author  chaieb 
Thu, 09 Jul 2009 10:34:51 +0200  
changeset 31968  0314441a53a6 
parent 31790  05c92381363c 
child 32042  df28ead1cf19 
child 32157  adea7a729c7a 
permissions  rwrr 
29687  1 
(* Title: 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 
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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] expand_fun_eq) 
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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, 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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unfolding fps_zero_def by simp 
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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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unfolding fps_one_def by simp 
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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 
189 
qed 

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instance fps :: (monoid_add) monoid_add 
29687  192 
proof 
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fix a :: "'a fps" show "0 + a = a " 

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194 
by (simp add: fps_ext) 
29687  195 
next 
196 
fix a :: "'a fps" show "a + 0 = a " 

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197 
by (simp add: fps_ext) 
29687  198 
qed 
199 

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200 
instance fps :: (comm_monoid_add) comm_monoid_add 
29687  201 
proof 
202 
fix a :: "'a fps" show "0 + a = a " 

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203 
by (simp add: fps_ext) 
29687  204 
qed 
205 

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206 
instance fps :: (semiring_1) monoid_mult 
29687  207 
proof 
208 
fix a :: "'a fps" show "1 * a = a" 

29913  209 
by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta) 
29687  210 
next 
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211 
fix a :: "'a fps" show "a * 1 = a" 
29913  212 
by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta') 
29687  213 
qed 
214 

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215 
instance fps :: (cancel_semigroup_add) cancel_semigroup_add 
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216 
proof 
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217 
fix a b c :: "'a fps" 
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218 
assume "a + b = a + c" then show "b = c" 
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219 
by (simp add: expand_fps_eq) 
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220 
next 
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221 
fix a b c :: "'a fps" 
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222 
assume "b + a = c + a" then show "b = c" 
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223 
by (simp add: expand_fps_eq) 
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224 
qed 
29687  225 

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226 
instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add 
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227 
proof 
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228 
fix a b c :: "'a fps" 
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229 
assume "a + b = a + c" then show "b = c" 
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230 
by (simp add: expand_fps_eq) 
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231 
qed 
29687  232 

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233 
instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add .. 
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234 

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235 
instance fps :: (group_add) group_add 
29687  236 
proof 
237 
fix a :: "'a fps" show " a + a = 0" 

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238 
by (simp add: fps_ext) 
29687  239 
next 
240 
fix a b :: "'a fps" show "a  b = a +  b" 

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241 
by (simp add: fps_ext diff_minus) 
29687  242 
qed 
243 

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244 
instance fps :: (ab_group_add) ab_group_add 
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245 
proof 
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246 
fix a :: "'a fps" 
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247 
show " a + a = 0" 
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248 
by (simp add: fps_ext) 
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249 
next 
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250 
fix a b :: "'a fps" 
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251 
show "a  b = a +  b" 
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252 
by (simp add: fps_ext) 
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253 
qed 
29687  254 

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255 
instance fps :: (zero_neq_one) zero_neq_one 
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256 
by default (simp add: expand_fps_eq) 
29687  257 

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258 
instance fps :: (semiring_0) semiring 
29687  259 
proof 
260 
fix a b c :: "'a fps" 

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261 
show "(a + b) * c = a * c + b * c" 
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262 
by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf) 
29687  263 
next 
264 
fix a b c :: "'a fps" 

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265 
show "a * (b + c) = a * b + a * c" 
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266 
by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf) 
29687  267 
qed 
268 

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269 
instance fps :: (semiring_0) semiring_0 
29687  270 
proof 
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271 
fix a:: "'a fps" show "0 * a = 0" 
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272 
by (simp add: fps_ext fps_mult_nth) 
29687  273 
next 
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274 
fix a:: "'a fps" show "a * 0 = 0" 
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275 
by (simp add: fps_ext fps_mult_nth) 
29687  276 
qed 
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277 

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278 
instance fps :: (semiring_0_cancel) semiring_0_cancel .. 
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279 

29906  280 
subsection {* Selection of the nth power of the implicit variable in the infinite sum*} 
29687  281 

282 
lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)" 

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283 
by (simp add: expand_fps_eq) 
29687  284 

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285 
lemma fps_nonzero_nth_minimal: 
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286 
"f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0))" 
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287 
proof 
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288 
let ?n = "LEAST n. f $ n \<noteq> 0" 
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289 
assume "f \<noteq> 0" 
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290 
then have "\<exists>n. f $ n \<noteq> 0" 
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291 
by (simp add: fps_nonzero_nth) 
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292 
then have "f $ ?n \<noteq> 0" 
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293 
by (rule LeastI_ex) 
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294 
moreover have "\<forall>m<?n. f $ m = 0" 
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295 
by (auto dest: not_less_Least) 
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296 
ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" .. 
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297 
then show "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" .. 
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298 
next 
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299 
assume "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" 
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300 
then show "f \<noteq> 0" by (auto simp add: expand_fps_eq) 
29687  301 
qed 
302 

303 
lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)" 

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304 
by (rule expand_fps_eq) 
29687  305 

30488  306 
lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S" 
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307 
proof (cases "finite S") 
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308 
assume "\<not> finite S" then show ?thesis by simp 
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309 
next 
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310 
assume "finite S" 
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311 
then show ?thesis by (induct set: finite) auto 
29687  312 
qed 
313 

29906  314 
subsection{* Injection of the basic ring elements and multiplication by scalars *} 
29687  315 

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316 
definition 
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317 
"fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)" 
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318 

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319 
lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)" 
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320 
unfolding fps_const_def by simp 
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321 

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322 
lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0" 
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323 
by (simp add: fps_ext) 
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324 

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325 
lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1" 
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326 
by (simp add: fps_ext) 
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327 

