author  wenzelm 
Fri, 10 Aug 2012 16:19:51 +0200  
changeset 48757  1232760e208e 
parent 47217  501b9bbd0d6e 
child 49834  b27bbb021df1 
permissions  rwrr 
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(* Title: HOL/Library/Formal_Power_Series.thy 
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Author: Amine Chaieb, University of Cambridge 
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*) 

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header{* A formalization of formal power series *} 

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theory Formal_Power_Series 

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imports Complex_Main Binomial 
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begin 
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subsection {* The type of formal power series*} 
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typedef (open) 'a fps = "{f :: nat \<Rightarrow> 'a. True}" 
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morphisms fps_nth Abs_fps 
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by simp 
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notation fps_nth (infixl "$" 75) 
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lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p $ n = q $ n)" 
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by (simp add: fps_nth_inject [symmetric] fun_eq_iff) 
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lemma fps_ext: "(\<And>n. p $ n = q $ n) \<Longrightarrow> p = q" 
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by (simp add: expand_fps_eq) 
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lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n" 
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by (simp add: Abs_fps_inverse) 
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text{* Definition of the basic elements 0 and 1 and the basic operations of addition, 
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negation and multiplication *} 

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instantiation fps :: (zero) zero 
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begin 
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definition fps_zero_def: 
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"0 = Abs_fps (\<lambda>n. 0)" 
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instance .. 
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end 

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lemma fps_zero_nth [simp]: "0 $ n = 0" 
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instantiation fps :: ("{one, zero}") one 
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begin 
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definition fps_one_def: 
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"1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)" 
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instance .. 
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end 

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lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)" 
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instantiation fps :: (plus) plus 
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begin 

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definition fps_plus_def: 
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"op + = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))" 
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instance .. 
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end 

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lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n" 
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unfolding fps_plus_def by simp 
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instantiation fps :: (minus) minus 
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begin 
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definition fps_minus_def: 
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"op  = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n  g $ n))" 
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instance .. 
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end 

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lemma fps_sub_nth [simp]: "(f  g) $ n = f $ n  g $ n" 
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unfolding fps_minus_def by simp 
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instantiation fps :: (uminus) uminus 
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begin 
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definition fps_uminus_def: 
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"uminus = (\<lambda>f. Abs_fps (\<lambda>n.  (f $ n)))" 
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instance .. 
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end 

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lemma fps_neg_nth [simp]: "( f) $ n =  (f $ n)" 
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unfolding fps_uminus_def by simp 
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instantiation fps :: ("{comm_monoid_add, times}") times 
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begin 

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definition fps_times_def: 
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"op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n  i)))" 
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instance .. 
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end 

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lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n  i))" 
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unfolding fps_times_def by simp 
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declare atLeastAtMost_iff[presburger] 
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declare Bex_def[presburger] 
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declare Ball_def[presburger] 

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lemma mult_delta_left: 
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fixes x y :: "'a::mult_zero" 

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shows "(if b then x else 0) * y = (if b then x * y else 0)" 

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by simp 

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lemma mult_delta_right: 

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fixes x y :: "'a::mult_zero" 

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shows "x * (if b then y else 0) = (if b then x * y else 0)" 

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by simp 

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lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)" 
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by auto 

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lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" 

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by auto 

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subsection{* Formal power series form a commutative ring with unity, if the range of sequences 
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they represent is a commutative ring with unity*} 
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instance fps :: (semigroup_add) semigroup_add 
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proof 
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fix a b c :: "'a fps" show "a + b + c = a + (b + c)" 

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by (simp add: fps_ext add_assoc) 
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qed 
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instance fps :: (ab_semigroup_add) ab_semigroup_add 
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proof 
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fix a b :: "'a fps" show "a + b = b + a" 
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by (simp add: fps_ext add_commute) 
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qed 
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lemma fps_mult_assoc_lemma: 
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fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" 
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shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j  i) (n  j)) = 
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(\<Sum>j=0..k. \<Sum>i=0..k  j. f j i (n  j  i))" 
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proof (induct k) 
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case 0 show ?case by simp 
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next 
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case (Suc k) thus ?case 
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by (simp add: Suc_diff_le setsum_addf add_assoc 
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cong: strong_setsum_cong) 
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qed 
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instance fps :: (semiring_0) semigroup_mult 
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proof 
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fix a b c :: "'a fps" 

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show "(a * b) * c = a * (b * c)" 
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proof (rule fps_ext) 
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fix n :: nat 
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have "(\<Sum>j=0..n. \<Sum>i=0..j. a$i * b$(j  i) * c$(n  j)) = 
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(\<Sum>j=0..n. \<Sum>i=0..n  j. a$j * b$i * c$(n  j  i))" 
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by (rule fps_mult_assoc_lemma) 
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thus "((a * b) * c) $ n = (a * (b * c)) $ n" 
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by (simp add: fps_mult_nth setsum_right_distrib 
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setsum_left_distrib mult_assoc) 
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qed 
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qed 
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lemma fps_mult_commute_lemma: 
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fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" 
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shows "(\<Sum>i=0..n. f i (n  i)) = (\<Sum>i=0..n. f (n  i) i)" 
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proof (rule setsum_reindex_cong) 
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show "inj_on (\<lambda>i. n  i) {0..n}" 
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by (rule inj_onI) simp 
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show "{0..n} = (\<lambda>i. n  i) ` {0..n}" 
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by (auto, rule_tac x="n  x" in image_eqI, simp_all) 
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next 
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fix i assume "i \<in> {0..n}" 
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hence "n  (n  i) = i" by simp 
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thus "f (n  i) i = f (n  i) (n  (n  i))" by simp 
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qed 
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instance fps :: (comm_semiring_0) ab_semigroup_mult 
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proof 
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fix a b :: "'a fps" 
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show "a * b = b * a" 
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proof (rule fps_ext) 
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fix n :: nat 
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have "(\<Sum>i=0..n. a$i * b$(n  i)) = (\<Sum>i=0..n. a$(n  i) * b$i)" 
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by (rule fps_mult_commute_lemma) 
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thus "(a * b) $ n = (b * a) $ n" 
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by (simp add: fps_mult_nth mult_commute) 
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qed 
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qed 

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instance fps :: (monoid_add) monoid_add 
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proof 
194 
fix a :: "'a fps" show "0 + a = a " 

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195 
by (simp add: fps_ext) 
29687  196 
next 
197 
fix a :: "'a fps" show "a + 0 = a " 

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198 
by (simp add: fps_ext) 
29687  199 
qed 
200 

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201 
instance fps :: (comm_monoid_add) comm_monoid_add 
29687  202 
proof 
203 
fix a :: "'a fps" show "0 + a = a " 

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204 
by (simp add: fps_ext) 
29687  205 
qed 
206 

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207 
instance fps :: (semiring_1) monoid_mult 
29687  208 
proof 
209 
fix a :: "'a fps" show "1 * a = a" 

29913  210 
by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta) 
29687  211 
next 
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212 
fix a :: "'a fps" show "a * 1 = a" 
29913  213 
by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta') 
29687  214 
qed 
215 

