src/HOL/Library/Formal_Power_Series.thy
author chaieb
Thu Jan 29 14:56:29 2009 +0000 (2009-01-29)
changeset 29687 4d934a895d11
child 29689 dd086f26ee4f
permissions -rwxr-xr-x
A formalization of formal power series
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(*  Title:      Formal_Power_Series.thy
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    ID:         
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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 Main Fact Parity
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begin
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section {* The type of formal power series*}
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typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
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  by simp
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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: "(0 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). 0)"
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instance ..
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end
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instantiation fps :: ("{one,zero}")  one
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begin
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definition fps_one_def: "(1 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). if n = 0 then 1 else 0)"
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instance ..
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end
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instantiation fps :: (plus)  plus
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begin
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definition fps_plus_def: "op + \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). Rep_fps (f) n + Rep_fps (g) n))"
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instance ..
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end
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instantiation fps :: (minus)  minus
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begin
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definition fps_minus_def: "op - \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). Rep_fps (f) n - Rep_fps (g) n))"
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instance ..
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end
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instantiation fps :: (uminus)  uminus
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begin
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definition fps_uminus_def: "uminus \<equiv> (\<lambda>(f::'a fps). Abs_fps (\<lambda>(n::nat). - Rep_fps (f) n))"
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instance ..
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end
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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 * \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). setsum (\<lambda>i. Rep_fps (f) i  * Rep_fps (g) (n - i)) {0.. n}))"
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instance ..
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end
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text{* Some useful theorems to get rid of Abs and Rep *}
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lemma mem_fps_set_trivial[intro, simp]: "f \<in> fps" unfolding fps_def by blast
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lemma Rep_fps_Abs_fps[simp]: "Rep_fps (Abs_fps f) = f" 
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  by (blast intro: Abs_fps_inverse) 
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lemma Abs_fps_Rep_fps[simp]: "Abs_fps (Rep_fps f) = f" 
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  by (blast intro: Rep_fps_inverse) 
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lemma Abs_fps_eq[simp]: "Abs_fps f = Abs_fps g \<longleftrightarrow> f = g"
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proof-
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  {assume "f = g" hence "Abs_fps f = Abs_fps g" by simp}
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  moreover
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  {assume a: "Abs_fps f = Abs_fps g"
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    from a have "Rep_fps (Abs_fps f) = Rep_fps (Abs_fps g)" by simp
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    hence "f = g" by simp}
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  ultimately show ?thesis by blast
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qed
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lemma Rep_fps_eq[simp]: "Rep_fps f = Rep_fps g \<longleftrightarrow> f = g"
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proof-
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  {assume "Rep_fps f = Rep_fps g" 
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    hence "Abs_fps (Rep_fps f) = Abs_fps (Rep_fps g)" by simp hence "f=g" by simp}
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  moreover
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  {assume "f = g" hence "Rep_fps f = Rep_fps g" by simp}
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  ultimately show ?thesis by blast
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qed
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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 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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section{* 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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instantiation fps :: (semigroup_add) semigroup_add
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begin
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instance
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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 (auto simp add: fps_plus_def expand_fun_eq add_assoc)
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qed
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end
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instantiation fps :: (ab_semigroup_add) ab_semigroup_add
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begin
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instance by (intro_classes, simp add: fps_plus_def expand_fun_eq add_commute)
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end
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instantiation fps :: (semiring_1) semigroup_mult
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begin
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instance
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proof
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  fix a b c :: "'a fps"
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  let ?a = "Rep_fps a"
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  let ?b = "Rep_fps b"
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  let ?c = "Rep_fps c"
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  let ?x = "\<lambda> i k. if k \<le> i then (1::'a) else 0" 
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  show "a*b*c = a* (b * c)"
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  proof(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
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    fix n::nat
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    let ?r = "\<lambda>i. n - i"
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    have i: "inj_on ?r {0..n}" by (auto simp add: inj_on_def)
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    have ri: "{0 .. n} = ?r ` {0..n}" apply (auto simp add: image_iff)
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      by presburger
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    let ?f = "\<lambda>i j. ?a j * ?b (i - j) * ?c (n -i)"
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    let ?g = "\<lambda>i j. ?a i * (?b j * ?c (n - (i + j)))"
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    have "setsum (\<lambda>i. setsum (?f i) {0..i}) {0..n} 
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      = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..i}) {0..n}"
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      by (rule setsum_cong2)+ auto
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    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..n}) {0..n}"
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    proof(rule setsum_cong2)
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      fix i assume i: "i \<in> {0..n}"
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      have eq: "{0 .. n} = {0 ..i} \<union> {i+1 .. n}" using i by auto
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      have d: "{0 ..i} \<inter> {i+1 .. n} = {}" using i by auto
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      have f: "finite {0..i}" "finite {i+1 ..n}" by auto
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      have s0: "setsum (\<lambda>j. ?f i j * ?x i j) {i+1 ..n} = 0" by simp
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      show "setsum (\<lambda>j. ?f i j * ?x i j) {0..i} = setsum (\<lambda>j. ?f i j * ?x i j) {0..n}"
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	unfolding eq setsum_Un_disjoint[OF f d] s0
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	by simp
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    qed
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    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {0 .. n}) {0 .. n}"
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      by (rule setsum_commute)
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    also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {i .. n}) {0 .. n}"
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      apply(rule setsum_cong2)
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      apply (rule setsum_mono_zero_right)
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      apply auto
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      done
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    also have "\<dots> = setsum (\<lambda>i. setsum (?g i) {0..n - i}) {0..n}"
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      apply (rule setsum_cong2)
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      apply (rule_tac f="\<lambda>i. i + x" in setsum_reindex_cong)
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      apply (simp add: inj_on_def)
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      apply (rule set_ext)
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      apply (presburger add: image_iff)
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      by (simp add: add_ac mult_assoc)
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    finally  show "setsum (\<lambda>i. setsum (\<lambda>j. ?a j * ?b (i - j) * ?c (n -i)) {0..i}) {0..n} 
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      = setsum (\<lambda>i. setsum (\<lambda>j. ?a i * (?b j * ?c (n - (i + j)))) {0..n - i}) {0..n}".
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  qed
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qed
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end
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instantiation fps :: (comm_semiring_1) ab_semigroup_mult
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begin
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instance
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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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  apply(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
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  apply (rule_tac f = "\<lambda>i. n - i" in setsum_reindex_cong)
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  apply (simp add: inj_on_def)
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  apply presburger
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  apply (rule set_ext)
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  apply (presburger add: image_iff)
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  by (simp add: mult_commute)
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qed
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end
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instantiation fps :: (monoid_add) monoid_add
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begin
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instance
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proof
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  fix a :: "'a fps" show "0 + a = a "
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    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
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next
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  fix a :: "'a fps" show "a + 0 = a "
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    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
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qed
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end
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instantiation fps :: (comm_monoid_add) comm_monoid_add
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begin
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instance
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proof
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  fix a :: "'a fps" show "0 + a = a "
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    by (auto simp add: fps_plus_def fps_zero_def intro: ext)
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qed
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end
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instantiation fps :: (semiring_1) monoid_mult
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begin
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instance
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proof
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  fix a :: "'a fps" show "1 * a = a"
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    apply (auto simp add: fps_one_def fps_times_def)
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    apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
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    unfolding Abs_fps_eq
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    apply (rule ext)
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    by (simp add: cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
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next
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  fix a :: "'a fps" show "a*1 = a"
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    apply (auto simp add: fps_one_def fps_times_def)
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    apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
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    unfolding Abs_fps_eq
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    apply (rule ext)
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    by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
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qed
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end
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instantiation fps :: (cancel_semigroup_add) cancel_semigroup_add
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begin
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instance by (intro_classes) (auto simp add: fps_plus_def expand_fun_eq Rep_fps_eq[symmetric])
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end
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instantiation fps :: (group_add) group_add
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begin
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instance
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proof
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  fix a :: "'a fps" show "- a + a = 0"
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    by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def intro: ext)
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next
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  fix a b :: "'a fps" show "a - b = a + - b"
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    by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def 
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      fps_minus_def expand_fun_eq diff_minus)
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qed
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end
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context comm_ring_1
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begin
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subclass group_add proof qed
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end
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instantiation fps :: (zero_neq_one) zero_neq_one
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begin
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instance by (intro_classes, auto simp add: zero_neq_one 
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  fps_one_def fps_zero_def expand_fun_eq)
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end
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instantiation fps :: (semiring_1) semiring
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begin
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instance
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proof
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  fix a b c :: "'a fps"
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  show "(a + b) * c = a * c + b*c"
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    apply (auto simp add: fps_plus_def fps_times_def, rule ext)
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    unfolding setsum_addf[symmetric]
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    apply (simp add: ring_simps)
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    done
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next
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  fix a b c :: "'a fps"
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  show "a * (b + c) = a * b + a*c"
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    apply (auto simp add: fps_plus_def fps_times_def, rule ext)
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    unfolding setsum_addf[symmetric]
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    apply (simp add: ring_simps)
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    done
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qed
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end
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instantiation fps :: (semiring_1) semiring_0
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begin
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instance
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proof
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  fix a:: "'a fps" show "0 * a = 0" by (simp add: fps_zero_def fps_times_def)
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next
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  fix a:: "'a fps" show "a*0 = 0" by (simp add: fps_zero_def fps_times_def)
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qed
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end
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section {* Selection of the nth power of the implicit variable in the infinite sum*}
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definition fps_nth:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" (infixl "$" 75)
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  where "f $ n = Rep_fps f n"
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lemma fps_nth_Abs_fps[simp]: "Abs_fps a $ n = a n" by (simp add: fps_nth_def)
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lemma fps_zero_nth[simp]: "0 $ n = 0" by (simp add: fps_zero_def)
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lemma fps_one_nth[simp]: "1 $ n = (if n = 0 then 1 else 0)" 
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  by (simp add: fps_one_def)
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lemma fps_add_nth[simp]: "(f + g) $ n = f$n + g$n" by (simp add: fps_plus_def fps_nth_def)
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lemma fps_mult_nth: "(f * g) $ n = setsum (\<lambda>i. f$i * g$(n - i)) {0..n}"
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  by (simp add: fps_times_def fps_nth_def)
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lemma fps_neg_nth[simp]: "(- f) $n = - (f $n)" by (simp add: fps_nth_def fps_uminus_def)
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lemma fps_sub_nth[simp]: "(f - g)$n = f$n - g$n" by (simp add: fps_nth_def fps_minus_def)
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lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
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proof-
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  {assume "f \<noteq> 0"
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    hence "Rep_fps f \<noteq> Rep_fps 0" by simp 
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    hence "\<exists>n. f $n \<noteq> 0" by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
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  moreover
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  {assume "\<exists>n. f$n \<noteq> 0" and "f = 0" 
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    then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
chaieb@29687
   319
  ultimately show ?thesis by blast
chaieb@29687
   320
qed
chaieb@29687
   321
chaieb@29687
   322
lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0))"
chaieb@29687
   323
proof-
chaieb@29687
   324
  let ?S = "{n. f$n \<noteq> 0}"
chaieb@29687
   325
  {assume "\<exists>n. f$n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" and "f = 0" 
chaieb@29687
   326
    then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
chaieb@29687
   327
  moreover
chaieb@29687
   328
  {assume f0: "f \<noteq> 0"
chaieb@29687
   329
    from f0 fps_nonzero_nth have ex: "\<exists>n. f$n \<noteq> 0" by blast
chaieb@29687
   330
    hence Se: "?S\<noteq> {}" by blast
chaieb@29687
   331
    from ex obtain n where n: "f$n \<noteq> 0" by blast
chaieb@29687
   332
    from n have nS: "n \<in> ?S" by blast
chaieb@29687
   333
        let ?U = "?S \<inter> {0..n}"
chaieb@29687
   334
    have fU: "finite ?U" by auto
chaieb@29687
   335
    from n have Ue: "?U \<noteq> {}" by auto
chaieb@29687
   336
    let ?m = "Min ?U" 
chaieb@29687
   337
    have mU: "?m \<in> ?U" using Min_in[OF fU Ue] .
chaieb@29687
   338
    hence mn: "?m \<le> n" by simp
chaieb@29687
   339
    from mU have mf: "f $ ?m \<noteq> 0" by blast
chaieb@29687
   340
    {fix m assume m: "m < ?m" and f: "f $m \<noteq> 0"
chaieb@29687
   341
      from m mn have mn': "m < n" by arith
chaieb@29687
   342
      with f have mU': "m \<in> ?U" by simp
chaieb@29687
   343
      from Min_le[OF fU mU'] m have False by arith}
chaieb@29687
   344
    hence "\<forall>m <?m. f$m = 0" by blast
chaieb@29687
   345
    with mf have "\<exists> n. f $n \<noteq> 0 \<and> (\<forall>m <n. f $m = 0)" by blast}
chaieb@29687
   346
  ultimately show ?thesis by blast
chaieb@29687
   347
qed
chaieb@29687
   348
chaieb@29687
   349
lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
chaieb@29687
   350
  by (auto simp add: fps_nth_def Rep_fps_eq[unfolded expand_fun_eq])
chaieb@29687
   351
chaieb@29687
   352
lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S" 
chaieb@29687
   353
proof-
chaieb@29687
   354
  {assume "\<not> finite S" hence ?thesis by simp}
chaieb@29687
   355
  moreover
chaieb@29687
   356
  {assume fS: "finite S"
chaieb@29687
   357
    have ?thesis by(induct rule: finite_induct[OF fS]) auto}
chaieb@29687
   358
  ultimately show ?thesis by blast
chaieb@29687
   359
qed
chaieb@29687
   360
chaieb@29687
   361
section{* Injection of the basic ring elements and multiplication by scalars *}
chaieb@29687
   362
chaieb@29687
   363
definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
chaieb@29687
   364
lemma fps_const_0_eq_0[simp]: "fps_const 0 = 0" by (simp add: fps_const_def fps_eq_iff)
chaieb@29687
   365
lemma fps_const_1_eq_1[simp]: "fps_const 1 = 1" by (simp add: fps_const_def fps_eq_iff)
chaieb@29687
   366
lemma fps_const_neg[simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
chaieb@29687
   367
  by (simp add: fps_uminus_def fps_const_def fps_eq_iff)
chaieb@29687
   368
lemma fps_const_add[simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
chaieb@29687
   369
  by (simp add: fps_plus_def fps_const_def fps_eq_iff)
chaieb@29687
   370
lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
chaieb@29687
   371
  by (auto simp add: fps_times_def fps_const_def fps_eq_iff intro: setsum_0')
chaieb@29687
   372
chaieb@29687
   373
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)"
chaieb@29687
   374
  unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)
chaieb@29687
   375
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)"
chaieb@29687
   376
  unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)
chaieb@29687
   377
chaieb@29687
   378
lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"
chaieb@29687
   379
  unfolding fps_eq_iff fps_mult_nth 
chaieb@29687
   380
  by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
chaieb@29687
   381
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)"
chaieb@29687
   382
  unfolding fps_eq_iff fps_mult_nth 
chaieb@29687
   383
  by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
chaieb@29687
   384
chaieb@29687
   385
lemma fps_const_nth[simp]: "(fps_const c) $n = (if n = 0 then c else 0)"
chaieb@29687
   386
  by (simp add: fps_const_def)
chaieb@29687
   387
chaieb@29687
   388
lemma fps_mult_left_const_nth[simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"
chaieb@29687
   389
  by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
chaieb@29687
   390
chaieb@29687
   391
lemma fps_mult_right_const_nth[simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
chaieb@29687
   392
  by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
chaieb@29687
   393
chaieb@29687
   394
section {* Formal power series form an integral domain*}
chaieb@29687
   395
chaieb@29687
   396
instantiation fps :: (ring_1) ring_1
chaieb@29687
   397
begin
chaieb@29687
   398
chaieb@29687
   399
instance by (intro_classes, auto simp add: diff_minus left_distrib)
chaieb@29687
   400
end
chaieb@29687
   401
chaieb@29687
   402
instantiation fps :: (comm_ring_1) comm_ring_1
chaieb@29687
   403
begin
chaieb@29687
   404
chaieb@29687
   405
instance by (intro_classes, auto simp add: diff_minus left_distrib)
chaieb@29687
   406
end
chaieb@29687
   407
instantiation fps :: ("{ring_no_zero_divisors, comm_ring_1}") ring_no_zero_divisors
chaieb@29687
   408
begin
chaieb@29687
   409
chaieb@29687
   410
instance 
chaieb@29687
   411
proof
chaieb@29687
   412
  fix a b :: "'a fps"
chaieb@29687
   413
  assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
chaieb@29687
   414
  then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0"
chaieb@29687
   415
    and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal
chaieb@29687
   416
    by blast+
chaieb@29687
   417
  have eq: "({0..i+j} -{i}) \<union> {i} = {0..i+j}" by auto
chaieb@29687
   418
  have d: "({0..i+j} -{i}) \<inter> {i} = {}" by auto
chaieb@29687
   419
  have f: "finite ({0..i+j} -{i})" "finite {i}" by auto
chaieb@29687
   420
  have th0: "setsum (\<lambda>k. a$k * b$(i+j - k)) ({0..i+j} -{i}) = 0"
chaieb@29687
   421
    apply (rule setsum_0')
chaieb@29687
   422
    apply auto
chaieb@29687
   423
    apply (case_tac "aa < i")
chaieb@29687
   424
    using i
chaieb@29687
   425
    apply auto
chaieb@29687
   426
    apply (subgoal_tac "b $ (i+j - aa) = 0")
chaieb@29687
   427
    apply blast
chaieb@29687
   428
    apply (rule j(2)[rule_format])
chaieb@29687
   429
    by arith
chaieb@29687
   430
  have "(a*b) $ (i+j) =  setsum (\<lambda>k. a$k * b$(i+j - k)) {0..i+j}"
chaieb@29687
   431
    by (rule fps_mult_nth)
chaieb@29687
   432
  hence "(a*b) $ (i+j) = a$i * b$j"
chaieb@29687
   433
    unfolding setsum_Un_disjoint[OF f d, unfolded eq] th0 by simp
chaieb@29687
   434
  with i j have "(a*b) $ (i+j) \<noteq> 0" by simp
chaieb@29687
   435
  then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
chaieb@29687
   436
qed
chaieb@29687
   437
end
chaieb@29687
   438
chaieb@29687
   439
instantiation fps :: (idom) idom
chaieb@29687
   440
begin
chaieb@29687
   441
chaieb@29687
   442
instance ..