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328 
lemma fps_const_neg [simp]: " (fps_const (c::'a::ring)) = fps_const ( c)" 
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329 
by (simp add: fps_ext) 
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330 

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331 
lemma fps_const_add [simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)" 
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332 
by (simp add: fps_ext) 
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
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333 
lemma fps_const_sub [simp]: "fps_const (c::'a\<Colon>group_add)  fps_const d = fps_const (c  d)" 
8b460fd12100
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334 
by (simp add: fps_ext) 
29687  335 
lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)" 
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336 
by (simp add: fps_eq_iff fps_mult_nth setsum_0') 
29687  337 

338 
lemma fps_const_add_left: "fps_const (c::'a\<Colon>monoid_add) + f = Abs_fps (\<lambda>n. if n = 0 then c + f$0 else f$n)" 

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339 
by (simp add: fps_ext) 
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340 

29687  341 
lemma fps_const_add_right: "f + fps_const (c::'a\<Colon>monoid_add) = Abs_fps (\<lambda>n. if n = 0 then f$0 + c else f$n)" 
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342 
by (simp add: fps_ext) 
29687  343 

344 
lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)" 

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345 
unfolding fps_eq_iff fps_mult_nth 
29913  346 
by (simp add: fps_const_def mult_delta_left setsum_delta) 
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347 

29687  348 
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)" 
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349 
unfolding fps_eq_iff fps_mult_nth 
29913  350 
by (simp add: fps_const_def mult_delta_right setsum_delta') 
29687  351 

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352 
lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n" 
29913  353 
by (simp add: fps_mult_nth mult_delta_left setsum_delta) 
29687  354 

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

29906  358 
subsection {* Formal power series form an integral domain*} 
29687  359 

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360 
instance fps :: (ring) ring .. 
29687  361 

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362 
instance fps :: (ring_1) ring_1 
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363 
by (intro_classes, auto simp add: diff_minus left_distrib) 
29687  364 

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

365 
instance fps :: (comm_ring_1) comm_ring_1 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
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diff
changeset

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

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

368 
instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors 
29687  369 
proof 
370 
fix a b :: "'a fps" 

371 
assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0" 

372 
then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0" 

373 
and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal 

374 
by blast+ 

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

375 
have "(a * b) $ (i+j) = (\<Sum>k=0..i+j. a$k * b$(i+jk))" 
29687  376 
by (rule fps_mult_nth) 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

377 
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:
29906
diff
changeset

378 
by (rule setsum_diff1') simp_all 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

379 
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

380 
proof (rule setsum_0' [rule_format]) 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

381 
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

382 
then have "k < i \<or> i+jk < j" by auto 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

383 
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:
29906
diff
changeset

384 
qed 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

385 
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:
29906
diff
changeset

386 
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

387 
finally have "(a*b) $ (i+j) \<noteq> 0" . 
29687  388 
then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast 
389 
qed 

390 

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

391 
instance fps :: (idom) idom .. 
29687  392 

30746  393 
instantiation fps :: (comm_ring_1) number_ring 
394 
begin 

395 
definition number_of_fps_def: "(number_of k::'a fps) = of_int k" 

396 

31273  397 
instance proof 
398 
qed (rule number_of_fps_def) 

30746  399 
end 
400 

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

401 
lemma number_of_fps_const: "(number_of k::('a::comm_ring_1) fps) = fps_const (of_int k)" 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

402 

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

403 
proof(induct k rule: int_induct[where k=0]) 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

404 
case base thus ?case unfolding number_of_fps_def of_int_0 by simp 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

405 
next 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

406 
case (step1 i) thus ?case unfolding number_of_fps_def 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

407 
by (simp add: fps_const_add[symmetric] del: fps_const_add) 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

408 
next 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

409 
case (step2 i) thus ?case unfolding number_of_fps_def 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

410 
by (simp add: fps_const_sub[symmetric] del: fps_const_sub) 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

411 
qed 
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

412 
subsection{* The eXtractor series X*} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

413 

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

414 
lemma minus_one_power_iff: "( (1::'a :: {comm_ring_1})) ^ n = (if even n then 1 else  1)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

415 
by (induct n, auto) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

416 

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

417 
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:
31790
diff
changeset

418 
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:
31790
diff
changeset

419 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

420 
{assume n: "n \<noteq> 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

421 
have fN: "finite {0 .. n}" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

422 
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

423 
also have "\<dots> = f $ (n  1)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

424 
using n by (simp add: X_def mult_delta_left setsum_delta [OF fN]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

425 
finally have ?thesis using n by simp } 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

426 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

427 
{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

428 
ultimately show ?thesis by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

429 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

430 

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

431 
lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n  1))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

432 
by (metis X_mult_nth mult_commute) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

433 

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

434 
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:
31790
diff
changeset

435 
proof(induct k) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

436 
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:
31790
diff
changeset

437 
next 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

438 
case (Suc k) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

439 
{fix m 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

440 
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:
31790
diff
changeset

441 
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

442 
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

443 
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

444 
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

445 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

446 

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

447 
lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n  k))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

448 
apply (induct k arbitrary: n) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

449 
apply (simp) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

450 
unfolding power_Suc mult_assoc 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

451 
by (case_tac n, auto) 
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 
lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n  k))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

454 
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

455 

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

456 

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

457 

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

458 

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

459 
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

460 

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

461 
definition (in dist) ball_def: "ball x r = {y. dist y x < r}" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

462 
instantiation fps :: (comm_ring_1) dist 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

463 
begin 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

464 

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

465 
definition dist_fps_def: "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)" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