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216 
instance fps :: (cancel_semigroup_add) cancel_semigroup_add 
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217 
proof 
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218 
fix a b c :: "'a fps" 
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219 
assume "a + b = a + c" then show "b = c" 
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220 
by (simp add: expand_fps_eq) 
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221 
next 
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222 
fix a b c :: "'a fps" 
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223 
assume "b + a = c + a" then show "b = c" 
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224 
by (simp add: expand_fps_eq) 
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225 
qed 
29687  226 

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227 
instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add 
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228 
proof 
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229 
fix a b c :: "'a fps" 
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230 
assume "a + b = a + c" then show "b = c" 
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231 
by (simp add: expand_fps_eq) 
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232 
qed 
29687  233 

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234 
instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add .. 
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235 

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236 
instance fps :: (group_add) group_add 
29687  237 
proof 
238 
fix a :: "'a fps" show " a + a = 0" 

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239 
by (simp add: fps_ext) 
29687  240 
next 
241 
fix a b :: "'a fps" show "a  b = a +  b" 

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242 
by (simp add: fps_ext diff_minus) 
29687  243 
qed 
244 

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245 
instance fps :: (ab_group_add) ab_group_add 
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246 
proof 
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247 
fix a :: "'a fps" 
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248 
show " a + a = 0" 
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249 
by (simp add: fps_ext) 
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250 
next 
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251 
fix a b :: "'a fps" 
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252 
show "a  b = a +  b" 
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253 
by (simp add: fps_ext) 
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254 
qed 
29687  255 

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256 
instance fps :: (zero_neq_one) zero_neq_one 
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257 
by default (simp add: expand_fps_eq) 
29687  258 

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259 
instance fps :: (semiring_0) semiring 
29687  260 
proof 
261 
fix a b c :: "'a fps" 

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262 
show "(a + b) * c = a * c + b * c" 
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263 
by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf) 
29687  264 
next 
265 
fix a b c :: "'a fps" 

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266 
show "a * (b + c) = a * b + a * c" 
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267 
by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf) 
29687  268 
qed 
269 

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270 
instance fps :: (semiring_0) semiring_0 
29687  271 
proof 
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272 
fix a:: "'a fps" show "0 * a = 0" 
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273 
by (simp add: fps_ext fps_mult_nth) 
29687  274 
next 
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275 
fix a:: "'a fps" show "a * 0 = 0" 
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276 
by (simp add: fps_ext fps_mult_nth) 
29687  277 
qed 
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278 

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279 
instance fps :: (semiring_0_cancel) semiring_0_cancel .. 
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280 

29906  281 
subsection {* Selection of the nth power of the implicit variable in the infinite sum*} 
29687  282 

283 
lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)" 

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284 
by (simp add: expand_fps_eq) 
29687  285 

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286 
lemma fps_nonzero_nth_minimal: 
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287 
"f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0))" 
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288 
proof 
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289 
let ?n = "LEAST n. f $ n \<noteq> 0" 
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290 
assume "f \<noteq> 0" 
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291 
then have "\<exists>n. f $ n \<noteq> 0" 
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292 
by (simp add: fps_nonzero_nth) 
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293 
then have "f $ ?n \<noteq> 0" 
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294 
by (rule LeastI_ex) 
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295 
moreover have "\<forall>m<?n. f $ m = 0" 
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296 
by (auto dest: not_less_Least) 
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297 
ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" .. 
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298 
then show "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" .. 
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299 
next 
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300 
assume "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" 
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301 
then show "f \<noteq> 0" by (auto simp add: expand_fps_eq) 
29687  302 
qed 
303 

304 
lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)" 

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305 
by (rule expand_fps_eq) 
29687  306 

30488  307 
lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S" 
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308 
proof (cases "finite S") 
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309 
assume "\<not> finite S" then show ?thesis by simp 
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310 
next 
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311 
assume "finite S" 
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312 
then show ?thesis by (induct set: finite) auto 
29687  313 
qed 
314 

29906  315 
subsection{* Injection of the basic ring elements and multiplication by scalars *} 
29687  316 

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317 
definition 
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318 
"fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)" 
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319 

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320 
lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)" 
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321 
unfolding fps_const_def by simp 
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322 

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323 
lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0" 
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324 
by (simp add: fps_ext) 
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325 

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326 
lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1" 
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327 
by (simp add: fps_ext) 
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328 

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329 
lemma fps_const_neg [simp]: " (fps_const (c::'a::ring)) = fps_const ( c)" 
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330 
by (simp add: fps_ext) 
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331 

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332 
lemma fps_const_add [simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)" 
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333 
by (simp add: fps_ext) 
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
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334 
lemma fps_const_sub [simp]: "fps_const (c::'a\<Colon>group_add)  fps_const d = fps_const (c  d)" 
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335 
by (simp add: fps_ext) 
29687  336 
lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)" 
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337 
by (simp add: fps_eq_iff fps_mult_nth setsum_0') 
29687  338 

48757  339 
lemma fps_const_add_left: "fps_const (c::'a\<Colon>monoid_add) + f = 
340 
Abs_fps (\<lambda>n. if n = 0 then c + f$0 else f$n)" 

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341 
by (simp add: fps_ext) 
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342 

48757  343 
lemma fps_const_add_right: "f + fps_const (c::'a\<Colon>monoid_add) = 
344 
Abs_fps (\<lambda>n. if n = 0 then f$0 + c else f$n)" 

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345 
by (simp add: fps_ext) 
29687  346 

347 
lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)" 

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348 
unfolding fps_eq_iff fps_mult_nth 
29913  349 
by (simp add: fps_const_def mult_delta_left setsum_delta) 
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350 

29687  351 
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)" 
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352 
unfolding fps_eq_iff fps_mult_nth 
29913  353 
by (simp add: fps_const_def mult_delta_right setsum_delta') 
29687  354 

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355 
lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n" 
29913  356 
by (simp add: fps_mult_nth mult_delta_left setsum_delta) 
29687  357 

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358 
lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c" 
29913  359 
by (simp add: fps_mult_nth mult_delta_right setsum_delta') 
29687  360 

29906  361 
subsection {* Formal power series form an integral domain*} 
29687  362 

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363 
instance fps :: (ring) ring .. 
29687  364 

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365 
instance fps :: (ring_1) ring_1 
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huffman
parents:
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diff
changeset

366 
by (intro_classes, auto simp add: diff_minus left_distrib) 
29687  367 

29911
c790a70a3d19
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huffman
parents:
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diff
changeset

368 
instance fps :: (comm_ring_1) comm_ring_1 
c790a70a3d19
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huffman
parents:
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diff
changeset

369 
by (intro_classes, auto simp add: diff_minus left_distrib) 
29687  370 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

371 
instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors 
29687  372 
proof 
373 
fix a b :: "'a fps" 

374 
assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0" 

375 
then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0" 

