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
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Sumbool.
Require Import BinInt.
Require Import Zorder.
Require Import Zcompare.
Local Open Scope Z_scope.
(* begin hide *)
(* Trivial, to deprecate? *)
Lemma Dcompare_inf : forall r:comparison, {r = Eq} + {r = Lt} + {r = Gt}.
Proof.
induction r; auto.
Defined.
(* end hide *)
Lemma Zcompare_rect (P:Type) (n m:Z) :
((n ?= m) = Eq -> P) -> ((n ?= m) = Lt -> P) -> ((n ?= m) = Gt -> P) -> P.
Proof.
intros H1 H2 H3.
destruct (n ?= m); auto.
Defined.
Lemma Zcompare_rec (P:Set) (n m:Z) :
((n ?= m) = Eq -> P) -> ((n ?= m) = Lt -> P) -> ((n ?= m) = Gt -> P) -> P.
Proof. apply Zcompare_rect. Defined.
Notation Z_eq_dec := Z.eq_dec (compat "8.3").
Section decidability.
Variables x y : Z.
(** * Decidability of order on binary integers *)
Definition Z_lt_dec : {x < y} + {~ x < y}.
Proof.
unfold Z.lt; case Z.compare; (now left) || (now right).
Defined.
Definition Z_le_dec : {x <= y} + {~ x <= y}.
Proof.
unfold Z.le; case Z.compare; (now left) || (right; tauto).
Defined.
Definition Z_gt_dec : {x > y} + {~ x > y}.
Proof.
unfold Z.gt; case Z.compare; (now left) || (now right).
Defined.
Definition Z_ge_dec : {x >= y} + {~ x >= y}.
Proof.
unfold Z.ge; case Z.compare; (now left) || (right; tauto).
Defined.
Definition Z_lt_ge_dec : {x < y} + {x >= y}.
Proof.
exact Z_lt_dec.
Defined.
Lemma Z_lt_le_dec : {x < y} + {y <= x}.
Proof.
elim Z_lt_ge_dec.
* now left.
* right; now apply Z.ge_le.
Defined.
Definition Z_le_gt_dec : {x <= y} + {x > y}.
Proof.
elim Z_le_dec; auto with arith.
intro. right. Z.swap_greater. now apply Z.nle_gt.
Defined.
Definition Z_gt_le_dec : {x > y} + {x <= y}.
Proof.
exact Z_gt_dec.
Defined.
Definition Z_ge_lt_dec : {x >= y} + {x < y}.
Proof.
elim Z_ge_dec; auto with arith.
intro. right. Z.swap_greater. now apply Z.lt_nge.
Defined.
Definition Z_le_lt_eq_dec : x <= y -> {x < y} + {x = y}.
Proof.
intro H.
apply Zcompare_rec with (n := x) (m := y).
intro. right. elim (Z.compare_eq_iff x y); auto with arith.
intro. left. elim (Z.compare_eq_iff x y); auto with arith.
intro H1. absurd (x > y); auto with arith.
Defined.
End decidability.
(** * Cotransitivity of order on binary integers *)
Lemma Zlt_cotrans : forall n m:Z, n < m -> forall p:Z, {n < p} + {p < m}.
Proof.
intros x y H z.
case (Z_lt_ge_dec x z).
intro.
left.
assumption.
intro.
right.
apply Z.le_lt_trans with (m := x).
apply Z.ge_le.
assumption.
assumption.
Defined.
Lemma Zlt_cotrans_pos : forall n m:Z, 0 < n + m -> {0 < n} + {0 < m}.
Proof.
intros x y H.
case (Zlt_cotrans 0 (x + y) H x).
- now left.
- right.
apply Z.add_lt_mono_l with (p := x).
now rewrite Z.add_0_r.
Defined.
Lemma Zlt_cotrans_neg : forall n m:Z, n + m < 0 -> {n < 0} + {m < 0}.
Proof.
intros x y H; case (Zlt_cotrans (x + y) 0 H x); intro Hxy;
[ right; apply Z.add_lt_mono_l with (p := x); rewrite Z.add_0_r | left ];
assumption.
Defined.
Lemma not_Zeq_inf : forall n m:Z, n <> m -> {n < m} + {m < n}.
Proof.
intros x y H.
case Z_lt_ge_dec with x y.
intro.
left.
assumption.
intro H0.
generalize (Z.ge_le _ _ H0).
intro.
case (Z_le_lt_eq_dec _ _ H1).
intro.
right.
assumption.
intro.
apply False_rec.
apply H.
symmetry .
assumption.
Defined.
Lemma Z_dec : forall n m:Z, {n < m} + {n > m} + {n = m}.
Proof.
intros x y.
case (Z_lt_ge_dec x y).
intro H.
left.
left.
assumption.
intro H.
generalize (Z.ge_le _ _ H).
intro H0.
case (Z_le_lt_eq_dec y x H0).
intro H1.
left.
right.
apply Z.lt_gt.
assumption.
intro.
right.
symmetry .
assumption.
Defined.
Lemma Z_dec' : forall n m:Z, {n < m} + {m < n} + {n = m}.
Proof.
intros x y.
case (Z.eq_dec x y); intro H;
[ right; assumption | left; apply (not_Zeq_inf _ _ H) ].
Defined.
(* begin hide *)
(* To deprecate ? *)
Corollary Z_zerop : forall x:Z, {x = 0} + {x <> 0}.
Proof.
exact (fun x:Z => Z.eq_dec x 0).
Defined.
Corollary Z_notzerop : forall (x:Z), {x <> 0} + {x = 0}.
Proof (fun x => sumbool_not _ _ (Z_zerop x)).
Corollary Z_noteq_dec : forall (x y:Z), {x <> y} + {x = y}.
Proof (fun x y => sumbool_not _ _ (Z.eq_dec x y)).
(* end hide *)
|