chaieb@29687
   443
end
chaieb@29687
   444
chaieb@29687
   445
section{* Inverses of formal power series *}
chaieb@29687
   446
chaieb@29687
   447
declare setsum_cong[fundef_cong]
chaieb@29687
   448
chaieb@29687
   449
chaieb@29687
   450
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse
chaieb@29687
   451
begin
chaieb@29687
   452
chaieb@29687
   453
fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where 
chaieb@29687
   454
  "natfun_inverse f 0 = inverse (f$0)"
chaieb@29687
   455
| "natfun_inverse f n = - inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}" 
chaieb@29687
   456
chaieb@29687
   457
definition fps_inverse_def: 
chaieb@29687
   458
  "inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))"
chaieb@29687
   459
definition fps_divide_def: "divide \<equiv> (\<lambda>(f::'a fps) g. f * inverse g)"
chaieb@29687
   460
instance ..
chaieb@29687
   461
end
chaieb@29687
   462
chaieb@29687
   463
lemma fps_inverse_zero[simp]: 
chaieb@29687
   464
  "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
chaieb@29687
   465
  by (simp add: fps_zero_def fps_inverse_def)
chaieb@29687
   466
chaieb@29687
   467
lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
chaieb@29687
   468
  apply (auto simp add: fps_one_def fps_inverse_def expand_fun_eq)
chaieb@29687
   469
  by (case_tac x, auto)
chaieb@29687
   470
chaieb@29687
   471
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}")  division_by_zero
chaieb@29687
   472
begin
chaieb@29687
   473
instance
chaieb@29687
   474
  apply (intro_classes)
chaieb@29687
   475
  by (rule fps_inverse_zero)
chaieb@29687
   476
end
chaieb@29687
   477
chaieb@29687
   478
lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
chaieb@29687
   479
  shows "inverse f * f = 1"
chaieb@29687
   480
proof-
chaieb@29687
   481
  have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
chaieb@29687
   482
  from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n" 
chaieb@29687
   483
    by (simp add: fps_inverse_def)
chaieb@29687
   484
  from f0 have th0: "(inverse f * f) $ 0 = 1"
chaieb@29687
   485
    by (simp add: fps_inverse_def fps_one_def fps_mult_nth)
chaieb@29687
   486
  {fix n::nat assume np: "n >0 "
chaieb@29687
   487
    from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
chaieb@29687
   488
    have d: "{0} \<inter> {1 .. n} = {}" by auto
chaieb@29687
   489
    have f: "finite {0::nat}" "finite {1..n}" by auto
chaieb@29687
   490
    from f0 np have th0: "- (inverse f$n) = 
chaieb@29687
   491
      (setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)"
chaieb@29687
   492
      by (cases n, simp_all add: divide_inverse fps_inverse_def fps_nth_def ring_simps)
chaieb@29687
   493
    from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
chaieb@29687
   494
    have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n} = 
chaieb@29687
   495
      - (f$0) * (inverse f)$n" 
chaieb@29687
   496
      by (simp add: ring_simps)
chaieb@29687
   497
    have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n - i))" 
chaieb@29687
   498
      unfolding fps_mult_nth ifn ..
chaieb@29687
   499
    also have "\<dots> = f$0 * natfun_inverse f n 
chaieb@29687
   500
      + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
chaieb@29687
   501
      unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
chaieb@29687
   502
      by simp
chaieb@29687
   503
    also have "\<dots> = 0" unfolding th1 ifn by simp
chaieb@29687
   504
    finally have "(inverse f * f)$n = 0" unfolding c . }
chaieb@29687
   505
  with th0 show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
   506
qed
chaieb@29687
   507
chaieb@29687
   508
lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"
chaieb@29687
   509
  apply (simp add: fps_inverse_def)
chaieb@29687
   510
  by (metis fps_nth_def fps_nth_def inverse_zero_imp_zero)
chaieb@29687
   511
chaieb@29687
   512
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0"
chaieb@29687
   513
proof-
chaieb@29687
   514
  {assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
chaieb@29687
   515
  moreover
chaieb@29687
   516
  {assume h: "inverse f = 0" and c: "f $0 \<noteq> 0"
chaieb@29687
   517
    from inverse_mult_eq_1[OF c] h have False by simp}
chaieb@29687
   518
  ultimately show ?thesis by blast
chaieb@29687
   519
qed
chaieb@29687
   520
chaieb@29687
   521
lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
chaieb@29687
   522
  shows "inverse (inverse f) = f"
chaieb@29687
   523
proof-
chaieb@29687
   524
  from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp
chaieb@29687
   525
  from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] 
chaieb@29687
   526
  have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
chaieb@29687
   527
  then show ?thesis using f0 unfolding mult_cancel_left by simp
chaieb@29687
   528
qed
chaieb@29687
   529
chaieb@29687
   530
lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1" 
chaieb@29687
   531
  shows "inverse f = g"
chaieb@29687
   532
proof-
chaieb@29687
   533
  from inverse_mult_eq_1[OF f0] fg
chaieb@29687
   534
  have th0: "inverse f * f = g * f" by (simp add: mult_ac)
chaieb@29687
   535
  then show ?thesis using f0  unfolding mult_cancel_right
chaieb@29687
   536
    unfolding Rep_fps_eq[of f 0, symmetric]
chaieb@29687
   537
    by (auto simp add: fps_zero_def expand_fun_eq fps_nth_def)
chaieb@29687
   538
qed
chaieb@29687
   539
chaieb@29687
   540
lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
chaieb@29687
   541
  = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
chaieb@29687
   542
  apply (rule fps_inverse_unique)
chaieb@29687
   543
  apply simp
chaieb@29687
   544
  apply (simp add: fps_eq_iff fps_nth_def fps_times_def fps_one_def)
chaieb@29687
   545
proof(clarsimp)
chaieb@29687
   546
  fix n::nat assume n: "n > 0"
chaieb@29687
   547
  let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
chaieb@29687
   548
  let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
chaieb@29687
   549
  let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
chaieb@29687
   550
  have th1: "setsum ?f {0..n} = setsum ?g {0..n}"  
chaieb@29687
   551
    by (rule setsum_cong2) auto
chaieb@29687
   552
  have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"  
chaieb@29687
   553
    using n apply - by (rule setsum_cong2) auto
chaieb@29687
   554
  have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
chaieb@29687
   555
  from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto 
chaieb@29687
   556
  have f: "finite {0.. n - 1}" "finite {n}" by auto
chaieb@29687
   557
  show "setsum ?f {0..n} = 0"
chaieb@29687
   558
    unfolding th1 
chaieb@29687
   559
    apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
chaieb@29687
   560
    unfolding th2
chaieb@29687
   561
    by(simp add: setsum_delta)
chaieb@29687
   562
qed
chaieb@29687
   563
chaieb@29687
   564
section{* Formal Derivatives, and the McLauren theorem around 0*}
chaieb@29687
   565
chaieb@29687
   566
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
chaieb@29687
   567
chaieb@29687
   568
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def)
chaieb@29687
   569
chaieb@29687
   570
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"
chaieb@29687
   571
  unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: ring_simps)
chaieb@29687
   572
chaieb@29687
   573
lemma fps_deriv_mult[simp]: 
chaieb@29687
   574
  fixes f :: "('a :: comm_ring_1) fps"
chaieb@29687
   575
  shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
chaieb@29687
   576
proof-
chaieb@29687
   577
  let ?D = "fps_deriv"
chaieb@29687
   578
  {fix n::nat
chaieb@29687
   579
    let ?Zn = "{0 ..n}"
chaieb@29687
   580
    let ?Zn1 = "{0 .. n + 1}"
chaieb@29687
   581
    let ?f = "\<lambda>i. i + 1"
chaieb@29687
   582
    have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
chaieb@29687
   583
    have eq: "{1.. n+1} = ?f ` {0..n}" by auto
chaieb@29687
   584
    let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n - i) +
chaieb@29687
   585
        of_nat (i+1)* f $ (i+1) * g $ (n - i)"
chaieb@29687
   586
    let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1) - i) +
chaieb@29687
   587
        of_nat i* f $ i * g $ ((n + 1) - i)"
chaieb@29687
   588
    {fix k assume k: "k \<in> {0..n}"
chaieb@29687
   589
      have "?h (k + 1) = ?g k" using k by auto}
chaieb@29687
   590
    note th0 = this
chaieb@29687
   591
    have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
chaieb@29687
   592
    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"
chaieb@29687
   593
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
chaieb@29687
   594
      apply (simp add: inj_on_def Ball_def)
chaieb@29687
   595
      apply presburger
chaieb@29687
   596
      apply (rule set_ext)
chaieb@29687
   597
      apply (presburger add: image_iff)
chaieb@29687
   598
      by simp
chaieb@29687
   599
    have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
chaieb@29687
   600
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
chaieb@29687
   601
      apply (simp add: inj_on_def Ball_def)
chaieb@29687
   602
      apply presburger
chaieb@29687
   603
      apply (rule set_ext)
chaieb@29687
   604
      apply (presburger add: image_iff)
chaieb@29687
   605
      by simp
chaieb@29687
   606
    have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute)
chaieb@29687
   607
    also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
chaieb@29687
   608
      by (simp add: fps_mult_nth setsum_addf[symmetric])
chaieb@29687
   609
    also have "\<dots> = setsum ?h {1..n+1}"
chaieb@29687
   610
      using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
chaieb@29687
   611
    also have "\<dots> = setsum ?h {0..n+1}"
chaieb@29687
   612
      apply (rule setsum_mono_zero_left)
chaieb@29687
   613
      apply simp
chaieb@29687
   614
      apply (simp add: subset_eq)
chaieb@29687
   615
      unfolding eq'
chaieb@29687
   616
      by simp
chaieb@29687
   617
    also have "\<dots> = (fps_deriv (f * g)) $ n"
chaieb@29687
   618
      apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
chaieb@29687
   619
      unfolding s0 s1
chaieb@29687
   620
      unfolding setsum_addf[symmetric] setsum_right_distrib
chaieb@29687
   621
      apply (rule setsum_cong2)
chaieb@29687
   622
      by (auto simp add: of_nat_diff ring_simps)
chaieb@29687
   623
    finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .}
chaieb@29687
   624
  then show ?thesis unfolding fps_eq_iff by auto 
chaieb@29687
   625
qed
chaieb@29687
   626
chaieb@29687
   627
lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
chaieb@29687
   628
  by (simp add: fps_uminus_def fps_eq_iff fps_deriv_def fps_nth_def)
chaieb@29687
   629
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
chaieb@29687
   630
  using fps_deriv_linear[of 1 f 1 g] by simp
chaieb@29687
   631
chaieb@29687
   632
lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
chaieb@29687
   633
  unfolding diff_minus by simp 
chaieb@29687
   634
chaieb@29687
   635
lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
chaieb@29687
   636
  by (simp add: fps_deriv_def fps_const_def fps_zero_def)
chaieb@29687
   637
chaieb@29687
   638
lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
chaieb@29687
   639
  by simp
chaieb@29687
   640
chaieb@29687
   641
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
chaieb@29687
   642
  by (simp add: fps_deriv_def fps_eq_iff)
chaieb@29687
   643
chaieb@29687
   644
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
chaieb@29687
   645
  by (simp add: fps_deriv_def fps_eq_iff )
chaieb@29687
   646
chaieb@29687
   647
lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
chaieb@29687
   648
  by simp
chaieb@29687
   649
chaieb@29687
   650
lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
chaieb@29687
   651
proof-
chaieb@29687
   652
  {assume "\<not> finite S" hence ?thesis by simp}
chaieb@29687
   653
  moreover
chaieb@29687
   654
  {assume fS: "finite S"
chaieb@29687
   655
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
chaieb@29687
   656
  ultimately show ?thesis by blast
chaieb@29687
   657
qed
chaieb@29687
   658
chaieb@29687
   659
lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))"
chaieb@29687
   660
proof-
chaieb@29687
   661
  {assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
chaieb@29687
   662
    hence "fps_deriv f = 0" by simp }
chaieb@29687
   663
  moreover
chaieb@29687
   664
  {assume z: "fps_deriv f = 0"
chaieb@29687
   665
    hence "\<forall>n. (fps_deriv f)$n = 0" by simp
chaieb@29687
   666
    hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
chaieb@29687
   667
    hence "f = fps_const (f$0)"
chaieb@29687
   668
      apply (clarsimp simp add: fps_eq_iff fps_const_def)
chaieb@29687
   669
      apply (erule_tac x="n - 1" in allE)
chaieb@29687
   670
      by simp}
chaieb@29687
   671
  ultimately show ?thesis by blast
chaieb@29687
   672
qed
chaieb@29687
   673
chaieb@29687
   674
lemma fps_deriv_eq_iff: 
chaieb@29687
   675
  fixes f:: "('a::{idom,semiring_char_0}) fps"
chaieb@29687
   676
  shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
chaieb@29687
   677
proof-
chaieb@29687
   678
  have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
chaieb@29687
   679
  also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)$0)" unfolding fps_deriv_eq_0_iff ..
chaieb@29687
   680
  finally show ?thesis by (simp add: ring_simps)
chaieb@29687
   681
qed
chaieb@29687
   682
chaieb@29687
   683
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)"
chaieb@29687
   684
  apply auto unfolding fps_deriv_eq_iff by blast
chaieb@29687
   685
  
chaieb@29687
   686
chaieb@29687
   687
fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
chaieb@29687
   688
  "fps_nth_deriv 0 f = f"
chaieb@29687
   689
| "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
chaieb@29687
   690
chaieb@29687
   691
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
chaieb@29687
   692
  by (induct n arbitrary: f, auto)
chaieb@29687
   693
chaieb@29687
   694
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"
chaieb@29687
   695
  by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)
chaieb@29687
   696
chaieb@29687
   697
lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
chaieb@29687
   698
  by (induct n arbitrary: f, simp_all)
chaieb@29687
   699
chaieb@29687
   700
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"
chaieb@29687
   701
  using fps_nth_deriv_linear[of n 1 f 1 g] by simp
chaieb@29687
   702
chaieb@29687
   703
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"
chaieb@29687
   704
  unfolding diff_minus fps_nth_deriv_add by simp 
chaieb@29687
   705
chaieb@29687
   706
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
chaieb@29687
   707
  by (induct n, simp_all )
chaieb@29687
   708
chaieb@29687
   709
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
chaieb@29687
   710
  by (induct n, simp_all )
chaieb@29687
   711
chaieb@29687
   712
lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
chaieb@29687
   713
  by (cases n, simp_all)
chaieb@29687
   714
chaieb@29687
   715
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"
chaieb@29687
   716
  using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
chaieb@29687
   717
chaieb@29687
   718
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"
chaieb@29687
   719
  using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
chaieb@29687
   720
chaieb@29687
   721
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"
chaieb@29687
   722
proof-
chaieb@29687
   723
  {assume "\<not> finite S" hence ?thesis by simp}
chaieb@29687
   724
  moreover
chaieb@29687
   725
  {assume fS: "finite S"
chaieb@29687
   726
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
chaieb@29687
   727
  ultimately show ?thesis by blast
chaieb@29687
   728
qed
chaieb@29687
   729
chaieb@29687
   730
lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"
chaieb@29687
   731
  by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)
chaieb@29687
   732
chaieb@29687
   733
section {* Powers*}
chaieb@29687
   734
chaieb@29687
   735
instantiation fps :: (semiring_1) power
chaieb@29687
   736
begin
chaieb@29687
   737
chaieb@29687
   738
fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
chaieb@29687
   739
  "fps_pow 0 f = 1"
chaieb@29687
   740
| "fps_pow (Suc n) f = f * fps_pow n f"
chaieb@29687
   741
chaieb@29687
   742
definition fps_power_def: "power (f::'a fps) n = fps_pow n f"
chaieb@29687
   743
instance ..
