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(************************************************************************)
(* 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 $Id$ 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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