1.1 --- a/src/HOL/Library/Formal_Power_Series.thy Sun Apr 26 20:23:09 2009 +0200
1.2 +++ b/src/HOL/Library/Formal_Power_Series.thy Sun Apr 26 23:40:22 2009 +0100
1.3 @@ -963,7 +963,7 @@
1.4 (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
1.5
1.6 lemma fps_power_mult_eq_shift:
1.7 - "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k}" (is "?lhs = ?rhs")
1.8 + "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: comm_ring_1) * X^i) {0 .. k}" (is "?lhs = ?rhs")
1.9 proof-
1.10 {fix n:: nat
1.11 have "?lhs $ n = (if n < Suc k then 0 else a n)"
1.12 @@ -974,7 +974,7 @@
1.13 next
1.14 case (Suc k)
1.15 note th = Suc.hyps[symmetric]
1.16 - have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
1.17 + have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
1.18 also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
1.19 using th
1.20 unfolding fps_sub_nth by simp
1.21 @@ -1012,8 +1012,9 @@
1.22
1.23 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
1.24
1.25 +
1.26 lemma fps_mult_XD_shift:
1.27 - "(XD ^^ k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
1.28 + "(XD ^^ k) (a:: ('a::{comm_ring_1, recpower}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
1.29 by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
1.30
1.31 subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}