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-rw-r--r--theories/Reals/RIneq.v3
-rw-r--r--theories/Reals/Rpower.v25
2 files changed, 19 insertions, 9 deletions
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..301fe20b0 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.
@@ -488,11 +488,13 @@ 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 +709,18 @@ 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.