author huffman Thu Sep 12 15:08:46 2013 -0700 (2013-09-12) changeset 53599 78ea983f7987 parent 53598 2bd8d459ebae child 53600 8fda7ad57466
generalize lemmas
```     1.1 --- a/src/HOL/Transcendental.thy	Thu Sep 12 15:08:07 2013 -0700
1.2 +++ b/src/HOL/Transcendental.thy	Thu Sep 12 15:08:46 2013 -0700
1.3 @@ -57,7 +57,7 @@
1.4    x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
1.5
1.6  lemma powser_insidea:
1.7 -  fixes x z :: "'a::real_normed_field"
1.8 +  fixes x z :: "'a::real_normed_div_algebra"
1.9    assumes 1: "summable (\<lambda>n. f n * x ^ n)"
1.10      and 2: "norm z < norm x"
1.11    shows "summable (\<lambda>n. norm (f n * z ^ n))"
1.12 @@ -95,7 +95,7 @@
1.13    proof -
1.14      from 2 have "norm (norm (z * inverse x)) < 1"
1.15        using x_neq_0
1.16 -      by (simp add: nonzero_norm_divide divide_inverse [symmetric])
1.17 +      by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
1.18      hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
1.19        by (rule summable_geometric)
1.20      hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
1.21 @@ -110,7 +110,7 @@
1.22  qed
1.23
1.24  lemma powser_inside:
1.25 -  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
1.26 +  fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
1.27    shows
1.28      "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
1.29        summable (\<lambda>n. f n * (z ^ n))"
```