466 

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

467 
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

468 
by (simp add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

469 

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

470 
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

471 
apply (auto simp add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

472 
thm cong[OF refl] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

473 
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

474 
apply (rule ext) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

475 
by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

476 
instance .. 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

477 
end 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

478 

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

479 
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

480 
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

481 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

482 
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

483 
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

484 
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

485 
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

486 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

487 
{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

488 
then have "m = n" using ln 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

489 
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

490 
apply (erule allE[where x=n]) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

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

492 
by simp} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

493 
ultimately show ?thesis by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

494 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

495 

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

496 
lemma fps_eq_least_unique: assumes ab: "(a::('a::ab_group_add) fps) \<noteq> b" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

497 
shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> b$n) n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

498 
using fps_nonzero_least_unique[of "a  b"] ab 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

499 
by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

500 

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

501 
instantiation fps :: (comm_ring_1) metric_space 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

502 
begin 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

503 

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

504 
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

505 

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

506 
instance 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

507 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

508 
fix S :: "'a fps set" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

509 
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

510 
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

511 
next 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

512 
{ fix a b :: "'a fps" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

513 
{assume ab: "a = b" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

514 
then have "\<not> (\<exists>n. a$n \<noteq> b$n)" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

515 
then have "dist a b = 0" by (simp add: dist_fps_def)} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

516 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

517 
{assume d: "dist a b = 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

518 
then have "\<forall>n. a$n = b$n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

519 
by  (rule ccontr, simp add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

520 
then have "a = b" by (simp add: fps_eq_iff)} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

521 
ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

522 
note th = this 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

523 
from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

524 
fix a b c :: "'a fps" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

525 
{assume ab: "a = b" then have d0: "dist a b = 0" unfolding th . 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

526 
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

527 
using dist_fps_ge0[of a c] dist_fps_ge0[of b c] by simp} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

528 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

529 
{assume c: "c = a \<or> c = b" 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

530 
by (cases "c=a", simp_all add: th dist_fps_sym) } 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

531 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

532 
{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

533 
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

534 
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

535 
fps_eq_least_unique[OF bc] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

536 
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

537 
and nac: "leastP (?P a c) nac" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

538 
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

539 
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

540 
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

541 
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

542 
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

543 
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

544 
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

545 

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

546 
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

547 
by (simp add: fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

548 
from ab ac bc nab nac nbc 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

549 
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

550 
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

551 
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

552 
unfolding th0 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

553 
apply (simp_all add: dist_fps_def) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

554 
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

555 
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

556 
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

557 
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

558 
unfolding th by simp_all 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

559 
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

560 
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

561 
by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

562 
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

563 
{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

564 
then have gt: "dist a b > dist a c" "dist a b > dist b c" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

565 
using pos by auto 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

566 
from gt have gtn: "nab < nbc" "nab < nac" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

567 
unfolding dab dbc dac by (auto simp add: th1) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

568 
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

569 
have "a$nab = b$nab" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

570 
with nab'(2) have False by simp} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

571 
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

572 
by (auto simp add: not_le[symmetric]) } 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

573 
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

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

575 

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

576 
end 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

577 

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

578 
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

579 

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

580 
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

581 
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

582 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

583 
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

584 
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

585 
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

586 
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

587 
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

588 
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

589 
using ln_gt_zero_iff[OF yp] y1 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

590 
by (simp add: minus_divide_left field_simps del:minus_divide_left[symmetric] ) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

591 
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

592 
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

593 
by (simp add: mult_ac) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

594 
then have "y ^ k * x > 1" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

595 
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

596 
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

597 
then have "x > (1/y)^k" using yp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

598 
by (simp add: field_simps nonzero_power_divide ) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

599 
then show ?thesis using kp by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

600 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

601 
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

602 
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

603 
by (simp add: X_power_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

604 

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

605 

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

606 
lemma fps_sum_rep_nth: "(setsum (%i. fps_const(a$i)*X^i) {0..m})$n = (if n \<le> m then a$n else (0::'a::comm_ring_1))" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

607 
apply (auto simp add: fps_eq_iff fps_setsum_nth X_power_nth cond_application_beta cond_value_iff cong del: if_weak_cong) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

608 
by (simp add: setsum_delta') 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

609 

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

610 
lemma fps_notation: 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

611 
"(%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

612 
proof 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

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

614 
assume rp: "r > 0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

615 
have th0: "(2::real) > 1" by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

616 
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

617 
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

618 
{fix n::nat 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

619 
assume nn0: "n \<ge> n0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

620 
then have thnn0: "(1/2)^n <= (1/2 :: real)^n0" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

621 
by (auto intro: power_decreasing) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

622 
{assume "?s n = a" then have "dist (?s n) a < r" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

623 
unfolding dist_eq_0_iff[of "?s n" a, symmetric] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

624 
using rp by (simp del: dist_eq_0_iff)} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

625 
moreover 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

626 
{assume neq: "?s n \<noteq> a" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

627 
from fps_eq_least_unique[OF neq] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

628 
obtain k where k: "leastP (\<lambda>i. ?s n $ i \<noteq> a$i) k" by blast 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

629 
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

630 
by (simp add: fps_eq_iff) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

631 
from neq have dth: "dist (?s n) a = (1/2)^k" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

632 
unfolding th0 dist_fps_def 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

633 
unfolding the1_equality[OF fps_eq_least_unique[OF neq], OF k] 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

634 
by (auto simp add: inverse_eq_divide power_divide) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

635 

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

636 
from k have kn: "k > n" 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

637 
apply (simp add: leastP_def setge_def fps_sum_rep_nth) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

638 
by (cases "k \<le> n", auto) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

639 
then have "dist (?s n) a < (1/2)^n" unfolding dth 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

640 
by (auto intro: power_strict_decreasing) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

641 
also have "\<dots> <= (1/2)^n0" using nn0 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

642 
by (auto intro: power_decreasing) 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

