diff options
Diffstat (limited to 'theories/Reals/Rpower.v')
| -rw-r--r-- | theories/Reals/Rpower.v | 35 |
1 files changed, 22 insertions, 13 deletions
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index a646104cd..c6fac951b 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *) +(* * 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 *) +(* // * 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) *) (************************************************************************) (*i Due to L.Thery i*) @@ -431,9 +433,9 @@ Proof. Qed. Theorem Rpower_lt : - forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z. + forall x y z:R, 1 < x -> y < z -> x ^R y < x ^R z. Proof. - intros x y z H H0 H1. + intros x y z H H1. unfold Rpower. apply exp_increasing. apply Rmult_lt_compat_r. @@ -488,11 +490,13 @@ Proof. Qed. Theorem Rle_Rpower : - forall e n m:R, 1 < e -> 0 <= n -> n <= m -> e ^R n <= e ^R m. + forall e n m:R, 1 <= e -> n <= m -> e ^R n <= e ^R m. Proof. - intros e n m H H0 H1; case H1. - intros H2; left; apply Rpower_lt; assumption. - intros H2; rewrite H2; right; reflexivity. + intros e n m [H | H]; intros H1. + case H1. + intros H2; left; apply Rpower_lt; assumption. + intros H2; rewrite H2; right; reflexivity. + now rewrite <- H; unfold Rpower; rewrite ln_1, !Rmult_0_r; apply Rle_refl. Qed. Theorem ln_lt_2 : / 2 < ln 2. @@ -707,13 +711,18 @@ intros x y z x0 y0; unfold Rpower. rewrite <- exp_plus, ln_mult, Rmult_plus_distr_l; auto. Qed. -Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> Rpower a c <= Rpower b c. +Lemma Rlt_Rpower_l a b c: 0 < c -> 0 < a < b -> a ^R c < b ^R c. +Proof. +intros c0 [a0 ab]; apply exp_increasing. +now apply Rmult_lt_compat_l; auto; apply ln_increasing; fourier. +Qed. + +Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> a ^R c <= b ^R c. Proof. intros [c0 | c0]; [ | intros; rewrite <- c0, !Rpower_O; [apply Rle_refl | |] ]. intros [a0 [ab|ab]]. - left; apply exp_increasing. - now apply Rmult_lt_compat_l; auto; apply ln_increasing; fourier. + now apply Rlt_le, Rlt_Rpower_l;[ | split]; fourier. rewrite ab; apply Rle_refl. apply Rlt_le_trans with a; tauto. tauto. |