--- a/src/HOL/Library/Formal_Power_Series.thy Wed May 13 17:13:33 2009 +0100
+++ b/src/HOL/Library/Formal_Power_Series.thy Tue May 19 14:13:23 2009 +0100
@@ -2182,6 +2182,29 @@
show "?dia = inverse ?d" by simp
qed
+lemma fps_ginv_deriv:
+ assumes a0:"a$0 = (0::'a::{field})" and a1: "a$1 \<noteq> 0"
+ shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv X a"
+proof-
+ let ?ia = "fps_ginv b a"
+ let ?iXa = "fps_ginv X a"
+ let ?d = "fps_deriv"
+ let ?dia = "?d ?ia"
+ have iXa0: "?iXa $ 0 = 0" by (simp add: fps_ginv_def)
+ have da0: "?d a $ 0 \<noteq> 0" using a1 by simp
+ from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" by simp
+ then have "(?d ?ia oo a) * ?d a = ?d b" unfolding fps_compose_deriv[OF a0] .
+ then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" by simp
+ then have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
+ by (simp add: fps_divide_def)
+ then have "(?d ?ia oo a) oo ?iXa = (?d b / ?d a) oo ?iXa "
+ unfolding inverse_mult_eq_1[OF da0] by simp
+ then have "?d ?ia oo (a oo ?iXa) = (?d b / ?d a) oo ?iXa"
+ unfolding fps_compose_assoc[OF iXa0 a0] .
+ then show ?thesis unfolding fps_inv_ginv[symmetric]
+ unfolding fps_inv_right[OF a0 a1] by simp
+qed
+
subsection{* Elementary series *}
subsubsection{* Exponential series *}