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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(*                              Le_lt_compl.v                               *)
(*                                                                          *)
(* Author: Pierre Casteran.                                                 *)
(*    LABRI, URA CNRS 1304,                                                 *)
(*    Departement d'Informatique, Universite Bordeaux I,                    *)
(*    33405 Talence CEDEX,                                                  *)
(*    e-mail:  casteran@labri.u-bordeaux.fr                                 *)
(* This file gives some additionnal lemmas to the modules Le and Lt of the distribution: ($COQTH/ARITH/[Le,Lt] )  Please notice that some lemmas and proofs use the constants one,two,three,four declared in the "Constants" module *)

(* Specifications: *)
Lemma lt_or_Zero : forall q : nat, 0 < q \/ q = 0.
(************************************)
Lemma one_cases : forall q : nat, 1 < q \/ q = 0 \/ q = 1.
(************************************************)
Lemma enum1 : forall n : nat, n < two -> n = 0 \/ n = 1.
(******************************************)
Lemma le5 :
forall P : nat -> Prop,
P 0 ->
P one ->
P two ->
P three ->
P four -> (forall n : nat, four < n -> P n) -> forall n : nat, P n.