chaieb@29687
   744
end
chaieb@29687
   745
chaieb@29687
   746
instantiation fps :: (comm_ring_1) recpower
chaieb@29687
   747
begin
chaieb@29687
   748
instance
chaieb@29687
   749
  apply (intro_classes)
chaieb@29687
   750
  by (simp_all add: fps_power_def)
chaieb@29687
   751
end
chaieb@29687
   752
chaieb@29687
   753
lemma eq_neg_iff_add_eq_0: "(a::'a::ring) = -b \<longleftrightarrow> a + b = 0"
chaieb@29687
   754
proof-
chaieb@29687
   755
  {assume "a = -b" hence "b + a = b + -b" by simp
chaieb@29687
   756
    hence "a + b = 0" by (simp add: ring_simps)}
chaieb@29687
   757
  moreover
chaieb@29687
   758
  {assume "a + b = 0" hence "a + b - b = -b" by simp
chaieb@29687
   759
    hence "a = -b" by simp}
chaieb@29687
   760
  ultimately show ?thesis by blast
chaieb@29687
   761
qed
chaieb@29687
   762
chaieb@29687
   763
lemma fps_sqrare_eq_iff: "(a:: 'a::idom fps)^ 2 = b^2  \<longleftrightarrow> (a = b \<or> a = -b)"
chaieb@29687
   764
proof-
chaieb@29687
   765
  {assume "a = b \<or> a = -b" hence "a^2 = b^2" by auto}
chaieb@29687
   766
  moreover
chaieb@29687
   767
  {assume "a^2 = b^2 "
chaieb@29687
   768
    hence "a^2 - b^2 = 0" by simp
chaieb@29687
   769
    hence "(a-b) * (a+b) = 0" by (simp add: power2_eq_square ring_simps)
chaieb@29687
   770
    hence "a = b \<or> a = -b" by (simp add: eq_neg_iff_add_eq_0)}
chaieb@29687
   771
  ultimately show ?thesis by blast
chaieb@29687
   772
qed
chaieb@29687
   773
chaieb@29687
   774
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
chaieb@29687
   775
  by (induct n, auto simp add: fps_power_def fps_times_def fps_nth_def fps_one_def)
chaieb@29687
   776
chaieb@29687
   777
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
chaieb@29687
   778
proof(induct n)
chaieb@29687
   779
  case 0 thus ?case by (simp add: fps_power_def)
chaieb@29687
   780
next
chaieb@29687
   781
  case (Suc n)
chaieb@29687
   782
  note h = Suc.hyps[OF `a$0 = 1`]
chaieb@29687
   783
  show ?case unfolding power_Suc fps_mult_nth 
chaieb@29687
   784
    using h `a$0 = 1`  fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps)
chaieb@29687
   785
qed
chaieb@29687
   786
chaieb@29687
   787
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
chaieb@29687
   788
  by (induct n, auto simp add: fps_power_def fps_mult_nth)
chaieb@29687
   789
chaieb@29687
   790
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
chaieb@29687
   791
  by (induct n, auto simp add: fps_power_def fps_mult_nth)
chaieb@29687
   792
chaieb@29687
   793
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n $0 = v^n"
chaieb@29687
   794
  by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)
chaieb@29687
   795
chaieb@29687
   796
lemma startsby_zero_power_iff[simp]:
chaieb@29687
   797
  "a^n $0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"
chaieb@29687
   798
apply (rule iffI)
chaieb@29687
   799
apply (induct n, auto simp add: power_Suc fps_mult_nth)
chaieb@29687
   800
by (rule startsby_zero_power, simp_all)
chaieb@29687
   801
chaieb@29687
   802
lemma startsby_zero_power_prefix: 
chaieb@29687
   803
  assumes a0: "a $0 = (0::'a::idom)"
chaieb@29687
   804
  shows "\<forall>n < k. a ^ k $ n = 0"
chaieb@29687
   805
  using a0 
chaieb@29687
   806
proof(induct k rule: nat_less_induct)
chaieb@29687
   807
  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)"
chaieb@29687
   808
  let ?ths = "\<forall>m<k. a ^ k $ m = 0"
chaieb@29687
   809
  {assume "k = 0" then have ?ths by simp}
chaieb@29687
   810
  moreover
chaieb@29687
   811
  {fix l assume k: "k = Suc l"
chaieb@29687
   812
    {fix m assume mk: "m < k"
chaieb@29687
   813
      {assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0 
chaieb@29687
   814
	  by simp}
chaieb@29687
   815
      moreover
chaieb@29687
   816
      {assume m0: "m \<noteq> 0"
chaieb@29687
   817
	have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute)
chaieb@29687
   818
	also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
chaieb@29687
   819
	also have "\<dots> = 0" apply (rule setsum_0')
chaieb@29687
   820
	  apply auto
chaieb@29687
   821
	  apply (case_tac "aa = m")
chaieb@29687
   822
	  using a0
chaieb@29687
   823
	  apply simp
chaieb@29687
   824
	  apply (rule H[rule_format])
chaieb@29687
   825
	  using a0 k mk by auto 
chaieb@29687
   826
	finally have "a^k $ m = 0" .}
chaieb@29687
   827
    ultimately have "a^k $ m = 0" by blast}
chaieb@29687
   828
    hence ?ths by blast}
chaieb@29687
   829
  ultimately show ?ths by (cases k, auto)
chaieb@29687
   830
qed
chaieb@29687
   831
chaieb@29687
   832
lemma startsby_zero_setsum_depends: 
chaieb@29687
   833
  assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k"
chaieb@29687
   834
  shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}"
chaieb@29687
   835
  apply (rule setsum_mono_zero_right)
chaieb@29687
   836
  using kn apply auto
chaieb@29687
   837
  apply (rule startsby_zero_power_prefix[rule_format, OF a0])
chaieb@29687
   838
  by arith
chaieb@29687
   839
chaieb@29687
   840
lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{recpower, idom})"
chaieb@29687
   841
  shows "a^n $ n = (a$1) ^ n"
chaieb@29687
   842
proof(induct n)
chaieb@29687
   843
  case 0 thus ?case by (simp add: power_0)
chaieb@29687
   844
next
chaieb@29687
   845
  case (Suc n)
chaieb@29687
   846
  have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc)
chaieb@29687
   847
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
chaieb@29687
   848
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
chaieb@29687
   849
    apply (rule setsum_mono_zero_right)
chaieb@29687
   850
    apply simp
chaieb@29687
   851
    apply clarsimp
chaieb@29687
   852
    apply clarsimp
chaieb@29687
   853
    apply (rule startsby_zero_power_prefix[rule_format, OF a0])
chaieb@29687
   854
    apply arith
chaieb@29687
   855
    done
chaieb@29687
   856
  also have "\<dots> = a^n $ n * a$1" using a0 by simp
chaieb@29687
   857
  finally show ?case using Suc.hyps by (simp add: power_Suc)
chaieb@29687
   858
qed
chaieb@29687
   859
chaieb@29687
   860
lemma fps_inverse_power:
chaieb@29687
   861
  fixes a :: "('a::{field, recpower}) fps"
chaieb@29687
   862
  shows "inverse (a^n) = inverse a ^ n"
chaieb@29687
   863
proof-
chaieb@29687
   864
  {assume a0: "a$0 = 0"
chaieb@29687
   865
    hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
chaieb@29687
   866
    {assume "n = 0" hence ?thesis by simp}
chaieb@29687
   867
    moreover
chaieb@29687
   868
    {assume n: "n > 0"
chaieb@29687
   869
      from startsby_zero_power[OF a0 n] eq a0 n have ?thesis 
chaieb@29687
   870
	by (simp add: fps_inverse_def)}
chaieb@29687
   871
    ultimately have ?thesis by blast}
chaieb@29687
   872
  moreover
chaieb@29687
   873
  {assume a0: "a$0 \<noteq> 0"
chaieb@29687
   874
    have ?thesis
chaieb@29687
   875
      apply (rule fps_inverse_unique)
chaieb@29687
   876
      apply (simp add: a0)
chaieb@29687
   877
      unfolding power_mult_distrib[symmetric]
chaieb@29687
   878
      apply (rule ssubst[where t = "a * inverse a" and s= 1])
chaieb@29687
   879
      apply simp_all
chaieb@29687
   880
      apply (subst mult_commute)
chaieb@29687
   881
      by (rule inverse_mult_eq_1[OF a0])}
chaieb@29687
   882
  ultimately show ?thesis by blast
chaieb@29687
   883
qed
chaieb@29687
   884
chaieb@29687
   885
lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
chaieb@29687
   886
  apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)
chaieb@29687
   887
  by (case_tac n, auto simp add: power_Suc ring_simps)
chaieb@29687
   888
chaieb@29687
   889
lemma fps_inverse_deriv: 
chaieb@29687
   890
  fixes a:: "('a :: field) fps"
chaieb@29687
   891
  assumes a0: "a$0 \<noteq> 0"
chaieb@29687
   892
  shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
chaieb@29687
   893
proof-
chaieb@29687
   894
  from inverse_mult_eq_1[OF a0]
chaieb@29687
   895
  have "fps_deriv (inverse a * a) = 0" by simp
chaieb@29687
   896
  hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
chaieb@29687
   897
  hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
chaieb@29687
   898
  with inverse_mult_eq_1[OF a0]
chaieb@29687
   899
  have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
chaieb@29687
   900
    unfolding power2_eq_square
chaieb@29687
   901
    apply (simp add: ring_simps)
chaieb@29687
   902
    by (simp add: mult_assoc[symmetric])
chaieb@29687
   903
  hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
chaieb@29687
   904
    by simp
chaieb@29687
   905
  then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)
chaieb@29687
   906
qed
chaieb@29687
   907
chaieb@29687
   908
lemma fps_inverse_mult: 
chaieb@29687
   909
  fixes a::"('a :: field) fps"
chaieb@29687
   910
  shows "inverse (a * b) = inverse a * inverse b"
chaieb@29687
   911
proof-
chaieb@29687
   912
  {assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
chaieb@29687
   913
    from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
chaieb@29687
   914
    have ?thesis unfolding th by simp}
chaieb@29687
   915
  moreover
chaieb@29687
   916
  {assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
chaieb@29687
   917
    from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
chaieb@29687
   918
    have ?thesis unfolding th by simp}
chaieb@29687
   919
  moreover
chaieb@29687
   920
  {assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"
chaieb@29687
   921
    from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
chaieb@29687
   922
    from inverse_mult_eq_1[OF ab0] 
chaieb@29687
   923
    have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
chaieb@29687
   924
    then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
chaieb@29687
   925
      by (simp add: ring_simps)
chaieb@29687
   926
    then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
chaieb@29687
   927
ultimately show ?thesis by blast
chaieb@29687
   928
qed
chaieb@29687
   929
chaieb@29687
   930
lemma fps_inverse_deriv': 
chaieb@29687
   931
  fixes a:: "('a :: field) fps"
chaieb@29687
   932
  assumes a0: "a$0 \<noteq> 0"
chaieb@29687
   933
  shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
chaieb@29687
   934
  using fps_inverse_deriv[OF a0]
chaieb@29687
   935
  unfolding power2_eq_square fps_divide_def
chaieb@29687
   936
    fps_inverse_mult by simp
chaieb@29687
   937
chaieb@29687
   938
lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)"
chaieb@29687
   939
  shows "f * inverse f= 1"
chaieb@29687
   940
  by (metis mult_commute inverse_mult_eq_1 f0)
chaieb@29687
   941
chaieb@29687
   942
lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
chaieb@29687
   943
  assumes a0: "b$0 \<noteq> 0"
chaieb@29687
   944
  shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
chaieb@29687
   945
  using fps_inverse_deriv[OF a0]
chaieb@29687
   946
  by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
chaieb@29687
   947
  
chaieb@29687
   948
section{* The eXtractor series X*}
chaieb@29687
   949
chaieb@29687
   950
lemma minus_one_power_iff: "(- (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else - 1)"
chaieb@29687
   951
  by (induct n, auto)
chaieb@29687
   952
chaieb@29687
   953
definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
chaieb@29687
   954
chaieb@29687
   955
lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field))) 
chaieb@29687
   956
  = 1 - X"
chaieb@29687
   957
  by (simp add: fps_inverse_gp fps_eq_iff X_def fps_minus_def fps_one_def)
chaieb@29687
   958
chaieb@29687
   959
lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n - 1))"
chaieb@29687
   960
proof-
chaieb@29687
   961
  {assume n: "n \<noteq> 0"
chaieb@29687
   962
    have fN: "finite {0 .. n}" by simp
chaieb@29687
   963
    have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n - i))" by (simp add: fps_mult_nth)
chaieb@29687
   964
    also have "\<dots> = f $ (n - 1)" 
chaieb@29687
   965
      using n by (simp add: X_def cond_value_iff cond_application_beta setsum_delta[OF fN] 
chaieb@29687
   966
	del: One_nat_def cong del:  if_weak_cong)
chaieb@29687
   967
  finally have ?thesis using n by simp }
chaieb@29687
   968
  moreover
chaieb@29687
   969
  {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
chaieb@29687
   970
  ultimately show ?thesis by blast
chaieb@29687
   971
qed
chaieb@29687
   972
chaieb@29687
   973
lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))"
chaieb@29687
   974
  by (metis X_mult_nth mult_commute)
chaieb@29687
   975
chaieb@29687
   976
lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
chaieb@29687
   977
proof(induct k)
chaieb@29687
   978
  case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)
chaieb@29687
   979
next
chaieb@29687
   980
  case (Suc k)
chaieb@29687
   981
  {fix m 
chaieb@29687
   982
    have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))"
chaieb@29687
   983
      by (simp add: power_Suc del: One_nat_def)
chaieb@29687
   984
    then     have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"
chaieb@29687
   985
      using Suc.hyps by (auto cong del: if_weak_cong)}
chaieb@29687
   986
  then show ?case by (simp add: fps_eq_iff)
chaieb@29687
   987
qed
chaieb@29687
   988
chaieb@29687
   989
lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n - k))"
chaieb@29687
   990
  apply (induct k arbitrary: n)
chaieb@29687
   991
  apply (simp)
chaieb@29687
   992
  unfolding power_Suc mult_assoc 
chaieb@29687
   993
  by (case_tac n, auto)
chaieb@29687
   994
chaieb@29687
   995
lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
chaieb@29687
   996
  by (metis X_power_mult_nth mult_commute)
chaieb@29687
   997
lemma fps_deriv_X[simp]: "fps_deriv X = 1"
chaieb@29687
   998
  by (simp add: fps_deriv_def X_def fps_eq_iff)
chaieb@29687
   999
chaieb@29687
  1000
lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
chaieb@29687
  1001
  by (cases "n", simp_all)
chaieb@29687
  1002
chaieb@29687
  1003
lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
chaieb@29687
  1004
lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"
chaieb@29687
  1005
  by (simp add: X_power_iff)
chaieb@29687
  1006
chaieb@29687
  1007
lemma fps_inverse_X_plus1:
chaieb@29687
  1008
  "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{recpower, field})) ^ n)" (is "_ = ?r")
chaieb@29687
  1009
proof-
chaieb@29687
  1010
  have eq: "(1 + X) * ?r = 1"
chaieb@29687
  1011
    unfolding minus_one_power_iff
chaieb@29687
  1012
    apply (auto simp add: ring_simps fps_eq_iff)
chaieb@29687
  1013
    by presburger+
chaieb@29687
  1014
  show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
chaieb@29687
  1015
qed
chaieb@29687
  1016
chaieb@29687
  1017
  
chaieb@29687
  1018
section{* Integration *}
chaieb@29687
  1019
definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
chaieb@29687
  1020
chaieb@29687
  1021
lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
chaieb@29687
  1022
  by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
chaieb@29687
  1023
chaieb@29687
  1024
lemma fps_integral_linear: "fps_integral (fps_const (a::'a::{field, ring_char_0}) * f + fps_const b * g) (a*a0 + b*b0) = fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" (is "?l = ?r")
chaieb@29687
  1025
proof-
chaieb@29687
  1026
  have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
chaieb@29687
  1027
  moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
chaieb@29687
  1028
  ultimately show ?thesis
chaieb@29687
  1029
    unfolding fps_deriv_eq_iff by auto
chaieb@29687
  1030
qed
chaieb@29687
  1031
  
chaieb@29687
  1032
section {* Composition of FPSs *}
chaieb@29687
  1033
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
chaieb@29687
  1034
  fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"
chaieb@29687
  1035
chaieb@29687
  1036
lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def)
chaieb@29687
  1037
chaieb@29687
  1038
lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
chaieb@29687
  1039
  by (auto simp add: fps_compose_def X_power_iff fps_eq_iff cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
chaieb@29687
  1040
 
chaieb@29687
  1041
lemma fps_const_compose[simp]: 
chaieb@29687
  1042
  "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
chaieb@29687
  1043
  apply (auto simp add: fps_eq_iff fps_compose_nth fps_mult_nth
chaieb@29687
  1044
  cond_application_beta cond_value_iff cong del: if_weak_cong)
chaieb@29687
  1045
  by (simp add: setsum_delta )
chaieb@29687
  1046
chaieb@29687
  1047
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
chaieb@29687
  1048
  apply (auto simp add: fps_compose_def fps_eq_iff cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
chaieb@29687
  1049
  apply (simp add: power_Suc)
chaieb@29687
  1050
  apply (subgoal_tac "n = 0")
chaieb@29687
  1051
  by simp_all
chaieb@29687
  1052
chaieb@29687
  1053
chaieb@29687
  1054
section {* Rules from Herbert Wilf's Generatingfunctionology*}
chaieb@29687
  1055
chaieb@29687
  1056
subsection {* Rule 1 *}
chaieb@29687
  1057
  (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
chaieb@29687
  1058
chaieb@29687
  1059
lemma fps_power_mult_eq_shift: 
chaieb@29687
  1060
  "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k}" (is "?lhs = ?rhs")
chaieb@29687
  1061
proof-
chaieb@29687
  1062
  {fix n:: nat
chaieb@29687
  1063
    have "?lhs $ n = (if n < Suc k then 0 else a n)" 
chaieb@29687
  1064
      unfolding X_power_mult_nth by auto
chaieb@29687
  1065
    also have "\<dots> = ?rhs $ n"
chaieb@29687
  1066
    proof(induct k)
chaieb@29687
  1067
      case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
chaieb@29687
  1068
    next
chaieb@29687
  1069
      case (Suc k)
chaieb@29687
  1070
      note th = Suc.hyps[symmetric]
chaieb@29687
  1071
      have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
chaieb@29687
  1072
      also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
chaieb@29687
  1073
	using th 
chaieb@29687
  1074
	unfolding fps_sub_nth by simp
chaieb@29687
  1075
      also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
chaieb@29687
  1076
