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(*         *   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)         *)
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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.
intros; simpl; rewrite Rmult_1_r, sqrt_square; auto.
Qed.
intros x y H H0; try assumption.
replace 0 with (x * 0).
apply Rmult_lt_compat_l; auto with real.
ring.
Qed.
intros x y H H0; try assumption.
case H; intros.
red; left.
apply Rlt_mult_inv_pos; auto with real.
rewrite <- H1.
red; right; ring.
Qed.

Lemma sqrt_cauchy :
forall a b c d:R,
a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d).