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(* 02110-1301 USA                                                     *)
Set Asymmetric Patterns.
(* 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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                Sorting.v                                 *)
(****************************************************************************)
(* These sorting examples have first been compiled by P. Manoury using the  *)
(* ProPre tool to build recursive functions by cases                        *)
(* == Quelques fonctions utiles == *)
(* Inductive Set bool := true : bool | false : bool. *)
Definition si (X : Set) (b : bool) (x y : X) :=
match b with
| true => x
| _ => y
end.
(* Inductive Set nat : O:nat | S:nat->nat. *)
Fixpoint egal_nat (n : nat) : nat -> bool :=
fun m : nat =>
match n, m with
| O, O => true
| S n, S m => egal_nat n m
| _, _ => false
end.
Fixpoint inf_egal (n : nat) : nat -> bool :=
fun m : nat =>
match n, m with
| O, m => true
| S n, O => false
| S n, S m => inf_egal n m
end.

Lemma inf_egal_eq3 : forall n m : nat, inf_egal (S n) (S m) = inf_egal n m.