	unfolding X_power_mult_right_nth
chaieb@29687
  1077
	apply (auto simp add: not_less fps_const_def)
chaieb@29687
  1078
	apply (rule cong[of a a, OF refl])
chaieb@29687
  1079
	by arith
chaieb@29687
  1080
      finally show ?case by simp
chaieb@29687
  1081
    qed
chaieb@29687
  1082
    finally have "?lhs $ n = ?rhs $ n"  .}
chaieb@29687
  1083
  then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  1084
qed
chaieb@29687
  1085
chaieb@29687
  1086
subsection{* Rule 2*}
chaieb@29687
  1087
chaieb@29687
  1088
  (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
chaieb@29687
  1089
  (* If f reprents {a_n} and P is a polynomial, then 
chaieb@29687
  1090
        P(xD) f represents {P(n) a_n}*)
chaieb@29687
  1091
chaieb@29687
  1092
definition "XD = op * X o fps_deriv"
chaieb@29687
  1093
chaieb@29687
  1094
lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
chaieb@29687
  1095
  by (simp add: XD_def ring_simps)
chaieb@29687
  1096
chaieb@29687
  1097
lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
chaieb@29687
  1098
  by (simp add: XD_def ring_simps)
chaieb@29687
  1099
chaieb@29687
  1100
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)"
chaieb@29687
  1101
  by simp
chaieb@29687
  1102
chaieb@29687
  1103
chaieb@29687
  1104
fun funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
chaieb@29687
  1105
  "funpow 0 f = id"
chaieb@29687
  1106
  | "funpow (Suc n) f = f o funpow n f"
chaieb@29687
  1107
chaieb@29687
  1108
lemma XDN_linear: "(funpow n XD) (fps_const c * a + fps_const d * b) = fps_const c * (funpow n XD) a + fps_const d * (funpow n XD) (b :: ('a::comm_ring_1) fps)"
chaieb@29687
  1109
  by (induct n, simp_all)
chaieb@29687
  1110
chaieb@29687
  1111
lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
chaieb@29687
  1112
chaieb@29687
  1113
lemma fps_mult_XD_shift: "funpow k XD (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
chaieb@29687
  1114
by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
chaieb@29687
  1115
chaieb@29687
  1116
subsection{* Rule 3 is trivial and is given by fps_times_def*}
chaieb@29687
  1117
subsection{* Rule 5 --- summation and "division" by (1 - X)*}
chaieb@29687
  1118
chaieb@29687
  1119
lemma fps_divide_X_minus1_setsum_lemma:
chaieb@29687
  1120
  "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
chaieb@29687
  1121
proof-
chaieb@29687
  1122
  let ?X = "X::('a::comm_ring_1) fps"
chaieb@29687
  1123
  let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
chaieb@29687
  1124
  have th0: "\<And>i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
chaieb@29687
  1125
  {fix n:: nat
chaieb@29687
  1126
    {assume "n=0" hence "a$n = ((1 - ?X) * ?sa) $ n" 
chaieb@29687
  1127
	by (simp add: fps_mult_nth)}
chaieb@29687
  1128
    moreover
chaieb@29687
  1129
    {assume n0: "n \<noteq> 0"
chaieb@29687
  1130
      then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
chaieb@29687
  1131
	"{0..n - 1}\<union>{n} = {0..n}"
chaieb@29687
  1132
	apply (simp_all add: expand_set_eq) by presburger+
chaieb@29687
  1133
      have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" 
chaieb@29687
  1134
	"{0..n - 1}\<inter>{n} ={}" using n0
chaieb@29687
  1135
	by (simp_all add: expand_set_eq, presburger+)
chaieb@29687
  1136
      have f: "finite {0}" "finite {1}" "finite {2 .. n}" 
chaieb@29687
  1137
	"finite {0 .. n - 1}" "finite {n}" by simp_all 
chaieb@29687
  1138
    have "((1 - ?X) * ?sa) $ n = setsum (\<lambda>i. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}"
chaieb@29687
  1139
      by (simp add: fps_mult_nth)
chaieb@29687
  1140
    also have "\<dots> = a$n" unfolding th0
chaieb@29687
  1141
      unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
chaieb@29687
  1142
      unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
chaieb@29687
  1143
      apply (simp)
chaieb@29687
  1144
      unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
chaieb@29687
  1145
      by simp
chaieb@29687
  1146
    finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp}
chaieb@29687
  1147
  ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast}
chaieb@29687
  1148
then show ?thesis 
chaieb@29687
  1149
  unfolding fps_eq_iff by blast
chaieb@29687
  1150
qed
chaieb@29687
  1151
chaieb@29687
  1152
lemma fps_divide_X_minus1_setsum:
chaieb@29687
  1153
  "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
chaieb@29687
  1154
proof-
chaieb@29687
  1155
  let ?X = "1 - (X::('a::field) fps)"
chaieb@29687
  1156
  have th0: "?X $ 0 \<noteq> 0" by simp
chaieb@29687
  1157
  have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X"
chaieb@29687
  1158
    using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
chaieb@29687
  1159
    by (simp add: fps_divide_def mult_assoc)
chaieb@29687
  1160
  also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) "
chaieb@29687
  1161
    by (simp add: mult_ac)
chaieb@29687
  1162
  finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
chaieb@29687
  1163
qed
chaieb@29687
  1164
chaieb@29687
  1165
subsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary 
chaieb@29687
  1166
  finite product of FPS, also the relvant instance of powers of a FPS*}
chaieb@29687
  1167
chaieb@29687
  1168
definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
chaieb@29687
  1169
chaieb@29687
  1170
lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
chaieb@29687
  1171
  apply (auto simp add: natpermute_def)
chaieb@29687
  1172
  apply (case_tac x, auto)
chaieb@29687
  1173
  done
chaieb@29687
  1174
chaieb@29687
  1175
lemma foldl_add_start0: 
chaieb@29687
  1176
  "foldl op + x xs = x + foldl op + (0::nat) xs"
chaieb@29687
  1177
  apply (induct xs arbitrary: x)
chaieb@29687
  1178
  apply simp
chaieb@29687
  1179
  unfolding foldl.simps
chaieb@29687
  1180
  apply atomize
chaieb@29687
  1181
  apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
chaieb@29687
  1182
  apply (erule_tac x="x + a" in allE)
chaieb@29687
  1183
  apply (erule_tac x="a" in allE)
chaieb@29687
  1184
  apply simp
chaieb@29687
  1185
  apply assumption
chaieb@29687
  1186
  done
chaieb@29687
  1187
chaieb@29687
  1188
lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
chaieb@29687
  1189
  apply (induct ys arbitrary: x xs)
chaieb@29687
  1190
  apply auto
chaieb@29687
  1191
  apply (subst (2) foldl_add_start0)
chaieb@29687
  1192
  apply simp
chaieb@29687
  1193
  apply (subst (2) foldl_add_start0)
chaieb@29687
  1194
  by simp
chaieb@29687
  1195
chaieb@29687
  1196
lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
chaieb@29687
  1197
proof(induct xs arbitrary: x)
chaieb@29687
  1198
  case Nil thus ?case by simp
chaieb@29687
  1199
next
chaieb@29687
  1200
  case (Cons a as x)
chaieb@29687
  1201
  have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
chaieb@29687
  1202
    apply (rule setsum_reindex_cong [where f=Suc])
chaieb@29687
  1203
    by (simp_all add: inj_on_def)
chaieb@29687
  1204
  have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
chaieb@29687
  1205
  have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
chaieb@29687
  1206
  have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
chaieb@29687
  1207
  have "foldl op + x (a#as) = x + foldl op + a as "
chaieb@29687
  1208
    apply (subst foldl_add_start0)    by simp
chaieb@29687
  1209
  also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
chaieb@29687
  1210
  also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
chaieb@29687
  1211
    unfolding eq[symmetric] 
chaieb@29687
  1212
    unfolding setsum_Un_disjoint[OF f d, unfolded seq]
chaieb@29687
  1213
    by simp
chaieb@29687
  1214
  finally show ?case  .
chaieb@29687
  1215
qed
chaieb@29687
  1216
chaieb@29687
  1217
chaieb@29687
  1218
lemma append_natpermute_less_eq:
chaieb@29687
  1219
  assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
chaieb@29687
  1220
proof-
chaieb@29687
  1221
  {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
chaieb@29687
  1222
    hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
chaieb@29687
  1223
  note th = this
chaieb@29687
  1224
  {from th show "foldl op + 0 xs \<le> n" by simp}
chaieb@29687
  1225
  {from th show "foldl op + 0 ys \<le> n" by simp}
chaieb@29687
  1226
qed
chaieb@29687
  1227
chaieb@29687
  1228
lemma natpermute_split:
chaieb@29687
  1229
  assumes mn: "h \<le> k"
chaieb@29687
  1230
  shows "natpermute n k = (\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})" (is "?L = ?R" is "?L = (\<Union>m \<in>{0..n}. ?S m)")
chaieb@29687
  1231
proof-
chaieb@29687
  1232
  {fix l assume l: "l \<in> ?R" 
chaieb@29687
  1233
    from l obtain m xs ys where h: "m \<in> {0..n}" and xs: "xs \<in> natpermute m h" and ys: "ys \<in> natpermute (n - m) (k - h)"  and leq: "l = xs@ys" by blast
chaieb@29687
  1234
    from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
chaieb@29687
  1235
    from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
chaieb@29687
  1236
    have "l \<in> ?L" using leq xs ys h 
chaieb@29687
  1237
      apply simp
chaieb@29687
  1238
      apply (clarsimp simp add: natpermute_def simp del: foldl_append)
chaieb@29687
  1239
      apply (simp add: foldl_add_append[unfolded foldl_append])
chaieb@29687
  1240
      unfolding xs' ys'
chaieb@29687
  1241
      using mn xs ys 
chaieb@29687
  1242
      unfolding natpermute_def by simp}
chaieb@29687
  1243
  moreover
chaieb@29687
  1244
  {fix l assume l: "l \<in> natpermute n k"
chaieb@29687
  1245
    let ?xs = "take h l"
chaieb@29687
  1246
    let ?ys = "drop h l"
chaieb@29687
  1247
    let ?m = "foldl op + 0 ?xs"
chaieb@29687
  1248
    from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
chaieb@29687
  1249
    have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)     
chaieb@29687
  1250
    have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
chaieb@29687
  1251
      by (simp add: natpermute_def)
chaieb@29687
  1252
    from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
chaieb@29687
  1253
    from xs ys ls have "l \<in> ?R" 
chaieb@29687
  1254
      apply auto
chaieb@29687
  1255
      apply (rule bexI[where x = "?m"])
chaieb@29687
  1256
      apply (rule exI[where x = "?xs"])
chaieb@29687
  1257
      apply (rule exI[where x = "?ys"])
chaieb@29687
  1258
      using ls l unfolding foldl_add_append 
chaieb@29687
  1259
      by (auto simp add: natpermute_def)}
chaieb@29687
  1260
  ultimately show ?thesis by blast
chaieb@29687
  1261
qed
chaieb@29687
  1262
chaieb@29687
  1263
lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
chaieb@29687
  1264
  by (auto simp add: natpermute_def)
chaieb@29687
  1265
lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
chaieb@29687
  1266
  apply (auto simp add: set_replicate_conv_if natpermute_def)
chaieb@29687
  1267
  apply (rule nth_equalityI)
chaieb@29687
  1268
  by simp_all
chaieb@29687
  1269
chaieb@29687
  1270
lemma natpermute_finite: "finite (natpermute n k)"
chaieb@29687
  1271
proof(induct k arbitrary: n)
chaieb@29687
  1272
  case 0 thus ?case 
chaieb@29687
  1273
    apply (subst natpermute_split[of 0 0, simplified])
chaieb@29687
  1274
    by (simp add: natpermute_0)
chaieb@29687
  1275
next
chaieb@29687
  1276
  case (Suc k)
chaieb@29687
  1277
  then show ?case unfolding natpermute_split[of k "Suc k", simplified]
chaieb@29687
  1278
    apply -
chaieb@29687
  1279
    apply (rule finite_UN_I)
chaieb@29687
  1280
    apply simp
chaieb@29687
  1281
    unfolding One_nat_def[symmetric] natlist_trivial_1
chaieb@29687
  1282
    apply simp
chaieb@29687
  1283
    unfolding image_Collect[symmetric]
chaieb@29687
  1284
    unfolding Collect_def mem_def
chaieb@29687
  1285
    apply (rule finite_imageI)
chaieb@29687
  1286
    apply blast
chaieb@29687
  1287
    done
chaieb@29687
  1288
qed
chaieb@29687
  1289
chaieb@29687
  1290
lemma natpermute_contain_maximal:
chaieb@29687
  1291
  "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
chaieb@29687
  1292
  (is "?A = ?B")
chaieb@29687
  1293
proof-
chaieb@29687
  1294
  {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
chaieb@29687
  1295
    from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
chaieb@29687
  1296
      unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def) 
chaieb@29687
  1297
    have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
chaieb@29687
  1298
    have f: "finite({0..k} - {i})" "finite {i}" by auto
chaieb@29687
  1299
    have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
chaieb@29687
  1300
    from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
chaieb@29687
  1301
      unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
chaieb@29687
  1302
    also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
chaieb@29687
  1303
      unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
chaieb@29687
  1304
    finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
chaieb@29687
  1305
    from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
chaieb@29687
  1306
    from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
chaieb@29687
  1307
      unfolding length_replicate  by arith+
chaieb@29687
  1308
    have "xs = replicate (k+1) 0 [i := n]"
chaieb@29687
  1309
      apply (rule nth_equalityI)
chaieb@29687
  1310
      unfolding xsl length_list_update length_replicate
chaieb@29687
  1311
      apply simp
chaieb@29687
  1312
      apply clarify
chaieb@29687
  1313
      unfolding nth_list_update[OF i'(1)]
chaieb@29687
  1314
      using i zxs
chaieb@29687
  1315
      by (case_tac "ia=i", auto simp del: replicate.simps)
chaieb@29687
  1316
    then have "xs \<in> ?B" using i by blast}
chaieb@29687
  1317
  moreover
chaieb@29687
  1318
  {fix i assume i: "i \<in> {0..k}"
chaieb@29687
  1319
    let ?xs = "replicate (k+1) 0 [i:=n]"
chaieb@29687
  1320
    have nxs: "n \<in> set ?xs"
chaieb@29687
  1321
      apply (rule set_update_memI) using i by simp
chaieb@29687
  1322
    have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
chaieb@29687
  1323
    have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
chaieb@29687
  1324
      unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
chaieb@29687
  1325
    also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
chaieb@29687
  1326
      apply (rule setsum_cong2) by (simp del: replicate.simps)
chaieb@29687
  1327
    also have "\<dots> = n" using i by (simp add: setsum_delta)
chaieb@29687
  1328
    finally 
chaieb@29687
  1329
    have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def
chaieb@29687
  1330
      by blast
chaieb@29687
  1331
    then have "?xs \<in> ?A"  using nxs  by blast}
chaieb@29687
  1332
  ultimately show ?thesis by auto
chaieb@29687
  1333
qed
chaieb@29687
  1334
chaieb@29687
  1335
    (* The general form *)	
chaieb@29687
  1336
lemma fps_setprod_nth:
chaieb@29687
  1337
  fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
chaieb@29687
  1338
  shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
chaieb@29687
  1339
  (is "?P m n")
chaieb@29687
  1340
proof(induct m arbitrary: n rule: nat_less_induct)
chaieb@29687
  1341
  fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
chaieb@29687
  1342
  {assume m0: "m = 0"
chaieb@29687
  1343
    hence "?P m n" apply simp
chaieb@29687
  1344
      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
chaieb@29687
  1345
  moreover
chaieb@29687
  1346
  {fix k assume k: "m = Suc k"
chaieb@29687
  1347
    have km: "k < m" using k by arith
chaieb@29687
  1348
    have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger
chaieb@29687
  1349
    have f0: "finite {0 .. k}" "finite {m}" by auto
chaieb@29687
  1350
    have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
chaieb@29687
  1351
    have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
chaieb@29687
  1352
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
chaieb@29687
  1353
    also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j $ v ! j) * a m $ (n - i))"
chaieb@29687
  1354
      unfolding fps_mult_nth H[rule_format, OF km] ..
chaieb@29687
  1355
    also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
chaieb@29687
  1356
      apply (simp add: k)
chaieb@29687
  1357
      unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
chaieb@29687
  1358
      apply (subst setsum_UN_disjoint)
chaieb@29687
  1359
      apply simp 
chaieb@29687
  1360
      apply simp
chaieb@29687
  1361
      unfolding image_Collect[symmetric]
chaieb@29687
  1362
      apply clarsimp
chaieb@29687
  1363
      apply (rule finite_imageI)
chaieb@29687
  1364
      apply (rule natpermute_finite)
chaieb@29687
  1365
      apply (clarsimp simp add: expand_set_eq)
chaieb@29687
  1366
      apply auto
chaieb@29687
  1367
      apply (rule setsum_cong2)
chaieb@29687
  1368
      unfolding setsum_left_distrib
chaieb@29687
  1369
      apply (rule sym)
chaieb@29687
  1370
      apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
chaieb@29687
  1371
      apply (simp add: inj_on_def)
chaieb@29687
  1372
      apply auto
chaieb@29687
  1373
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
chaieb@29687
  1374
      apply (clarsimp simp add: natpermute_def nth_append)
chaieb@29687
  1375
      apply (rule_tac f="\<lambda>x. x * a (Suc k) $ (n - foldl op + 0 aa)" in cong[OF refl])
chaieb@29687
  1376
      apply (rule setprod_cong)
chaieb@29687
  1377
      apply simp
chaieb@29687
  1378
      apply simp
chaieb@29687
  1379
      done
chaieb@29687
  1380
    finally have "?P m n" .}
chaieb@29687
  1381
  ultimately show "?P m n " by (cases m, auto)
chaieb@29687
  1382
qed
chaieb@29687
  1383
chaieb@29687
  1384
text{* The special form for powers *}
chaieb@29687
  1385
lemma fps_power_nth_Suc:
chaieb@29687
  1386
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
chaieb@29687
  1387
  shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
chaieb@29687
  1388
proof-
chaieb@29687
  1389
  have f: "finite {0 ..m}" by simp
chaieb@29687
  1390
  have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
chaieb@29687
  1391
  show ?thesis unfolding th0 fps_setprod_nth ..
