--- a/src/HOL/Transcendental.thy Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Transcendental.thy Tue Mar 31 16:48:48 2015 +0100
@@ -4542,16 +4542,6 @@
(is "summable (?c x)")
by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
-lemma less_one_imp_sqr_less_one:
- fixes x :: real
- assumes "\<bar>x\<bar> < 1"
- shows "x\<^sup>2 < 1"
-proof -
- have "\<bar>x\<^sup>2\<bar> < 1"
- by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff)
- thus ?thesis using zero_le_power2 by auto
-qed
-
lemma DERIV_arctan_series:
assumes "\<bar> x \<bar> < 1"
shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))"
@@ -4568,7 +4558,7 @@
{
fix x :: real
assume "\<bar>x\<bar> < 1"
- hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
+ hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`])
hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
@@ -4676,7 +4666,7 @@
hence "\<bar>x\<bar> < r" by auto
hence "\<bar>x\<bar> < 1" using `r < 1` by auto
have "\<bar> - (x\<^sup>2) \<bar> < 1"
- using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
+ using abs_square_less_1 `\<bar>x\<bar> < 1` by auto
hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
unfolding real_norm_def[symmetric] by (rule geometric_sums)
hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"