diff options
Diffstat (limited to 'theories/Arith/Peano_dec.v')
| -rw-r--r-- | theories/Arith/Peano_dec.v | 26 |
1 files changed, 22 insertions, 4 deletions
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. |