chaieb@29687
  1392
qed
chaieb@29687
  1393
lemma fps_power_nth:
chaieb@29687
  1394
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
chaieb@29687
  1395
  shows "(a ^m)$n = (if m=0 then 1$n else setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))"
chaieb@29687
  1396
  by (cases m, simp_all add: fps_power_nth_Suc)
chaieb@29687
  1397
chaieb@29687
  1398
lemma fps_nth_power_0: 
chaieb@29687
  1399
  fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps"
chaieb@29687
  1400
  shows "(a ^m)$0 = (a$0) ^ m"
chaieb@29687
  1401
proof-
chaieb@29687
  1402
  {assume "m=0" hence ?thesis by simp}
chaieb@29687
  1403
  moreover
chaieb@29687
  1404
  {fix n assume m: "m = Suc n"
chaieb@29687
  1405
    have c: "m = card {0..n}" using m by simp
chaieb@29687
  1406
   have "(a ^m)$0 = setprod (\<lambda>i. a$0) {0..n}"
chaieb@29687
  1407
     apply (simp add: m fps_power_nth del: replicate.simps)
chaieb@29687
  1408
     apply (rule setprod_cong)
chaieb@29687
  1409
     by (simp_all del: replicate.simps)
chaieb@29687
  1410
   also have "\<dots> = (a$0) ^ m"
chaieb@29687
  1411
     unfolding c by (rule setprod_constant, simp)
chaieb@29687
  1412
   finally have ?thesis .}
chaieb@29687
  1413
 ultimately show ?thesis by (cases m, auto)
chaieb@29687
  1414
qed
chaieb@29687
  1415
chaieb@29687
  1416
lemma fps_compose_inj_right: 
chaieb@29687
  1417
  assumes a0: "a$0 = (0::'a::{recpower,idom})"
chaieb@29687
  1418
  and a1: "a$1 \<noteq> 0"
chaieb@29687
  1419
  shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
chaieb@29687
  1420
proof-
chaieb@29687
  1421
  {assume ?rhs then have "?lhs" by simp}
chaieb@29687
  1422
  moreover
chaieb@29687
  1423
  {assume h: ?lhs
chaieb@29687
  1424
    {fix n have "b$n = c$n" 
chaieb@29687
  1425
      proof(induct n rule: nat_less_induct)
chaieb@29687
  1426
	fix n assume H: "\<forall>m<n. b$m = c$m"
chaieb@29687
  1427
	{assume n0: "n=0"
chaieb@29687
  1428
	  from h have "(b oo a)$n = (c oo a)$n" by simp
chaieb@29687
  1429
	  hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)}
chaieb@29687
  1430
	moreover
chaieb@29687
  1431
	{fix n1 assume n1: "n = Suc n1"
chaieb@29687
  1432
	  have f: "finite {0 .. n1}" "finite {n}" by simp_all
chaieb@29687
  1433
	  have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
chaieb@29687
  1434
	  have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
chaieb@29687
  1435
	  have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
chaieb@29687
  1436
	    apply (rule setsum_cong2)
chaieb@29687
  1437
	    using H n1 by auto
chaieb@29687
  1438
	  have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
chaieb@29687
  1439
	    unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
chaieb@29687
  1440
	    using startsby_zero_power_nth_same[OF a0]
chaieb@29687
  1441
	    by simp
chaieb@29687
  1442
	  have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
chaieb@29687
  1443
	    unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
chaieb@29687
  1444
	    using startsby_zero_power_nth_same[OF a0]
chaieb@29687
  1445
	    by simp
chaieb@29687
  1446
	  from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
chaieb@29687
  1447
	  have "b$n = c$n" by auto}
chaieb@29687
  1448
	ultimately show "b$n = c$n" by (cases n, auto)
chaieb@29687
  1449
      qed}
chaieb@29687
  1450
    then have ?rhs by (simp add: fps_eq_iff)}
chaieb@29687
  1451
  ultimately show ?thesis by blast
chaieb@29687
  1452
qed
chaieb@29687
  1453
chaieb@29687
  1454
chaieb@29687
  1455
section {* Radicals *}
chaieb@29687
  1456
chaieb@29687
  1457
declare setprod_cong[fundef_cong]
chaieb@29687
  1458
function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a::{field, recpower}) fps \<Rightarrow> nat \<Rightarrow> 'a" where
chaieb@29687
  1459
  "radical r 0 a 0 = 1"
chaieb@29687
  1460
| "radical r 0 a (Suc n) = 0"
chaieb@29687
  1461
| "radical r (Suc k) a 0 = r (Suc k) (a$0)"
chaieb@29687
  1462
| "radical r (Suc k) a (Suc n) = (a$ Suc n - setsum (\<lambda>xs. setprod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k}) {xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) / (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
chaieb@29687
  1463
by pat_completeness auto
chaieb@29687
  1464
chaieb@29687
  1465
termination radical
chaieb@29687
  1466
proof
chaieb@29687
  1467
  let ?R = "measure (\<lambda>(r, k, a, n). n)"
chaieb@29687
  1468
  {
chaieb@29687
  1469
    show "wf ?R" by auto}
chaieb@29687
  1470
  {fix r k a n xs i
chaieb@29687
  1471
    assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
chaieb@29687
  1472
    {assume c: "Suc n \<le> xs ! i"
chaieb@29687
  1473
      from xs i have "xs !i \<noteq> Suc n" by (auto simp add: in_set_conv_nth natpermute_def)
chaieb@29687
  1474
      with c have c': "Suc n < xs!i" by arith
chaieb@29687
  1475
      have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
chaieb@29687
  1476
      have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
chaieb@29687
  1477
      have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
chaieb@29687
  1478
      from xs have "Suc n = foldl op + 0 xs" by (simp add: natpermute_def)
chaieb@29687
  1479
      also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
chaieb@29687
  1480
	by (simp add: natpermute_def)
chaieb@29687
  1481
      also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
chaieb@29687
  1482
	unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
chaieb@29687
  1483
	unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
chaieb@29687
  1484
	by simp
chaieb@29687
  1485
      finally have False using c' by simp}
chaieb@29687
  1486
    then show "((r,Suc k,a,xs!i), r, Suc k, a, Suc n) \<in> ?R" 
chaieb@29687
  1487
      apply auto by (metis not_less)}
chaieb@29687
  1488
  {fix r k a n 
chaieb@29687
  1489
    show "((r,Suc k, a, 0),r, Suc k, a, Suc n) \<in> ?R" by simp}
chaieb@29687
  1490
qed
chaieb@29687
  1491
chaieb@29687
  1492
definition "fps_radical r n a = Abs_fps (radical r n a)"
chaieb@29687
  1493
chaieb@29687
  1494
lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
chaieb@29687
  1495
  apply (auto simp add: fps_eq_iff fps_radical_def)  by (case_tac n, auto)
chaieb@29687
  1496
chaieb@29687
  1497
lemma fps_radical_nth_0[simp]: "fps_radical r n a $ 0 = (if n=0 then 1 else r n (a$0))"
chaieb@29687
  1498
  by (cases n, simp_all add: fps_radical_def)
chaieb@29687
  1499
chaieb@29687
  1500
lemma fps_radical_power_nth[simp]: 
chaieb@29687
  1501
  assumes r: "(r k (a$0)) ^ k = a$0"
chaieb@29687
  1502
  shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)"
chaieb@29687
  1503
proof-
chaieb@29687
  1504
  {assume "k=0" hence ?thesis by simp }
chaieb@29687
  1505
  moreover
chaieb@29687
  1506
  {fix h assume h: "k = Suc h" 
chaieb@29687
  1507
    have fh: "finite {0..h}" by simp
chaieb@29687
  1508
    have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
chaieb@29687
  1509
      unfolding fps_power_nth h by simp
chaieb@29687
  1510
    also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))"
chaieb@29687
  1511
      apply (rule setprod_cong)
chaieb@29687
  1512
      apply simp
chaieb@29687
  1513
      using h
chaieb@29687
  1514
      apply (subgoal_tac "replicate k (0::nat) ! x = 0")
chaieb@29687
  1515
      by (auto intro: nth_replicate simp del: replicate.simps)
chaieb@29687
  1516
    also have "\<dots> = a$0"
chaieb@29687
  1517
      unfolding setprod_constant[OF fh] using r by (simp add: h)
chaieb@29687
  1518
    finally have ?thesis using h by simp}
chaieb@29687
  1519
  ultimately show ?thesis by (cases k, auto)
chaieb@29687
  1520
qed 
chaieb@29687
  1521
chaieb@29687
  1522
lemma natpermute_max_card: assumes n0: "n\<noteq>0" 
chaieb@29687
  1523
  shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k+1"
chaieb@29687
  1524
  unfolding natpermute_contain_maximal
chaieb@29687
  1525
proof-
chaieb@29687
  1526
  let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
chaieb@29687
  1527
  let ?K = "{0 ..k}"
chaieb@29687
  1528
  have fK: "finite ?K" by simp
chaieb@29687
  1529
  have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
chaieb@29687
  1530
  have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow> {replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
chaieb@29687
  1531
  proof(clarify)
chaieb@29687
  1532
    fix i j assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
chaieb@29687
  1533
    {assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
chaieb@29687
  1534
      have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps)
chaieb@29687
  1535
      moreover
chaieb@29687
  1536
      have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps)
chaieb@29687
  1537
      ultimately have False using eq n0 by (simp del: replicate.simps)}
chaieb@29687
  1538
    then show "{replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
chaieb@29687
  1539
      by auto
chaieb@29687
  1540
  qed
chaieb@29687
  1541
  from card_UN_disjoint[OF fK fAK d] 
chaieb@29687
  1542
  show "card (\<Union>i\<in>{0..k}. {replicate (k + 1) 0[i := n]}) = k+1" by simp
chaieb@29687
  1543
qed
chaieb@29687
  1544
  
chaieb@29687
  1545
lemma power_radical: 
chaieb@29687
  1546
  fixes a:: "'a ::{field, ring_char_0, recpower} fps"
chaieb@29687
  1547
  assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
chaieb@29687
  1548
  shows "(fps_radical r (Suc k) a) ^ (Suc k) = a" 
chaieb@29687
  1549
proof-
chaieb@29687
  1550
  let ?r = "fps_radical r (Suc k) a"
chaieb@29687
  1551
  from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto
chaieb@29687
  1552
  {fix z have "?r ^ Suc k $ z = a$z"
chaieb@29687
  1553
    proof(induct z rule: nat_less_induct)
chaieb@29687
  1554
      fix n assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
chaieb@29687
  1555
      {assume "n = 0" hence "?r ^ Suc k $ n = a $n"
chaieb@29687
  1556
	  using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
chaieb@29687
  1557
      moreover
chaieb@29687
  1558
      {fix n1 assume n1: "n = Suc n1"
chaieb@29687
  1559
	have fK: "finite {0..k}" by simp
chaieb@29687
  1560
	have nz: "n \<noteq> 0" using n1 by arith
chaieb@29687
  1561
	let ?Pnk = "natpermute n (k + 1)"
chaieb@29687
  1562
	let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
chaieb@29687
  1563
	let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
chaieb@29687
  1564
	have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
chaieb@29687
  1565
	have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
chaieb@29687
  1566
	have f: "finite ?Pnkn" "finite ?Pnknn" 
chaieb@29687
  1567
	  using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
chaieb@29687
  1568
	  by (metis natpermute_finite)+
chaieb@29687
  1569
	let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
chaieb@29687
  1570
	have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" 
chaieb@29687
  1571
	proof(rule setsum_cong2)
chaieb@29687
  1572
	  fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
chaieb@29687
  1573
	  let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
chaieb@29687
  1574
	  from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
chaieb@29687
  1575
	    unfolding natpermute_contain_maximal by auto
chaieb@29687
  1576
	  have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
chaieb@29687
  1577
	    apply (rule setprod_cong, simp)
chaieb@29687
  1578
	    using i r0 by (simp del: replicate.simps)
chaieb@29687
  1579
	  also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
chaieb@29687
  1580
	    unfolding setprod_gen_delta[OF fK] using i r0 by simp
chaieb@29687
  1581
	  finally show ?ths .
chaieb@29687
  1582
	qed
chaieb@29687
  1583
	then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"  
chaieb@29687
  1584
	  by (simp add: natpermute_max_card[OF nz, simplified]) 
chaieb@29687
  1585
	also have "\<dots> = a$n - setsum ?f ?Pnknn"
chaieb@29687
  1586
	  unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
chaieb@29687
  1587
	finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
chaieb@29687
  1588
	have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn" 
chaieb@29687
  1589
	  unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
chaieb@29687
  1590
	also have "\<dots> = a$n" unfolding fn by simp
chaieb@29687
  1591
	finally have "?r ^ Suc k $ n = a $n" .}
chaieb@29687
  1592
      ultimately  show "?r ^ Suc k $ n = a $n" by (cases n, auto)
chaieb@29687
  1593
  qed }
chaieb@29687
  1594
  then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  1595
qed
chaieb@29687
  1596
chaieb@29687
  1597
lemma eq_divide_imp': assumes c0: "(c::'a::field) ~= 0" and eq: "a * c = b"
chaieb@29687
  1598
  shows "a = b / c" 
chaieb@29687
  1599
proof-
chaieb@29687
  1600
  from eq have "a * c * inverse c = b * inverse c" by simp
chaieb@29687
  1601
  hence "a * (inverse c * c) = b/c" by (simp only: field_simps divide_inverse)
chaieb@29687
  1602
  then show "a = b/c" unfolding  field_inverse[OF c0] by simp
chaieb@29687
  1603
qed
chaieb@29687
  1604
chaieb@29687
  1605
lemma radical_unique:  
chaieb@29687
  1606
  assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0" 
chaieb@29687
  1607
  and a0: "r (Suc k) (b$0 ::'a::{field, ring_char_0, recpower}) = a$0" and b0: "b$0 \<noteq> 0"
chaieb@29687
  1608
  shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
chaieb@29687
  1609
proof-
chaieb@29687
  1610
  let ?r = "fps_radical r (Suc k) b"
chaieb@29687
  1611
  have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
chaieb@29687
  1612
  {assume H: "a = ?r"
chaieb@29687
  1613
    from H have "a^Suc k = b" using power_radical[of r k, OF r0 b0] by simp}
chaieb@29687
  1614
  moreover
chaieb@29687
  1615
  {assume H: "a^Suc k = b"
chaieb@29687
  1616
    (* Generally a$0 would need to be the k+1 st root of b$0 *)
chaieb@29687
  1617
    have ceq: "card {0..k} = Suc k" by simp
chaieb@29687
  1618
    have fk: "finite {0..k}" by simp
chaieb@29687
  1619
    from a0 have a0r0: "a$0 = ?r$0" by simp
chaieb@29687
  1620
    {fix n have "a $ n = ?r $ n"
chaieb@29687
  1621
      proof(induct n rule: nat_less_induct)
chaieb@29687
  1622
	fix n assume h: "\<forall>m<n. a$m = ?r $m"
chaieb@29687
  1623
	{assume "n = 0" hence "a$n = ?r $n" using a0 by simp }
chaieb@29687
  1624
	moreover
chaieb@29687
  1625
	{fix n1 assume n1: "n = Suc n1"
chaieb@29687
  1626
	  have fK: "finite {0..k}" by simp
chaieb@29687
  1627
	have nz: "n \<noteq> 0" using n1 by arith
chaieb@29687
  1628
	let ?Pnk = "natpermute n (Suc k)"
chaieb@29687
  1629
	let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
chaieb@29687
  1630
	let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
chaieb@29687
  1631
	have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
chaieb@29687
  1632
	have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
chaieb@29687
  1633
	have f: "finite ?Pnkn" "finite ?Pnknn" 
chaieb@29687
  1634
	  using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
chaieb@29687
  1635
	  by (metis natpermute_finite)+
chaieb@29687
  1636
	let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
chaieb@29687
  1637
	let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a $ v ! j"
chaieb@29687
  1638
	have "setsum ?g ?Pnkn = setsum (\<lambda>v. a $ n * (?r$0)^k) ?Pnkn" 
chaieb@29687
  1639
	proof(rule setsum_cong2)
chaieb@29687
  1640
	  fix v assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
chaieb@29687
  1641
	  let ?ths = "(\<Prod>j\<in>{0..k}. a $ v ! j) = a $ n * (?r$0)^k"
chaieb@29687
  1642
	  from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
chaieb@29687
  1643
	    unfolding Suc_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps)
chaieb@29687
  1644
	  have "(\<Prod>j\<in>{0..k}. a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then a $ n else r (Suc k) (b$0))"
chaieb@29687
  1645
	    apply (rule setprod_cong, simp)
chaieb@29687
  1646
	    using i a0 by (simp del: replicate.simps)
chaieb@29687
  1647
	  also have "\<dots> = a $ n * (?r $ 0)^k"
chaieb@29687
  1648
	    unfolding  setprod_gen_delta[OF fK] using i by simp
chaieb@29687
  1649
	  finally show ?ths .
