diff options
Diffstat (limited to 'theories/Reals/RIneq.v')
| -rw-r--r-- | theories/Reals/RIneq.v | 24 |
1 files changed, 14 insertions, 10 deletions
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index bfa975aaba..3adea5a10d 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -1935,18 +1935,22 @@ Proof. apply (Rmult_le_compat_l x 0 y H H0). Qed. +Lemma Rinv_le_contravar : + forall x y, 0 < x -> x <= y -> / y <= / x. +Proof. + intros x y H1 [H2|H2]. + apply Rlt_le. + apply Rinv_lt_contravar with (2 := H2). + apply Rmult_lt_0_compat with (1 := H1). + now apply Rlt_trans with x. + rewrite H2. + apply Rle_refl. +Qed. + Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x. Proof. - intros; apply Rmult_le_reg_l with x. - apply H. - rewrite <- Rinv_r_sym. - apply Rmult_le_reg_l with y. - apply H0. - rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; apply H1. - red; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0). - red; intro; rewrite H2 in H; elim (Rlt_irrefl _ H). + intros x y H _. + apply Rinv_le_contravar with (1 := H). Qed. Lemma double : forall r1, 2 * r1 = r1 + r1. |