376 
and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal 

377 
by blast+ 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

378 
have "(a * b) $ (i+j) = (\<Sum>k=0..i+j. a$k * b$(i+jk))" 
29687  379 
by (rule fps_mult_nth) 
29911
c790a70a3d19
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huffman
parents:
29906
diff
changeset

380 
also have "\<dots> = (a$i * b$(i+ji)) + (\<Sum>k\<in>{0..i+j}{i}. a$k * b$(i+jk))" 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

381 
by (rule setsum_diff1') simp_all 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

382 
also have "(\<Sum>k\<in>{0..i+j}{i}. a$k * b$(i+jk)) = 0" 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

383 
proof (rule setsum_0' [rule_format]) 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

384 
fix k assume "k \<in> {0..i+j}  {i}" 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

385 
then have "k < i \<or> i+jk < j" by auto 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

386 
then show "a$k * b$(i+jk) = 0" using i j by auto 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

387 
qed 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

388 
also have "a$i * b$(i+ji) + 0 = a$i * b$j" by simp 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

389 
also have "a$i * b$j \<noteq> 0" using i j by simp 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

390 
finally have "(a*b) $ (i+j) \<noteq> 0" . 
29687  391 
then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast 
392 
qed 

393 

36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
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diff
changeset

394 
instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. 
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

395 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

396 
instance fps :: (idom) idom .. 
29687  397 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
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diff
changeset

398 
lemma numeral_fps_const: "numeral k = fps_const (numeral k)" 
48757  399 
by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

400 
fps_const_add [symmetric]) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

401 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

402 
lemma neg_numeral_fps_const: "neg_numeral k = fps_const (neg_numeral k)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

403 
by (simp only: neg_numeral_def numeral_fps_const fps_const_neg) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

404 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

405 
subsection{* The eXtractor series X*} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

406 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

407 
lemma minus_one_power_iff: "( (1::'a :: {comm_ring_1})) ^ n = (if even n then 1 else  1)" 
48757  408 
by (induct n) auto 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

409 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

410 
definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

411 
lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n  1))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

412 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

413 
{assume n: "n \<noteq> 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

414 
have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n  i))" by (simp add: fps_mult_nth) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

415 
also have "\<dots> = f $ (n  1)" 
46757  416 
using n by (simp add: X_def mult_delta_left setsum_delta) 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

417 
finally have ?thesis using n by simp } 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

418 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

419 
{assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

420 
ultimately show ?thesis by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

421 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

422 

48757  423 
lemma X_mult_right_nth[simp]: 
424 
"((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n  1))" 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

425 
by (metis X_mult_nth mult_commute) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

426 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

427 
lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

428 
proof(induct k) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

429 
case 0 thus ?case by (simp add: X_def fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

430 
next 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

431 
case (Suc k) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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diff
changeset

432 
{fix m 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

433 
have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m  1))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
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changeset

434 
by (simp add: power_Suc del: One_nat_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

435 
then have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

436 
using Suc.hyps by (auto cong del: if_weak_cong)} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

437 
then show ?case by (simp add: fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

438 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

439 

48757  440 
lemma X_power_mult_nth: 
441 
"(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n  k))" 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

442 
apply (induct k arbitrary: n) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

443 
apply (simp) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

444 
unfolding power_Suc mult_assoc 
48757  445 
apply (case_tac n) 
446 
apply auto 

447 
done 

448 

449 
lemma X_power_mult_right_nth: 

450 
"((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n  k))" 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

451 
by (metis X_power_mult_nth mult_commute) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

452 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

453 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

454 

31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

455 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

456 
subsection{* Formal Power series form a metric space *} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

457 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

458 
definition (in dist) ball_def: "ball x r = {y. dist y x < r}" 
48757  459 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

460 
instantiation fps :: (comm_ring_1) dist 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

461 
begin 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

462 

48757  463 
definition dist_fps_def: 
464 
"dist (a::'a fps) b = (if (\<exists>n. a$n \<noteq> b$n) then inverse (2 ^ The (leastP (\<lambda>n. a$n \<noteq> b$n))) else 0)" 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

465 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

466 
lemma dist_fps_ge0: "dist (a::'a fps) b \<ge> 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

467 
by (simp add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

468 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

469 
lemma dist_fps_sym: "dist (a::'a fps) b = dist b a" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

470 
apply (auto simp add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

471 
apply (rule cong[OF refl, where x="(\<lambda>n\<Colon>nat. a $ n \<noteq> b $ n)"]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

472 
apply (rule ext) 
48757  473 
apply auto 
474 
done 

475 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

476 
instance .. 
48757  477 

30746  478 
end 
479 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

480 
lemma fps_nonzero_least_unique: assumes a0: "a \<noteq> 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

481 
shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> 0) n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

482 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

483 
from fps_nonzero_nth_minimal[of a] a0 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

484 
obtain n where n: "a$n \<noteq> 0" "\<forall>m < n. a$m = 0" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

485 
from n have ln: "leastP (\<lambda>n. a$n \<noteq> 0) n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

486 
by (auto simp add: leastP_def setge_def not_le[symmetric]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

487 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

488 
{fix m assume "leastP (\<lambda>n. a$n \<noteq> 0) m" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

489 
then have "m = n" using ln 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

490 
apply (auto simp add: leastP_def setge_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

491 
apply (erule allE[where x=n]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

492 
apply (erule allE[where x=m]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

493 
by simp} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

494 
ultimately show ?thesis by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

495 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

496 

48757  497 
lemma fps_eq_least_unique: 
498 
assumes ab: "(a::('a::ab_group_add) fps) \<noteq> b" 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

499 
shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> b$n) n" 
48757  500 
using fps_nonzero_least_unique[of "a  b"] ab 
501 
by auto 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

502 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

503 
instantiation fps :: (comm_ring_1) metric_space 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

504 
begin 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

505 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

506 
definition open_fps_def: "open (S :: 'a fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> ball a r \<subseteq> S)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

507 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

508 
instance 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

509 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

510 
fix S :: "'a fps set" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

511 
show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

512 
by (auto simp add: open_fps_def ball_def subset_eq) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

513 
next 
48757  514 
{ 
515 
fix a b :: "'a fps" 

516 
{ 

517 
assume ab: "a = b" 

518 
then have "\<not> (\<exists>n. a$n \<noteq> b$n)" by simp 

519 
then have "dist a b = 0" by (simp add: dist_fps_def) 

520 
} 

521 
moreover 

522 
{ 

523 
assume d: "dist a b = 0" 

524 
then have "\<forall>n. a$n = b$n" 

525 
by  (rule ccontr, simp add: dist_fps_def) 

526 
then have "a = b" by (simp add: fps_eq_iff) 

527 
} 

528 
ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast 

529 
} 

530 
note th = this 

531 
from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

532 
fix a b c :: "'a fps" 
48757  533 
{ 
534 
assume ab: "a = b" then have d0: "dist a b = 0" unfolding th . 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

535 
then have "dist a b \<le> dist a c + dist b c" 
48757  536 
using dist_fps_ge0[of a c] dist_fps_ge0[of b c] by simp 
537 
} 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