chaieb@29687
  1650
	qed
chaieb@29687
  1651
	then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k"  
chaieb@29687
  1652
	  by (simp add: natpermute_max_card[OF nz, simplified])
chaieb@29687
  1653
	have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
chaieb@29687
  1654
	proof (rule setsum_cong2, rule setprod_cong, simp)
chaieb@29687
  1655
	  fix xs i assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
chaieb@29687
  1656
	  {assume c: "n \<le> xs ! i"
chaieb@29687
  1657
	    from xs i have "xs !i \<noteq> n" by (auto simp add: in_set_conv_nth natpermute_def)
chaieb@29687
  1658
	    with c have c': "n < xs!i" by arith
chaieb@29687
  1659
	    have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
chaieb@29687
  1660
	    have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
chaieb@29687
  1661
	    have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
chaieb@29687
  1662
	    from xs have "n = foldl op + 0 xs" by (simp add: natpermute_def)
chaieb@29687
  1663
	    also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
chaieb@29687
  1664
	      by (simp add: natpermute_def)
chaieb@29687
  1665
	    also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
chaieb@29687
  1666
	      unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
chaieb@29687
  1667
	      unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
chaieb@29687
  1668
	      by simp
chaieb@29687
  1669
	    finally have False using c' by simp}
chaieb@29687
  1670
	  then have thn: "xs!i < n" by arith
chaieb@29687
  1671
	  from h[rule_format, OF thn]  
chaieb@29687
  1672
	  show "a$(xs !i) = ?r$(xs!i)" .
chaieb@29687
  1673
	qed
chaieb@29687
  1674
	have th00: "\<And>(x::'a). of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
chaieb@29687
  1675
	  by (simp add: field_simps del: of_nat_Suc)
chaieb@29687
  1676
	from H have "b$n = a^Suc k $ n" by (simp add: fps_eq_iff)
chaieb@29687
  1677
	also have "a ^ Suc k$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
chaieb@29687
  1678
	  unfolding fps_power_nth_Suc 
chaieb@29687
  1679
	  using setsum_Un_disjoint[OF f d, unfolded Suc_plus1[symmetric], 
chaieb@29687
  1680
	    unfolded eq, of ?g] by simp
chaieb@29687
  1681
	also have "\<dots> = of_nat (k+1) * a $ n * (?r $ 0)^k + setsum ?f ?Pnknn" unfolding th0 th1 ..
chaieb@29687
  1682
	finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - setsum ?f ?Pnknn" by simp
chaieb@29687
  1683
	then have "a$n = (b$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
chaieb@29687
  1684
	  apply - 
chaieb@29687
  1685
	  apply (rule eq_divide_imp')
chaieb@29687
  1686
	  using r00
chaieb@29687
  1687
	  apply (simp del: of_nat_Suc)
chaieb@29687
  1688
	  by (simp add: mult_ac)
chaieb@29687
  1689
	then have "a$n = ?r $n"
chaieb@29687
  1690
	  apply (simp del: of_nat_Suc)
chaieb@29687
  1691
	  unfolding fps_radical_def n1
chaieb@29687
  1692
	  by (simp add: field_simps n1 fps_nth_def th00 del: of_nat_Suc)}
chaieb@29687
  1693
	ultimately show "a$n = ?r $ n" by (cases n, auto)
chaieb@29687
  1694
      qed}
chaieb@29687
  1695
    then have "a = ?r" by (simp add: fps_eq_iff)}
chaieb@29687
  1696
  ultimately show ?thesis by blast
chaieb@29687
  1697
qed
chaieb@29687
  1698
chaieb@29687
  1699
chaieb@29687
  1700
lemma radical_power: 
chaieb@29687
  1701
  assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0" 
chaieb@29687
  1702
  and a0: "(a$0 ::'a::{field, ring_char_0, recpower}) \<noteq> 0"
chaieb@29687
  1703
  shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
chaieb@29687
  1704
proof-
chaieb@29687
  1705
  let ?ak = "a^ Suc k"
chaieb@29687
  1706
  have ak0: "?ak $ 0 = (a$0) ^ Suc k" by (simp add: fps_nth_power_0)
chaieb@29687
  1707
  from r0 have th0: "r (Suc k) (a ^ Suc k $ 0) ^ Suc k = a ^ Suc k $ 0" using ak0 by auto
chaieb@29687
  1708
  from r0 ak0 have th1: "r (Suc k) (a ^ Suc k $ 0) = a $ 0" by auto
chaieb@29687
  1709
  from ak0 a0 have ak00: "?ak $ 0 \<noteq>0 " by auto
chaieb@29687
  1710
  from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis by metis
chaieb@29687
  1711
qed
chaieb@29687
  1712
chaieb@29687
  1713
lemma fps_deriv_radical: 
chaieb@29687
  1714
  fixes a:: "'a ::{field, ring_char_0, recpower} fps"
chaieb@29687
  1715
  assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
chaieb@29687
  1716
  shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
chaieb@29687
  1717
proof-
chaieb@29687
  1718
  let ?r= "fps_radical r (Suc k) a"
chaieb@29687
  1719
  let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
chaieb@29687
  1720
  from a0 r0 have r0': "r (Suc k) (a$0) \<noteq> 0" by auto
chaieb@29687
  1721
  from r0' have w0: "?w $ 0 \<noteq> 0" by (simp del: of_nat_Suc)
chaieb@29687
  1722
  note th0 = inverse_mult_eq_1[OF w0]
chaieb@29687
  1723
  let ?iw = "inverse ?w"
chaieb@29687
  1724
  from power_radical[of r, OF r0 a0]
chaieb@29687
  1725
  have "fps_deriv (?r ^ Suc k) = fps_deriv a" by simp
chaieb@29687
  1726
  hence "fps_deriv ?r * ?w = fps_deriv a"
chaieb@29687
  1727
    by (simp add: fps_deriv_power mult_ac)
chaieb@29687
  1728
  hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp
chaieb@29687
  1729
  hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
chaieb@29687
  1730
    by (simp add: fps_divide_def)
chaieb@29687
  1731
  then show ?thesis unfolding th0 by simp 
chaieb@29687
  1732
qed
chaieb@29687
  1733
chaieb@29687
  1734
lemma radical_mult_distrib: 
chaieb@29687
  1735
  fixes a:: "'a ::{field, ring_char_0, recpower} fps"
chaieb@29687
  1736
  assumes 
chaieb@29687
  1737
  ra0: "r (k) (a $ 0) ^ k = a $ 0" 
chaieb@29687
  1738
  and rb0: "r (k) (b $ 0) ^ k = b $ 0"
chaieb@29687
  1739
  and r0': "r (k) ((a * b) $ 0) = r (k) (a $ 0) * r (k) (b $ 0)"
chaieb@29687
  1740
  and a0: "a$0 \<noteq> 0"
chaieb@29687
  1741
  and b0: "b$0 \<noteq> 0"
chaieb@29687
  1742
  shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
chaieb@29687
  1743
proof-
chaieb@29687
  1744
  from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
chaieb@29687
  1745
    by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
chaieb@29687
  1746
  {assume "k=0" hence ?thesis by simp}
chaieb@29687
  1747
  moreover
chaieb@29687
  1748
  {fix h assume k: "k = Suc h"
chaieb@29687
  1749
  let ?ra = "fps_radical r (Suc h) a"
chaieb@29687
  1750
  let ?rb = "fps_radical r (Suc h) b"
chaieb@29687
  1751
  have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" 
chaieb@29687
  1752
    using r0' k by (simp add: fps_mult_nth)
chaieb@29687
  1753
  have ab0: "(a*b) $ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
chaieb@29687
  1754
  from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric] 
chaieb@29687
  1755
    power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
chaieb@29687
  1756
  have ?thesis by (auto simp add: power_mult_distrib)}
chaieb@29687
  1757
ultimately show ?thesis by (cases k, auto)
chaieb@29687
  1758
qed
chaieb@29687
  1759
chaieb@29687
  1760
lemma radical_inverse:
chaieb@29687
  1761
  fixes a:: "'a ::{field, ring_char_0, recpower} fps"
chaieb@29687
  1762
  assumes 
chaieb@29687
  1763
  ra0: "r (k) (a $ 0) ^ k = a $ 0" 
chaieb@29687
  1764
  and ria0: "r (k) (inverse (a $ 0)) = inverse (r (k) (a $ 0))"
chaieb@29687
  1765
  and r1: "(r (k) 1) = 1" 
chaieb@29687
  1766
  and a0: "a$0 \<noteq> 0"
chaieb@29687
  1767
  shows "fps_radical r (k) (inverse a) = inverse (fps_radical r (k) a)"
chaieb@29687
  1768
proof-
chaieb@29687
  1769
  {assume "k=0" then have ?thesis by simp}
chaieb@29687
  1770
  moreover
chaieb@29687
  1771
  {fix h assume k[simp]: "k = Suc h"
chaieb@29687
  1772
    let ?ra = "fps_radical r (Suc h) a"
chaieb@29687
  1773
    let ?ria = "fps_radical r (Suc h) (inverse a)"
chaieb@29687
  1774
    from ra0 a0 have th00: "r (Suc h) (a$0) \<noteq> 0" by auto
chaieb@29687
  1775
    have ria0': "r (Suc h) (inverse a $ 0) ^ Suc h = inverse a$0"
chaieb@29687
  1776
    using ria0 ra0 a0
chaieb@29687
  1777
    by (simp add: fps_inverse_def  nonzero_power_inverse[OF th00, symmetric])
chaieb@29687
  1778
  from inverse_mult_eq_1[OF a0] have th0: "a * inverse a = 1" 
chaieb@29687
  1779
    by (simp add: mult_commute)
chaieb@29687
  1780
  from radical_unique[where a=1 and b=1 and r=r and k=h, simplified, OF r1[unfolded k]]
chaieb@29687
  1781
  have th01: "fps_radical r (Suc h) 1 = 1" .
chaieb@29687
  1782
  have th1: "r (Suc h) ((a * inverse a) $ 0) ^ Suc h = (a * inverse a) $ 0"
chaieb@29687
  1783
    "r (Suc h) ((a * inverse a) $ 0) =
chaieb@29687
  1784
r (Suc h) (a $ 0) * r (Suc h) (inverse a $ 0)"
chaieb@29687
  1785
    using r1 unfolding th0  apply (simp_all add: ria0[symmetric])
chaieb@29687
  1786
    apply (simp add: fps_inverse_def a0)
chaieb@29687
  1787
    unfolding ria0[unfolded k]
chaieb@29687
  1788
    using th00 by simp
chaieb@29687
  1789
  from nonzero_imp_inverse_nonzero[OF a0] a0
chaieb@29687
  1790
  have th2: "inverse a $ 0 \<noteq> 0" by (simp add: fps_inverse_def)
chaieb@29687
  1791
  from radical_mult_distrib[of r "Suc h" a "inverse a", OF ra0[unfolded k] ria0' th1(2) a0 th2]
chaieb@29687
  1792
  have th3: "?ra * ?ria = 1" unfolding th0 th01 by simp
chaieb@29687
  1793
  from th00 have ra0: "?ra $ 0 \<noteq> 0" by simp
chaieb@29687
  1794
  from fps_inverse_unique[OF ra0 th3] have ?thesis by simp}
chaieb@29687
  1795
ultimately show ?thesis by (cases k, auto)
chaieb@29687
  1796
qed
chaieb@29687
  1797
chaieb@29687
  1798
lemma fps_divide_inverse: "(a::('a::field) fps) / b = a * inverse b"
chaieb@29687
  1799
  by (simp add: fps_divide_def)
chaieb@29687
  1800
chaieb@29687
  1801
lemma radical_divide:
chaieb@29687
  1802
  fixes a:: "'a ::{field, ring_char_0, recpower} fps"
chaieb@29687
  1803
  assumes 
chaieb@29687
  1804
      ra0: "r k (a $ 0) ^ k = a $ 0" 
chaieb@29687
  1805
  and rb0: "r k (b $ 0) ^ k = b $ 0"
chaieb@29687
  1806
  and r1: "r k 1 = 1"
chaieb@29687
  1807
  and rb0': "r k (inverse (b $ 0)) = inverse (r k (b $ 0))" 
chaieb@29687
  1808
  and raib': "r k (a$0 / (b$0)) = r k (a$0) / r k (b$0)"
chaieb@29687
  1809
  and a0: "a$0 \<noteq> 0" 
chaieb@29687
  1810
  and b0: "b$0 \<noteq> 0"
chaieb@29687
  1811
  shows "fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
chaieb@29687
  1812
proof-
chaieb@29687
  1813
  from raib'
chaieb@29687
  1814
  have raib: "r k (a$0 / (b$0)) = r k (a$0) * r k (inverse (b$0))"
chaieb@29687
  1815
    by (simp add: divide_inverse rb0'[symmetric])
chaieb@29687
  1816
chaieb@29687
  1817
  {assume "k=0" hence ?thesis by (simp add: fps_divide_def)}
chaieb@29687
  1818
  moreover
chaieb@29687
  1819
  {assume k0: "k\<noteq> 0"
chaieb@29687
  1820
    from b0 k0 rb0 have rbn0: "r k (b $0) \<noteq> 0"
chaieb@29687
  1821
      by (auto simp add: power_0_left)
chaieb@29687
  1822
    
chaieb@29687
  1823
    from rb0 rb0' have rib0: "(r k (inverse (b $ 0)))^k = inverse (b$0)"
chaieb@29687
  1824
    by (simp add: nonzero_power_inverse[OF rbn0, symmetric])
chaieb@29687
  1825
  from rib0 have th0: "r k (inverse b $ 0) ^ k = inverse b $ 0"
chaieb@29687
  1826
    by (simp add:fps_inverse_def b0)
chaieb@29687
  1827
  from raib 
chaieb@29687
  1828
  have th1: "r k ((a * inverse b) $ 0) = r k (a $ 0) * r k (inverse b $ 0)"
chaieb@29687
  1829
    by (simp add: divide_inverse fps_inverse_def  b0 fps_mult_nth)
chaieb@29687
  1830
  from nonzero_imp_inverse_nonzero[OF b0] b0 have th2: "inverse b $ 0 \<noteq> 0"
chaieb@29687
  1831
    by (simp add: fps_inverse_def)
chaieb@29687
  1832
  from radical_mult_distrib[of r k a "inverse b", OF ra0 th0 th1 a0 th2]
chaieb@29687
  1833
  have th: "fps_radical r k (a/b) = fps_radical r k a * fps_radical r k (inverse b)"
chaieb@29687
  1834
    by (simp add: fps_divide_def)
chaieb@29687
  1835
  with radical_inverse[of r k b, OF rb0 rb0' r1 b0]
chaieb@29687
  1836
  have ?thesis by (simp add: fps_divide_def)}
chaieb@29687
  1837
ultimately show ?thesis by blast
chaieb@29687
  1838
qed
chaieb@29687
  1839
chaieb@29687
  1840
section{* Derivative of composition *}
chaieb@29687
  1841
chaieb@29687
  1842
lemma fps_compose_deriv: 
chaieb@29687
  1843
  fixes a:: "('a::idom) fps"
chaieb@29687
  1844
  assumes b0: "b$0 = 0"
chaieb@29687
  1845
  shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)"
chaieb@29687
  1846
proof-
chaieb@29687
  1847
  {fix n
chaieb@29687
  1848
    have "(fps_deriv (a oo b))$n = setsum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
chaieb@29687
  1849
      by (simp add: fps_compose_def ring_simps setsum_right_distrib del: of_nat_Suc)
chaieb@29687
  1850
    also have "\<dots> = setsum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
chaieb@29687
  1851
      by (simp add: ring_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
chaieb@29687
  1852
  also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
chaieb@29687
  1853
    unfolding fps_mult_left_const_nth  by (simp add: ring_simps)
chaieb@29687
  1854
  also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (setsum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}"
chaieb@29687
  1855
    unfolding fps_mult_nth ..
chaieb@29687
  1856
  also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (setsum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}"
chaieb@29687
  1857
    apply (rule setsum_mono_zero_right)
chaieb@29687
  1858
    by (auto simp add: cond_value_iff cond_application_beta setsum_delta 
chaieb@29687
  1859
      not_le cong del: if_weak_cong)
chaieb@29687
  1860
  also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
chaieb@29687
  1861
    unfolding fps_deriv_nth
chaieb@29687
  1862
    apply (rule setsum_reindex_cong[where f="Suc"])
chaieb@29687
  1863
    by (auto simp add: mult_assoc)
chaieb@29687
  1864
  finally have th0: "(fps_deriv (a oo b))$n = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" .
