(*#* #stop file *)

Require Export terms_defs.

      Inductive Set C := CSort: nat -> C
                      |  CTail: C -> K -> T -> C.

      Hint f3CKT : ltlc := Resolve (f_equal3 C K T).

      Tactic Definition CGenBase :=
         Match Context With
            | [ H: (CSort ?) = (CSort ?) |- ? ]         -> Inversion H; Clear H
            | [ H: (CTail ? ? ?) = (CTail ? ? ?) |- ? ] -> Inversion H; Clear H
	    | _                                         -> TGenBase.

      Definition r: K -> nat -> nat := [k;i] Cases k of
         | (Bind _) => i
         | (Flat _) => (S i)
      end.

      Fixpoint app [c:C] : nat -> T -> T := [j;t]Cases j c of
         | (0)   _             => t
         | _     (CSort _)     => t
         | (S i) (CTail c k u) => (app c (r k i) (TTail k u t))
      end.

   Section r_props. (********************************************************)

      Theorem r_S: (k:?; i:?) (r k (S i)) = (S (r k i)).
      XElim k; XAuto.
      Qed.

      Theorem r_plus_sym: (k:?; i,j:?) (r k (plus i j)) = (plus i (r k j)).
      XElim k; Intros; Simpl; XAuto.
      Qed.

      Theorem r_minus: (i,n:?) (lt n i) ->
                       (k:?) (minus (r k i) (S n)) = (r k (minus i (S n))).
      XElim k; Intros; Simpl; XEAuto.
      Qed.

      Theorem r_dis: (k:?; P:Prop)
                     (((i:?) (r k i) = i) -> P) ->
                     (((i:?) (r k i) = (S i)) -> P) -> P.
      XElim k; XAuto.
      Qed.

   End r_props.

      Tactic Definition RRw :=
         Repeat (Rewrite r_S Orelse Rewrite r_plus_sym).

   Section r_arith. (********************************************************)

      Theorem r_arith0: (k:?; i:?) (minus (r k (S i)) (1)) = (r k i).
      Intros; RRw; Rewrite minus_Sx_SO; XAuto.
      Qed.

      Theorem r_arith1: (k:?; i,j:?) (minus (r k (S i)) (S j)) = (minus (r k i) j).
      Intros; RRw; XAuto.
      Qed.

   End r_arith.

   Section app_props. (******************************************************)

      Theorem app_csort: (t:?; i,n:?) (app (CSort n) i t) = t.
      XElim i; Intros; Simpl; XAuto.
      Qed.

      Theorem app_O: (c:?; t:?) (app c (0) t) = t.
      XElim c; XAuto.
      Qed.

   End app_props.

      Hints Resolve app_csort app_O : ltlc.