1.1 --- a/src/HOL/Library/Formal_Power_Series.thy Sun Apr 19 17:27:43 2009 +0200
1.2 +++ b/src/HOL/Library/Formal_Power_Series.thy Fri Apr 24 19:29:14 2009 +0100
1.3 @@ -979,7 +979,7 @@
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 @@ -990,7 +990,7 @@
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)$n"
1.19 using th
1.20 unfolding fps_sub_nth by simp
1.21 @@ -1027,7 +1027,7 @@
1.22
1.23 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
1.24
1.25 -lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
1.26 +lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
1.27 by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
1.28
1.29 subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}