538 
moreover 
48757  539 
{ 
540 
assume c: "c = a \<or> c = b" 

541 
then have "dist a b \<le> dist a c + dist b c" 

542 
by (cases "c=a") (simp_all add: th dist_fps_sym) 

543 
} 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

544 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

545 
{assume ab: "a \<noteq> b" and ac: "a \<noteq> c" and bc: "b \<noteq> c" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

546 
let ?P = "\<lambda>a b n. a$n \<noteq> b$n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

547 
from fps_eq_least_unique[OF ab] fps_eq_least_unique[OF ac] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

548 
fps_eq_least_unique[OF bc] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

549 
obtain nab nac nbc where nab: "leastP (?P a b) nab" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

550 
and nac: "leastP (?P a c) nac" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

551 
and nbc: "leastP (?P b c) nbc" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

552 
from nab have nab': "\<And>m. m < nab \<Longrightarrow> a$m = b$m" "a$nab \<noteq> b$nab" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

553 
by (auto simp add: leastP_def setge_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

554 
from nac have nac': "\<And>m. m < nac \<Longrightarrow> a$m = c$m" "a$nac \<noteq> c$nac" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

555 
by (auto simp add: leastP_def setge_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

556 
from nbc have nbc': "\<And>m. m < nbc \<Longrightarrow> b$m = c$m" "b$nbc \<noteq> c$nbc" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

557 
by (auto simp add: leastP_def setge_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

558 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

559 
have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a$n \<noteq> b$n)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

560 
by (simp add: fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

561 
from ab ac bc nab nac nbc 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

562 
have dab: "dist a b = inverse (2 ^ nab)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

563 
and dac: "dist a c = inverse (2 ^ nac)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

564 
and dbc: "dist b c = inverse (2 ^ nbc)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

565 
unfolding th0 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

566 
apply (simp_all add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

567 
apply (erule the1_equality[OF fps_eq_least_unique[OF ab]]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

568 
apply (erule the1_equality[OF fps_eq_least_unique[OF ac]]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

569 
by (erule the1_equality[OF fps_eq_least_unique[OF bc]]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

570 
from ab ac bc have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

571 
unfolding th by simp_all 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

572 
from nz have pos: "dist a b > 0" "dist a c > 0" "dist b c > 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

573 
using dist_fps_ge0[of a b] dist_fps_ge0[of a c] dist_fps_ge0[of b c] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

574 
by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

575 
have th1: "\<And>n. (2::real)^n >0" by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

576 
{assume h: "dist a b > dist a c + dist b c" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

577 
then have gt: "dist a b > dist a c" "dist a b > dist b c" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

578 
using pos by auto 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

579 
from gt have gtn: "nab < nbc" "nab < nac" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

580 
unfolding dab dbc dac by (auto simp add: th1) 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

581 
from nac'(1)[OF gtn(2)] nbc'(1)[OF gtn(1)] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

582 
have "a$nab = b$nab" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

583 
with nab'(2) have False by simp} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

584 
then have "dist a b \<le> dist a c + dist b c" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

585 
by (auto simp add: not_le[symmetric]) } 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

586 
ultimately show "dist a b \<le> dist a c + dist b c" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

587 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

588 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

589 
end 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

590 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

591 
text{* The infinite sums and justification of the notation in textbooks*} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

592 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

593 
lemma reals_power_lt_ex: assumes xp: "x > 0" and y1: "(y::real) > 1" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

594 
shows "\<exists>k>0. (1/y)^k < x" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

595 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

596 
have yp: "y > 0" using y1 by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

597 
from reals_Archimedean2[of "max 0 ( log y x) + 1"] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

598 
obtain k::nat where k: "real k > max 0 ( log y x) + 1" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

599 
from k have kp: "k > 0" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

600 
from k have "real k >  log y x" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

601 
then have "ln y * real k >  ln x" unfolding log_def 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

602 
using ln_gt_zero_iff[OF yp] y1 
36350  603 
by (simp add: minus_divide_left field_simps del:minus_divide_left[symmetric]) 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

604 
then have "ln y * real k + ln x > 0" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

605 
then have "exp (real k * ln y + ln x) > exp 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

606 
by (simp add: mult_ac) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

607 
then have "y ^ k * x > 1" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

608 
unfolding exp_zero exp_add exp_real_of_nat_mult 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

609 
exp_ln[OF xp] exp_ln[OF yp] by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

610 
then have "x > (1/y)^k" using yp 
36350  611 
by (simp add: field_simps nonzero_power_divide) 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

612 
then show ?thesis using kp by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

613 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

614 
lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

615 
lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

616 
by (simp add: X_power_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

617 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

618 

48757  619 
lemma fps_sum_rep_nth: "(setsum (%i. fps_const(a$i)*X^i) {0..m})$n = 
620 
(if n \<le> m then a$n else (0::'a::comm_ring_1))" 

621 
apply (auto simp add: fps_eq_iff fps_setsum_nth X_power_nth cond_application_beta cond_value_iff 

622 
cong del: if_weak_cong) 

623 
apply (simp add: setsum_delta') 

624 
done 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

625 

0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

626 
lemma fps_notation: 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

627 
"(%n. setsum (%i. fps_const(a$i) * X^i) {0..n}) > a" (is "?s > a") 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

628 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

629 
{fix r:: real 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

630 
assume rp: "r > 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

631 
have th0: "(2::real) > 1" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

632 
from reals_power_lt_ex[OF rp th0] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

633 
obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

634 
{fix n::nat 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

635 
assume nn0: "n \<ge> n0" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

636 
then have thnn0: "(1/2)^n <= (1/2 :: real)^n0" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

637 
by (auto intro: power_decreasing) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

638 
{assume "?s n = a" then have "dist (?s n) a < r" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

639 
unfolding dist_eq_0_iff[of "?s n" a, symmetric] 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

640 
using rp by (simp del: dist_eq_0_iff)} 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

641 
moreover 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

642 
{assume neq: "?s n \<noteq> a" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

643 
from fps_eq_least_unique[OF neq] 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

644 
obtain k where k: "leastP (\<lambda>i. ?s n $ i \<noteq> a$i) k" by blast 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

645 
have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a$n \<noteq> b$n)" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

646 
by (simp add: fps_eq_iff) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

647 
from neq have dth: "dist (?s n) a = (1/2)^k" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

648 
unfolding th0 dist_fps_def 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

649 
unfolding the1_equality[OF fps_eq_least_unique[OF neq], OF k] 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

650 
by (auto simp add: inverse_eq_divide power_divide) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

651 

69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

652 
from k have kn: "k > n" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

653 
by (simp add: leastP_def setge_def fps_sum_rep_nth split:split_if_asm) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

654 
then have "dist (?s n) a < (1/2)^n" unfolding dth 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

655 
by (auto intro: power_strict_decreasing) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

656 
also have "\<dots> <= (1/2)^n0" using nn0 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

657 
by (auto intro: power_decreasing) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

