Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Reals.R_sqrt

Require Import Rbase.
Require Import Rfunctions.
Require Import Rsqrt_def.
Local Open Scope R_scope.

# 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_pow2 : forall x, 0 <= x -> sqrt (x ^ 2) = x.
Lemma pow2_sqrt x : 0 <= x -> sqrt x ^ 2 = 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 Rlt_mult_inv_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x * / y.
Lemma Rle_mult_inv_pos : forall x y:R, 0 <= x -> 0 < y -> 0 <= x * / y.
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_neg_0 x : x <= 0 -> sqrt x = 0.
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_inv x : sqrt (/ x) = / sqrt x.
Lemma inv_sqrt_depr x : 0 < x -> / sqrt x = sqrt (/ x).
#[deprecated(since="8.16",note="Use sqrt_inv.")]
Notation inv_sqrt := inv_sqrt_depr.
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

Definition Delta (a:nonzeroreal) (b c:R) : R := Rsqr b - 4 * a * c.
Definition Delta_is_pos (a:nonzeroreal) (b c:R) : Prop := 0 <= Delta a b c.
Definition sol_x1 (a:nonzeroreal) (b c:R) : R :=
(- b + sqrt (Delta a b c)) / (2 * a).
Definition sol_x2 (a:nonzeroreal) (b c:R) : R :=
(- b - sqrt (Delta a b c)) / (2 * a).
Lemma Rsqr_sol_eq_0_1 :
forall (a:nonzeroreal) (b c x:R),
Delta_is_pos a b c ->
x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0.
Lemma Rsqr_sol_eq_0_0 :
forall (a:nonzeroreal) (b c x:R),
Delta_is_pos a b c ->
a * Rsqr x + b * x + c = 0 -> x = sol_x1 a b c \/ x = sol_x2 a b c.