From 3ef7797ef6fc605dfafb32523261fe1b023aeecb Mon Sep 17 00:00:00 2001
From: Samuel Mimram <smimram@debian.org>
Date: Fri, 28 Apr 2006 14:59:16 +0000
Subject: Imported Upstream version 8.0pl3+8.1alpha

---
 theories/ZArith/Zminmax.v | 82 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 82 insertions(+)
 create mode 100644 theories/ZArith/Zminmax.v

(limited to 'theories/ZArith/Zminmax.v')

diff --git a/theories/ZArith/Zminmax.v b/theories/ZArith/Zminmax.v
new file mode 100644
index 00000000..ebe9318e
--- /dev/null
+++ b/theories/ZArith/Zminmax.v
@@ -0,0 +1,82 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+(*i $Id: Zminmax.v 8034 2006-02-12 22:08:04Z herbelin $ i*)
+
+Require Import Zmin Zmax.
+Require Import BinInt Zorder.
+
+Open Local Scope Z_scope.
+
+(** *** Lattice properties of min and max on Z *)
+
+(** Absorption *)
+
+Lemma Zmin_max_absorption_r_r : forall n m, Zmax n (Zmin n m) = n.
+Proof.
+intros; apply Zmin_case_strong; intro; apply Zmax_case_strong; intro; 
+  reflexivity || apply Zle_antisym; trivial.
+Qed.
+
+Lemma Zmax_min_absorption_r_r : forall n m, Zmin n (Zmax n m) = n.
+Proof.
+intros; apply Zmax_case_strong; intro; apply Zmin_case_strong; intro; 
+  reflexivity || apply Zle_antisym; trivial.
+Qed.
+
+(** Distributivity *)
+
+Lemma Zmax_min_distr_r : 
+  forall n m p, Zmax n (Zmin m p) = Zmin (Zmax n m) (Zmax n p).
+Proof.
+intros.
+repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros; 
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+Lemma Zmin_max_distr_r : 
+  forall n m p, Zmin n (Zmax m p) = Zmax (Zmin n m) (Zmin n p).
+Proof.
+intros.
+repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros; 
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+(** Modularity *)
+
+Lemma Zmax_min_modular_r :
+  forall n m p, Zmax n (Zmin m (Zmax n p)) = Zmin (Zmax n m) (Zmax n p).
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+Lemma Zmin_max_modular_r :
+  forall n m p, Zmin n (Zmax m (Zmin n p)) = Zmax (Zmin n m) (Zmin n p).
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+(** Disassociativity *)
+
+Lemma max_min_disassoc : forall n m p, Zmin n (Zmax m p) <= Zmax (Zmin n m) p.
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  apply Zle_refl || (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+
+
+
+
+
+
-- 
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