src/HOL/Tools/Meson/meson_clausify.ML
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(*  Title:      HOL/Tools/Meson/meson_clausify.ML
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    Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA
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    Author:     Jasmin Blanchette, TU Muenchen
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Transformation of HOL theorems into CNF forms.
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*)
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signature MESON_CLAUSIFY =
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sig
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  val new_skolem_var_prefix : string
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  val extensionalize_theorem : thm -> thm
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  val introduce_combinators_in_cterm : cterm -> thm
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  val introduce_combinators_in_theorem : thm -> thm
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  val to_definitional_cnf_with_quantifiers : theory -> thm -> thm
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  val cluster_of_zapped_var_name : string -> (int * (int * int)) * bool
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  val cnf_axiom :
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    Proof.context -> bool -> int -> thm -> (thm * term) option * thm list
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end;
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structure Meson_Clausify : MESON_CLAUSIFY =
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struct
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open Meson
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(* the extra "Meson" helps prevent clashes (FIXME) *)
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val new_skolem_var_prefix = "MesonSK"
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val new_nonskolem_var_prefix = "MesonV"
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(**** Transformation of Elimination Rules into First-Order Formulas****)
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val cfalse = cterm_of @{theory HOL} HOLogic.false_const;
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val ctp_false = cterm_of @{theory HOL} (HOLogic.mk_Trueprop HOLogic.false_const);
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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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   "Sledgehammer_Util".) *)
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fun transform_elim_theorem th =
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  case concl_of th of    (*conclusion variable*)
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       @{const Trueprop} $ (v as Var (_, @{typ bool})) =>
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           Thm.instantiate ([], [(cterm_of @{theory HOL} v, cfalse)]) th
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    | v as Var(_, @{typ prop}) =>
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           Thm.instantiate ([], [(cterm_of @{theory HOL} v, ctp_false)]) th
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    | _ => th
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(**** SKOLEMIZATION BY INFERENCE (lcp) ****)
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fun mk_old_skolem_term_wrapper t =
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  let val T = fastype_of t in
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    Const (@{const_name Meson.skolem}, T --> T) $ t
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  end
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fun beta_eta_in_abs_body (Abs (s, T, t')) = Abs (s, T, beta_eta_in_abs_body t')
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  | beta_eta_in_abs_body t = Envir.beta_eta_contract t
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(*Traverse a theorem, accumulating Skolem function definitions.*)
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fun old_skolem_defs th =
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  let
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    fun dec_sko (Const (@{const_name Ex}, _) $ (body as Abs (_, T, p))) rhss =
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        (*Existential: declare a Skolem function, then insert into body and continue*)
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        let
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          val args = OldTerm.term_frees body
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          (* Forms a lambda-abstraction over the formal parameters *)
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          val rhs =
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            list_abs_free (map dest_Free args,
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                           HOLogic.choice_const T $ beta_eta_in_abs_body body)
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            |> mk_old_skolem_term_wrapper
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          val comb = list_comb (rhs, args)
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        in dec_sko (subst_bound (comb, p)) (rhs :: rhss) end
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      | dec_sko (Const (@{const_name All},_) $ Abs (a, T, p)) rhss =
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        (*Universal quant: insert a free variable into body and continue*)
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        let val fname = Name.variant (OldTerm.add_term_names (p,[])) a
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        in dec_sko (subst_bound (Free(fname,T), p)) rhss end
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      | dec_sko (@{const conj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
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      | dec_sko (@{const disj} $ p $ q) rhss = rhss |> dec_sko p |> dec_sko q
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      | dec_sko (@{const Trueprop} $ p) rhss = dec_sko p rhss
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      | dec_sko _ rhss = rhss
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  in  dec_sko (prop_of th) []  end;
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(**** REPLACING ABSTRACTIONS BY COMBINATORS ****)
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val fun_cong_all = @{thm fun_eq_iff [THEN iffD1]}
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(* Removes the lambdas from an equation of the form "t = (%x. u)".
