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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 Orders NatOrderedType GenericMinMax.
(** * Maximum and Minimum of two natural numbers *)
Fixpoint max n m : nat :=
match n, m with
| O, _ => m
| S n', O => n
| S n', S m' => S (max n' m')
end.
Fixpoint min n m : nat :=
match n, m with
| O, _ => 0
| S n', O => 0
| S n', S m' => S (min n' m')
end.
(** These functions implement indeed a maximum and a minimum *)
Lemma max_l : forall x y, y<=x -> max x y = x.
Proof.
induction x; destruct y; simpl; auto with arith.
Qed.
Lemma max_r : forall x y, x<=y -> max x y = y.
Proof.
induction x; destruct y; simpl; auto with arith.
Qed.
Lemma min_l : forall x y, x<=y -> min x y = x.
Proof.
induction x; destruct y; simpl; auto with arith.
Qed.
Lemma min_r : forall x y, y<=x -> min x y = y.
Proof.
induction x; destruct y; simpl; auto with arith.
Qed.
Module NatHasMinMax <: HasMinMax Nat_as_OT.
Definition max := max.
Definition min := min.
Definition max_l := max_l.
Definition max_r := max_r.
Definition min_l := min_l.
Definition min_r := min_r.
End NatHasMinMax.
(** We obtain hence all the generic properties of [max] and [min],
see file [GenericMinMax] or use SearchAbout. *)
Module Export MMP := UsualMinMaxProperties Nat_as_OT NatHasMinMax.
(** * 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 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 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 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 min_monotone with (f:=fun x => x + p).
repeat red; auto with arith.
Qed.
Hint Resolve
max_l max_r le_max_l le_max_r
min_l min_r le_min_l le_min_r : arith v62.
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