diff options
author | Guillaume Melquiond <guillaume.melquiond@inria.fr> | 2013-12-03 13:40:51 +0100 |
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committer | Guillaume Melquiond <guillaume.melquiond@inria.fr> | 2013-12-03 13:40:51 +0100 |
commit | e0ff9328c51cec3bd65d4893af5da5c9f8ba2570 (patch) | |
tree | 56faa40ecf27fcd735c0315d450931bb96e73069 /theories/Reals | |
parent | a0383d4ce6d69b086331310b32c7262223a659e5 (diff) |
Remove a useless hypothesis from Rle_Rinv.
Diffstat (limited to 'theories/Reals')
-rw-r--r-- | theories/Reals/Exp_prop.v | 6 | ||||
-rw-r--r-- | theories/Reals/RIneq.v | 24 | ||||
-rw-r--r-- | theories/Reals/Ratan.v | 6 | ||||
-rw-r--r-- | theories/Reals/RiemannInt.v | 6 |
4 files changed, 20 insertions, 22 deletions
diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index b65ab045b..40064fd51 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.v @@ -173,8 +173,7 @@ Proof. apply pow_le; apply Rabs_pos. rewrite (Rmult_comm (/ INR (fact (S n0)))); apply Rmult_le_compat_l. apply pow_le; apply Rabs_pos. - apply Rle_Rinv. - apply INR_fact_lt_0. + apply Rinv_le_contravar. apply INR_fact_lt_0. apply le_INR; apply fact_le; apply le_n_S. apply le_plus_l. @@ -254,8 +253,7 @@ Proof. do 2 rewrite <- (Rmult_comm (/ INR (fact (N - n0)))). apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. - apply Rle_Rinv. - apply INR_fact_lt_0. + apply Rinv_le_contravar. apply INR_fact_lt_0. apply le_INR. apply fact_le. diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index bfa975aab..3adea5a10 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. diff --git a/theories/Reals/Ratan.v b/theories/Reals/Ratan.v index 1a0ea969e..b1eb17b21 100644 --- a/theories/Reals/Ratan.v +++ b/theories/Reals/Ratan.v @@ -810,13 +810,11 @@ intros x Hx eps Heps. apply Rle_lt_trans with (/ INR (2 * N + 1))%R. unfold Rdiv. rewrite Rmult_1_l. - apply Rle_Rinv. + apply Rinv_le_contravar. apply lt_INR_0. omega. - replace 0 with (INR 0) by intuition. - apply lt_INR. + apply le_INR. omega. - intuition. rewrite <- (Rinv_involutive eps). apply Rinv_lt_contravar. apply Rmult_lt_0_compat. diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index 0a00ca223..3fe3694f9 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.v @@ -388,11 +388,9 @@ Proof. [ idtac | left; change (0 < / (INR n + 1)); apply Rinv_0_lt_compat; assumption ]; apply Rle_lt_trans with (/ (INR x + 1)). - apply Rle_Rinv. + apply Rinv_le_contravar. apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. - assumption. - do 2 rewrite <- (Rplus_comm 1); apply Rplus_le_compat_l; apply le_INR; - apply H4. + apply Rplus_le_compat_r; apply le_INR; apply H4. rewrite <- (Rinv_involutive eps). apply Rinv_lt_contravar. apply Rmult_lt_0_compat. |