chaieb@29687
  1865
  
chaieb@29687
  1866
  have "(((fps_deriv a) oo b) * (fps_deriv b))$n = setsum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
chaieb@29687
  1867
    unfolding fps_mult_nth by (simp add: mult_ac)
chaieb@29687
  1868
  also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
chaieb@29687
  1869
    unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
chaieb@29687
  1870
    apply (rule setsum_cong2)
chaieb@29687
  1871
    apply (rule setsum_mono_zero_left)
chaieb@29687
  1872
    apply (simp_all add: subset_eq)
chaieb@29687
  1873
    apply clarify
chaieb@29687
  1874
    apply (subgoal_tac "b^i$x = 0")
chaieb@29687
  1875
    apply simp
chaieb@29687
  1876
    apply (rule startsby_zero_power_prefix[OF b0, rule_format])
chaieb@29687
  1877
    by simp
chaieb@29687
  1878
  also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
chaieb@29687
  1879
    unfolding setsum_right_distrib
chaieb@29687
  1880
    apply (subst setsum_commute)
chaieb@29687
  1881
    by ((rule setsum_cong2)+) simp
chaieb@29687
  1882
  finally have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n"
chaieb@29687
  1883
    unfolding th0 by simp}
chaieb@29687
  1884
then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  1885
qed
chaieb@29687
  1886
chaieb@29687
  1887
lemma fps_mult_X_plus_1_nth:
chaieb@29687
  1888
  "((1+X)*a) $n = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
chaieb@29687
  1889
proof-
chaieb@29687
  1890
  {assume "n = 0" hence ?thesis by (simp add: fps_mult_nth )}
chaieb@29687
  1891
  moreover
chaieb@29687
  1892
  {fix m assume m: "n = Suc m"
chaieb@29687
  1893
    have "((1+X)*a) $n = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0..n}"
chaieb@29687
  1894
      by (simp add: fps_mult_nth)
chaieb@29687
  1895
    also have "\<dots> = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0.. 1}"
chaieb@29687
  1896
      unfolding m
chaieb@29687
  1897
      apply (rule setsum_mono_zero_right)
chaieb@29687
  1898
      by (auto simp add: )
chaieb@29687
  1899
    also have "\<dots> = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
chaieb@29687
  1900
      unfolding m
chaieb@29687
  1901
      by (simp add: )
chaieb@29687
  1902
    finally have ?thesis .}
chaieb@29687
  1903
  ultimately show ?thesis by (cases n, auto)
chaieb@29687
  1904
qed
chaieb@29687
  1905
chaieb@29687
  1906
section{* Finite FPS (i.e. polynomials) and X *}
chaieb@29687
  1907
lemma fps_poly_sum_X:
chaieb@29687
  1908
  assumes z: "\<forall>i > n. a$i = (0::'a::comm_ring_1)" 
chaieb@29687
  1909
  shows "a = setsum (\<lambda>i. fps_const (a$i) * X^i) {0..n}" (is "a = ?r")
chaieb@29687
  1910
proof-
chaieb@29687
  1911
  {fix i
chaieb@29687
  1912
    have "a$i = ?r$i" 
chaieb@29687
  1913
      unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
chaieb@29687
  1914
      apply (simp add: cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
chaieb@29687
  1915
      using z by auto}
chaieb@29687
  1916
  then show ?thesis unfolding fps_eq_iff by blast
chaieb@29687
  1917
qed
chaieb@29687
  1918
chaieb@29687
  1919
section{* Compositional inverses *}
chaieb@29687
  1920
chaieb@29687
  1921
chaieb@29687
  1922
fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::{recpower,field}" where
chaieb@29687
  1923
  "compinv a 0 = X$0"
chaieb@29687
  1924
| "compinv a (Suc n) = (X$ Suc n - setsum (\<lambda>i. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
chaieb@29687
  1925
chaieb@29687
  1926
definition "fps_inv a = Abs_fps (compinv a)"
chaieb@29687
  1927
chaieb@29687
  1928
lemma fps_inv: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
chaieb@29687
  1929
  shows "fps_inv a oo a = X"
chaieb@29687
  1930
proof-
chaieb@29687
  1931
  let ?i = "fps_inv a oo a"
chaieb@29687
  1932
  {fix n
chaieb@29687
  1933
    have "?i $n = X$n" 
chaieb@29687
  1934
    proof(induct n rule: nat_less_induct)
chaieb@29687
  1935
      fix n assume h: "\<forall>m<n. ?i$m = X$m"
chaieb@29687
  1936
      {assume "n=0" hence "?i $n = X$n" using a0 
chaieb@29687
  1937
	  by (simp add: fps_compose_nth fps_inv_def)}
chaieb@29687
  1938
      moreover
chaieb@29687
  1939
      {fix n1 assume n1: "n = Suc n1"
chaieb@29687
  1940
	have "?i $ n = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
chaieb@29687
  1941
	  by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0])
chaieb@29687
  1942
	also have "\<dots> = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + (X$ Suc n1 - setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
chaieb@29687
  1943
	  using a0 a1 n1 by (simp add: fps_inv_def fps_nth_def)
chaieb@29687
  1944
	also have "\<dots> = X$n" using n1 by simp 
chaieb@29687
  1945
	finally have "?i $ n = X$n" .}
chaieb@29687
  1946
      ultimately show "?i $ n = X$n" by (cases n, auto)
chaieb@29687
  1947
    qed}
chaieb@29687
  1948
  then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  1949
qed
chaieb@29687
  1950
chaieb@29687
  1951
chaieb@29687
  1952
fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::{recpower,field}" where
chaieb@29687
  1953
  "gcompinv b a 0 = b$0"
chaieb@29687
  1954
| "gcompinv b a (Suc n) = (b$ Suc n - setsum (\<lambda>i. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
chaieb@29687
  1955
chaieb@29687
  1956
definition "fps_ginv b a = Abs_fps (gcompinv b a)"
chaieb@29687
  1957
chaieb@29687
  1958
lemma fps_ginv: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
chaieb@29687
  1959
  shows "fps_ginv b a oo a = b"
chaieb@29687
  1960
proof-
chaieb@29687
  1961
  let ?i = "fps_ginv b a oo a"
chaieb@29687
  1962
  {fix n
chaieb@29687
  1963
    have "?i $n = b$n" 
chaieb@29687
  1964
    proof(induct n rule: nat_less_induct)
chaieb@29687
  1965
      fix n assume h: "\<forall>m<n. ?i$m = b$m"
chaieb@29687
  1966
      {assume "n=0" hence "?i $n = b$n" using a0 
chaieb@29687
  1967
	  by (simp add: fps_compose_nth fps_ginv_def)}
chaieb@29687
  1968
      moreover
chaieb@29687
  1969
      {fix n1 assume n1: "n = Suc n1"
chaieb@29687
  1970
	have "?i $ n = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
chaieb@29687
  1971
	  by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0])
chaieb@29687
  1972
	also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + (b$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
chaieb@29687
  1973
	  using a0 a1 n1 by (simp add: fps_ginv_def fps_nth_def)
chaieb@29687
  1974
	also have "\<dots> = b$n" using n1 by simp 
chaieb@29687
  1975
	finally have "?i $ n = b$n" .}
chaieb@29687
  1976
      ultimately show "?i $ n = b$n" by (cases n, auto)
chaieb@29687
  1977
    qed}
chaieb@29687
  1978
  then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  1979
qed
chaieb@29687
  1980
chaieb@29687
  1981
lemma fps_inv_ginv: "fps_inv = fps_ginv X"
chaieb@29687
  1982
  apply (auto simp add: expand_fun_eq fps_eq_iff fps_inv_def fps_ginv_def)
chaieb@29687
  1983
  apply (induct_tac n rule: nat_less_induct, auto)
chaieb@29687
  1984
  apply (case_tac na)
chaieb@29687
  1985
  apply simp
chaieb@29687
  1986
  apply simp
chaieb@29687
  1987
  done
chaieb@29687
  1988
chaieb@29687
  1989
lemma fps_compose_1[simp]: "1 oo a = 1"
chaieb@29687
  1990
  apply (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
chaieb@29687
  1991
  apply (simp add: setsum_delta)
chaieb@29687
  1992
  done
chaieb@29687
  1993
chaieb@29687
  1994
lemma fps_compose_0[simp]: "0 oo a = 0"
chaieb@29687
  1995
  by (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
chaieb@29687
  1996
chaieb@29687
  1997
lemma fps_pow_0: "fps_pow n 0 = (if n = 0 then 1 else 0)"
chaieb@29687
  1998
  by (induct n, simp_all)
chaieb@29687
  1999
chaieb@29687
  2000
lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a$0)"
chaieb@29687
  2001
  apply (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
chaieb@29687
  2002
  by (case_tac n, auto simp add: fps_pow_0 intro: setsum_0')
chaieb@29687
  2003
chaieb@29687
  2004
lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
chaieb@29687
  2005
  by (simp add: fps_eq_iff fps_compose_nth  ring_simps setsum_addf)
chaieb@29687
  2006
chaieb@29687
  2007
lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
chaieb@29687
  2008
proof-
chaieb@29687
  2009
  {assume "\<not> finite S" hence ?thesis by simp}
chaieb@29687
  2010
  moreover
chaieb@29687
  2011
  {assume fS: "finite S"
chaieb@29687
  2012
    have ?thesis
chaieb@29687
  2013
    proof(rule finite_induct[OF fS])
chaieb@29687
  2014
      show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
chaieb@29687
  2015
    next
chaieb@29687
  2016
      fix x F assume fF: "finite F" and xF: "x \<notin> F" and h: "setsum f F oo a = setsum (\<lambda>i. f i oo a) F"
chaieb@29687
  2017
      show "setsum f (insert x F) oo a  = setsum (\<lambda>i. f i oo a) (insert x F)"
chaieb@29687
  2018
	using fF xF h by (simp add: fps_compose_add_distrib)
chaieb@29687
  2019
    qed}
chaieb@29687
  2020
  ultimately show ?thesis by blast 
chaieb@29687
  2021
qed
chaieb@29687
  2022
chaieb@29687
  2023
lemma convolution_eq: 
chaieb@29687
  2024
  "setsum (%i. a (i :: nat) * b (n - i)) {0 .. n} = setsum (%(i,j). a i * b j) {(i,j). i <= n \<and> j \<le> n \<and> i + j = n}"
chaieb@29687
  2025
  apply (rule setsum_reindex_cong[where f=fst])
chaieb@29687
  2026
  apply (clarsimp simp add: inj_on_def)
chaieb@29687
  2027
  apply (auto simp add: expand_set_eq image_iff)
chaieb@29687
  2028
  apply (rule_tac x= "x" in exI)
chaieb@29687
  2029
  apply clarsimp
chaieb@29687
  2030
  apply (rule_tac x="n - x" in exI)
chaieb@29687
  2031
  apply arith
chaieb@29687
  2032
  done
chaieb@29687
  2033
chaieb@29687
  2034
lemma product_composition_lemma:
chaieb@29687
  2035
  assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0"
chaieb@29687
  2036
  shows "((a oo c) * (b oo d))$n = setsum (%(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m \<le> n}" (is "?l = ?r")
chaieb@29687
  2037
proof-
chaieb@29687
  2038
  let ?S = "{(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
chaieb@29687
  2039
  have s: "?S \<subseteq> {0..n} <*> {0..n}" by (auto simp add: subset_eq)  
chaieb@29687
  2040
  have f: "finite {(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}" 
chaieb@29687
  2041
    apply (rule finite_subset[OF s])
chaieb@29687
  2042
    by auto
chaieb@29687
  2043
  have "?r =  setsum (%i. setsum (%(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
chaieb@29687
  2044
    apply (simp add: fps_mult_nth setsum_right_distrib)
chaieb@29687
  2045
    apply (subst setsum_commute)
chaieb@29687
  2046
    apply (rule setsum_cong2)
chaieb@29687
  2047
    by (auto simp add: ring_simps)
chaieb@29687
  2048
  also have "\<dots> = ?l" 
chaieb@29687
  2049
    apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
chaieb@29687
  2050
    apply (rule setsum_cong2)
chaieb@29687
  2051
    apply (simp add: setsum_cartesian_product mult_assoc)
chaieb@29687
  2052
    apply (rule setsum_mono_zero_right[OF f])
chaieb@29687
  2053
    apply (simp add: subset_eq) apply presburger
chaieb@29687
  2054
    apply clarsimp
chaieb@29687
  2055
    apply (rule ccontr)
chaieb@29687
  2056
    apply (clarsimp simp add: not_le)
chaieb@29687
  2057
    apply (case_tac "x < aa")
chaieb@29687
  2058
    apply simp
chaieb@29687
  2059
    apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
chaieb@29687
  2060
    apply blast
chaieb@29687
  2061
    apply simp
chaieb@29687
  2062
    apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
chaieb@29687
  2063
    apply blast
chaieb@29687
  2064
    done
chaieb@29687
  2065
  finally show ?thesis by simp
chaieb@29687
  2066
qed
chaieb@29687
  2067
chaieb@29687
  2068
lemma product_composition_lemma':
chaieb@29687
  2069
  assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0"
chaieb@29687
  2070
  shows "((a oo c) * (b oo d))$n = setsum (%k. setsum (%m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}" (is "?l = ?r")
chaieb@29687
  2071
  unfolding product_composition_lemma[OF c0 d0]
chaieb@29687
  2072
  unfolding setsum_cartesian_product
chaieb@29687
  2073
  apply (rule setsum_mono_zero_left)
chaieb@29687
  2074
  apply simp
chaieb@29687
  2075
  apply (clarsimp simp add: subset_eq)
chaieb@29687
  2076
  apply clarsimp
chaieb@29687
  2077
  apply (rule ccontr)
chaieb@29687
  2078
  apply (subgoal_tac "(c^aa * d^ba) $ n = 0")
chaieb@29687
  2079
  apply simp
chaieb@29687
  2080
  unfolding fps_mult_nth
chaieb@29687
  2081
  apply (rule setsum_0')
chaieb@29687
  2082
  apply (clarsimp simp add: not_le)
chaieb@29687
  2083
  apply (case_tac "aaa < aa")
chaieb@29687
  2084
  apply (rule startsby_zero_power_prefix[OF c0, rule_format])
chaieb@29687
  2085
  apply simp
chaieb@29687
  2086
  apply (subgoal_tac "n - aaa < ba")
chaieb@29687
  2087
  apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
chaieb@29687
  2088
  apply simp
chaieb@29687
  2089
  apply arith
chaieb@29687
  2090
  done
chaieb@29687
  2091
  
chaieb@29687
  2092
chaieb@29687
  2093
lemma setsum_pair_less_iff: 
chaieb@29687
  2094
  "setsum (%((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} = setsum (%s. setsum (%i. a i * b (s - i) * c s) {0..s}) {0..n}" (is "?l = ?r")
chaieb@29687
  2095
proof-
chaieb@29687
  2096
  let ?KM=  "{(k,m). k + m \<le> n}"
chaieb@29687
  2097
  let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})"
chaieb@29687
  2098
  have th0: "?KM = UNION {0..n} ?f"
chaieb@29687
  2099
    apply (simp add: expand_set_eq)
chaieb@29687
  2100
    apply arith
chaieb@29687
  2101
    done
chaieb@29687
  2102
  show "?l = ?r "
chaieb@29687
  2103
    unfolding th0
chaieb@29687
  2104
    apply (subst setsum_UN_disjoint)
chaieb@29687
  2105
    apply auto
chaieb@29687
  2106
    apply (subst setsum_UN_disjoint)
chaieb@29687
  2107
    apply auto
chaieb@29687
  2108
    done
chaieb@29687
  2109
qed
chaieb@29687
  2110
chaieb@29687
  2111
lemma fps_compose_mult_distrib_lemma:
chaieb@29687
  2112
  assumes c0: "c$0 = (0::'a::idom)"
chaieb@29687
  2113
  shows "((a oo c) * (b oo c))$n = setsum (%s. setsum (%i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}" (is "?l = ?r")
chaieb@29687
  2114
  unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
chaieb@29687
  2115
  unfolding setsum_pair_less_iff[where a = "%k. a$k" and b="%m. b$m" and c="%s. (c ^ s)$n" and n = n] ..
