From 38dac30c6877122634e7b34ec7cd1b6ab2b67ebb Mon Sep 17 00:00:00 2001
From: letouzey
Date: Tue, 9 Feb 2010 17:44:50 +0000
Subject: NSub: a missing lemma (sub usually decreases)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12716 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Numbers/Natural/Abstract/NSub.v | 9 +++++++++
 1 file changed, 9 insertions(+)

diff --git a/theories/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v
index 35d3b8aa22..de0c2da1b0 100644
--- a/theories/Numbers/Natural/Abstract/NSub.v
+++ b/theories/Numbers/Natural/Abstract/NSub.v
@@ -205,6 +205,15 @@ Proof.
 intros n m p. rewrite add_comm; apply lt_add_lt_sub_r.
 Qed.
 
+Theorem sub_lt : forall n m, m <= n -> 0 < m -> n - m < n.
+Proof.
+intros n m LE LT.
+assert (LE' := le_sub_l n m). rewrite lt_eq_cases in LE'.
+destruct LE' as [LT'|EQ]. assumption.
+apply add_sub_eq_nz in EQ; [|order].
+rewrite (add_lt_mono_r _ _ n), add_0_l in LT. order.
+Qed.
+
 (** Sub and mul *)
 
 Theorem mul_pred_r : forall n m, n * (P m) == n * m - n.
-- 
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