From 2d137819572c4307b34bfdbdcf4ad0dc9ab240f3 Mon Sep 17 00:00:00 2001 From: Yves Bertot Date: Wed, 6 Sep 2017 08:39:06 +0200 Subject: changed statements of Rpower_lt and Rle_power and added Rlt_Rpower_l and pow_INR. Unfortunately theorems Rpower_lt and Rle_power are named inconsistently, in spite of their similarity. --- theories/Reals/RIneq.v | 3 +++ theories/Reals/Rpower.v | 19 ++++++++++++------- 2 files changed, 15 insertions(+), 7 deletions(-) (limited to 'theories') diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index 7bcd2799a..bc82c3712 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -1611,6 +1611,9 @@ Proof. Qed. Hint Resolve mult_INR: real. +Lemma pow_INR (m n: nat) : INR (m ^ n) = pow (INR m) n. +Proof. now induction n as [|n IHn];[ | simpl; rewrite mult_INR, IHn]. Qed. + (*********) Lemma lt_0_INR : forall n:nat, (0 < n)%nat -> 0 < INR n. Proof. diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index a646104cd..ddf7f568e 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -431,9 +431,9 @@ 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. @@ -488,9 +488,9 @@ 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 e n m H H1; case H1. intros H2; left; apply Rpower_lt; assumption. intros H2; rewrite H2; right; reflexivity. Qed. @@ -707,13 +707,18 @@ 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. -- cgit v1.2.3 From dc71c8b582552dcc1ea40af8a2894f7f0ec84597 Mon Sep 17 00:00:00 2001 From: Yves Bertot Date: Wed, 6 Sep 2017 13:01:23 +0200 Subject: weakens an hypothesis of Rle_Rpower --- theories/Reals/Rpower.v | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) (limited to 'theories') diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index ddf7f568e..301fe20b0 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -488,11 +488,13 @@ Proof. Qed. Theorem Rle_Rpower : - forall e n m:R, 1 < e -> 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 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. -- cgit v1.2.3