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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                              Ensf_couple.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*									*)
(*  first et second renvoient repsectivement le premier et le deuxieme 	*)
(*  element d'un couple.						*)
(*									*)
Definition first (x : Elt) : Elt :=
match x return Elt with
| natural n =>
(* natural *)  zero (* couple  *)
| couple a b => a
(* up      *)
| up e => zero
(* word    *)
| word w => zero
end.
Definition second (x : Elt) : Elt :=
match x return Elt with
| natural n =>
(* natural *)  zero (* couple  *)
| couple a b => b
(* up      *)
| up e => zero
(* word    *)
| word w => zero
end.
(* Grace a first et second on recupere facilement le lemme suivant : 	*)
intros x y z t H.
injection H; auto.
Qed.
intros a b c d H.
injection H; auto.
Qed.

Lemma couple_couple_inv2 :
forall a b c d : Elt, couple a c = couple b d :>Elt -> c = d :>Elt.