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   (Cf. "extensionalize_term" in "Sledgehammer_Translate".) *)
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fun extensionalize_theorem th =
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  case prop_of th of
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    _ $ (Const (@{const_name HOL.eq}, Type (_, [Type (@{type_name fun}, _), _]))
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         $ _ $ Abs _) => extensionalize_theorem (th RS fun_cong_all)
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  | _ => th
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fun is_quasi_lambda_free (Const (@{const_name Meson.skolem}, _) $ _) = true
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  | is_quasi_lambda_free (t1 $ t2) =
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    is_quasi_lambda_free t1 andalso is_quasi_lambda_free t2
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  | is_quasi_lambda_free (Abs _) = false
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  | is_quasi_lambda_free _ = true
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val [f_B,g_B] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_B}));
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val [g_C,f_C] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_C}));
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val [f_S,g_S] = map (cterm_of @{theory}) (OldTerm.term_vars (prop_of @{thm abs_S}));
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(* FIXME: Requires more use of cterm constructors. *)
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fun abstract ct =
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  let
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      val thy = theory_of_cterm ct
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      val Abs(x,_,body) = term_of ct
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      val Type(@{type_name fun}, [xT,bodyT]) = typ_of (ctyp_of_term ct)
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      val cxT = ctyp_of thy xT
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      val cbodyT = ctyp_of thy bodyT
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      fun makeK () =
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        instantiate' [SOME cxT, SOME cbodyT] [SOME (cterm_of thy body)]
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                     @{thm abs_K}
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  in
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      case body of
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          Const _ => makeK()
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        | Free _ => makeK()
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        | Var _ => makeK()  (*though Var isn't expected*)
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        | Bound 0 => instantiate' [SOME cxT] [] @{thm abs_I} (*identity: I*)
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        | rator$rand =>
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            if loose_bvar1 (rator,0) then (*C or S*)
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               if loose_bvar1 (rand,0) then (*S*)
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                 let val crator = cterm_of thy (Abs(x,xT,rator))
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                     val crand = cterm_of thy (Abs(x,xT,rand))
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                     val abs_S' = cterm_instantiate [(f_S,crator),(g_S,crand)] @{thm abs_S}
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                     val (_,rhs) = Thm.dest_equals (cprop_of abs_S')
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                 in
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                   Thm.transitive abs_S' (Conv.binop_conv abstract rhs)
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                 end
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               else (*C*)
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                 let val crator = cterm_of thy (Abs(x,xT,rator))
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                     val abs_C' = cterm_instantiate [(f_C,crator),(g_C,cterm_of thy rand)] @{thm abs_C}
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                     val (_,rhs) = Thm.dest_equals (cprop_of abs_C')
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                 in
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                   Thm.transitive abs_C' (Conv.fun_conv (Conv.arg_conv abstract) rhs)
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                 end
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            else if loose_bvar1 (rand,0) then (*B or eta*)
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               if rand = Bound 0 then Thm.eta_conversion ct
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               else (*B*)
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                 let val crand = cterm_of thy (Abs(x,xT,rand))
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                     val crator = cterm_of thy rator
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                     val abs_B' = cterm_instantiate [(f_B,crator),(g_B,crand)] @{thm abs_B}
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                     val (_,rhs) = Thm.dest_equals (cprop_of abs_B')
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                 in Thm.transitive abs_B' (Conv.arg_conv abstract rhs) end
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            else makeK()
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        | _ => raise Fail "abstract: Bad term"
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  end;
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(* Traverse a theorem, remplacing lambda-abstractions with combinators. *)
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fun introduce_combinators_in_cterm ct =
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  if is_quasi_lambda_free (term_of ct) then
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    Thm.reflexive ct
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  else case term_of ct of
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    Abs _ =>
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    let
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      val (cv, cta) = Thm.dest_abs NONE ct
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      val (v, _) = dest_Free (term_of cv)
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      val u_th = introduce_combinators_in_cterm cta
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      val cu = Thm.rhs_of u_th
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      val comb_eq = abstract (Thm.cabs cv cu)
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    in Thm.transitive (Thm.abstract_rule v cv u_th) comb_eq end
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  | _ $ _ =>
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    let val (ct1, ct2) = Thm.dest_comb ct in
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        Thm.combination (introduce_combinators_in_cterm ct1)
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                        (introduce_combinators_in_cterm ct2)
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    end
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fun introduce_combinators_in_theorem th =
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  if is_quasi_lambda_free (prop_of th) then
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    th
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  else
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    let
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      val th = Drule.eta_contraction_rule th
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      val eqth = introduce_combinators_in_cterm (cprop_of th)
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    in Thm.equal_elim eqth th end
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    handle THM (msg, _, _) =>
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           (warning ("Error in the combinator translation of " ^
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                     Display.string_of_thm_without_context th ^
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                     "\nException message: " ^ msg ^ ".");
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            (* A type variable of sort "{}" will make abstraction fail. *)
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            TrueI)
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(*cterms are used throughout for efficiency*)
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val cTrueprop = cterm_of @{theory HOL} HOLogic.Trueprop;
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(*Given an abstraction over n variables, replace the bound variables by free
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  ones. Return the body, along with the list of free variables.*)
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fun c_variant_abs_multi (ct0, vars) =
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      let val (cv,ct) = Thm.dest_abs NONE ct0
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      in  c_variant_abs_multi (ct, cv::vars)  end
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      handle CTERM _ => (ct0, rev vars);
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val skolem_def_raw = @{thms skolem_def_raw}
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(* Given the definition of a Skolem function, return a theorem to replace
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   an existential formula by a use of that function.
