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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <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        *)
+(************************************************************************)
+
+Require Import NAxioms NSub GenericMinMax.
+
+(** * Properties of minimum and maximum specific to natural numbers *)
+
+Module Type NMaxMinProp (Import N : NAxiomsMiniSig').
+Include NSubProp N.
+
+(** Zero *)
+
+Lemma max_0_l : forall n, max 0 n == n.
+Proof.
+ intros. apply max_r. apply le_0_l.
+Qed.
+
+Lemma max_0_r : forall n, max n 0 == n.
+Proof.
+ intros. apply max_l. apply le_0_l.
+Qed.
+
+Lemma min_0_l : forall n, min 0 n == 0.
+Proof.
+ intros. apply min_l. apply le_0_l.
+Qed.
+
+Lemma min_0_r : forall n, min n 0 == 0.
+Proof.
+ intros. apply min_r. apply le_0_l.
+Qed.
+
+(** The following results are concrete instances of [max_monotone]
+    and similar lemmas. *)
+
+(** Succ *)
+
+Lemma succ_max_distr : forall n m, S (max n m) == max (S n) (S m).
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; now rewrite <- ?succ_le_mono.
+Qed.
+
+Lemma succ_min_distr : forall n m, S (min n m) == min (S n) (S m).
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; now rewrite <- ?succ_le_mono.
+Qed.
+
+(** Add *)
+
+Lemma add_max_distr_l : forall n m p, max (p + n) (p + m) == p + max n m.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; now rewrite <- ?add_le_mono_l.
+Qed.
+
+Lemma add_max_distr_r : forall n m p, max (n + p) (m + p) == max n m + p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; now rewrite <- ?add_le_mono_r.
+Qed.
+
+Lemma add_min_distr_l : forall n m p, min (p + n) (p + m) == p + min n m.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; now rewrite <- ?add_le_mono_l.
+Qed.
+
+Lemma add_min_distr_r : forall n m p, min (n + p) (m + p) == min n m + p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; now rewrite <- ?add_le_mono_r.
+Qed.
+
+(** Mul *)
+
+Lemma mul_max_distr_l : forall n m p, max (p * n) (p * m) == p * max n m.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; try order; now apply mul_le_mono_l.
+Qed.
+
+Lemma mul_max_distr_r : forall n m p, max (n * p) (m * p) == max n m * p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; try order; now apply mul_le_mono_r.
+Qed.
+
+Lemma mul_min_distr_l : forall n m p, min (p * n) (p * m) == p * min n m.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; try order; now apply mul_le_mono_l.
+Qed.
+
+Lemma mul_min_distr_r : forall n m p, min (n * p) (m * p) == min n m * p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; try order; now apply mul_le_mono_r.
+Qed.
+
+(** Sub *)
+
+Lemma sub_max_distr_l : forall n m p, max (p - n) (p - m) == p - min n m.
+Proof.
+ intros. destruct (le_ge_cases n m).
+ rewrite min_l by trivial. apply max_l. now apply sub_le_mono_l.
+ rewrite min_r by trivial. apply max_r. now apply sub_le_mono_l.
+Qed.
+
+Lemma sub_max_distr_r : forall n m p, max (n - p) (m - p) == max n m - p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 max_r | rewrite 2 max_l]; try order; now apply sub_le_mono_r.
+Qed.
+
+Lemma sub_min_distr_l : forall n m p, min (p - n) (p - m) == p - max n m.
+Proof.
+ intros. destruct (le_ge_cases n m).
+ rewrite max_r by trivial. apply min_r. now apply sub_le_mono_l.
+ rewrite max_l by trivial. apply min_l. now apply sub_le_mono_l.
+Qed.
+
+Lemma sub_min_distr_r : forall n m p, min (n - p) (m - p) == min n m - p.
+Proof.
+ intros. destruct (le_ge_cases n m);
+  [rewrite 2 min_l | rewrite 2 min_r]; try order; now apply sub_le_mono_r.
+Qed.
+
+End NMaxMinProp.