(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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)         *)
(************************************************************************)
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_lem_1 : forall x y:R, 0 <= x -> 0 <= y -> y * y = x -> sqrt x = y.