diff options
Diffstat (limited to 'theories/Reals/R_sqrt.v')
| -rw-r--r-- | theories/Reals/R_sqrt.v | 11 |
1 files changed, 8 insertions, 3 deletions
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index 38a38400..20319a2b 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -37,8 +37,8 @@ Lemma sqrt_sqrt : forall x:R, 0 <= x -> sqrt x * sqrt x = x. Proof. intros. unfold sqrt. - case (Rcase_abs x); intro. - elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). + case (Rcase_abs x) as [Hlt|Hge]. + elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ Hlt H)). rewrite Rsqrt_Rsqrt; reflexivity. Qed. @@ -94,6 +94,10 @@ Proof. intros; unfold Rsqr; apply sqrt_square; assumption. Qed. +Lemma sqrt_pow2 : forall x, 0 <= x -> sqrt (x ^ 2) = x. +intros; simpl; rewrite Rmult_1_r, sqrt_square; auto. +Qed. + Lemma sqrt_Rsqr_abs : forall x:R, sqrt (Rsqr x) = Rabs x. Proof. intro x; rewrite Rsqr_abs; apply sqrt_Rsqr; apply Rabs_pos. @@ -517,3 +521,4 @@ Proof. reflexivity. reflexivity. Qed. + |