aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Arith/Plus.v
blob: 3b823da6f37a9b89dc4d201501ba7bfc7d9d4785 (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
 (************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(** Properties of addition.

 This file is mostly OBSOLETE now, see module [PeanoNat.Nat] instead.

 [Nat.add] is defined in [Init/Nat.v] as:
<<
Fixpoint add (n m:nat) : nat :=
  match n with
  | O => m
  | S p => S (p + m)
  end
where "n + m" := (add n m) : nat_scope.
>>
*)

Require Import PeanoNat.

Local Open Scope nat_scope.

(** * Neutrality of 0, commutativity, associativity *)

Notation plus_0_l := Nat.add_0_l (compat "8.4").
Notation plus_0_r := Nat.add_0_r (compat "8.4").
Notation plus_comm := Nat.add_comm (compat "8.4").
Notation plus_assoc := Nat.add_assoc (compat "8.4").

Notation plus_permute := Nat.add_shuffle3 (compat "8.4").

Definition plus_Snm_nSm : forall n m, S n + m = n + S m :=
 Peano.plus_n_Sm.

Lemma plus_assoc_reverse n m p : n + m + p = n + (m + p).
Proof.
  symmetry. apply Nat.add_assoc.
Qed.

(** * Simplification *)

Lemma plus_reg_l n m p : p + n = p + m -> n = m.
Proof.
 apply Nat.add_cancel_l.
Qed.

Lemma plus_le_reg_l n m p : p + n <= p + m -> n <= m.
Proof.
 apply Nat.add_le_mono_l.
Qed.

Lemma plus_lt_reg_l n m p : p + n < p + m -> n < m.
Proof.
 apply Nat.add_lt_mono_l.
Qed.

(** * Compatibility with order *)

Lemma plus_le_compat_l n m p : n <= m -> p + n <= p + m.
Proof.
 apply Nat.add_le_mono_l.
Qed.

Lemma plus_le_compat_r n m p : n <= m -> n + p <= m + p.
Proof.
 apply Nat.add_le_mono_r.
Qed.

Lemma plus_lt_compat_l n m p : n < m -> p + n < p + m.
Proof.
 apply Nat.add_lt_mono_l.
Qed.

Lemma plus_lt_compat_r n m p : n < m -> n + p < m + p.
Proof.
 apply Nat.add_lt_mono_r.
Qed.

Lemma plus_le_compat n m p q : n <= m -> p <= q -> n + p <= m + q.
Proof.
 apply Nat.add_le_mono.
Qed.

Lemma plus_le_lt_compat n m p q : n <= m -> p < q -> n + p < m + q.
Proof.
 apply Nat.add_le_lt_mono.
Qed.

Lemma plus_lt_le_compat n m p q : n < m -> p <= q -> n + p < m + q.
Proof.
 apply Nat.add_lt_le_mono.
Qed.

Lemma plus_lt_compat n m p q : n < m -> p < q -> n + p < m + q.
Proof.
 apply Nat.add_lt_mono.
Qed.

Lemma le_plus_l n m : n <= n + m.
Proof.
 apply Nat.le_add_r.
Qed.

Lemma le_plus_r n m : m <= n + m.
Proof.
 rewrite Nat.add_comm. apply Nat.le_add_r.
Qed.

Theorem le_plus_trans n m p : n <= m -> n <= m + p.
Proof.
  intros. now rewrite <- Nat.le_add_r.
Qed.

Theorem lt_plus_trans n m p : n < m -> n < m + p.
Proof.
  intros. apply Nat.lt_le_trans with m. trivial. apply Nat.le_add_r.
Qed.

(** * Inversion lemmas *)

Lemma plus_is_O n m : n + m = 0 -> n = 0 /\ m = 0.
Proof.
  destruct n; now split.
Qed.

Definition plus_is_one m n :
  m + n = 1 -> {m = 0 /\ n = 1} + {m = 1 /\ n = 0}.
Proof.
  destruct m as [| m]; auto.
  destruct m; auto.
  discriminate.
Defined.

(** * Derived properties *)

Notation plus_permute_2_in_4 := Nat.add_shuffle1 (compat "8.4").

(** * Tail-recursive plus *)

(** [tail_plus] is an alternative definition for [plus] which is
    tail-recursive, whereas [plus] is not. This can be useful
    when extracting programs. *)

Fixpoint tail_plus n m : nat :=
  match n with
    | O => m
    | S n => tail_plus n (S m)
  end.

Lemma plus_tail_plus : forall n m, n + m = tail_plus n m.
Proof.
induction n as [| n IHn]; simpl; auto.
intro m; rewrite <- IHn; simpl; auto.
Qed.

(** * Discrimination *)

Lemma succ_plus_discr n m : n <> S (m+n).
Proof.
 apply Nat.succ_add_discr.
Qed.

Lemma n_SSn n : n <> S (S n).
Proof (succ_plus_discr n 1).

Lemma n_SSSn n : n <> S (S (S n)).
Proof (succ_plus_discr n 2).

Lemma n_SSSSn n : n <> S (S (S (S n))).
Proof (succ_plus_discr n 3).


(** * Compatibility Hints *)

Hint Immediate plus_comm : arith v62.
Hint Resolve plus_assoc plus_assoc_reverse : arith v62.
Hint Resolve plus_le_compat_l plus_le_compat_r : arith v62.
Hint Resolve le_plus_l le_plus_r le_plus_trans : arith v62.
Hint Immediate lt_plus_trans : arith v62.
Hint Resolve plus_lt_compat_l plus_lt_compat_r : arith v62.

(** For compatibility, we "Require" the same files as before *)

Require Import Le Lt.