643 
also have "\<dots> < r" using n0 by simp 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

644 
finally have "dist (?s n) a < r" .} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

645 
ultimately have "dist (?s n) a < r" by blast} 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

646 
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

647 
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

648 
qed 
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset

649 

29906  650 
subsection{* Inverses of formal power series *} 
29687  651 

652 
declare setsum_cong[fundef_cong] 

653 

654 

655 
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse 

656 
begin 

657 

30488  658 
fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where 
29687  659 
"natfun_inverse f 0 = inverse (f$0)" 
30488  660 
 "natfun_inverse f n =  inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n}" 
29687  661 

30488  662 
definition fps_inverse_def: 
29687  663 
"inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

664 
definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)" 
29687  665 
instance .. 
666 
end 

667 

30488  668 
lemma fps_inverse_zero[simp]: 
29687  669 
"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

670 
by (simp add: fps_ext fps_inverse_def) 
29687  671 

672 
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

673 
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

674 
by (case_tac n, auto) 
29687  675 

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

676 
instance fps :: ("{comm_monoid_add,inverse, times, uminus}") division_by_zero 
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

677 
by default (rule fps_inverse_zero) 
29687  678 

679 
lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)" 

680 
shows "inverse f * f = 1" 

681 
proof 

682 
have c: "inverse f * f = f * inverse f" by (simp add: mult_commute) 

30488  683 
from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n" 
29687  684 
by (simp add: fps_inverse_def) 
685 
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

686 
by (simp add: fps_mult_nth fps_inverse_def) 
29687  687 
{fix n::nat assume np: "n >0 " 
688 
from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto 

689 
have d: "{0} \<inter> {1 .. n} = {}" by auto 

690 
have f: "finite {0::nat}" "finite {1..n}" by auto 

30488  691 
from f0 np have th0: " (inverse f$n) = 
29687  692 
(setsum (\<lambda>i. f$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

693 
by (cases n, simp, simp add: divide_inverse fps_inverse_def) 
29687  694 
from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]] 
30488  695 
have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n  i)) {1..n} = 
696 
 (f$0) * (inverse f)$n" 

29687  697 
by (simp add: ring_simps) 
30488  698 
have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n  i))" 
29687  699 
unfolding fps_mult_nth ifn .. 
30488  700 
also have "\<dots> = f$0 * natfun_inverse f n 
29687  701 
+ (\<Sum>i = 1..n. f$i * natfun_inverse f (ni))" 
702 
unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] 

703 
by simp 

704 
also have "\<dots> = 0" unfolding th1 ifn by simp 

705 
finally have "(inverse f * f)$n = 0" unfolding c . } 

706 
with th0 show ?thesis by (simp add: fps_eq_iff) 

707 
qed 

708 

709 
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

710 
by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero) 
29687  711 

712 
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0" 

713 
proof 

714 
{assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)} 

715 
moreover 

716 
{assume h: "inverse f = 0" and c: "f $0 \<noteq> 0" 

717 
from inverse_mult_eq_1[OF c] h have False by simp} 

718 
ultimately show ?thesis by blast 

719 
qed 

720 

721 
lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)" 

722 
shows "inverse (inverse f) = f" 

723 
proof 

724 
from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp 

30488  725 
from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] 
29687  726 
have th0: "inverse f * f = inverse f * inverse (inverse f)" by (simp add: mult_ac) 
727 
then show ?thesis using f0 unfolding mult_cancel_left by simp 

728 
qed 

729 

30488  730 
lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1" 
29687  731 
shows "inverse f = g" 
732 
proof 

733 
from inverse_mult_eq_1[OF f0] fg 

734 
have th0: "inverse f * f = g * f" by (simp add: mult_ac) 

735 
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

736 
by (auto simp add: expand_fps_eq) 
29687  737 
qed 
738 

30488  739 
lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
29687  740 
= Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then  1 else 0)" 
741 
apply (rule fps_inverse_unique) 

742 
apply simp 

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

743 
apply (simp add: fps_eq_iff fps_mult_nth) 
29687  744 
proof(clarsimp) 
745 
fix n::nat assume n: "n > 0" 

746 
let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n  i = 1 then  1 else 0" 

747 
let ?g = "\<lambda>i. if i = n then 1 else if i=n  1 then  1 else 0" 

748 
let ?h = "\<lambda>i. if i=n  1 then  1 else 0" 

30488  749 
have th1: "setsum ?f {0..n} = setsum ?g {0..n}" 
29687  750 
by (rule setsum_cong2) auto 
30488  751 
have th2: "setsum ?g {0..n  1} = setsum ?h {0..n  1}" 
29687  752 
using n apply  by (rule setsum_cong2) auto 
753 
have eq: "{0 .. n} = {0.. n  1} \<union> {n}" by auto 

30488  754 
from n have d: "{0.. n  1} \<inter> {n} = {}" by auto 
29687  755 
have f: "finite {0.. n  1}" "finite {n}" by auto 
756 
show "setsum ?f {0..n} = 0" 

30488  757 
unfolding th1 
29687  758 
apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def) 
759 
unfolding th2 

760 
by(simp add: setsum_delta) 

761 
qed 

762 

29912  763 
subsection{* Formal Derivatives, and the MacLaurin theorem around 0*} 
29687  764 

765 
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))" 

766 

767 
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def) 

768 

769 
lemma fps_deriv_linear[simp]: "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g" 

770 
unfolding fps_eq_iff fps_add_nth fps_const_mult_left fps_deriv_nth by (simp add: ring_simps) 

771 

30488  772 
lemma fps_deriv_mult[simp]: 
29687  773 
fixes f :: "('a :: comm_ring_1) fps" 
774 
shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g" 

775 
proof 

776 
let ?D = "fps_deriv" 

