@38282
   566
          [t1, t2] => ATerm (name, [aux NONE t1, aux (SOME t1) t2])
blanchet@41138
   567
        | _ => raise Fail "malformed type tag"
blanchet@38282
   568
      else
blanchet@38282
   569
        let
blanchet@38282
   570
          val ts = map (aux NONE) ts
blanchet@41134
   571
          val (ts1, ts2) = chop (min_arity_of thy type_sys const_tab s) ts
blanchet@38282
   572
        in list_hAPP_rev opt_ty (ATerm (name, ts1)) (rev ts2) end
blanchet@38282
   573
  in aux NONE end
blanchet@38282
   574
blanchet@38282
   575
fun boolify t = ATerm (`I "hBOOL", [t])
blanchet@38282
   576
blanchet@38282
   577
(* True if the constant ever appears outside of the top-level position in
blanchet@38282
   578
   literals, or if it appears with different arities (e.g., because of different
blanchet@38282
   579
   type instantiations). If false, the constant always receives all of its
blanchet@38282
   580
   arguments and is used as a predicate. *)
blanchet@38282
   581
fun is_predicate NONE s =
blanchet@38589
   582
    s = "equal" orelse s = "$false" orelse s = "$true" orelse
blanchet@38589
   583
    String.isPrefix type_const_prefix s orelse String.isPrefix class_prefix s
blanchet@38282
   584
  | is_predicate (SOME the_const_tab) s =
blanchet@38282
   585
    case Symtab.lookup the_const_tab s of
blanchet@38282
   586
      SOME {min_arity, max_arity, sub_level} =>
blanchet@38282
   587
      not sub_level andalso min_arity = max_arity
blanchet@38282
   588
    | NONE => false
blanchet@38282
   589
blanchet@41140
   590
fun repair_predicates_in_term pred_const_tab (t as ATerm ((s, _), ts)) =
blanchet@41138
   591
  if s = type_tag_name then
blanchet@38282
   592
    case ts of
blanchet@38282
   593
      [_, t' as ATerm ((s', _), _)] =>
blanchet@41140
   594
      if is_predicate pred_const_tab s' then t' else boolify t
blanchet@41138
   595
    | _ => raise Fail "malformed type tag"
blanchet@38282
   596
  else
blanchet@41140
   597
    t |> not (is_predicate pred_const_tab s) ? boolify
blanchet@38282
   598
blanchet@41134
   599
fun repair_formula thy explicit_forall type_sys const_tab =
blanchet@38282
   600
  let
blanchet@41140
   601
    val pred_const_tab = case type_sys of Tags _ => NONE | _ => const_tab
blanchet@38282
   602
    fun aux (AQuant (q, xs, phi)) = AQuant (q, xs, aux phi)
blanchet@38282
   603
      | aux (AConn (c, phis)) = AConn (c, map aux phis)
blanchet@38282
   604
      | aux (AAtom tm) =
blanchet@41134
   605
        AAtom (tm |> repair_applications_in_term thy type_sys const_tab
blanchet@41140
   606
                  |> repair_predicates_in_term pred_const_tab)
blanchet@38282
   607
  in aux #> explicit_forall ? close_universally end
blanchet@38282
   608
blanchet@41134
   609
fun repair_problem_line thy explicit_forall type_sys const_tab
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   610
                        (Fof (ident, kind, phi)) =
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  Fof (ident, kind, repair_formula thy explicit_forall type_sys const_tab phi)
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   612
fun repair_problem thy = map o apsnd o map ooo repair_problem_line thy
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   613
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   614
fun dest_Fof (Fof z) = z
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   615
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   616
fun prepare_atp_problem ctxt readable_names explicit_forall type_sys
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                        explicit_apply hyp_ts concl_t facts =
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  let
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    val thy = ProofContext.theory_of ctxt
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    val (fact_names, (conjectures, facts, class_rel_clauses, arity_clauses)) =
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   621
      translate_formulas ctxt type_sys hyp_ts concl_t facts
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   622
    val fact_lines = map (problem_line_for_fact ctxt fact_prefix type_sys) facts
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   623
    val conjecture_lines =
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   624
      map (problem_line_for_conjecture ctxt type_sys) conjectures
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   625
    val tfree_lines = problem_lines_for_free_types type_sys conjectures
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   626
    val class_rel_lines =
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   627
      map problem_line_for_class_rel_clause class_rel_clauses
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   628
    val arity_lines = map problem_line_for_arity_clause arity_clauses
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   629
    (* Reordering these might or might not confuse the proof reconstruction
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   630
       code or the SPASS Flotter hack. *)
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   631
    val problem =
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      [("Relevant facts", fact_lines),
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   633
       ("Class relationships", class_rel_lines),
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   634
       ("Arity declarations", arity_lines),
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   635
       ("Helper facts", []),
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   636
       ("Conjectures", conjecture_lines),
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   637
       ("Type variables", tfree_lines)]
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   638
    val const_tab = const_table_for_problem explicit_apply problem
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   639
    val problem =
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   640
      problem |> repair_problem thy explicit_forall type_sys const_tab
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    val (helper_facts, raw_helper_lines) =
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      get_helper_facts ctxt explicit_forall type_sys
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   643
                       (maps (map (#3 o dest_Fof) o snd) problem)
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   644
    val helper_lines =
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   645
      (helper_facts
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   646
       |> map (problem_line_for_fact ctxt helper_prefix type_sys
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   647
               #> repair_problem_line thy explicit_forall type_sys const_tab)) @
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   648
      raw_helper_lines
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   649
    val (problem, pool) =
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   650
      problem |> AList.update (op =) ("Helper facts", helper_lines)
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   651
              |> nice_atp_problem readable_names
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   652
    val conjecture_offset =
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   653
      length fact_lines + length class_rel_lines + length arity_lines
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   654
      + length helper_lines
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   655
  in
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    (problem,
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   657
     case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
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     conjecture_offset, fact_names)
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   659
  end
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   660
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   661
end;