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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 *)
(************************************************************************)
Require Import Orders NPeano.
(** The Definition of maximum and minimum of two natural numbers
is now in NPeano, as well as generic properties. *)
(** * Properties specific to the [nat] domain *)
(** Simplifications *)
Lemma max_0_l : forall n, max 0 n = n.
Proof. reflexivity. Qed.
Lemma max_0_r : forall n, max n 0 = n.
Proof. destruct n; auto. Qed.
Lemma min_0_l : forall n, min 0 n = 0.
Proof. reflexivity. Qed.
Lemma min_0_r : forall n, min n 0 = 0.
Proof. destruct n; auto. Qed.
(** Compatibilities (consequences of monotonicity) *)
Lemma succ_max_distr : forall n m, S (max n m) = max (S n) (S m).
Proof. auto. Qed.
Lemma succ_min_distr : forall n m, S (min n m) = min (S n) (S m).
Proof. auto. Qed.
Lemma plus_max_distr_l : forall n m p, max (p + n) (p + m) = p + max n m.
Proof.
intros. apply Nat.max_monotone. repeat red; auto with arith.
Qed.
Lemma plus_max_distr_r : forall n m p, max (n + p) (m + p) = max n m + p.
Proof.
intros. apply Nat.max_monotone with (f:=fun x => x + p).
repeat red; auto with arith.
Qed.
Lemma plus_min_distr_l : forall n m p, min (p + n) (p + m) = p + min n m.
Proof.
intros. apply Nat.min_monotone. repeat red; auto with arith.
Qed.
Lemma plus_min_distr_r : forall n m p, min (n + p) (m + p) = min n m + p.
Proof.
intros. apply Nat.min_monotone with (f:=fun x => x + p).
repeat red; auto with arith.
Qed.
Hint Resolve
Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r
Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62.
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