--- a/src/HOL/Library/Formal_Power_Series.thy Fri Apr 24 08:24:54 2009 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy Fri Apr 24 17:45:15 2009 +0200
@@ -1007,13 +1007,13 @@
by simp
lemma XDN_linear:
- "(XD o^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD o^ n) a + fps_const d * (XD o^ n) (b :: ('a::comm_ring_1) fps)"
+ "(XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)"
by (induct n, simp_all)
lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
lemma fps_mult_XD_shift:
- "(XD o^ k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
+ "(XD ^^ k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}