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   Example: "EX x. x : A & x ~: B ==> sko A B : A & sko A B ~: B"  [.] *)
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fun old_skolem_theorem_from_def thy rhs0 =
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  let
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    val rhs = rhs0 |> Type.legacy_freeze_thaw |> #1 |> cterm_of thy
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    val rhs' = rhs |> Thm.dest_comb |> snd
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    val (ch, frees) = c_variant_abs_multi (rhs', [])
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    val (hilbert, cabs) = ch |> Thm.dest_comb |>> term_of
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    val T =
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      case hilbert of
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        Const (_, Type (@{type_name fun}, [_, T])) => T
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      | _ => raise TERM ("old_skolem_theorem_from_def: expected \"Eps\"",
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                         [hilbert])
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    val cex = cterm_of thy (HOLogic.exists_const T)
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    val ex_tm = Thm.capply cTrueprop (Thm.capply cex cabs)
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    val conc =
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      Drule.list_comb (rhs, frees)
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      |> Drule.beta_conv cabs |> Thm.capply cTrueprop
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    fun tacf [prem] =
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      rewrite_goals_tac skolem_def_raw
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      THEN rtac ((prem |> rewrite_rule skolem_def_raw)
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                 RS Global_Theory.get_thm thy "Hilbert_Choice.someI_ex") 1
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  in
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    Goal.prove_internal [ex_tm] conc tacf
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    |> forall_intr_list frees
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    |> Thm.forall_elim_vars 0  (*Introduce Vars, but don't discharge defs.*)
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    |> Thm.varifyT_global
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  end
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fun to_definitional_cnf_with_quantifiers thy th =
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  let
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    val eqth = cnf.make_cnfx_thm thy (HOLogic.dest_Trueprop (prop_of th))
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    val eqth = eqth RS @{thm eq_reflection}
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    val eqth = eqth RS @{thm TruepropI}
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  in Thm.equal_elim eqth th end
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fun zapped_var_name ((ax_no, cluster_no), skolem) index_no s =
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  (if skolem then new_skolem_var_prefix else new_nonskolem_var_prefix) ^
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  "_" ^ string_of_int ax_no ^ "_" ^ string_of_int cluster_no ^ "_" ^
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  string_of_int index_no ^ "_" ^ Name.desymbolize false s
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fun cluster_of_zapped_var_name s =
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  let val get_int = the o Int.fromString o nth (space_explode "_" s) in
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    ((get_int 1, (get_int 2, get_int 3)),
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     String.isPrefix new_skolem_var_prefix s)
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  end
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fun rename_bound_vars_to_be_zapped ax_no =
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  let
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    fun aux (cluster as (cluster_no, cluster_skolem)) index_no pos t =
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      case t of
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        (t1 as Const (s, _)) $ Abs (s', T, t') =>
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        if s = @{const_name all} orelse s = @{const_name All} orelse
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           s = @{const_name Ex} then
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          let
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            val skolem = (pos = (s = @{const_name Ex}))
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            val (cluster, index_no) =
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              if skolem = cluster_skolem then (cluster, index_no)
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              else ((cluster_no ||> cluster_skolem ? Integer.add 1, skolem), 0)
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            val s' = zapped_var_name cluster index_no s'
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          in t1 $ Abs (s', T, aux cluster (index_no + 1) pos t') end
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        else
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          t
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      | (t1 as Const (s, _)) $ t2 $ t3 =>
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        if s = @{const_name "==>"} orelse s = @{const_name HOL.implies} then
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          t1 $ aux cluster index_no (not pos) t2 $ aux cluster index_no pos t3
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        else if s = @{const_name HOL.conj} orelse
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                s = @{const_name HOL.disj} then
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          t1 $ aux cluster index_no pos t2 $ aux cluster index_no pos t3
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        else
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          t
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      | (t1 as Const (s, _)) $ t2 =>
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        if s = @{const_name Trueprop} then
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          t1 $ aux cluster index_no pos t2
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        else if s = @{const_name Not} then
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          t1 $ aux cluster index_no (not pos) t2
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        else
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          t
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      | _ => t
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  in aux ((ax_no, 0), true) 0 true end
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fun zap pos ct =
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  ct
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  |> (case term_of ct of
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        Const (s, _) $ Abs (s', _, _) =>
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        if s = @{const_name all} orelse s = @{const_name All} orelse
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           s = @{const_name Ex} then
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          Thm.dest_comb #> snd #> Thm.dest_abs (SOME s') #> snd #> zap pos
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        else
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          Conv.all_conv
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      | Const (s, _) $ _ $ _ =>
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        if s = @{const_name "==>"} orelse s = @{const_name implies} then
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          Conv.combination_conv (Conv.arg_conv (zap (not pos))) (zap pos)
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        else if s = @{const_name conj} orelse s = @{const_name disj} then
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          Conv.combination_conv (Conv.arg_conv (zap pos)) (zap pos)
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        else
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          Conv.all_conv
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      | Const (s, _) $ _ =>
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        if s = @{const_name Trueprop} then Conv.arg_conv (zap pos)
