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
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(*i $Id: Compare_dec.v,v 1.13.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
Require Import Le.
Require Import Lt.
Require Import Gt.
Require Import Decidable.
Open Local Scope nat_scope.
Implicit Types m n x y : nat.
Definition zerop : forall n, {n = 0} + {0 < n}.
destruct n; auto with arith.
Defined.
Definition lt_eq_lt_dec : forall n m, {n < m} + {n = m} + {m < n}.
Proof.
induction n; simple destruct m; auto with arith.
intros m0; elim (IHn m0); auto with arith.
induction 1; auto with arith.
Defined.
Lemma gt_eq_gt_dec : forall n m, {m > n} + {n = m} + {n > m}.
Proof lt_eq_lt_dec.
Lemma le_lt_dec : forall n m, {n <= m} + {m < n}.
Proof.
induction n.
auto with arith.
induction m.
auto with arith.
elim (IHn m); auto with arith.
Defined.
Definition le_le_S_dec : forall n m, {n <= m} + {S m <= n}.
Proof.
exact le_lt_dec.
Defined.
Definition le_ge_dec : forall n m, {n <= m} + {n >= m}.
Proof.
intros; elim (le_lt_dec n m); auto with arith.
Defined.
Definition le_gt_dec : forall n m, {n <= m} + {n > m}.
Proof.
exact le_lt_dec.
Defined.
Definition le_lt_eq_dec : forall n m, n <= m -> {n < m} + {n = m}.
Proof.
intros; elim (lt_eq_lt_dec n m); auto with arith.
intros; absurd (m < n); auto with arith.
Defined.
(** Proofs of decidability *)
Theorem dec_le : forall n m, decidable (n <= m).
intros x y; unfold decidable in |- *; elim (le_gt_dec x y);
[ auto with arith | intro; right; apply gt_not_le; assumption ].
Qed.
Theorem dec_lt : forall n m, decidable (n < m).
intros x y; unfold lt in |- *; apply dec_le.
Qed.
Theorem dec_gt : forall n m, decidable (n > m).
intros x y; unfold gt in |- *; apply dec_lt.
Qed.
Theorem dec_ge : forall n m, decidable (n >= m).
intros x y; unfold ge in |- *; apply dec_le.
Qed.
Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.
intros x y H; elim (lt_eq_lt_dec x y);
[ intros H1; elim H1;
[ auto with arith | intros H2; absurd (x = y); assumption ]
| auto with arith ].
Qed.
Theorem not_le : forall n m, ~ n <= m -> n > m.
intros x y H; elim (le_gt_dec x y);
[ intros H1; absurd (x <= y); assumption | trivial with arith ].
Qed.
Theorem not_gt : forall n m, ~ n > m -> n <= m.
intros x y H; elim (le_gt_dec x y);
[ trivial with arith | intros H1; absurd (x > y); assumption ].
Qed.
Theorem not_ge : forall n m, ~ n >= m -> n < m.
intros x y H; exact (not_le y x H).
Qed.
Theorem not_lt : forall n m, ~ n < m -> n >= m.
intros x y H; exact (not_gt y x H).
Qed.
|
