1.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Mon Apr 27 00:29:54 2009 +0200
     1.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Sun Apr 26 23:41:18 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}*}