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
|
(************************************************************************)
(* 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: Min.v,v 1.10.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
Require Import Arith.
Open Local Scope nat_scope.
Implicit Types m n : nat.
(** minimum of two natural numbers *)
Fixpoint min n m {struct n} : nat :=
match n, m with
| O, _ => 0
| S n', O => 0
| S n', S m' => S (min n' m')
end.
(** Simplifications of [min] *)
Lemma min_SS : forall n m, S (min n m) = min (S n) (S m).
Proof.
auto with arith.
Qed.
Lemma min_comm : forall n m, min n m = min m n.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
(** [min] and [le] *)
Lemma min_l : forall n m, n <= m -> min n m = n.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
Lemma min_r : forall n m, m <= n -> min n m = m.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
Lemma le_min_l : forall n m, min n m <= n.
Proof.
induction n; intros; simpl in |- *; auto with arith.
elim m; intros; simpl in |- *; auto with arith.
Qed.
Lemma le_min_r : forall n m, min n m <= m.
Proof.
induction n; simpl in |- *; auto with arith.
induction m; simpl in |- *; auto with arith.
Qed.
Hint Resolve min_l min_r le_min_l le_min_r: arith v62.
(** [min n m] is equal to [n] or [m] *)
Lemma min_dec : forall n m, {min n m = n} + {min n m = m}.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
elim (IHn m); intro H; elim H; auto.
Qed.
Lemma min_case : forall n m (P:nat -> Set), P n -> P m -> P (min n m).
Proof.
induction n; simpl in |- *; auto with arith.
induction m; intros; simpl in |- *; auto with arith.
pattern (min n m) in |- *; apply IHn; auto with arith.
Qed.
Lemma min_case2 : forall n m (P:nat -> Prop), P n -> P m -> P (min n m).
Proof.
induction n; simpl in |- *; auto with arith.
induction m; intros; simpl in |- *; auto with arith.
pattern (min n m) in |- *; apply IHn; auto with arith.
Qed.
|