777 
{fix n::nat 

778 
let ?Zn = "{0 ..n}" 

779 
let ?Zn1 = "{0 .. n + 1}" 

780 
let ?f = "\<lambda>i. i + 1" 

781 
have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def) 

782 
have eq: "{1.. n+1} = ?f ` {0..n}" by auto 

783 
let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n  i) + 

784 
of_nat (i+1)* f $ (i+1) * g $ (n  i)" 

785 
let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1)  i) + 

786 
of_nat i* f $ i * g $ ((n + 1)  i)" 

787 
{fix k assume k: "k \<in> {0..n}" 

788 
have "?h (k + 1) = ?g k" using k by auto} 

789 
note th0 = this 

790 
have eq': "{0..n +1} {1 .. n+1} = {0}" by auto 

791 
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" 

792 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"]) 

793 
apply (simp add: inj_on_def Ball_def) 

794 
apply presburger 

795 
apply (rule set_ext) 

796 
apply (presburger add: image_iff) 

797 
by simp 

798 
have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1  i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1  i) * g $ i) ?Zn1" 

799 
apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1  i"]) 

800 
apply (simp add: inj_on_def Ball_def) 

801 
apply presburger 

802 
apply (rule set_ext) 

803 
apply (presburger add: image_iff) 

804 
by simp 

805 
have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute) 

806 
also have "\<dots> = (\<Sum>i = 0..n. ?g i)" 

807 
by (simp add: fps_mult_nth setsum_addf[symmetric]) 

808 
also have "\<dots> = setsum ?h {1..n+1}" 

809 
using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto 

810 
also have "\<dots> = setsum ?h {0..n+1}" 

811 
apply (rule setsum_mono_zero_left) 

812 
apply simp 

813 
apply (simp add: subset_eq) 

814 
unfolding eq' 

815 
by simp 

816 
also have "\<dots> = (fps_deriv (f * g)) $ n" 

817 
apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf) 

818 
unfolding s0 s1 

819 
unfolding setsum_addf[symmetric] setsum_right_distrib 

820 
apply (rule setsum_cong2) 

821 
by (auto simp add: of_nat_diff ring_simps) 

822 
finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .} 

30488  823 
then show ?thesis unfolding fps_eq_iff by auto 
29687  824 
qed 
825 

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

826 
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

827 
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

828 

29687  829 
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

830 
by (simp add: fps_eq_iff fps_deriv_def) 
29687  831 
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g" 
832 
using fps_deriv_linear[of 1 f 1 g] by simp 

833 

834 
lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps)  g) = fps_deriv f  fps_deriv g" 

30488  835 
unfolding diff_minus by simp 
29687  836 

837 
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

838 
by (simp add: fps_ext fps_deriv_def fps_const_def) 
29687  839 

840 
lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f" 

841 
by simp 

842 

843 
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0" 

844 
by (simp add: fps_deriv_def fps_eq_iff) 

845 

846 
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0" 

847 
by (simp add: fps_deriv_def fps_eq_iff ) 

848 

849 
lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c" 

850 
by simp 

851 

852 
lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S" 

853 
proof 

854 
{assume "\<not> finite S" hence ?thesis by simp} 

855 
moreover 

856 
{assume fS: "finite S" 

857 
have ?thesis by (induct rule: finite_induct[OF fS], simp_all)} 

858 
ultimately show ?thesis by blast 

859 
qed 

860 

861 
lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))" 

862 
proof 

863 
{assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp 

864 
hence "fps_deriv f = 0" by simp } 

865 
moreover 

866 
{assume z: "fps_deriv f = 0" 

867 
hence "\<forall>n. (fps_deriv f)$n = 0" by simp 

868 
hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def) 

869 
hence "f = fps_const (f$0)" 

870 
apply (clarsimp simp add: fps_eq_iff fps_const_def) 

871 
apply (erule_tac x="n  1" in allE) 

872 
by simp} 

873 
ultimately show ?thesis by blast 

874 
qed 

875 

30488  876 
lemma fps_deriv_eq_iff: 
29687  877 
fixes f:: "('a::{idom,semiring_char_0}) fps" 
878 
shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0  g$0) + g)" 

879 
proof 

880 
have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f  g) = 0" by simp 

881 
also have "\<dots> \<longleftrightarrow> f  g = fps_const ((fg)$0)" unfolding fps_deriv_eq_0_iff .. 

882 
finally show ?thesis by (simp add: ring_simps) 

883 
qed 

884 

885 
lemma fps_deriv_eq_iff_ex: "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>(c::'a::{idom,semiring_char_0}). f = fps_const c + g)" 

886 
apply auto unfolding fps_deriv_eq_iff by blast 

30488  887 

29687  888 

889 
fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where 

890 
"fps_nth_deriv 0 f = f" 

891 
 "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)" 

892 

893 
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)" 

894 
by (induct n arbitrary: f, auto) 

895 

896 
lemma fps_nth_deriv_linear[simp]: "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g" 

897 
by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute) 

898 

899 
lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n ( (f:: ('a:: comm_ring_1) fps)) =  (fps_nth_deriv n f)" 

900 
by (induct n arbitrary: f, simp_all) 

901 

902 
lemma fps_nth_deriv_add[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g" 

903 
using fps_nth_deriv_linear[of n 1 f 1 g] by simp 

904 

905 
lemma fps_nth_deriv_sub[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps)  g) = fps_nth_deriv n f  fps_nth_deriv n g" 

30488  906 
unfolding diff_minus fps_nth_deriv_add by simp 
29687  907 

908 
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0" 

909 
by (induct n, simp_all ) 

910 

911 
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)" 

912 
by (induct n, simp_all ) 

913 

914 
lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)" 

