summaryrefslogtreecommitdiff
path: root/theories/Init/Peano.v
blob: 43b1f6349c949cafaabdb9d6684de73a3c7f4476 (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
213
214
215
216
217
218
219
220
221
222
223
(************************************************************************)
(*  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: Peano.v 11735 2009-01-02 17:22:31Z herbelin $ i*)

(** The type [nat] of Peano natural numbers (built from [O] and [S])
    is defined in [Datatypes.v] *)

(** This module defines the following operations on natural numbers :
    - predecessor [pred]
    - addition [plus]
    - multiplication [mult]
    - less or equal order [le]
    - less [lt]
    - greater or equal [ge]
    - greater [gt]

   It states various lemmas and theorems about natural numbers,
   including Peano's axioms of arithmetic (in Coq, these are provable).
   Case analysis on [nat] and induction on [nat * nat] are provided too
 *)

Require Import Notations.
Require Import Datatypes.
Require Import Logic.
Unset Boxed Definitions.

Open Scope nat_scope.

Definition eq_S := f_equal S.

Hint Resolve (f_equal S): v62.
Hint Resolve (f_equal (A:=nat)): core.

(** The predecessor function *)

Definition pred (n:nat) : nat := match n with
                                 | O => n
                                 | S u => u
                                 end.
Hint Resolve (f_equal pred): v62.

Theorem pred_Sn : forall n:nat, n = pred (S n).
Proof.
  simpl; reflexivity.
Qed.

(** Injectivity of successor *)

Theorem eq_add_S : forall n m:nat, S n = S m -> n = m.
Proof.
  intros n m Sn_eq_Sm.
  replace (n=m) with (pred (S n) = pred (S m)) by auto using pred_Sn.
  rewrite Sn_eq_Sm; trivial.
Qed.

Hint Immediate eq_add_S: core.

Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m.
Proof.
  red in |- *; auto.
Qed.
Hint Resolve not_eq_S: core.

Definition IsSucc (n:nat) : Prop :=
  match n with
  | O => False
  | S p => True
  end.

(** Zero is not the successor of a number *)

Theorem O_S : forall n:nat, 0 <> S n.
Proof.
  unfold not; intros n H.
  inversion H.
Qed.
Hint Resolve O_S: core.

Theorem n_Sn : forall n:nat, n <> S n.
Proof.
  induction n; auto.
Qed.
Hint Resolve n_Sn: core.

(** Addition *)

Fixpoint plus (n m:nat) {struct n} : nat :=
  match n with
  | O => m
  | S p => S (p + m)
  end

where "n + m" := (plus n m) : nat_scope.

Hint Resolve (f_equal2 plus): v62.
Hint Resolve (f_equal2 (A1:=nat) (A2:=nat)): core.

Lemma plus_n_O : forall n:nat, n = n + 0.
Proof.
  induction n; simpl in |- *; auto.
Qed.
Hint Resolve plus_n_O: core.

Lemma plus_O_n : forall n:nat, 0 + n = n.
Proof.
  auto.
Qed.

Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
Proof.
  intros n m; induction n; simpl in |- *; auto.
Qed.
Hint Resolve plus_n_Sm: core.

Lemma plus_Sn_m : forall n m:nat, S n + m = S (n + m).
Proof.
  auto.
Qed.

(** Standard associated names *)

Notation plus_0_r_reverse := plus_n_O (only parsing).
Notation plus_succ_r_reverse := plus_n_Sm (only parsing).

(** Multiplication *)

Fixpoint mult (n m:nat) {struct n} : nat :=
  match n with
  | O => 0
  | S p => m + p * m
  end

where "n * m" := (mult n m) : nat_scope.

Hint Resolve (f_equal2 mult): core.

Lemma mult_n_O : forall n:nat, 0 = n * 0.
Proof.
  induction n; simpl in |- *; auto.
Qed.
Hint Resolve mult_n_O: core.

Lemma mult_n_Sm : forall n m:nat, n * m + n = n * S m.
Proof.
  intros; induction n as [| p H]; simpl in |- *; auto.
  destruct H; rewrite <- plus_n_Sm; apply (f_equal S).
  pattern m at 1 3 in |- *; elim m; simpl in |- *; auto.
Qed.
Hint Resolve mult_n_Sm: core.

(** Standard associated names *)

Notation mult_0_r_reverse := mult_n_O (only parsing).
Notation mult_succ_r_reverse := mult_n_Sm (only parsing).

(** Truncated subtraction: [m-n] is [0] if [n>=m] *)

Fixpoint minus (n m:nat) {struct n} : nat :=
  match n, m with
  | O, _ => n
  | S k, O => n
  | S k, S l => k - l
  end

where "n - m" := (minus n m) : nat_scope.

(** Definition of the usual orders, the basic properties of [le] and [lt]
    can be found in files Le and Lt *)

Inductive le (n:nat) : nat -> Prop :=
  | le_n : n <= n
  | le_S : forall m:nat, n <= m -> n <= S m

where "n <= m" := (le n m) : nat_scope.

Hint Constructors le: core.
(*i equivalent to : "Hints Resolve le_n le_S : core." i*)

Definition lt (n m:nat) := S n <= m.
Hint Unfold lt: core.

Infix "<" := lt : nat_scope.

Definition ge (n m:nat) := m <= n.
Hint Unfold ge: core.

Infix ">=" := ge : nat_scope.

Definition gt (n m:nat) := m < n.
Hint Unfold gt: core.

Infix ">" := gt : nat_scope.

Notation "x <= y <= z" := (x <= y /\ y <= z) : nat_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : nat_scope.
Notation "x < y < z" := (x < y /\ y < z) : nat_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : nat_scope.

(** Case analysis *)

Theorem nat_case :
 forall (n:nat) (P:nat -> Prop), P 0 -> (forall m:nat, P (S m)) -> P n.
Proof.
  induction n; auto.
Qed.

(** Principle of double induction *)

Theorem nat_double_ind :
 forall R:nat -> nat -> Prop,
   (forall n:nat, R 0 n) ->
   (forall n:nat, R (S n) 0) ->
   (forall n m:nat, R n m -> R (S n) (S m)) -> forall n m:nat, R n m.
Proof.
  induction n; auto.
  destruct m as [| n0]; auto.
Qed.