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
|
Require Export NOrder.
Module NTimesOrderPropFunct (Import NAxiomsMod : NAxiomsSig).
Module Export NOrderPropMod := NOrderPropFunct NAxiomsMod.
Open Local Scope NatIntScope.
Theorem plus_lt_mono_l : forall n m p : N, n < m <-> p + n < p + m.
Proof NZplus_lt_mono_l.
Theorem plus_lt_mono_r : forall n m p : N, n < m <-> n + p < m + p.
Proof NZplus_lt_mono_r.
Theorem plus_lt_mono : forall n m p q : N, n < m -> p < q -> n + p < m + q.
Proof NZplus_lt_mono.
Theorem plus_le_mono_l : forall n m p : N, n <= m <-> p + n <= p + m.
Proof NZplus_le_mono_l.
Theorem plus_le_mono_r : forall n m p : N, n <= m <-> n + p <= m + p.
Proof NZplus_le_mono_r.
Theorem plus_le_mono : forall n m p q : N, n <= m -> p <= q -> n + p <= m + q.
Proof NZplus_le_mono.
Theorem plus_lt_le_mono : forall n m p q : N, n < m -> p <= q -> n + p < m + q.
Proof NZplus_lt_le_mono.
Theorem plus_le_lt_mono : forall n m p q : N, n <= m -> p < q -> n + p < m + q.
Proof NZplus_le_lt_mono.
Theorem plus_le_lt_mono_opp : forall n m p q : N, n <= m -> p + m < q + n -> p < q.
Proof NZplus_le_lt_mono_opp.
Theorem plus_lt_inv : forall n m p q : N, n + m < p + q -> n < p \/ m < q.
Proof NZplus_lt_inv.
Theorem plus_gt_inv_0 : forall n m : N, 0 < n + m -> 0 < n \/ 0 < m.
Proof NZplus_gt_inv_0.
(** Theorems true for natural numbers *)
Theorem le_plus_r : forall n m : N, n <= n + m.
Proof.
intro n; induct m.
rewrite plus_0_r; le_equal.
intros m IH. rewrite plus_succ_r; now apply le_le_succ.
Qed.
Theorem lt_plus_r : forall n m : N, m ~= 0 -> n < n + m.
Proof.
intros n m; cases m.
intro H; elimtype False; now apply H.
intros. rewrite plus_succ_r. apply <- lt_succ_le. apply le_plus_r.
Qed.
Theorem lt_lt_plus : forall n m p : N, n < m -> n < m + p.
Proof.
intros n m p H; rewrite <- (plus_0_r n).
apply plus_lt_le_mono; [assumption | apply le_0_l].
Qed.
(* The following property is similar to plus_repl_pair in NPlus.v
and is used to prove the correctness of the definition of order
on integers constructed from pairs of natural numbers *)
Theorem plus_lt_repl_pair : forall n m n' m' u v,
n + u < m + v -> n + m' == n' + m -> n' + u < m' + v.
Proof.
intros n m n' m' u v H1 H2.
symmetry in H2. assert (H3 : n' + m <= n + m'); [now le_equal |].
assert (H4 : n + u + (n' + m) < m + v + (n + m')); [now apply plus_lt_le_mono |].
stepl (n + (m + (n' + u))) in H4 by ring. stepr (n + (m + (m' + v))) in H4 by ring.
now do 2 apply <- plus_lt_mono_l in H4.
Qed.
(** Multiplication and order *)
Theorem times_lt_pred :
forall p q n m : N, S p == q -> (p * n < p * m <-> q * n + m < q * m + n).
Proof NZtimes_lt_pred.
Theorem times_lt_mono_pos_l : forall p n m : N, 0 < p -> (n < m <-> p * n < p * m).
Proof NZtimes_lt_mono_pos_l.
Theorem times_lt_mono_pos_r : forall p n m : N, 0 < p -> (n < m <-> n * p < m * p).
Proof NZtimes_lt_mono_pos_r.
Theorem times_le_mono_l : forall n m p : N, n <= m -> p * n <= p * m.
Proof.
intros; apply NZtimes_le_mono_nonneg_l. apply le_0_l. assumption.
Qed.
Theorem times_le_mono_r : forall n m p : N, n <= m -> n * p <= m * p.
Proof.
intros; apply NZtimes_le_mono_nonneg_r. apply le_0_l. assumption.
Qed.
Theorem times_cancel_l : forall n m p : N, p ~= 0 -> (p * n == p * m <-> n == m).
Proof NZtimes_cancel_l.
Theorem times_le_mono_pos_l : forall n m p : N, 0 < p -> (n <= m <-> p * n <= p * m).
Proof NZtimes_le_mono_pos_l.
Theorem times_le_mono_pos_r : forall n m p : N, 0 < p -> (n <= m <-> n * p <= m * p).
Proof NZtimes_le_mono_pos_r.
Theorem times_lt_mono : forall n m p q : N, n < m -> p < q -> n * p < m * q.
Proof.
intros; apply NZtimes_lt_mono; try assumption; apply le_0_l.
Qed.
Theorem times_le_mono : forall n m p q : N, n <= m -> p <= q -> n * p <= m * q.
Proof.
intros; apply NZtimes_le_mono; try assumption; apply le_0_l.
Qed.
Theorem times_pos_pos : forall n m : N, 0 < n -> 0 < m -> 0 < n * m.
Proof NZtimes_pos_pos.
Theorem times_eq_0 : forall n m : N, n * m == 0 -> n == 0 \/ m == 0.
Proof NZtimes_eq_0.
Theorem times_neq_0 : forall n m : N, n ~= 0 /\ m ~= 0 <-> n * m ~= 0.
Proof NZtimes_neq_0.
End NTimesOrderPropFunct.
(*
Local Variables:
tags-file-name: "~/coq/trunk/theories/Numbers/TAGS"
End:
*)
|