915 
by (cases n, simp_all) 

916 

917 
lemma fps_nth_deriv_mult_const_left[simp]: "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f" 

918 
using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp 

919 

920 
lemma fps_nth_deriv_mult_const_right[simp]: "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c" 

921 
using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute) 

922 

923 
lemma fps_nth_deriv_setsum: "fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: ('a::comm_ring_1) fps)) S" 

924 
proof 

925 
{assume "\<not> finite S" hence ?thesis by simp} 

926 
moreover 

927 
{assume fS: "finite S" 

928 
have ?thesis by (induct rule: finite_induct[OF fS], simp_all)} 

929 
ultimately show ?thesis by blast 

930 
qed 

931 

932 
lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)" 

933 
by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult) 

934 

29906  935 
subsection {* Powers*} 
29687  936 

937 
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)" 

30960  938 
by (induct n, auto simp add: expand_fps_eq fps_mult_nth) 
29687  939 

940 
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1" 

941 
proof(induct n) 

30960  942 
case 0 thus ?case by simp 
29687  943 
next 
944 
case (Suc n) 

945 
note h = Suc.hyps[OF `a$0 = 1`] 

30488  946 
show ?case unfolding power_Suc fps_mult_nth 
29687  947 
using h `a$0 = 1` fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps) 
948 
qed 

949 

950 
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1" 

30960  951 
by (induct n, auto simp add: fps_mult_nth) 
29687  952 

953 
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0" 

30960  954 
by (induct n, auto simp add: fps_mult_nth) 
29687  955 

31021  956 
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1}) \<Longrightarrow> a^n $0 = v^n" 
30960  957 
by (induct n, auto simp add: fps_mult_nth power_Suc) 
29687  958 

959 
lemma startsby_zero_power_iff[simp]: 

31021  960 
"a^n $0 = (0::'a::{idom}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)" 
29687  961 
apply (rule iffI) 
962 
apply (induct n, auto simp add: power_Suc fps_mult_nth) 

963 
by (rule startsby_zero_power, simp_all) 

964 

30488  965 
lemma startsby_zero_power_prefix: 
29687  966 
assumes a0: "a $0 = (0::'a::idom)" 
967 
shows "\<forall>n < k. a ^ k $ n = 0" 

30488  968 
using a0 
29687  969 
proof(induct k rule: nat_less_induct) 
970 
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)" 

971 
let ?ths = "\<forall>m<k. a ^ k $ m = 0" 

972 
{assume "k = 0" then have ?ths by simp} 

973 
moreover 

974 
{fix l assume k: "k = Suc l" 

975 
{fix m assume mk: "m < k" 

30488  976 
{assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0 
29687  977 
by simp} 
978 
moreover 

979 
{assume m0: "m \<noteq> 0" 

980 
have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute) 

981 
also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m  i))" by (simp add: fps_mult_nth) 

982 
also have "\<dots> = 0" apply (rule setsum_0') 

983 
apply auto 

984 
apply (case_tac "aa = m") 

985 
using a0 

986 
apply simp 

987 
apply (rule H[rule_format]) 

30488  988 
using a0 k mk by auto 
29687  989 
finally have "a^k $ m = 0" .} 
990 
ultimately have "a^k $ m = 0" by blast} 

991 
hence ?ths by blast} 

992 
ultimately show ?ths by (cases k, auto) 

993 
qed 

994 

30488  995 
lemma startsby_zero_setsum_depends: 
29687  996 
assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k" 
997 
shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}" 

998 
apply (rule setsum_mono_zero_right) 

999 
using kn apply auto 

1000 
apply (rule startsby_zero_power_prefix[rule_format, OF a0]) 

1001 
by arith 

1002 

31021  1003 
lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{idom})" 
29687  1004 
shows "a^n $ n = (a$1) ^ n" 
1005 
proof(induct n) 

1006 
case 0 thus ?case by (simp add: power_0) 

1007 
next 

1008 
case (Suc n) 

1009 
have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc) 

1010 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {0.. Suc n}" by (simp add: fps_mult_nth) 

1011 
also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n  i)) {n .. Suc n}" 

1012 
apply (rule setsum_mono_zero_right) 

1013 
apply simp 

1014 
apply clarsimp 

1015 
apply clarsimp 

1016 
apply (rule startsby_zero_power_prefix[rule_format, OF a0]) 

1017 
apply arith 

1018 
done 

1019 
also have "\<dots> = a^n $ n * a$1" using a0 by simp 

1020 
finally show ?case using Suc.hyps by (simp add: power_Suc) 

1021 
qed 

1022 

1023 
lemma fps_inverse_power: 

31021  1024 
fixes a :: "('a::{field}) fps" 
29687  1025 
shows "inverse (a^n) = inverse a ^ n" 
1026 
proof 

1027 
{assume a0: "a$0 = 0" 

1028 
hence eq: "inverse a = 0" by (simp add: fps_inverse_def) 

1029 
{assume "n = 0" hence ?thesis by simp} 

1030 
moreover 

1031 
{assume n: "n > 0" 

30488  1032 
from startsby_zero_power[OF a0 n] eq a0 n have ?thesis 
29687  1033 
by (simp add: fps_inverse_def)} 
1034 
ultimately have ?thesis by blast} 

1035 
moreover 

1036 
{assume a0: "a$0 \<noteq> 0" 

1037 
have ?thesis 

1038 
apply (rule fps_inverse_unique) 

1039 
apply (simp add: a0) 

1040 
unfolding power_mult_distrib[symmetric] 

1041 
apply (rule ssubst[where t = "a * inverse a" and s= 1]) 

1042 
apply simp_all 

1043 
apply (subst mult_commute) 