658 
also have "\<dots> < r" using n0 by simp 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

659 
finally have "dist (?s n) a < r" .} 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

660 
ultimately have "dist (?s n) a < r" by blast} 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

661 
then have "\<exists>n0. \<forall> n \<ge> n0. dist (?s n) a < r " by blast} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

662 
then show ?thesis unfolding LIMSEQ_def by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

663 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

664 

29906  665 
subsection{* Inverses of formal power series *} 
29687  666 

667 
declare setsum_cong[fundef_cong] 

668 

36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

669 
instantiation fps :: ("{comm_monoid_add, inverse, times, uminus}") inverse 
29687  670 
begin 
671 

30488  672 
fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where 
29687  673 
"natfun_inverse f 0 = inverse (f$0)" 
30488  674 
 "natfun_inverse f n =  inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n}" 
29687  675 

30488  676 
definition fps_inverse_def: 
36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

677 
"inverse f = (if f $ 0 = 0 then 0 else Abs_fps (natfun_inverse f))" 
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

678 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

679 
definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)" 
36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

680 

29687  681 
instance .. 
36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

682 

29687  683 
end 
684 

30488  685 
lemma fps_inverse_zero[simp]: 
29687  686 
"inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

687 
by (simp add: fps_ext fps_inverse_def) 
29687  688 

689 
lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1" 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

690 
apply (auto simp add: expand_fps_eq fps_inverse_def) 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

691 
by (case_tac n, auto) 
29687  692 

36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset

693 
lemma inverse_mult_eq_1 [intro]: assumes f0: "f$0 \<noteq> (0::'a::field)" 
29687  694 
shows "inverse f * f = 1" 
695 
proof 

696 
have c: "inverse f * f = f * inverse f" by (simp add: mult_commute) 

30488  697 
from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n" 
29687  698 
by (simp add: fps_inverse_def) 
699 
from f0 have th0: "(inverse f * f) $ 0 = 1" 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

700 
by (simp add: fps_mult_nth fps_inverse_def) 
29687  701 
{fix n::nat assume np: "n >0 " 
702 
from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto 

703 
have d: "{0} \<inter> {1 .. n} = {}" by auto 

30488  704 
from f0 np have th0: " (inverse f$n) = 
29687  705 
(setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n}) / (f$0)" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

706 
by (cases n, simp, simp add: divide_inverse fps_inverse_def) 
29687  707 
from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]] 
30488  708 
have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n} = 
709 
 (f$0) * (inverse f)$n" 

36350  710 
by (simp add: field_simps) 
30488  711 
have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n  i))" 
29687  712 
unfolding fps_mult_nth ifn .. 
30488  713 
also have "\<dots> = f$0 * natfun_inverse f n 
29687  714 
+ (\<Sum>i = 1..n. f$i * natfun_inverse f (ni))" 
46757  715 
by (simp add: eq) 
29687  716 
also have "\<dots> = 0" unfolding th1 ifn by simp 
717 
finally have "(inverse f * f)$n = 0" unfolding c . } 

718 
with th0 show ?thesis by (simp add: fps_eq_iff) 

719 
qed 

720 

721 
lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0" 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

722 
by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero) 
29687  723 

724 
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0" 

725 
proof 

726 
{assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)} 

727 
moreover 

728 
{assume h: "inverse f = 0" and c: "f $0 \<noteq> 0" 

729 
from inverse_mult_eq_1[OF c] h have False by simp} 

730 
ultimately show ?thesis by blast 

731 
qed 

732 

48757  733 
lemma fps_inverse_idempotent[intro]: 
734 
assumes f0: "f$0 \<noteq> (0::'a::field)" 

29687  735 
shows "inverse (inverse f) = f" 
736 
proof 

737 
from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp 

30488  738 
from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] 
29687  739 
have th0: "inverse f * f = inverse f * inverse (inverse f)" by (simp add: mult_ac) 
740 
then show ?thesis using f0 unfolding mult_cancel_left by simp 

741 
qed 

742 

48757  743 
lemma fps_inverse_unique: 
744 
assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1" 

29687  745 
shows "inverse f = g" 
746 
proof 

747 
from inverse_mult_eq_1[OF f0] fg 

748 
have th0: "inverse f * f = g * f" by (simp add: mult_ac) 

749 
then show ?thesis using f0 unfolding mult_cancel_right 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

750 
by (auto simp add: expand_fps_eq) 
29687  751 
qed 
752 

30488  753 
lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
29687  754 
= Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then  1 else 0)" 
755 
apply (rule fps_inverse_unique) 

756 
apply simp 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

757 
apply (simp add: fps_eq_iff fps_mult_nth) 
29687  758 
proof(clarsimp) 
759 
fix n::nat assume n: "n > 0" 

760 
let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n  i = 1 then  1 else 0" 

761 
let ?g = "\<lambda>i. if i = n then 1 else if i=n  1 then  1 else 0" 

762 
let ?h = "\<lambda>i. if i=n  1 then  1 else 0" 

30488  763 
have th1: "setsum ?f {0..n} = setsum ?g {0..n}" 
29687  764 
by (rule setsum_cong2) auto 
30488  765 
have th2: "setsum ?g {0..n  1} = setsum ?h {0..n  1}" 
29687  766 
using n apply  by (rule setsum_cong2) auto 
767 
have eq: "{0 .. n} = {0.. n  1} \<union> {n}" by auto 

30488  768 
from n have d: "{0.. n  1} \<inter> {n} = {}" by auto 
29687  769 
have f: "finite {0.. n  1}" "finite {n}" by auto 
770 
show "setsum ?f {0..n} = 0" 

30488  771 
unfolding th1 
29687  772 
apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def) 
773 
unfolding th2 

774 
by(simp add: setsum_delta) 

775 
qed 

776 

29912  777 
subsection{* Formal Derivatives, and the MacLaurin theorem around 0*} 
29687  778 

779 
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))" 

780 

48757  781 
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" 
782 
by (simp add: fps_deriv_def) 

783 

784 
lemma fps_deriv_linear[simp]: 

785 
"fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = 

786 
fps_const a * fps_deriv f + fps_const b * fps_deriv g" 

36350  787 
unfolding fps_eq_iff fps_add_nth fps_const_mult_left fps_deriv_nth by (simp add: field_simps) 
29687  788 

30488  789 
lemma fps_deriv_mult[simp]: 
29687  790 
fixes f :: "('a :: comm_ring_1) fps" 
791 
shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g" 

792 
proof 

793 
let ?D = "fps_deriv" 

794 
{fix n::nat 

795 
let ?Zn = "{0 ..n}" 

796 
let ?Zn1 = "{0 .. n + 1}" 

797 
let ?f = "\<lambda>i. i + 1" 

798 
have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def) 

799 
have eq: "{1.. n+1} = ?f ` {0..n}" by auto 

800 
let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n  i) + 

801 
of_nat (i+1)* f $ (i+1) * g $ (n  i)" 

802 
let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1)  i) + 

803 
of_nat i* f $ i * g $ ((n + 1)  i)" 

