From 97fefe1fcca363a1317e066e7f4b99b9c1e9987b Mon Sep 17 00:00:00 2001
From: Stephane Glondu <steph@glondu.net>
Date: Thu, 12 Jan 2012 16:02:20 +0100
Subject: Imported Upstream version 8.4~beta

---
 theories/Arith/Peano_dec.v | 26 ++++++++++++++++++++++----
 1 file changed, 22 insertions(+), 4 deletions(-)

(limited to 'theories/Arith/Peano_dec.v')

diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v
index 5cceab8b..6eb667c1 100644
--- a/theories/Arith/Peano_dec.v
+++ b/theories/Arith/Peano_dec.v
@@ -1,15 +1,14 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-(*i $Id: Peano_dec.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Import Decidable.
-
+Require Eqdep_dec.
+Require Import Le Lt.
 Open Local Scope nat_scope.
 
 Implicit Types m n x y : nat.
@@ -32,3 +31,22 @@ Hint Resolve O_or_S eq_nat_dec: arith.
 Theorem dec_eq_nat : forall n m, decidable (n = m).
   intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith.
 Defined.
+
+Definition UIP_nat:= Eqdep_dec.UIP_dec eq_nat_dec.
+
+Lemma le_unique: forall m n (h1 h2: m <= n), h1 = h2.
+Proof.
+fix 3.
+refine (fun m _ h1 => match h1 as h' in _ <= k return forall hh: m <= k, h' = hh
+  with le_n => _ |le_S i H => _ end).
+refine (fun hh => match hh as h' in _ <= k return forall eq: m = k, 
+  le_n m = match eq in _ = p return m <= p -> m <= m with |eq_refl => fun bli => bli end h' with
+    |le_n => fun eq => _ |le_S j H' => fun eq => _ end eq_refl).
+rewrite (UIP_nat _ _ eq eq_refl). reflexivity.
+subst m. destruct (Lt.lt_irrefl j H').
+refine (fun hh => match hh as h' in _ <= k return match k as k' return m <= k' -> Prop
+  with |0 => fun _ => True |S i' => fun h'' => forall H':m <= i', le_S m i' H' = h'' end h'
+    with |le_n => _ |le_S j H2 => fun H' => _ end H).
+destruct m. exact I. intros; destruct (Lt.lt_irrefl m H').
+f_equal. apply le_unique.
+Qed.
-- 
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