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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 BinPos Pnat POrderedType GenericMinMax.
(** * Maximum and Minimum of two positive numbers *)
Local Open Scope positive_scope.
(** The functions [Pmax] and [Pmin] implement indeed
a maximum and a minimum *)
Lemma Pmax_l : forall x y, y<=x -> Pmax x y = x.
Proof.
unfold Ple, Pmax. intros x y.
rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y).
destruct ((x ?= y) Eq); intuition.
Qed.
Lemma Pmax_r : forall x y, x<=y -> Pmax x y = y.
Proof.
unfold Ple, Pmax. intros x y. destruct ((x ?= y) Eq); intuition.
Qed.
Lemma Pmin_l : forall x y, x<=y -> Pmin x y = x.
Proof.
unfold Ple, Pmin. intros x y. destruct ((x ?= y) Eq); intuition.
Qed.
Lemma Pmin_r : forall x y, y<=x -> Pmin x y = y.
Proof.
unfold Ple, Pmin. intros x y.
rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y).
destruct ((x ?= y) Eq); intuition.
Qed.
Module PositiveHasMinMax <: HasMinMax Positive_as_OT.
Definition max := Pmax.
Definition min := Pmin.
Definition max_l := Pmax_l.
Definition max_r := Pmax_r.
Definition min_l := Pmin_l.
Definition min_r := Pmin_r.
End PositiveHasMinMax.
Module P.
(** We obtain hence all the generic properties of max and min. *)
Include UsualMinMaxProperties Positive_as_OT PositiveHasMinMax.
(** * Properties specific to the [positive] domain *)
(** Simplifications *)
Lemma max_1_l : forall n, Pmax 1 n = n.
Proof.
intros. unfold Pmax. rewrite ZC4. generalize (Pcompare_1 n).
destruct (n ?= 1); intuition.
Qed.
Lemma max_1_r : forall n, Pmax n 1 = n.
Proof. intros. rewrite P.max_comm. apply max_1_l. Qed.
Lemma min_1_l : forall n, Pmin 1 n = 1.
Proof.
intros. unfold Pmin. rewrite ZC4. generalize (Pcompare_1 n).
destruct (n ?= 1); intuition.
Qed.
Lemma min_1_r : forall n, Pmin n 1 = 1.
Proof. intros. rewrite P.min_comm. apply min_1_l. Qed.
(** Compatibilities (consequences of monotonicity) *)
Lemma succ_max_distr :
forall n m, Psucc (Pmax n m) = Pmax (Psucc n) (Psucc m).
Proof.
intros. symmetry. apply max_monotone.
intros x x'. unfold Ple.
rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism.
simpl; auto.
Qed.
Lemma succ_min_distr : forall n m, Psucc (Pmin n m) = Pmin (Psucc n) (Psucc m).
Proof.
intros. symmetry. apply min_monotone.
intros x x'. unfold Ple.
rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism.
simpl; auto.
Qed.
Lemma plus_max_distr_l : forall n m p, Pmax (p + n) (p + m) = p + Pmax n m.
Proof.
intros. apply max_monotone.
intros x x'. unfold Ple.
rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism.
rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
Qed.
Lemma plus_max_distr_r : forall n m p, Pmax (n + p) (m + p) = Pmax n m + p.
Proof.
intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p).
apply plus_max_distr_l.
Qed.
Lemma plus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m.
Proof.
intros. apply min_monotone.
intros x x'. unfold Ple.
rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism.
rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
Qed.
Lemma plus_min_distr_r : forall n m p, Pmin (n + p) (m + p) = Pmin n m + p.
Proof.
intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p).
apply plus_min_distr_l.
Qed.
End P.
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