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
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Export QArith.
Require Export Qreduction.
Hint Resolve Qlt_le_weak : qarith.
Definition Qabs (x:Q) := let (n,d):=x in (Z.abs n#d).
Lemma Qabs_case : forall (x:Q) (P : Q -> Type), (0 <= x -> P x) -> (x <= 0 -> P (- x)) -> P (Qabs x).
Proof.
intros x P H1 H2.
destruct x as [[|xn|xn] xd];
[apply H1|apply H1|apply H2];
abstract (compute; discriminate).
Defined.
Add Morphism Qabs with signature Qeq ==> Qeq as Qabs_wd.
intros [xn xd] [yn yd] H.
simpl.
unfold Qeq in *.
simpl in *.
change (' yd)%Z with (Z.abs (' yd)).
change (' xd)%Z with (Z.abs (' xd)).
repeat rewrite <- Z.abs_mul.
congruence.
Qed.
Lemma Qabs_pos : forall x, 0 <= x -> Qabs x == x.
Proof.
intros x H.
apply Qabs_case.
reflexivity.
intros H0.
setoid_replace x with 0.
reflexivity.
apply Qle_antisym; assumption.
Qed.
Lemma Qabs_neg : forall x, x <= 0 -> Qabs x == - x.
Proof.
intros x H.
apply Qabs_case.
intros H0.
setoid_replace x with 0.
reflexivity.
apply Qle_antisym; assumption.
reflexivity.
Qed.
Lemma Qabs_nonneg : forall x, 0 <= (Qabs x).
intros x.
apply Qabs_case.
auto.
apply (Qopp_le_compat x 0).
Qed.
Lemma Zabs_Qabs : forall n d, (Z.abs n#d)==Qabs (n#d).
Proof.
intros [|n|n]; reflexivity.
Qed.
Lemma Qabs_opp : forall x, Qabs (-x) == Qabs x.
Proof.
intros x.
do 2 apply Qabs_case; try (intros; ring);
(intros H0 H1;
setoid_replace x with 0;[reflexivity|];
apply Qle_antisym);try assumption;
rewrite Qle_minus_iff in *;
ring_simplify;
ring_simplify in H1;
assumption.
Qed.
Lemma Qabs_triangle : forall x y, Qabs (x+y) <= Qabs x + Qabs y.
Proof.
intros [xn xd] [yn yd].
unfold Qplus.
unfold Qle.
simpl.
apply Z.mul_le_mono_nonneg_r;auto with *.
change (' yd)%Z with (Z.abs (' yd)).
change (' xd)%Z with (Z.abs (' xd)).
repeat rewrite <- Z.abs_mul.
apply Z.abs_triangle.
Qed.
Lemma Qabs_Qmult : forall a b, Qabs (a*b) == (Qabs a)*(Qabs b).
Proof.
intros [an ad] [bn bd].
simpl.
rewrite Z.abs_mul.
reflexivity.
Qed.
Lemma Qabs_Qminus x y: Qabs (x - y) = Qabs (y - x).
Proof.
unfold Qminus, Qopp. simpl.
rewrite Pos.mul_comm, <- Z.abs_opp.
do 2 f_equal. ring.
Qed.
Lemma Qle_Qabs : forall a, a <= Qabs a.
Proof.
intros a.
apply Qabs_case; auto with *.
intros H.
apply Qle_trans with 0; try assumption.
change 0 with (-0).
apply Qopp_le_compat.
assumption.
Qed.
Lemma Qabs_triangle_reverse : forall x y, Qabs x - Qabs y <= Qabs (x - y).
Proof.
intros x y.
rewrite Qle_minus_iff.
setoid_replace (Qabs (x - y) + - (Qabs x - Qabs y)) with ((Qabs (x - y) + Qabs y) + - Qabs x) by ring.
rewrite <- Qle_minus_iff.
setoid_replace (Qabs x) with (Qabs (x-y+y)).
apply Qabs_triangle.
apply Qabs_wd.
ring.
Qed.
Lemma Qabs_Qle_condition x y: Qabs x <= y <-> -y <= x <= y.
Proof.
split.
split.
rewrite <- (Qopp_opp x).
apply Qopp_le_compat.
apply Qle_trans with (Qabs (-x)).
apply Qle_Qabs.
now rewrite Qabs_opp.
apply Qle_trans with (Qabs x); auto using Qle_Qabs.
intros (H,H').
apply Qabs_case; trivial.
intros. rewrite <- (Qopp_opp y). now apply Qopp_le_compat.
Qed.
Lemma Qabs_diff_Qle_condition x y r: Qabs (x - y) <= r <-> x - r <= y <= x + r.
Proof.
intros. unfold Qminus.
rewrite Qabs_Qle_condition.
rewrite <- (Qplus_le_l (-r) (x+-y) (y+r)).
rewrite <- (Qplus_le_l (x+-y) r (y-r)).
setoid_replace (-r + (y + r)) with y by ring.
setoid_replace (r + (y - r)) with y by ring.
setoid_replace (x + - y + (y + r)) with (x + r) by ring.
setoid_replace (x + - y + (y - r)) with (x - r) by ring.
intuition.
Qed.
|
