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
202
203
204
205
206
207
208
209
210
211
|
(***********************************************************************)
(* 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*)
Require Export Plus.
Require Export Minus.
Require Export Lt.
Require Export Le.
Open Local Scope nat_scope.
Implicit Types m n p : nat.
(** Zero property *)
Lemma mult_0_r : forall n, n * 0 = 0.
Proof.
intro; symmetry in |- *; apply mult_n_O.
Qed.
Lemma mult_0_l : forall n, 0 * n = 0.
Proof.
reflexivity.
Qed.
(** Distributivity *)
Lemma mult_plus_distr_r : forall n m p, (n + m) * p = n * p + m * p.
Proof.
intros; elim n; simpl in |- *; intros; auto with arith.
elim plus_assoc; elim H; auto with arith.
Qed.
Hint Resolve mult_plus_distr_r: arith v62.
Lemma mult_plus_distr_l : forall n m p, n * (m + p) = n * m + n * p.
Proof.
induction n. trivial.
intros. simpl in |- *. rewrite (IHn m p). apply sym_eq. apply plus_permute_2_in_4.
Qed.
Lemma mult_minus_distr_r : forall n m p, (n - m) * p = n * p - m * p.
Proof.
intros; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; intros;
auto with arith.
elim minus_plus_simpl_l_reverse; auto with arith.
Qed.
Hint Resolve mult_minus_distr_r: arith v62.
(** Associativity *)
Lemma mult_assoc_reverse : forall n m p, n * m * p = n * (m * p).
Proof.
intros; elim n; intros; simpl in |- *; auto with arith.
rewrite mult_plus_distr_r.
elim H; auto with arith.
Qed.
Hint Resolve mult_assoc_reverse: arith v62.
Lemma mult_assoc : forall n m p, n * (m * p) = n * m * p.
Proof.
auto with arith.
Qed.
Hint Resolve mult_assoc: arith v62.
(** Commutativity *)
Lemma mult_comm : forall n m, n * m = m * n.
Proof.
intros; elim n; intros; simpl in |- *; auto with arith.
elim mult_n_Sm.
elim H; apply plus_comm.
Qed.
Hint Resolve mult_comm: arith v62.
(** 1 is neutral *)
Lemma mult_1_l : forall n, 1 * n = n.
Proof.
simpl in |- *; auto with arith.
Qed.
Hint Resolve mult_1_l: arith v62.
Lemma mult_1_r : forall n, n * 1 = n.
Proof.
intro; elim mult_comm; auto with arith.
Qed.
Hint Resolve mult_1_r: arith v62.
(** Compatibility with orders *)
Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n.
Proof.
induction m; simpl in |- *; auto with arith.
Qed.
Hint Resolve mult_O_le: arith v62.
Lemma mult_le_compat_l : forall n m p, n <= m -> p * n <= p * m.
Proof.
induction p as [| p IHp]. intros. simpl in |- *. apply le_n.
intros. simpl in |- *. apply plus_le_compat. assumption.
apply IHp. assumption.
Qed.
Hint Resolve mult_le_compat_l: arith.
Lemma mult_le_compat_r : forall n m p, n <= m -> n * p <= m * p.
intros m n p H.
rewrite mult_comm. rewrite (mult_comm n).
auto with arith.
Qed.
Lemma mult_le_compat :
forall n m p (q:nat), n <= m -> p <= q -> n * p <= m * q.
Proof.
intros m n p q Hmn Hpq; induction Hmn.
induction Hpq.
(* m*p<=m*p *)
apply le_n.
(* m*p<=m*m0 -> m*p<=m*(S m0) *)
rewrite <- mult_n_Sm; apply le_trans with (m * m0).
assumption.
apply le_plus_l.
(* m*p<=m0*q -> m*p<=(S m0)*q *)
simpl in |- *; apply le_trans with (m0 * q).
assumption.
apply le_plus_r.
Qed.
Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p.
Proof.
intro m; induction m. intros. simpl in |- *. rewrite <- plus_n_O. rewrite <- plus_n_O. assumption.
intros. exact (plus_lt_compat _ _ _ _ H (IHm _ _ H)).
Qed.
Hint Resolve mult_S_lt_compat_l: arith.
Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p.
intros m n p H H0.
induction p.
elim (lt_irrefl _ H0).
rewrite mult_comm.
replace (n * S p) with (S p * n); auto with arith.
Qed.
Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p.
Proof.
intros m n p H. elim (le_or_lt n p). trivial.
intro H0. cut (S m * n < S m * n). intro. elim (lt_irrefl _ H1).
apply le_lt_trans with (m := S m * p). assumption.
apply mult_S_lt_compat_l. assumption.
Qed.
(** n|->2*n and n|->2n+1 have disjoint image *)
Theorem odd_even_lem : forall p q, 2 * p + 1 <> 2 * q.
intros p; elim p; auto.
intros q; case q; simpl in |- *.
red in |- *; intros; discriminate.
intros q'; rewrite (fun x y => plus_comm x (S y)); simpl in |- *; red in |- *;
intros; discriminate.
intros p' H q; case q.
simpl in |- *; red in |- *; intros; discriminate.
intros q'; red in |- *; intros H0; case (H q').
replace (2 * q') with (2 * S q' - 2).
rewrite <- H0; simpl in |- *; auto.
repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *; auto.
simpl in |- *; repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *;
auto.
case q'; simpl in |- *; auto.
Qed.
(** Tail-recursive mult *)
(** [tail_mult] is an alternative definition for [mult] which is
tail-recursive, whereas [mult] is not. This can be useful
when extracting programs. *)
Fixpoint mult_acc (s:nat) m n {struct n} : nat :=
match n with
| O => s
| S p => mult_acc (tail_plus m s) m p
end.
Lemma mult_acc_aux : forall n m p, m + n * p = mult_acc m p n.
Proof.
induction n as [| p IHp]; simpl in |- *; auto.
intros s m; rewrite <- plus_tail_plus; rewrite <- IHp.
rewrite <- plus_assoc_reverse; apply (f_equal2 (A1:=nat) (A2:=nat)); auto.
rewrite plus_comm; auto.
Qed.
Definition tail_mult n m := mult_acc 0 m n.
Lemma mult_tail_mult : forall n m, n * m = tail_mult n m.
Proof.
intros; unfold tail_mult in |- *; rewrite <- mult_acc_aux; auto.
Qed.
(** [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus]
and [mult] and simplify *)
Ltac tail_simpl :=
repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult;
simpl in |- *.
|