1044 
by (rule inverse_mult_eq_1[OF a0])} 

1045 
ultimately show ?thesis by blast 

1046 
qed 

1047 

1048 
lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n  1)" 

1049 
apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add) 

1050 
by (case_tac n, auto simp add: power_Suc ring_simps) 

1051 

30488  1052 
lemma fps_inverse_deriv: 
29687  1053 
fixes a:: "('a :: field) fps" 
1054 
assumes a0: "a$0 \<noteq> 0" 

1055 
shows "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2" 

1056 
proof 

1057 
from inverse_mult_eq_1[OF a0] 

1058 
have "fps_deriv (inverse a * a) = 0" by simp 

1059 
hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp 

1060 
hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0" by simp 

1061 
with inverse_mult_eq_1[OF a0] 

1062 
have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0" 

1063 
unfolding power2_eq_square 

1064 
apply (simp add: ring_simps) 

1065 
by (simp add: mult_assoc[symmetric]) 

1066 
hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a)  fps_deriv a * inverse a ^ 2 = 0  fps_deriv a * inverse a ^ 2" 

1067 
by simp 

1068 
then show "fps_deriv (inverse a) =  fps_deriv a * inverse a ^ 2" by (simp add: ring_simps) 

1069 
qed 

1070 

30488  1071 
lemma fps_inverse_mult: 
29687  1072 
fixes a::"('a :: field) fps" 
1073 
shows "inverse (a * b) = inverse a * inverse b" 

1074 
proof 

1075 
{assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) 

1076 
from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all 

1077 
have ?thesis unfolding th by simp} 

1078 
moreover 

1079 
{assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) 

1080 
from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all 

1081 
have ?thesis unfolding th by simp} 

1082 
moreover 

1083 
{assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0" 

1084 
from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp add: fps_mult_nth) 

30488  1085 
from inverse_mult_eq_1[OF ab0] 
29687  1086 
have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp 
1087 
then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b" 

1088 
by (simp add: ring_simps) 

1089 
then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp} 

1090 
ultimately show ?thesis by blast 

1091 
qed 

1092 

30488  1093 
lemma fps_inverse_deriv': 
29687  1094 
fixes a:: "('a :: field) fps" 
1095 
assumes a0: "a$0 \<noteq> 0" 

1096 
shows "fps_deriv (inverse a) =  fps_deriv a / a ^ 2" 

1097 
using fps_inverse_deriv[OF a0] 

1098 
unfolding power2_eq_square fps_divide_def 

1099 
fps_inverse_mult by simp 

1100 

1101 
lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)" 

1102 
shows "f * inverse f= 1" 

1103 
by (metis mult_commute inverse_mult_eq_1 f0) 

1104 

1105 
lemma fps_divide_deriv: fixes a:: "('a :: field) fps" 

1106 
assumes a0: "b$0 \<noteq> 0" 

1107 
shows "fps_deriv (a / b) = (fps_deriv a * b  a * fps_deriv b) / b ^ 2" 

1108 
using fps_inverse_deriv[OF a0] 

1109 
by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0]) 

30488  1110 

29687  1111 

30488  1112 
lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
29687  1113 
= 1  X" 
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset

1114 
by (simp add: fps_inverse_gp fps_eq_iff X_def) 
29687  1115 

1116 
lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)" 

1117 
by (cases "n", simp_all) 

1118 

1119 

1120 
lemma fps_inverse_X_plus1: 

31021  1121 
"inverse (1 + X) = Abs_fps (\<lambda>n. ( (1::'a::{field})) ^ n)" (is "_ = ?r") 
29687  1122 
proof 
1123 
have eq: "(1 + X) * ?r = 1" 

1124 
unfolding minus_one_power_iff 

31148  1125 
by (auto simp add: ring_simps fps_eq_iff) 
29687  1126 
show ?thesis by (auto simp add: eq intro: fps_inverse_unique) 
1127 
qed 

1128 

30488  1129 

29906  1130 
subsection{* Integration *} 
31273  1131 

1132 
definition 

1133 
fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps" where 

1134 
"fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n  1) / of_nat n))" 

29687  1135 

31273  1136 
lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a" 
1137 
unfolding fps_integral_def fps_deriv_def 

1138 
by (simp add: fps_eq_iff del: of_nat_Suc) 

29687  1139 

31273  1140 
lemma fps_integral_linear: 
1141 
"fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) = 

1142 
fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" 

1143 
(is "?l = ?r") 

29687  1144 
proof 
1145 
have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral) 

1146 
moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def) 

1147 
ultimately show ?thesis 

1148 
unfolding fps_deriv_eq_iff by auto 

1149 
qed 

30488  1150 

29906  1151 
subsection {* Composition of FPSs *} 
29687  1152 
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where 
1153 
fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})" 

1154 

1155 
lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def) 

1156 

1157 
lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)" 

29913  1158 
by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta') 
30488  1159 

1160 
lemma fps_const_compose[simp]: 

29687  1161 
"fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)" 
29913  1162 
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta) 
29687  1163 

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

1164 
lemma number_of_compose[simp]: "(number_of k::('a::{comm_ring_1}) fps) oo b = number_of k" 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

1165 
unfolding number_of_fps_const by simp 
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset

1166 

29687  1167 
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)" 
29913  1168 
by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta 
1169 
power_Suc not_le) 

29687  1170 

1171 

29906  1172 
subsection {* Rules from Herbert Wilf's Generatingfunctionology*} 
29687  1173 

29906  1174 
subsubsection {* Rule 1 *} 
29687  1175 
(* {a_{n+k}}_0^infty Corresponds to (f  setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*) 
1176 

30488  1177 
lemma fps_power_mult_eq_shift: 
30992  1178 
"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  1179 
proof 
1180 
{fix n:: nat 

