diff options
Diffstat (limited to 'theories/ZArith/Zwf.v')
| -rw-r--r-- | theories/ZArith/Zwf.v | 27 |
1 files changed, 12 insertions, 15 deletions
diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v index 30802f82..e07fc715 100644 --- a/theories/ZArith/Zwf.v +++ b/theories/ZArith/Zwf.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -9,7 +9,7 @@ Require Import ZArith_base. Require Export Wf_nat. Require Import Omega. -Open Local Scope Z_scope. +Local Open Scope Z_scope. (** Well-founded relations on Z. *) @@ -29,28 +29,28 @@ Section wf_proof. (** The proof of well-foundness is classic: we do the proof by induction on a measure in nat, which is here [|x-c|] *) - Let f (z:Z) := Zabs_nat (z - c). + Let f (z:Z) := Z.abs_nat (z - c). Lemma Zwf_well_founded : well_founded (Zwf c). - red in |- *; intros. + red; intros. assert (forall (n:nat) (a:Z), (f a < n)%nat \/ a < c -> Acc (Zwf c) a). clear a; simple induction n; intros. (** n= 0 *) case H; intros. case (lt_n_O (f a)); auto. - apply Acc_intro; unfold Zwf in |- *; intros. + apply Acc_intro; unfold Zwf; intros. assert False; omega || contradiction. (** inductive case *) case H0; clear H0; intro; auto. apply Acc_intro; intros. apply H. unfold Zwf in H1. - case (Zle_or_lt c y); intro; auto with zarith. + case (Z.le_gt_cases c y); intro; auto with zarith. left. red in H0. apply lt_le_trans with (f a); auto with arith. - unfold f in |- *. - apply Zabs.Zabs_nat_lt; omega. + unfold f. + apply Zabs2Nat.inj_lt; omega. apply (H (S (f a))); auto. Qed. @@ -75,18 +75,15 @@ Section wf_proof_up. (** The proof of well-foundness is classic: we do the proof by induction on a measure in nat, which is here [|c-x|] *) - Let f (z:Z) := Zabs_nat (c - z). + Let f (z:Z) := Z.abs_nat (c - z). Lemma Zwf_up_well_founded : well_founded (Zwf_up c). Proof. apply well_founded_lt_compat with (f := f). - unfold Zwf_up, f in |- *. + unfold Zwf_up, f. intros. - apply Zabs.Zabs_nat_lt. - unfold Zminus in |- *. split. - apply Zle_left; intuition. - apply Zplus_lt_compat_l; unfold Zlt in |- *; rewrite <- Zcompare_opp; - intuition. + apply Zabs2Nat.inj_lt; try (apply Z.le_0_sub; intuition). + now apply Z.sub_lt_mono_l. Qed. End wf_proof_up. |