diff options
author | Benjamin Barenblat <bbaren@debian.org> | 2019-02-02 19:29:23 -0500 |
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committer | Benjamin Barenblat <bbaren@debian.org> | 2019-02-02 19:29:23 -0500 |
commit | 9ebf44d84754adc5b64fcf612c6816c02c80462d (patch) | |
tree | bf5e06a28488e0e06a2f2011ff0d110e2e02f8fc /theories/Reals/R_sqrt.v | |
parent | 9043add656177eeac1491a73d2f3ab92bec0013c (diff) |
Imported Upstream version 8.9.0upstream/8.9.0upstream
Diffstat (limited to 'theories/Reals/R_sqrt.v')
-rw-r--r-- | theories/Reals/R_sqrt.v | 20 |
1 files changed, 18 insertions, 2 deletions
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index d4035fad..6991923b 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -155,6 +155,22 @@ Proof. | apply (sqrt_positivity x (Rlt_le 0 x H1)) ]. Qed. +Lemma Rlt_mult_inv_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x * / y. +intros x y H H0; try assumption. +replace 0 with (x * 0). +apply Rmult_lt_compat_l; auto with real. +ring. +Qed. + +Lemma Rle_mult_inv_pos : forall x y:R, 0 <= x -> 0 < y -> 0 <= x * / y. +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_div_alt : forall x y : R, 0 < y -> sqrt (x / y) = sqrt x / sqrt y. Proof. @@ -176,14 +192,14 @@ Proof. clearbody Hx'. clear Hx. apply Rsqr_inj. apply sqrt_pos. - apply Fourier_util.Rle_mult_inv_pos. + apply Rle_mult_inv_pos. apply Rsqrt_positivity. now apply sqrt_lt_R0. rewrite Rsqr_div, 2!Rsqr_sqrt. unfold Rsqr. now rewrite Rsqrt_Rsqrt. now apply Rlt_le. - now apply Fourier_util.Rle_mult_inv_pos. + now apply Rle_mult_inv_pos. apply Rgt_not_eq. now apply sqrt_lt_R0. Qed. |