30488  1181 
have "?lhs $ n = (if n < Suc k then 0 else a n)" 
29687  1182 
unfolding X_power_mult_nth by auto 
1183 
also have "\<dots> = ?rhs $ n" 

1184 
proof(induct k) 

1185 
case 0 thus ?case by (simp add: fps_setsum_nth power_Suc) 

1186 
next 

1187 
case (Suc k) 

1188 
note th = Suc.hyps[symmetric] 

30992  1189 
have "(Abs_fps a  setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n = (Abs_fps a  setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k}  fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps) 
29687  1190 
also have "\<dots> = (if n < Suc k then 0 else a n)  (fps_const (a (Suc k)) * X^ Suc k)$n" 
30488  1191 
using th 
29687  1192 
unfolding fps_sub_nth by simp 
1193 
also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)" 

1194 
unfolding X_power_mult_right_nth 

1195 
apply (auto simp add: not_less fps_const_def) 

1196 
apply (rule cong[of a a, OF refl]) 

1197 
by arith 

1198 
finally show ?case by simp 

1199 
qed 

1200 
finally have "?lhs $ n = ?rhs $ n" .} 

1201 
then show ?thesis by (simp add: fps_eq_iff) 

1202 
qed 

1203 

29906  1204 
subsubsection{* Rule 2*} 
29687  1205 

1206 
(* We can not reach the form of Wilf, but still near to it using rewrite rules*) 

30488  1207 
(* If f reprents {a_n} and P is a polynomial, then 
29687  1208 
P(xD) f represents {P(n) a_n}*) 
1209 

1210 
definition "XD = op * X o fps_deriv" 

1211 

1212 
lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)" 

1213 
by (simp add: XD_def ring_simps) 

1214 

1215 
lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a" 

1216 
by (simp add: XD_def ring_simps) 

1217 

1218 
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)" 

1219 
by simp 

1220 

30952
7ab2716dd93b
power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents:
30837
diff
changeset

1221 
lemma XDN_linear: 
30971  1222 
"(XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)" 
29687  1223 
by (induct n, simp_all) 
1224 

1225 
lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff) 

1226 

30994  1227 

30952
7ab2716dd93b
power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents:
30837
diff
changeset

1228 
lemma fps_mult_XD_shift: 
31021  1229 
"(XD ^^ k) (a:: ('a::{comm_ring_1}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)" 
30952
7ab2716dd93b
power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents:
30837
diff
changeset

1230 
by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def) 
29687  1231 

29906  1232 
subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*} 
1233 
subsubsection{* Rule 5  summation and "division" by (1  X)*} 

29687  1234 

1235 
lemma fps_divide_X_minus1_setsum_lemma: 

1236 
"a = ((1::('a::comm_ring_1) fps)  X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})" 

1237 
proof 

1238 
let ?X = "X::('a::comm_ring_1) fps" 

1239 
let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})" 

1240 
have th0: "\<And>i. (1  (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then  1 else 0)" by simp 

1241 
{fix n:: nat 

30488  1242 
{assume "n=0" hence "a$n = ((1  ?X) * ?sa) $ n" 
29687  1243 
by (simp add: fps_mult_nth)} 
1244 
moreover 

1245 
{assume n0: "n \<noteq> 0" 

1246 
then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}" 

1247 
"{0..n  1}\<union>{n} = {0..n}" 

1248 
apply (simp_all add: expand_set_eq) by presburger+ 

30488  1249 
have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" 
29687  1250 
"{0..n  1}\<inter>{n} ={}" using n0 
1251 
by (simp_all add: expand_set_eq, presburger+) 

30488  1252 
have f: "finite {0}" "finite {1}" "finite {2 .. n}" 
1253 
"finite {0 .. n  1}" "finite {n}" by simp_all 

29687  1254 
have "((1  ?X) * ?sa) $ n = setsum (\<lambda>i. (1  ?X)$ i * ?sa $ (n  i)) {0 .. n}" 
1255 
by (simp add: fps_mult_nth) 

1256 
also have "\<dots> = a$n" unfolding th0 

1257 
unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)] 

1258 
unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)] 

1259 
apply (simp) 

1260 
unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)] 

1261 
by simp 

1262 
finally have "a$n = ((1  ?X) * ?sa) $ n" by simp} 

1263 
ultimately have "a$n = ((1  ?X) * ?sa) $ n" by blast} 

30488  1264 
then show ?thesis 
29687  1265 
unfolding fps_eq_iff by blast 
1266 
qed 

1267 

1268 
lemma fps_divide_X_minus1_setsum: 

1269 
"a /((1::('a::field) fps)  X) = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})" 

1270 
proof 

1271 
let ?X = "1  (X::('a::field) fps)" 

1272 
have th0: "?X $ 0 \<noteq> 0" by simp 

1273 
have "a /?X = ?X * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X" 

1274 
using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0 

1275 
by (simp add: fps_divide_def mult_assoc) 

1276 
also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) " 

1277 
by (simp add: mult_ac) 

1278 
finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0]) 

1279 
qed 

1280 

30488  1281 
subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary 
29687  1282 
finite product of FPS, also the relvant instance of powers of a FPS*} 
1283 

1284 
definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}" 

1285 

1286 
lemma natlist_trivial_1: "natpermute n 1 = {[n]}" 

1287 
apply (auto simp add: natpermute_def) 

1288 
apply (case_tac x, auto) 

1289 
done 

1290 

30488  1291 
lemma foldl_add_start0: 
29687  1292 
"foldl op + x xs = x + foldl op + (0::nat) xs" 
1293 
apply (induct xs arbitrary: x) 

1294 
apply simp 

1295 
unfolding foldl.simps 

4d934a895d11
A formalization o 