804 
{fix k assume k: "k \<in> {0..n}" 

805 
have "?h (k + 1) = ?g k" using k by auto} 

806 
note th0 = this 

807 
have eq': "{0..n +1} {1 .. n+1} = {0}" by auto 

808 
have s0: "setsum (\<lambda>i. of_nat i * f $ i * g $ (n + 1  i)) ?Zn1 = setsum (\<lambda>i. of_nat (n + 1  i) * f $ (n + 1  i) * g $ i) ?Zn1" 

809 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"]) 

810 
apply (simp add: inj_on_def Ball_def) 

811 
apply presburger 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

812 
apply (rule set_eqI) 
29687  813 
apply (presburger add: image_iff) 
814 
by simp 

815 
have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1  i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1  i) * g $ i) ?Zn1" 

816 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"]) 

817 
apply (simp add: inj_on_def Ball_def) 

818 
apply presburger 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

819 
apply (rule set_eqI) 
29687  820 
apply (presburger add: image_iff) 
821 
by simp 

822 
have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute) 

823 
also have "\<dots> = (\<Sum>i = 0..n. ?g i)" 

824 
by (simp add: fps_mult_nth setsum_addf[symmetric]) 

825 
also have "\<dots> = setsum ?h {1..n+1}" 

826 
using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto 

827 
also have "\<dots> = setsum ?h {0..n+1}" 

828 
apply (rule setsum_mono_zero_left) 

829 
apply simp 

830 
apply (simp add: subset_eq) 

831 
unfolding eq' 

832 
by simp 

833 
also have "\<dots> = (fps_deriv (f * g)) $ n" 

834 
apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf) 

835 
unfolding s0 s1 

836 
unfolding setsum_addf[symmetric] setsum_right_distrib 

837 
apply (rule setsum_cong2) 

36350  838 
by (auto simp add: of_nat_diff field_simps) 
29687  839 
finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .} 
30488  840 
then show ?thesis unfolding fps_eq_iff by auto 
29687  841 
qed 
842 

31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

843 
lemma fps_deriv_X[simp]: "fps_deriv X = 1" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

844 
by (simp add: fps_deriv_def X_def fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

845 

29687  846 
lemma fps_deriv_neg[simp]: "fps_deriv ( (f:: ('a:: comm_ring_1) fps)) =  (fps_deriv f)" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

847 
by (simp add: fps_eq_iff fps_deriv_def) 
29687  848 
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g" 
849 
using fps_deriv_linear[of 1 f 1 g] by simp 

850 

851 
lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps)  g) = fps_deriv f  fps_deriv g" 

30488  852 
unfolding diff_minus by simp 
29687  853 

854 
lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0" 

29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

855 
by (simp add: fps_ext fps_deriv_def fps_const_def) 
29687  856 

48757  857 
lemma fps_deriv_mult_const_left[simp]: 
858 
"fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f" 

29687  859 
by simp 
860 

861 
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0" 

862 
by (simp add: fps_deriv_def fps_eq_iff) 

863 

864 
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0" 

865 
by (simp add: fps_deriv_def fps_eq_iff ) 

866 

48757  867 
lemma fps_deriv_mult_const_right[simp]: 
868 
"fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c" 

29687  869 
by simp 
870 

48757  871 
lemma fps_deriv_setsum: 
872 
"fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S" 

29687  873 
proof 
48757  874 
{ assume "\<not> finite S" hence ?thesis by simp } 
29687  875 
moreover 
48757  876 
{ 
877 
assume fS: "finite S" 

878 
have ?thesis by (induct rule: finite_induct[OF fS]) simp_all 

879 
} 

29687  880 
ultimately show ?thesis by blast 
881 
qed 

882 

48757  883 
lemma fps_deriv_eq_0_iff[simp]: 
884 
"fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))" 

29687  885 
proof 
886 
{assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp 

887 
hence "fps_deriv f = 0" by simp } 

888 
moreover 

889 
{assume z: "fps_deriv f = 0" 

890 
hence "\<forall>n. (fps_deriv f)$n = 0" by simp 

891 
hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def) 

892 
hence "f = fps_const (f$0)" 

893 
apply (clarsimp simp add: fps_eq_iff fps_const_def) 

894 
apply (erule_tac x="n  1" in allE) 

895 
by simp} 

896 
ultimately show ?thesis by blast 

897 
qed 

898 

30488  899 
lemma fps_deriv_eq_iff: 
29687  900 
fixes f:: "('a::{idom,semiring_char_0}) fps" 
901 
shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0  g$0) + g)" 

902 
proof 

903 
have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f  g) = 0" by simp 

904 
also have "\<dots> \<longleftrightarrow> f  g = fps_const ((fg)$0)" unfolding fps_deriv_eq_0_iff .. 

36350  905 
finally show ?thesis by (simp add: field_simps) 
29687  906 
qed 
907 

48757  908 
lemma fps_deriv_eq_iff_ex: 
909 
"(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>(c::'a::{idom,semiring_char_0}). f = fps_const c + g)" 

910 
apply auto unfolding fps_deriv_eq_iff 

911 
apply blast 

912 
done 

913 

914 

915 
fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" 

916 
where 

29687  917 
"fps_nth_deriv 0 f = f" 
918 
 "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)" 

919 

920 
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)" 

48757  921 
by (induct n arbitrary: f) auto 
922 

923 
lemma fps_nth_deriv_linear[simp]: 

924 
"fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = 

925 
fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g" 

926 
by (induct n arbitrary: f g) (auto simp add: fps_nth_deriv_commute) 

927 

928 
lemma fps_nth_deriv_neg[simp]: 

929 
"fps_nth_deriv n ( (f:: ('a:: comm_ring_1) fps)) =  (fps_nth_deriv n f)" 

930 
by (induct n arbitrary: f) simp_all 

931 

932 
lemma fps_nth_deriv_add[simp]: 

933 
"fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g" 

29687  934 
using fps_nth_deriv_linear[of n 1 f 1 g] by simp 
935 

48757  936 
lemma fps_nth_deriv_sub[simp]: 
937 
"fps_nth_deriv n ((f:: ('a::comm_ring_1) fps)  g) = fps_nth_deriv n f  fps_nth_deriv n g" 

30488  938 
unfolding diff_minus fps_nth_deriv_add by simp 
29687  939 

940 
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0" 

48757  941 
by (induct n) simp_all 
29687  942 

943 
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)" 

48757  944 
by (induct n) simp_all 
945 

946 
lemma fps_nth_deriv_const[simp]: 

947 
"fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)" 

948 
by (cases n) simp_all 

949 

950 
lemma fps_nth_deriv_mult_const_left[simp]: 

951 
"fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f" 

29687  952 
using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp 
953 

48757  954 
lemma fps_nth_deriv_mult_const_right[simp]: 
955 
"fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c" 

29687  956 
using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute) 
957 

48757  958 
lemma fps_nth_deriv_setsum: 
959 
"fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: ('a::comm_ring_1) fps)) S" 