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        else if s = @{const_name Not} then Conv.arg_conv (zap (not pos))
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        else Conv.all_conv
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      | _ => Conv.all_conv)
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fun ss_only ths = Simplifier.clear_ss HOL_basic_ss addsimps ths
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val cheat_choice =
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  @{prop "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)"}
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  |> Logic.varify_global
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  |> Skip_Proof.make_thm @{theory}
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(* Converts an Isabelle theorem into NNF. *)
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fun nnf_axiom choice_ths new_skolemizer ax_no th ctxt =
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  let
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    val thy = ProofContext.theory_of ctxt
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    val th =
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      th |> transform_elim_theorem
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         |> zero_var_indexes
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         |> new_skolemizer ? forall_intr_vars
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    val (th, ctxt) = Variable.import true [th] ctxt |>> snd |>> the_single
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    val th = th |> Conv.fconv_rule Object_Logic.atomize
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                |> extensionalize_theorem
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                |> make_nnf ctxt
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  in
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    if new_skolemizer then
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      let
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        fun skolemize choice_ths =
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          skolemize_with_choice_theorems ctxt choice_ths
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          #> simplify (ss_only @{thms all_simps[symmetric]})
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        val pull_out =
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          simplify (ss_only @{thms all_simps[symmetric] ex_simps[symmetric]})
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        val no_choice = null choice_ths
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        val discharger_th =
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          th |> (if no_choice then pull_out else skolemize choice_ths)
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        val fully_skolemized_t =
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          discharger_th |> prop_of |> rename_bound_vars_to_be_zapped ax_no
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          |> (if no_choice then
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                Skip_Proof.make_thm thy #> skolemize [cheat_choice] #> cprop_of
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              else
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                cterm_of thy)
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          |> zap true |> Drule.export_without_context
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          |> cprop_of |> Thm.dest_equals |> snd |> term_of
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      in
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        if exists_subterm (fn Var ((s, _), _) =>
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                              String.isPrefix new_skolem_var_prefix s
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                            | _ => false) fully_skolemized_t then
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          let
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            val (fully_skolemized_ct, ctxt) =
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              Variable.import_terms true [fully_skolemized_t] ctxt
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              |>> the_single |>> cterm_of thy
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          in
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            (SOME (discharger_th, fully_skolemized_ct),
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             (Thm.assume fully_skolemized_ct, ctxt))
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          end
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       else
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         (NONE, (th, ctxt))
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      end
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    else
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      (NONE, (th, ctxt))
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  end
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(* Convert a theorem to CNF, with additional premises due to skolemization. *)
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fun cnf_axiom ctxt0 new_skolemizer ax_no th =
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  let
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    val thy = ProofContext.theory_of ctxt0
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    val choice_ths = choice_theorems thy
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    val (opt, (nnf_th, ctxt)) =
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      nnf_axiom choice_ths new_skolemizer ax_no th ctxt0
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    fun clausify th =
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      make_cnf (if new_skolemizer orelse null choice_ths then
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                  []
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                else
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                  map (old_skolem_theorem_from_def thy)
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                      (old_skolem_defs th)) th ctxt
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    val (cnf_ths, ctxt) =
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      clausify nnf_th
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      |> (fn ([], _) =>
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             (* FIXME: This probably doesn't work with the new Skolemizer *)
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             clausify (to_definitional_cnf_with_quantifiers thy nnf_th)
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           | p => p)
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    fun intr_imp ct th =
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      Thm.instantiate ([], map (pairself (cterm_of thy))
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                               [(Var (("i", 0), @{typ nat}),
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                                 HOLogic.mk_nat ax_no)])
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                      (zero_var_indexes @{thm skolem_COMBK_D})
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      RS Thm.implies_intr ct th
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  in
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    (opt |> Option.map (I #>> singleton (Variable.export ctxt ctxt0)
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                        ##> (term_of #> HOLogic.dest_Trueprop
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                             #> singleton (Variable.export_terms ctxt ctxt0))),
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     cnf_ths |> map (introduce_combinators_in_theorem
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                     #> (case opt of SOME (_, ct) => intr_imp ct | NONE => I))
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             |> Variable.export ctxt ctxt0
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             |> finish_cnf
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             |> map Thm.close_derivation)
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  end
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  handle THM _ => (NONE, [])
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b1483d423512 export just one setup function;
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d689ad772b2c skolem_cache_thm: Drule.close_derivation on clauses preserves some space;
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end;