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
|
(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
(**************************************************************************)
(* Properties of addition *)
(**************************************************************************)
Require Le.
Require Lt.
Lemma plus_sym : (n,m:nat)(plus n m)=(plus m n).
Proof.
Intros n m ; Elim n ; Simpl ; Auto with arith.
Intros y H ; Elim (plus_n_Sm m y) ; Auto with arith.
Qed.
Hints Immediate plus_sym : arith v62.
Lemma plus_Snm_nSm : (n,m:nat)(plus (S n) m)=(plus n (S m)).
Intros.
Simpl.
Rewrite -> (plus_sym n m).
Rewrite -> (plus_sym n (S m)).
Trivial with arith.
Qed.
Lemma simpl_plus_l : (n,m,p:nat)((plus n m)=(plus n p))->(m=p).
Proof.
Induction n ; Simpl ; Auto with arith.
Qed.
Lemma plus_assoc_l : (n,m,p:nat)((plus n (plus m p))=(plus (plus n m) p)).
Proof.
Intros n m p; Elim n; Simpl; Auto with arith.
Qed.
Hints Resolve plus_assoc_l : arith v62.
Lemma plus_permute : (n,m,p:nat) ((plus n (plus m p))=(plus m (plus n p))).
Proof.
Intros; Rewrite (plus_assoc_l m n p); Rewrite (plus_sym m n); Auto with arith.
Qed.
Lemma plus_assoc_r : (n,m,p:nat)((plus (plus n m) p)=(plus n (plus m p))).
Proof.
Auto with arith.
Qed.
Hints Resolve plus_assoc_r : arith v62.
Lemma simpl_le_plus_l : (p,n,m:nat)(le (plus p n) (plus p m))->(le n m).
Proof.
Induction p; Simpl; Auto with arith.
Qed.
Lemma le_reg_l : (n,m,p:nat)(le n m)->(le (plus p n) (plus p m)).
Proof.
Induction p; Simpl; Auto with arith.
Qed.
Hints Resolve le_reg_l : arith v62.
Lemma le_reg_r : (a,b,c:nat) (le a b)->(le (plus a c) (plus b c)).
Proof.
Induction 1 ; Simpl; Auto with arith.
Qed.
Hints Resolve le_reg_r : arith v62.
Lemma le_plus_plus :
(n,m,p,q:nat) (le n m)->(le p q)->(le (plus n p) (plus m q)).
Proof.
Intros n m p q H H0.
Elim H; Simpl; Auto with arith.
Qed.
Lemma le_plus_l : (n,m:nat)(le n (plus n m)).
Proof.
Induction n; Simpl; Auto with arith.
Qed.
Hints Resolve le_plus_l : arith v62.
Lemma le_plus_r : (n,m:nat)(le m (plus n m)).
Proof.
Intros n m; Elim n; Simpl; Auto with arith.
Qed.
Hints Resolve le_plus_r : arith v62.
Theorem le_plus_trans : (n,m,p:nat)(le n m)->(le n (plus m p)).
Proof.
Intros; Apply le_trans with m; Auto with arith.
Qed.
Hints Resolve le_plus_trans : arith v62.
Lemma simpl_lt_plus_l : (n,m,p:nat)(lt (plus p n) (plus p m))->(lt n m).
Proof.
Induction p; Simpl; Auto with arith.
Qed.
Lemma lt_reg_l : (n,m,p:nat)(lt n m)->(lt (plus p n) (plus p m)).
Proof.
Induction p; Simpl; Auto with arith.
Qed.
Hints Resolve lt_reg_l : arith v62.
Lemma lt_reg_r : (n,m,p:nat)(lt n m) -> (lt (plus n p) (plus m p)).
Proof.
Intros n m p H ; Rewrite (plus_sym n p) ; Rewrite (plus_sym m p).
Elim p; Auto with arith.
Qed.
Hints Resolve lt_reg_r : arith v62.
Theorem lt_plus_trans : (n,m,p:nat)(lt n m)->(lt n (plus m p)).
Proof.
Intros; Apply lt_le_trans with m; Auto with arith.
Qed.
Hints Immediate lt_plus_trans : arith v62.
Lemma le_lt_plus_plus : (n,m,p,q:nat) (le n m)->(lt p q)->(lt (plus n p) (plus m q)).
Proof.
Unfold lt. Intros. Change (le (plus (S n) p) (plus m q)). Rewrite plus_Snm_nSm.
Apply le_plus_plus; Assumption.
Qed.
Lemma lt_le_plus_plus : (n,m,p,q:nat) (lt n m)->(le p q)->(lt (plus n p) (plus m q)).
Proof.
Unfold lt. Intros. Change (le (plus (S n) p) (plus m q)). Apply le_plus_plus; Assumption.
Qed.
Lemma lt_plus_plus : (n,m,p,q:nat) (lt n m)->(lt p q)->(lt (plus n p) (plus m q)).
Proof.
Intros. Apply lt_le_plus_plus. Assumption.
Apply lt_le_weak. Assumption.
Qed.
Lemma plus_is_O : (m,n:nat) (plus m n)=O -> m=O /\ n=O.
Proof.
Destruct m; Auto.
Intros. Discriminate H.
Qed.
Lemma plus_is_one :
(m,n:nat) (plus m n)=(S O) -> {m=O /\ n=(S O)}+{m=(S O) /\ n=O}.
Proof.
Destruct m; Auto.
Destruct n; Auto.
Intros.
Simpl in H. Discriminate H.
Qed.
Lemma plus_permute_2_in_4 : (a,b,c,d:nat)
(plus (plus a b) (plus c d))=(plus (plus a c) (plus b d)).
Proof.
Intros.
Rewrite <- (plus_assoc_l a b (plus c d)). Rewrite (plus_assoc_l b c d).
Rewrite (plus_sym b c). Rewrite <- (plus_assoc_l c b d). Apply plus_assoc_l.
Qed.
|