29687  960 
proof 
48757  961 
{ assume "\<not> finite S" hence ?thesis by simp } 
29687  962 
moreover 
48757  963 
{ 
964 
assume fS: "finite S" 

965 
have ?thesis by (induct rule: finite_induct[OF fS]) simp_all 

966 
} 

29687  967 
ultimately show ?thesis by blast 
968 
qed 

969 

48757  970 
lemma fps_deriv_maclauren_0: 
971 
"(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)" 

36350  972 
by (induct k arbitrary: f) (auto simp add: field_simps of_nat_mult) 
29687  973 

29906  974 
subsection {* Powers*} 
29687  975 

976 
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)" 

48757  977 
by (induct n) (auto simp add: expand_fps_eq fps_mult_nth) 
29687  978 

979 
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1" 

980 
proof(induct n) 

30960  981 
case 0 thus ?case by simp 
29687  982 
next 
983 
case (Suc n) 

984 
note h = Suc.hyps[OF `a$0 = 1`] 

30488  985 
show ?case unfolding power_Suc fps_mult_nth 
36350  986 
using h `a$0 = 1` fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: field_simps) 
29687  987 
qed 
988 

989 
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1" 

48757  990 
by (induct n) (auto simp add: fps_mult_nth) 
29687  991 

992 
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0" 

48757  993 
by (induct n) (auto simp add: fps_mult_nth) 
29687  994 

31021  995 
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1}) \<Longrightarrow> a^n $0 = v^n" 
48757  996 
by (induct n) (auto simp add: fps_mult_nth power_Suc) 
29687  997 

998 
lemma startsby_zero_power_iff[simp]: 

31021  999 
"a^n $0 = (0::'a::{idom}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)" 
29687  1000 
apply (rule iffI) 
48757  1001 
apply (induct n) 
1002 
apply (auto simp add: fps_mult_nth) 

1003 
apply (rule startsby_zero_power, simp_all) 

1004 
done 

29687  1005 

30488  1006 
lemma startsby_zero_power_prefix: 
29687  1007 
assumes a0: "a $0 = (0::'a::idom)" 
1008 
shows "\<forall>n < k. a ^ k $ n = 0" 

30488  1009 
using a0 
29687  1010 
proof(induct k rule: nat_less_induct) 
1011 
fix k assume H: "\<forall>m<k. a $0 = 0 \<longrightarrow> (\<forall>n<m. a ^ m $ n = 0)" and a0: "a $0 = (0\<Colon>'a)" 

1012 
let ?ths = "\<forall>m<k. a ^ k $ m = 0" 

1013 
{assume "k = 0" then have ?ths by simp} 

1014 
moreover 

1015 
{fix l assume k: "k = Suc l" 

1016 
{fix m assume mk: "m < k" 

30488  1017 
{assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1018 
by simp} 
29687  1019 
moreover 
1020 
{assume m0: "m \<noteq> 0" 

32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1021 
have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1022 
also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m  i))" by (simp add: fps_mult_nth) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1023 
also have "\<dots> = 0" apply (rule setsum_0') 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1024 
apply auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1025 
apply (case_tac "aa = m") 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1026 
using a0 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1027 
apply simp 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1028 
apply (rule H[rule_format]) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1029 
using a0 k mk by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1030 
finally have "a^k $ m = 0" .} 
29687  1031 
ultimately have "a^k $ m = 0" by blast} 
1032 
hence ?ths by blast} 

1033 
ultimately show ?ths by (cases k, auto) 

1034 
qed 

1035 

30488  1036 
lemma startsby_zero_setsum_depends: 
29687  1037 
assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k" 
1038 
shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}" 

1039 
apply (rule setsum_mono_zero_right) 

1040 
using kn apply auto 

1041 
apply (rule startsby_zero_power_prefix[rule_format, OF a0]) 

1042 
by arith 

1043 

31021  1044 
lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{idom})" 
29687  1045 
shows "a^n $ n = (a$1) ^ n" 
1046 
proof(induct n) 

1047 
case 0 thus ?case by (simp add: power_0) 

1048 
next 

1049 
case (Suc n) 

36350  1050 
have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: field_simps power_Suc) 
29687  1051 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {0.. Suc n}" by (simp add: fps_mult_nth) 
1052 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {n .. Suc n}" 

1053 
apply (rule setsum_mono_zero_right) 

1054 
apply simp 

1055 
apply clarsimp 

1056 
apply clarsimp 

1057 
apply (rule startsby_zero_power_prefix[rule_format, OF a0]) 

1058 
apply arith 

1059 
done 

1060 
also have "\<dots> = a^n $ n * a$1" using a0 by simp 

1061 
finally show ?case using Suc.hyps by (simp add: power_Suc) 

1062 
qed 

1063 

1064 
lemma fps_inverse_power: 

31021  1065 
fixes a :: "('a::{field}) fps" 
29687  1066 
shows "inverse (a^n) = inverse a ^ n" 
1067 
proof 

1068 
{assume a0: "a$0 = 0" 

1069 
hence eq: "inverse a = 0" by (simp add: fps_inverse_def) 

1070 
{assume "n = 0" hence ?thesis by simp} 

1071 
moreover 

1072 
{assume n: "n > 0" 

30488  1073 
from startsby_zero_power[OF a0 n] eq a0 n have ?thesis 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1074 
by (simp add: fps_inverse_def)} 
29687  1075 
ultimately have ?thesis by blast} 
1076 
moreover 

1077 
{assume a0: "a$0 \<noteq> 0" 

1078 
have ?thesis 

1079 
apply (rule fps_inverse_unique) 

1080 
apply (simp add: a0) 

1081 
unfolding power_mult_distrib[symmetric] 

1082 
apply (rule ssubst[where t = "a * inverse a" and s= 1]) 

1083 
apply simp_all 

1084 
apply (subst mult_commute) 

1085 
by (rule inverse_mult_eq_1[OF a0])} 

1086 
ultimately show ?thesis by blast 

1087 
qed 

1088 

48757  1089 
lemma fps_deriv_power: 
1090 
"fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n  1)" 

1091 
apply (induct n) 

1092 
apply (auto simp add: power_Suc field_simps fps_const_add[symmetric] simp del: fps_const_add) 

1093 
apply (case_tac n) 

1094 
apply (auto simp add: power_Suc field_simps) 

1095 
done 

29687  1096 

30488  1097 
lemma fps_inverse_deriv: 
29687  1098 
fixes a:: "('a :: field) fps" 
1099 
assumes a0: "a$0 \<noteq> 0" 

1100 
shows "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2" 

1101 
proof 

1102 
from inverse_mult_eq_1[OF a0] 

1103 
have "fps_deriv (inverse a * a) = 0" by simp 

1104 
hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp 

1105 
hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0" by simp 

1106 
with inverse_mult_eq_1[OF a0] 

1107 
have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0" 

1108 
unfolding power2_eq_square 

36350  1109 
apply (simp add: field_simps) 
29687  1110 
by (simp add: mult_assoc[symmetric]) 
1111 
hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a)  fps_deriv a * inverse a ^ 2 = 0  fps_deriv a * inverse a ^ 2" 

