(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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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.

(* Specifications: *)
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.
intros; simpl; rewrite Rmult_1_r, sqrt_square; auto.