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
|
(************************************************************************)
(* 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 Square Root Function *)
Require Import NAxioms NSub NZSqrt.
Module NSqrtProp (Import A : NAxiomsSig')(Import B : NSubProp A).
Module Import NZSqrtP := Nop <+ NZSqrtProp A A B.
Ltac auto' := trivial; try rewrite <- neq_0_lt_0; auto using le_0_l.
Ltac wrap l := intros; apply l; auto'.
(** We redefine NZSqrt's results, without the non-negative hyps *)
Lemma sqrt_spec' : forall a, √a*√a <= a < S (√a) * S (√a).
Proof. wrap sqrt_spec. Qed.
Lemma sqrt_unique : forall a b, b*b<=a<(S b)*(S b) -> √a == b.
Proof. wrap sqrt_unique. Qed.
Lemma sqrt_square : forall a, √(a*a) == a.
Proof. wrap sqrt_square. Qed.
Lemma sqrt_le_mono : forall a b, a<=b -> √a <= √b.
Proof. wrap sqrt_le_mono. Qed.
Lemma sqrt_lt_cancel : forall a b, √a < √b -> a < b.
Proof. wrap sqrt_lt_cancel. Qed.
Lemma sqrt_le_square : forall a b, b*b<=a <-> b <= √a.
Proof. wrap sqrt_le_square. Qed.
Lemma sqrt_lt_square : forall a b, a<b*b <-> √a < b.
Proof. wrap sqrt_lt_square. Qed.
Definition sqrt_0 := sqrt_0.
Definition sqrt_1 := sqrt_1.
Definition sqrt_2 := sqrt_2.
Definition sqrt_lt_lin : forall a, 1<a -> √a<a := sqrt_lt_lin.
Lemma sqrt_le_lin : forall a, 0<=a -> √a<=a.
Proof. wrap sqrt_le_lin. Qed.
Lemma sqrt_mul_below : forall a b, √a * √b <= √(a*b).
Proof. wrap sqrt_mul_below. Qed.
Lemma sqrt_mul_above : forall a b, √(a*b) < S (√a) * S (√b).
Proof. wrap sqrt_mul_above. Qed.
Lemma sqrt_add_le : forall a b, √(a+b) <= √a + √b.
Proof. wrap sqrt_add_le. Qed.
Lemma add_sqrt_le : forall a b, √a + √b <= √(2*(a+b)).
Proof. wrap add_sqrt_le. Qed.
End NSqrtProp.
|