1112 
by simp 

36350  1113 
then show "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2" by (simp add: field_simps) 
29687  1114 
qed 
1115 

30488  1116 
lemma fps_inverse_mult: 
29687  1117 
fixes a::"('a :: field) fps" 
1118 
shows "inverse (a * b) = inverse a * inverse b" 

1119 
proof 

1120 
{assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) 

1121 
from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all 

1122 
have ?thesis unfolding th by simp} 

1123 
moreover 

1124 
{assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) 

1125 
from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all 

1126 
have ?thesis unfolding th by simp} 

1127 
moreover 

1128 
{assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0" 

1129 
from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp add: fps_mult_nth) 

30488  1130 
from inverse_mult_eq_1[OF ab0] 
29687  1131 
have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp 
1132 
then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b" 

36350  1133 
by (simp add: field_simps) 
29687  1134 
then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp} 
1135 
ultimately show ?thesis by blast 

1136 
qed 

1137 

30488  1138 
lemma fps_inverse_deriv': 
29687  1139 
fixes a:: "('a :: field) fps" 
1140 
assumes a0: "a$0 \<noteq> 0" 

1141 
shows "fps_deriv (inverse a) =  fps_deriv a / a ^ 2" 

1142 
using fps_inverse_deriv[OF a0] 

48757  1143 
unfolding power2_eq_square fps_divide_def fps_inverse_mult 
1144 
by simp 

29687  1145 

1146 
lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)" 

1147 
shows "f * inverse f= 1" 

1148 
by (metis mult_commute inverse_mult_eq_1 f0) 

1149 

1150 
lemma fps_divide_deriv: fixes a:: "('a :: field) fps" 

1151 
assumes a0: "b$0 \<noteq> 0" 

1152 
shows "fps_deriv (a / b) = (fps_deriv a * b  a * fps_deriv b) / b ^ 2" 

1153 
using fps_inverse_deriv[OF a0] 

48757  1154 
by (simp add: fps_divide_def field_simps 
1155 
power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0]) 

30488  1156 

29687  1157 

30488  1158 
lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
29687  1159 
= 1  X" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

1160 
by (simp add: fps_inverse_gp fps_eq_iff X_def) 
29687  1161 

1162 
lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)" 

1163 
by (cases "n", simp_all) 

1164 

1165 

1166 
lemma fps_inverse_X_plus1: 

31021  1167 
"inverse (1 + X) = Abs_fps (\<lambda>n. ( (1::'a::{field})) ^ n)" (is "_ = ?r") 
29687  1168 
proof 
1169 
have eq: "(1 + X) * ?r = 1" 

1170 
unfolding minus_one_power_iff 

36350  1171 
by (auto simp add: field_simps fps_eq_iff) 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1172 
show ?thesis by (auto simp add: eq intro: fps_inverse_unique simp del: minus_one) 
29687  1173 
qed 
1174 

30488  1175 

29906  1176 
subsection{* Integration *} 
31273  1177 

1178 
definition 

1179 
fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps" where 

1180 
"fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n  1) / of_nat n))" 

29687  1181 

31273  1182 
lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a" 
1183 
unfolding fps_integral_def fps_deriv_def 

1184 
by (simp add: fps_eq_iff del: of_nat_Suc) 

29687  1185 

31273  1186 
lemma fps_integral_linear: 
1187 
"fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) = 

1188 
fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" 

1189 
(is "?l = ?r") 

29687  1190 
proof 
1191 
have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral) 

1192 
moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def) 

1193 
ultimately show ?thesis 

1194 
unfolding fps_deriv_eq_iff by auto 

1195 
qed 

30488  1196 

29906  1197 
subsection {* Composition of FPSs *} 
29687  1198 
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where 
1199 
fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})" 

1200 

48757  1201 
lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. a$i * (b^i$n)) {0..n}" 
1202 
by (simp add: fps_compose_def) 

29687  1203 

1204 
lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)" 

29913  1205 
by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta') 
30488  1206 

1207 
lemma fps_const_compose[simp]: 

29687  1208 
"fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)" 
29913  1209 
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta) 
29687  1210 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1211 
lemma numeral_compose[simp]: "(numeral k::('a::{comm_ring_1}) fps) oo b = numeral k" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1212 
unfolding numeral_fps_const by simp 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1213 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1214 
lemma neg_numeral_compose[simp]: "(neg_numeral k::('a::{comm_ring_1}) fps) oo b = neg_numeral k" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset

1215 
unfolding neg_numeral_fps_const by simp 
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

1216 

29687  1217 
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)" 
29913  1218 
by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta 
1219 
power_Suc not_le) 

29687  1220 

1221 

29906  1222 
subsection {* Rules from Herbert Wilf's Generatingfunctionology*} 
29687  1223 

29906  1224 
subsubsection {* Rule 1 *} 
29687  1225 
(* {a_{n+k}}_0^infty Corresponds to (f  setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*) 
1226 

30488  1227 
lemma fps_power_mult_eq_shift: 
30992  1228 
"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") 
29687  1229 
proof 
1230 
{fix n:: nat 

30488  1231 
have "?lhs $ n = (if n < Suc k then 0 else a n)" 
29687  1232 
unfolding X_power_mult_nth by auto 
1233 
also have "\<dots> = ?rhs $ n" 

1234 
proof(induct k) 

1235 
case 0 thus ?case by (simp add: fps_setsum_nth power_Suc) 

1236 
next 

1237 
case (Suc k) 

1238 
note th = Suc.hyps[symmetric] 

36350  1239 
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: field_simps) 
29687  1240 
also have "\<dots> = (if n < Suc k then 0 else a n)  (fps_const (a (Suc k)) * X^ Suc k)$n" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1241 
using th 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1242 
unfolding fps_sub_nth by simp 
29687  1243 
also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1244 
unfolding X_power_mult_right_nth 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1245 
apply (auto simp add: not_less fps_const_def) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1246 
apply (rule cong[of a a, OF refl]) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32456
diff
changeset

1247 
by arith 
29687  1248 
finally show ?case by simp 
1249 
qed 

1250 
finally have "?lhs $ n = ?rhs $ n" .} 

1251 
then show ?thesis by (simp add: fps_eq_iff) 

1252 
qed 

1253 

29906  1254 
subsubsection{* Rule 2*} 
29687  1255 

1256 
(* We can not reach the form of Wilf, but still near to it using rewrite rules*) 

30488  1257 
(* If f reprents {a_n} and P is a polynomial, then 
29687  1258 
P(xD) f represents {P(n) a_n}*) 
1259 

1260 
definition "XD = op * X o fps_deriv" 

1261 

1262 
lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)" 

36350  1263 
by (simp add: XD_def field_simps) 
29687  1264 

1265 
lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a" 

36350  1266 
by (simp add: XD_def field_simps) 
29687  1267 

1268 
lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) = fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)" 

1269 
by simp 

