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
|
(************************************************************************)
(* 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 $Id$ i*)
Require Import Arith_base.
Require Import Compare_dec.
Require Import Sumbool.
Require Import Div2.
Require Import BinPos.
Require Import BinNat.
Require Import Pnat.
(** Translation from [N] to [nat] and back. *)
Definition nat_of_N (a:N) :=
match a with
| N0 => 0%nat
| Npos p => nat_of_P p
end.
Definition N_of_nat (n:nat) :=
match n with
| O => N0
| S n' => Npos (P_of_succ_nat n')
end.
Lemma N_of_nat_of_N : forall a:N, N_of_nat (nat_of_N a) = a.
Proof.
destruct a as [| p]. reflexivity.
simpl in |- *. elim (ZL4 p). intros n H. rewrite H. simpl in |- *.
rewrite <- nat_of_P_o_P_of_succ_nat_eq_succ in H.
rewrite nat_of_P_inj with (1 := H). reflexivity.
Qed.
Lemma nat_of_N_of_nat : forall n:nat, nat_of_N (N_of_nat n) = n.
Proof.
induction n. trivial.
intros. simpl in |- *. apply nat_of_P_o_P_of_succ_nat_eq_succ.
Qed.
(** Interaction of this translation and usual operations. *)
Lemma nat_of_Ndouble : forall a, nat_of_N (Ndouble a) = 2*(nat_of_N a).
Proof.
destruct a; simpl nat_of_N; auto.
apply nat_of_P_xO.
Qed.
Lemma N_of_double : forall n, N_of_nat (2*n) = Ndouble (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndouble.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ndouble_plus_one :
forall a, nat_of_N (Ndouble_plus_one a) = S (2*(nat_of_N a)).
Proof.
destruct a; simpl nat_of_N; auto.
apply nat_of_P_xI.
Qed.
Lemma N_of_double_plus_one :
forall n, N_of_nat (S (2*n)) = Ndouble_plus_one (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndouble_plus_one.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nsucc : forall a, nat_of_N (Nsucc a) = S (nat_of_N a).
Proof.
destruct a; simpl.
apply nat_of_P_xH.
apply nat_of_P_succ_morphism.
Qed.
Lemma N_of_S : forall n, N_of_nat (S n) = Nsucc (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Nsucc.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nplus :
forall a a', nat_of_N (Nplus a a') = (nat_of_N a)+(nat_of_N a').
Proof.
destruct a; destruct a'; simpl; auto.
apply nat_of_P_plus_morphism.
Qed.
Lemma N_of_plus :
forall n n', N_of_nat (n+n') = Nplus (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nplus.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nmult :
forall a a', nat_of_N (Nmult a a') = (nat_of_N a)*(nat_of_N a').
Proof.
destruct a; destruct a'; simpl; auto.
apply nat_of_P_mult_morphism.
Qed.
Lemma N_of_mult :
forall n n', N_of_nat (n*n') = Nmult (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nmult.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ndiv2 :
forall a, nat_of_N (Ndiv2 a) = div2 (nat_of_N a).
Proof.
destruct a; simpl in *; auto.
destruct p; auto.
rewrite nat_of_P_xI.
rewrite div2_double_plus_one; auto.
rewrite nat_of_P_xO.
rewrite div2_double; auto.
Qed.
Lemma N_of_div2 :
forall n, N_of_nat (div2 n) = Ndiv2 (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndiv2.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ncompare :
forall a a', Ncompare a a' = nat_compare (nat_of_N a) (nat_of_N a').
Proof.
destruct a; destruct a'; simpl.
compute; auto.
generalize (lt_O_nat_of_P p).
unfold nat_compare.
destruct (lt_eq_lt_dec 0 (nat_of_P p)) as [[H|H]|H]; auto.
rewrite <- H; inversion 1.
intros; generalize (lt_trans _ _ _ H0 H); inversion 1.
generalize (lt_O_nat_of_P p).
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_P p) 0) as [[H|H]|H]; auto.
intros; generalize (lt_trans _ _ _ H0 H); inversion 1.
rewrite H; inversion 1.
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_P p) (nat_of_P p0)) as [[H|H]|H]; auto.
apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
rewrite (nat_of_P_inj _ _ H); apply Pcompare_refl.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
Qed.
Lemma N_of_nat_compare :
forall n n', nat_compare n n' = Ncompare (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
symmetry; apply nat_of_Ncompare.
Qed.
|
