1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Orders BinNat Nnat NOrderedType GenericMinMax.
(** * Maximum and Minimum of two [N] numbers *)
Local Open Scope N_scope.
(** The functions [Nmax] and [Nmin] implement indeed
a maximum and a minimum *)
Lemma Nmax_l : forall x y, y<=x -> Nmax x y = x.
Proof.
unfold Nle, Nmax. intros x y.
generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
destruct (x ?= y); intuition.
Qed.
Lemma Nmax_r : forall x y, x<=y -> Nmax x y = y.
Proof.
unfold Nle, Nmax. intros x y. destruct (x ?= y); intuition.
Qed.
Lemma Nmin_l : forall x y, x<=y -> Nmin x y = x.
Proof.
unfold Nle, Nmin. intros x y. destruct (x ?= y); intuition.
Qed.
Lemma Nmin_r : forall x y, y<=x -> Nmin x y = y.
Proof.
unfold Nle, Nmin. intros x y.
generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
destruct (x ?= y); intuition.
Qed.
Module NHasMinMax <: HasMinMax N_as_OT.
Definition max := Nmax.
Definition min := Nmin.
Definition max_l := Nmax_l.
Definition max_r := Nmax_r.
Definition min_l := Nmin_l.
Definition min_r := Nmin_r.
End NHasMinMax.
Module N.
(** We obtain hence all the generic properties of max and min. *)
Include UsualMinMaxProperties N_as_OT NHasMinMax.
(** * Properties specific to the [positive] domain *)
(** Simplifications *)
Lemma max_0_l : forall n, Nmax 0 n = n.
Proof.
intros. unfold Nmax. rewrite <- Ncompare_antisym. generalize (Ncompare_0 n).
destruct (n ?= 0); intuition.
Qed.
Lemma max_0_r : forall n, Nmax n 0 = n.
Proof. intros. rewrite N.max_comm. apply max_0_l. Qed.
Lemma min_0_l : forall n, Nmin 0 n = 0.
Proof.
intros. unfold Nmin. rewrite <- Ncompare_antisym. generalize (Ncompare_0 n).
destruct (n ?= 0); intuition.
Qed.
Lemma min_0_r : forall n, Nmin n 0 = 0.
Proof. intros. rewrite N.min_comm. apply min_0_l. Qed.
(** Compatibilities (consequences of monotonicity) *)
Lemma succ_max_distr :
forall n m, Nsucc (Nmax n m) = Nmax (Nsucc n) (Nsucc m).
Proof.
intros. symmetry. apply max_monotone.
intros x x'. unfold Nle.
rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
simpl; auto.
Qed.
Lemma succ_min_distr : forall n m, Nsucc (Nmin n m) = Nmin (Nsucc n) (Nsucc m).
Proof.
intros. symmetry. apply min_monotone.
intros x x'. unfold Nle.
rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
simpl; auto.
Qed.
Lemma plus_max_distr_l : forall n m p, Nmax (p + n) (p + m) = p + Nmax n m.
Proof.
intros. apply max_monotone.
intros x x'. unfold Nle.
rewrite 2 nat_of_Ncompare, 2 nat_of_Nplus.
rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
Qed.
Lemma plus_max_distr_r : forall n m p, Nmax (n + p) (m + p) = Nmax n m + p.
Proof.
intros. rewrite (Nplus_comm n p), (Nplus_comm m p), (Nplus_comm _ p).
apply plus_max_distr_l.
Qed.
Lemma plus_min_distr_l : forall n m p, Nmin (p + n) (p + m) = p + Nmin n m.
Proof.
intros. apply min_monotone.
intros x x'. unfold Nle.
rewrite 2 nat_of_Ncompare, 2 nat_of_Nplus.
rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
Qed.
Lemma plus_min_distr_r : forall n m p, Nmin (n + p) (m + p) = Nmin n m + p.
Proof.
intros. rewrite (Nplus_comm n p), (Nplus_comm m p), (Nplus_comm _ p).
apply plus_min_distr_l.
Qed.
End N.
|
