Library Coq.Reals.R_sqrt
Continuous extension of Rsqrt on R
Definition sqrt (
x:
R) :
R :=
match Rcase_abs x with
|
left _ => 0
|
right a =>
Rsqrt (
mknonnegreal x (
Rge_le _ _ a))
end.
Lemma sqrt_pos :
forall x :
R, 0
<= sqrt x.
Lemma sqrt_positivity :
forall x:
R, 0
<= x -> 0
<= sqrt x.
Lemma sqrt_sqrt :
forall x:
R, 0
<= x ->
sqrt x * sqrt x = x.
Lemma sqrt_0 :
sqrt 0
= 0.
Lemma sqrt_1 :
sqrt 1
= 1.
Lemma sqrt_eq_0 :
forall x:
R, 0
<= x ->
sqrt x = 0 ->
x = 0.
Lemma sqrt_lem_0 :
forall x y:
R, 0
<= x -> 0
<= y ->
sqrt x = y ->
y * y = x.
Lemma sqrt_lem_1 :
forall x y:
R, 0
<= x -> 0
<= y ->
y * y = x ->
sqrt x = y.
Lemma sqrt_def :
forall x:
R, 0
<= x ->
sqrt x * sqrt x = x.
Lemma sqrt_square :
forall x:
R, 0
<= x ->
sqrt (
x * x)
= x.
Lemma sqrt_Rsqr :
forall x:
R, 0
<= x ->
sqrt (
Rsqr x)
= x.
Lemma sqrt_Rsqr_abs :
forall x:
R,
sqrt (
Rsqr x)
= Rabs x.
Lemma Rsqr_sqrt :
forall x:
R, 0
<= x ->
Rsqr (
sqrt x)
= x.
Lemma sqrt_mult_alt :
forall x y :
R, 0
<= x ->
sqrt (
x * y)
= sqrt x * sqrt y.
Lemma sqrt_mult :
forall x y:
R, 0
<= x -> 0
<= y ->
sqrt (
x * y)
= sqrt x * sqrt y.
Lemma sqrt_lt_R0 :
forall x:
R, 0
< x -> 0
< sqrt x.
Lemma sqrt_div_alt :
forall x y :
R, 0
< y ->
sqrt (
x / y)
= sqrt x / sqrt y.
Lemma sqrt_div :
forall x y:
R, 0
<= x -> 0
< y ->
sqrt (
x / y)
= sqrt x / sqrt y.
Lemma sqrt_lt_0_alt :
forall x y :
R,
sqrt x < sqrt y ->
x < y.
Lemma sqrt_lt_0 :
forall x y:
R, 0
<= x -> 0
<= y ->
sqrt x < sqrt y ->
x < y.
Lemma sqrt_lt_1_alt :
forall x y :
R, 0
<= x < y ->
sqrt x < sqrt y.
Lemma sqrt_lt_1 :
forall x y:
R, 0
<= x -> 0
<= y ->
x < y ->
sqrt x < sqrt y.
Lemma sqrt_le_0 :
forall x y:
R, 0
<= x -> 0
<= y ->
sqrt x <= sqrt y ->
x <= y.
Lemma sqrt_le_1_alt :
forall x y :
R,
x <= y ->
sqrt x <= sqrt y.
Lemma sqrt_le_1 :
forall x y:
R, 0
<= x -> 0
<= y ->
x <= y ->
sqrt x <= sqrt y.
Lemma sqrt_inj :
forall x y:
R, 0
<= x -> 0
<= y ->
sqrt x = sqrt y ->
x = y.
Lemma sqrt_less_alt :
forall x :
R, 1
< x ->
sqrt x < x.
Lemma sqrt_less :
forall x:
R, 0
<= x -> 1
< x ->
sqrt x < x.
Lemma sqrt_more :
forall x:
R, 0
< x ->
x < 1 ->
x < sqrt x.
Lemma sqrt_cauchy :
forall a b c d:
R,
a * c + b * d <= sqrt (
Rsqr a + Rsqr b)
* sqrt (
Rsqr c + Rsqr d).
Resolution of a*X^2+b*X+c=0