summaryrefslogtreecommitdiff
path: root/theories/NArith/BinNat.v
blob: e6a149382b3e654a5a90dae8204ce7cc8bb8baa4 (plain)
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
212
(************************************************************************)
(*  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: BinNat.v,v 1.7.2.1 2004/07/16 19:31:07 herbelin Exp $ i*)

Require Import BinPos.

(**********************************************************************)
(** Binary natural numbers *)

Inductive N : Set :=
  | N0 : N
  | Npos : positive -> N.

(** Declare binding key for scope positive_scope *)

Delimit Scope N_scope with N.

(** Automatically open scope N_scope for the constructors of N *)

Bind Scope N_scope with N.
Arguments Scope Npos [N_scope].

Open Local Scope N_scope.

(** Operation x -> 2*x+1 *)

Definition Ndouble_plus_one x :=
  match x with
  | N0 => Npos 1%positive
  | Npos p => Npos (xI p)
  end.

(** Operation x -> 2*x *)

Definition Ndouble n := match n with
                        | N0 => N0
                        | Npos p => Npos (xO p)
                        end.

(** Successor *)

Definition Nsucc n :=
  match n with
  | N0 => Npos 1%positive
  | Npos p => Npos (Psucc p)
  end.

(** Addition *)

Definition Nplus n m :=
  match n, m with
  | N0, _ => m
  | _, N0 => n
  | Npos p, Npos q => Npos (p + q)%positive
  end.

Infix "+" := Nplus : N_scope.

(** Multiplication *)

Definition Nmult n m :=
  match n, m with
  | N0, _ => N0
  | _, N0 => N0
  | Npos p, Npos q => Npos (p * q)%positive
  end.

Infix "*" := Nmult : N_scope.

(** Order *)

Definition Ncompare n m :=
  match n, m with
  | N0, N0 => Eq
  | N0, Npos m' => Lt
  | Npos n', N0 => Gt
  | Npos n', Npos m' => (n' ?= m')%positive Eq
  end.

Infix "?=" := Ncompare (at level 70, no associativity) : N_scope.

(** Peano induction on binary natural numbers *)

Theorem Nind :
 forall P:N -> Prop,
   P N0 -> (forall n:N, P n -> P (Nsucc n)) -> forall n:N, P n.
Proof.
destruct n.
  assumption.
  apply Pind with (P := fun p => P (Npos p)).
exact (H0 N0 H).
intro p'; exact (H0 (Npos p')).
Qed.

(** Properties of addition *)

Theorem Nplus_0_l : forall n:N, N0 + n = n.
Proof.
reflexivity.
Qed.

Theorem Nplus_0_r : forall n:N, n + N0 = n.
Proof.
destruct n; reflexivity.
Qed.

Theorem Nplus_comm : forall n m:N, n + m = m + n.
Proof.
intros.
destruct n; destruct m; simpl in |- *; try reflexivity.
rewrite Pplus_comm; reflexivity.
Qed.

Theorem Nplus_assoc : forall n m p:N, n + (m + p) = n + m + p.
Proof.
intros.
destruct n; try reflexivity.
destruct m; try reflexivity.
destruct p; try reflexivity.
simpl in |- *; rewrite Pplus_assoc; reflexivity.
Qed.

Theorem Nplus_succ : forall n m:N, Nsucc n + m = Nsucc (n + m).
Proof.
destruct n; destruct m.
  simpl in |- *; reflexivity.
  unfold Nsucc, Nplus in |- *; rewrite <- Pplus_one_succ_l; reflexivity.
  simpl in |- *; reflexivity.
  simpl in |- *; rewrite Pplus_succ_permute_l; reflexivity.
Qed.

Theorem Nsucc_inj : forall n m:N, Nsucc n = Nsucc m -> n = m.
Proof.
destruct n; destruct m; simpl in |- *; intro H; reflexivity || injection H;
 clear H; intro H.
  symmetry  in H; contradiction Psucc_not_one with p.
  contradiction Psucc_not_one with p.
  rewrite Psucc_inj with (1 := H); reflexivity.
Qed.

Theorem Nplus_reg_l : forall n m p:N, n + m = n + p -> m = p.
Proof.
intro n; pattern n in |- *; apply Nind; clear n; simpl in |- *.
  trivial.
  intros n IHn m p H0; do 2 rewrite Nplus_succ in H0.
  apply IHn; apply Nsucc_inj; assumption.
Qed.

(** Properties of multiplication *)

Theorem Nmult_1_l : forall n:N, Npos 1%positive * n = n.
Proof.
destruct n; reflexivity.
Qed.

Theorem Nmult_1_r : forall n:N, n * Npos 1%positive = n.
Proof.
destruct n; simpl in |- *; try reflexivity.
rewrite Pmult_1_r; reflexivity.
Qed.

Theorem Nmult_comm : forall n m:N, n * m = m * n.
Proof.
intros.
destruct n; destruct m; simpl in |- *; try reflexivity.
rewrite Pmult_comm; reflexivity.
Qed.

Theorem Nmult_assoc : forall n m p:N, n * (m * p) = n * m * p.
Proof.
intros.
destruct n; try reflexivity.
destruct m; try reflexivity.
destruct p; try reflexivity.
simpl in |- *; rewrite Pmult_assoc; reflexivity.
Qed.

Theorem Nmult_plus_distr_r : forall n m p:N, (n + m) * p = n * p + m * p.
Proof.
intros.
destruct n; try reflexivity.
destruct m; destruct p; try reflexivity.
simpl in |- *; rewrite Pmult_plus_distr_r; reflexivity.
Qed.

Theorem Nmult_reg_r : forall n m p:N, p <> N0 -> n * p = m * p -> n = m.
Proof.
destruct p; intros Hp H.
contradiction Hp; reflexivity.
destruct n; destruct m; reflexivity || (try discriminate H).
injection H; clear H; intro H; rewrite Pmult_reg_r with (1 := H); reflexivity.
Qed. 

Theorem Nmult_0_l : forall n:N, N0 * n = N0.
Proof.
reflexivity.
Qed.

(** Properties of comparison *)

Theorem Ncompare_Eq_eq : forall n m:N, (n ?= m) = Eq -> n = m.
Proof.
destruct n as [| n]; destruct m as [| m]; simpl in |- *; intro H;
 reflexivity || (try discriminate H).
  rewrite (Pcompare_Eq_eq n m H); reflexivity.
Qed.