chaieb@29687
  2116
chaieb@29687
  2117
chaieb@29687
  2118
lemma fps_compose_mult_distrib: 
chaieb@29687
  2119
  assumes c0: "c$0 = (0::'a::idom)"
chaieb@29687
  2120
  shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r")
chaieb@29687
  2121
  apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
chaieb@29687
  2122
  by (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
chaieb@29687
  2123
lemma fps_compose_setprod_distrib: 
chaieb@29687
  2124
  assumes c0: "c$0 = (0::'a::idom)"
chaieb@29687
  2125
  shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r")
chaieb@29687
  2126
  apply (cases "finite S")
chaieb@29687
  2127
  apply simp_all
chaieb@29687
  2128
  apply (induct S rule: finite_induct)
chaieb@29687
  2129
  apply simp
chaieb@29687
  2130
  apply (simp add: fps_compose_mult_distrib[OF c0])
chaieb@29687
  2131
  done
chaieb@29687
  2132
chaieb@29687
  2133
lemma fps_compose_power:   assumes c0: "c$0 = (0::'a::idom)"
chaieb@29687
  2134
  shows "(a oo c)^n = a^n oo c" (is "?l = ?r")
chaieb@29687
  2135
proof-
chaieb@29687
  2136
  {assume "n=0" then have ?thesis by simp}
chaieb@29687
  2137
  moreover
chaieb@29687
  2138
  {fix m assume m: "n = Suc m"
chaieb@29687
  2139
    have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}"
chaieb@29687
  2140
      by (simp_all add: setprod_constant m)
chaieb@29687
  2141
    then have ?thesis
chaieb@29687
  2142
      by (simp add: fps_compose_setprod_distrib[OF c0])}
chaieb@29687
  2143
  ultimately show ?thesis by (cases n, auto)
chaieb@29687
  2144
qed
chaieb@29687
  2145
chaieb@29687
  2146
lemma fps_const_mult_apply_left:
chaieb@29687
  2147
  "fps_const c * (a oo b) = (fps_const c * a) oo b"
chaieb@29687
  2148
  by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc)
chaieb@29687
  2149
chaieb@29687
  2150
lemma fps_const_mult_apply_right:
chaieb@29687
  2151
  "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
chaieb@29687
  2152
  by (auto simp add: fps_const_mult_apply_left mult_commute)
chaieb@29687
  2153
chaieb@29687
  2154
lemma fps_compose_assoc: 
chaieb@29687
  2155
  assumes c0: "c$0 = (0::'a::idom)" and b0: "b$0 = 0"
chaieb@29687
  2156
  shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
chaieb@29687
  2157
proof-
chaieb@29687
  2158
  {fix n
chaieb@29687
  2159
    have "?l$n = (setsum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n"
chaieb@29687
  2160
      by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left setsum_right_distrib mult_assoc fps_setsum_nth)
chaieb@29687
  2161
    also have "\<dots> = ((setsum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
chaieb@29687
  2162
      by (simp add: fps_compose_setsum_distrib)
chaieb@29687
  2163
    also have "\<dots> = ?r$n"
chaieb@29687
  2164
      apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
chaieb@29687
  2165
      apply (rule setsum_cong2)
chaieb@29687
  2166
      apply (rule setsum_mono_zero_right)
chaieb@29687
  2167
      apply (auto simp add: not_le)
chaieb@29687
  2168
      by (erule startsby_zero_power_prefix[OF b0, rule_format])
chaieb@29687
  2169
    finally have "?l$n = ?r$n" .}
chaieb@29687
  2170
  then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  2171
qed
chaieb@29687
  2172
chaieb@29687
  2173
chaieb@29687
  2174
lemma fps_X_power_compose:
chaieb@29687
  2175
  assumes a0: "a$0=0" shows "X^k oo a = (a::('a::idom fps))^k" (is "?l = ?r")
chaieb@29687
  2176
proof-
chaieb@29687
  2177
  {assume "k=0" hence ?thesis by simp}
chaieb@29687
  2178
  moreover
chaieb@29687
  2179
  {fix h assume h: "k = Suc h"
chaieb@29687
  2180
    {fix n
chaieb@29687
  2181
      {assume kn: "k>n" hence "?l $ n = ?r $n" using a0 startsby_zero_power_prefix[OF a0] h 
chaieb@29687
  2182
	  by (simp add: fps_compose_nth)}
chaieb@29687
  2183
      moreover
chaieb@29687
  2184
      {assume kn: "k \<le> n"
chaieb@29687
  2185
	hence "?l$n = ?r$n" apply (simp only: fps_compose_nth X_power_nth)
chaieb@29687
  2186
	  by (simp add: cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)}
chaieb@29687
  2187
      moreover have "k >n \<or> k\<le> n"  by arith
chaieb@29687
  2188
      ultimately have "?l$n = ?r$n"  by blast}
chaieb@29687
  2189
    then have ?thesis unfolding fps_eq_iff by blast}
chaieb@29687
  2190
  ultimately show ?thesis by (cases k, auto)
chaieb@29687
  2191
qed
chaieb@29687
  2192
chaieb@29687
  2193
lemma fps_inv_right: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
chaieb@29687
  2194
  shows "a oo fps_inv a = X"
chaieb@29687
  2195
proof-
chaieb@29687
  2196
  let ?ia = "fps_inv a"
chaieb@29687
  2197
  let ?iaa = "a oo fps_inv a"
chaieb@29687
  2198
  have th0: "?ia $ 0 = 0" by (simp add: fps_inv_def)
chaieb@29687
  2199
  have th1: "?iaa $ 0 = 0" using a0 a1 
chaieb@29687
  2200
    by (simp add: fps_inv_def fps_compose_nth)
chaieb@29687
  2201
  have th2: "X$0 = 0" by simp
chaieb@29687
  2202
  from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
chaieb@29687
  2203
  then have "(a oo fps_inv a) oo a = X oo a"
chaieb@29687
  2204
    by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0])
chaieb@29687
  2205
  with fps_compose_inj_right[OF a0 a1]
chaieb@29687
  2206
  show ?thesis by simp 
chaieb@29687
  2207
qed
chaieb@29687
  2208
chaieb@29687
  2209
lemma fps_inv_deriv:
chaieb@29687
  2210
  assumes a0:"a$0 = (0::'a::{recpower,field})" and a1: "a$1 \<noteq> 0"
chaieb@29687
  2211
  shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
chaieb@29687
  2212
proof-
chaieb@29687
  2213
  let ?ia = "fps_inv a"
chaieb@29687
  2214
  let ?d = "fps_deriv a oo ?ia"
chaieb@29687
  2215
  let ?dia = "fps_deriv ?ia"
chaieb@29687
  2216
  have ia0: "?ia$0 = 0" by (simp add: fps_inv_def)
chaieb@29687
  2217
  have th0: "?d$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth fps_deriv_nth)
chaieb@29687
  2218
  from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
chaieb@29687
  2219
    by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
chaieb@29687
  2220
  hence "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
chaieb@29687
  2221
  with inverse_mult_eq_1[OF th0]
chaieb@29687
  2222
  show "?dia = inverse ?d" by simp
chaieb@29687
  2223
qed
chaieb@29687
  2224
chaieb@29687
  2225
section{* Elementary series *}
chaieb@29687
  2226
chaieb@29687
  2227
subsection{* Exponential series *}
chaieb@29687
  2228
definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"   
chaieb@29687
  2229
chaieb@29687
  2230
lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::{field, recpower, ring_char_0}) * E a" (is "?l = ?r")
chaieb@29687
  2231
proof-
chaieb@29687
  2232
  {fix n
chaieb@29687
  2233
    have "?l$n = ?r $ n"
chaieb@29687
  2234
  apply (auto simp add: E_def field_simps power_Suc[symmetric]simp del: fact_Suc of_nat_Suc)
chaieb@29687
  2235
  by (simp add: of_nat_mult ring_simps)}
chaieb@29687
  2236
then show ?thesis by (simp add: fps_eq_iff)
chaieb@29687
  2237
qed
chaieb@29687
  2238
chaieb@29687
  2239
lemma E_unique_ODE: 
chaieb@29687
  2240
  "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * E (c :: 'a::{field, ring_char_0, recpower})"
chaieb@29687
  2241
  (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29687
  2242
proof-
chaieb@29687
  2243
  {assume d: ?lhs
chaieb@29687
  2244
  from d have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)" 
chaieb@29687
  2245
    by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
chaieb@29687
  2246
  {fix n have "a$n = a$0 * c ^ n/ (of_nat (fact n))"
chaieb@29687
  2247
      apply (induct n)
chaieb@29687
  2248
      apply simp
chaieb@29687
  2249
      unfolding th 
chaieb@29687
  2250
      using fact_gt_zero
chaieb@29687
  2251
      apply (simp add: field_simps del: of_nat_Suc fact.simps)
chaieb@29687
  2252
      apply (drule sym)
chaieb@29687
  2253
      by (simp add: ring_simps of_nat_mult power_Suc)}
chaieb@29687
  2254
  note th' = this
chaieb@29687
  2255
  have ?rhs 
chaieb@29687
  2256
    by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro : th')}
chaieb@29687
  2257
moreover
chaieb@29687
  2258
{assume h: ?rhs
chaieb@29687
  2259
  have ?lhs 
chaieb@29687
  2260
    apply (subst h)
chaieb@29687
  2261
    apply simp
chaieb@29687
  2262
    apply (simp only: h[symmetric])
chaieb@29687
  2263
  by simp}
chaieb@29687
  2264
ultimately show ?thesis by blast
chaieb@29687
  2265
qed
chaieb@29687
  2266
chaieb@29687
  2267
lemma E_add_mult: "E (a + b) = E (a::'a::{ring_char_0, field, recpower}) * E b" (is "?l = ?r")
chaieb@29687
  2268
proof-
chaieb@29687
  2269
  have "fps_deriv (?r) = fps_const (a+b) * ?r"
chaieb@29687
  2270
    by (simp add: fps_const_add[symmetric] ring_simps del: fps_const_add)
chaieb@29687
  2271
  then have "?r = ?l" apply (simp only: E_unique_ODE)
chaieb@29687
  2272
    by (simp add: fps_mult_nth E_def)
chaieb@29687
  2273
  then show ?thesis ..
chaieb@29687
  2274
qed
chaieb@29687
  2275
chaieb@29687
  2276
lemma E_nth[simp]: "E a $ n = a^n / of_nat (fact n)"
chaieb@29687
  2277
  by (simp add: E_def)
chaieb@29687
  2278
chaieb@29687
  2279
lemma E0[simp]: "E (0::'a::{field, recpower}) = 1"
chaieb@29687
  2280
  by (simp add: fps_eq_iff power_0_left)
chaieb@29687
  2281
chaieb@29687
  2282
lemma E_neg: "E (- a) = inverse (E (a::'a::{ring_char_0, field, recpower}))"
chaieb@29687
  2283
proof-
chaieb@29687
  2284
  from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
chaieb@29687
  2285
    by (simp )
chaieb@29687
  2286
  have th1: "E a $ 0 \<noteq> 0" by simp
chaieb@29687
  2287
  from fps_inverse_unique[OF th1 th0] show ?thesis by simp
chaieb@29687
  2288
qed
chaieb@29687
  2289
chaieb@29687
  2290
lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::{field, recpower, ring_char_0})) = (fps_const a)^n * (E a)"  
chaieb@29687
  2291
  by (induct n, auto simp add: power_Suc)
chaieb@29687
  2292
chaieb@29687
  2293
lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
chaieb@29687
  2294
  by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_negf[symmetric])
chaieb@29687
  2295
chaieb@29687
  2296
lemma fps_compose_sub_distrib: 
chaieb@29687
  2297
  shows "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
chaieb@29687
  2298
  unfolding diff_minus fps_compose_uminus fps_compose_add_distrib ..
chaieb@29687
  2299
chaieb@29687
  2300
lemma X_fps_compose:"X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a$n)"
chaieb@29687
  2301
  apply (simp add: fps_eq_iff fps_compose_nth)
chaieb@29687
  2302
  by (simp add: cond_value_iff cond_application_beta setsum_delta power_Suc cong del: if_weak_cong)
chaieb@29687
  2303
chaieb@29687
  2304
lemma X_compose_E[simp]: "X oo E (a::'a::{field, recpower}) = E a - 1"
chaieb@29687
  2305
  by (simp add: fps_eq_iff X_fps_compose)
chaieb@29687
  2306
chaieb@29687
  2307
lemma LE_compose: 
chaieb@29687
  2308
  assumes a: "a\<noteq>0" 
chaieb@29687
  2309
  shows "fps_inv (E a - 1) oo (E a - 1) = X"
chaieb@29687
  2310
  and "(E a - 1) oo fps_inv (E a - 1) = X"
chaieb@29687
  2311
proof-
chaieb@29687
  2312
  let ?b = "E a - 1"
chaieb@29687
  2313
  have b0: "?b $ 0 = 0" by simp
chaieb@29687
  2314
  have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
chaieb@29687
  2315
  from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" .
chaieb@29687
  2316
  from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" .
chaieb@29687
  2317
qed
chaieb@29687
  2318
chaieb@29687
  2319
chaieb@29687
  2320
lemma fps_const_inverse: 
chaieb@29687
  2321
  "inverse (fps_const (a::'a::{field, division_by_zero})) = fps_const (inverse a)"
chaieb@29687
  2322
  apply (auto simp add: fps_eq_iff fps_inverse_def) by (case_tac "n", auto)
chaieb@29687
  2323
chaieb@29687
  2324
chaieb@29687
  2325
lemma inverse_one_plus_X: 
chaieb@29687
  2326
  "inverse (1 + X) = Abs_fps (\<lambda>n. (- 1 ::'a::{field, recpower})^n)"
chaieb@29687
  2327
  (is "inverse ?l = ?r")
chaieb@29687
  2328
proof-
chaieb@29687
  2329
  have th: "?l * ?r = 1"
chaieb@29687
  2330
    apply (auto simp add: ring_simps fps_eq_iff X_mult_nth  minus_one_power_iff)
chaieb@29687
  2331
    apply presburger+
chaieb@29687
  2332
    done
chaieb@29687
  2333
  have th': "?l $ 0 \<noteq> 0" by (simp add: )
chaieb@29687
  2334
  from fps_inverse_unique[OF th' th] show ?thesis .
chaieb@29687
  2335
qed
chaieb@29687
  2336
chaieb@29687
  2337
lemma E_power_mult: "(E (c::'a::{field,recpower,ring_char_0}))^n = E (of_nat n * c)"
chaieb@29687
  2338
  by (induct n, auto simp add: ring_simps E_add_mult power_Suc)
chaieb@29687
  2339
chaieb@29687
  2340
subsection{* Logarithmic series *}  
chaieb@29687
  2341
definition "(L::'a::{field, ring_char_0,recpower} fps) 
chaieb@29687
  2342
  = Abs_fps (\<lambda>n. (- 1) ^ Suc n / of_nat n)"
chaieb@29687
  2343
chaieb@29687
  2344
lemma fps_deriv_L: "fps_deriv L = inverse (1 + X)"
chaieb@29687
  2345
  unfolding inverse_one_plus_X
chaieb@29687
  2346
  by (simp add: L_def fps_eq_iff power_Suc del: of_nat_Suc)
chaieb@29687
  2347
chaieb@29687
  2348
lemma L_nth: "L $ n = (- 1) ^ Suc n / of_nat n"
chaieb@29687
  2349
  by (simp add: L_def)
chaieb@29687
  2350
chaieb@29687
  2351
lemma L_E_inv:
chaieb@29687
  2352
  assumes a: "a\<noteq> (0::'a::{field,division_by_zero,ring_char_0,recpower})" 
chaieb@29687
  2353
  shows "L = fps_const a * fps_inv (E a - 1)" (is "?l = ?r")
chaieb@29687
  2354
proof-
chaieb@29687
  2355
  let ?b = "E a - 1"
chaieb@29687
  2356
  have b0: "?b $ 0 = 0" by simp
chaieb@29687
  2357
  have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
chaieb@29687
  2358
  have "fps_deriv (E a - 1) oo fps_inv (E a - 1) = (fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)"
chaieb@29687
  2359
    by (simp add: ring_simps)
chaieb@29687
  2360
  also have "\<dots> = fps_const a * (X + 1)" apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
chaieb@29687
  2361
    by (simp add: ring_simps)
chaieb@29687
  2362
  finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
chaieb@29687
  2363
  from fps_inv_deriv[OF b0 b1, unfolded eq]
chaieb@29687
  2364
  have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
chaieb@29687
  2365
    by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
chaieb@29687
  2366
  hence "fps_deriv (fps_const a * fps_inv ?b) = inverse (X + 1)"
chaieb@29687
  2367
    using a by (simp add: fps_divide_def field_simps)
chaieb@29687
  2368
  hence "fps_deriv ?l = fps_deriv ?r" 
chaieb@29687
  2369
    by (simp add: fps_deriv_L add_commute)
chaieb@29687
  2370
  then show ?thesis unfolding fps_deriv_eq_iff
chaieb@29687
  2371
    by (simp add: L_nth fps_inv_def)
chaieb@29687
  2372
qed
chaieb@29687
  2373
chaieb@29687
  2374
subsection{* Formal trigonometric functions  *}
chaieb@29687
  2375
chaieb@29687
  2376
definition "fps_sin (c::'a::{field, recpower, ring_char_0}) = 
chaieb@29687
  2377
  Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
chaieb@29687
  2378
chaieb@29687
  2379
definition "fps_cos (c::'a::{field, recpower, ring_char_0}) = Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
chaieb@29687
  2380
chaieb@29687
  2381
lemma fps_sin_deriv: 
chaieb@29687
  2382
  "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
chaieb@29687
  2383
  (is "?lhs = ?rhs")
chaieb@29687
  2384
proof-
chaieb@29687
  2385
  {fix n::nat
chaieb@29687
  2386
    {assume en: "even n"
chaieb@29687
  2387
      have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
chaieb@29687
  2388
      also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))" 
chaieb@29687
  2389
	using en by (simp add: fps_sin_def)
chaieb@29687
  2390
      also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
chaieb@29687
  2391
	unfolding fact_Suc of_nat_mult
chaieb@29687
  2392
	by (simp add: field_simps del: of_nat_add of_nat_Suc)
chaieb@29687
  2393
      also have "\<dots> = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
chaieb@29687
  2394
	by (simp add: field_simps del: of_nat_add of_nat_Suc)
chaieb@29687
  2395
      finally have "?lhs $n = ?rhs$n" using en 
chaieb@29687
  2396
	by (simp add: fps_cos_def ring_simps power_Suc )}
chaieb@29687
  2397
    then have "?lhs $ n = ?rhs $ n" 
chaieb@29687
  2398
      by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def) }
chaieb@29687
  2399
  then show ?thesis by (auto simp add: fps_eq_iff)
chaieb@29687
  2400
qed
chaieb@29687
  2401
chaieb@29687
  2402
lemma fps_cos_deriv: 
chaieb@29687
  2403
  "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
chaieb@29687
  2404
  (is "?lhs = ?rhs")
chaieb@29687
  2405
proof-
chaieb@29687
  2406
  have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc)
chaieb@29687
  2407
  have th1: "\<And>n. odd n\<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2" by presburger
chaieb@29687
  2408
  {fix n::nat
chaieb@29687
  2409
    {assume en: "odd n"
chaieb@29687
  2410
      from en have n0: "n \<noteq>0 " by presburger
chaieb@29687
  2411
      have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
chaieb@29687
  2412
      also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))" 
chaieb@29687
  2413
	using en by (simp add: fps_cos_def)
chaieb@29687
  2414
      also have "\<dots> = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
chaieb@29687
  2415
	unfolding fact_Suc of_nat_mult
chaieb@29687
  2416
	by (simp add: field_simps del: of_nat_add of_nat_Suc)
chaieb@29687
  2417
      also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
chaieb@29687
  2418
	by (simp add: field_simps del: of_nat_add of_nat_Suc)
chaieb@29687
  2419
      also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
chaieb@29687
  2420
	unfolding th0 unfolding th1[OF en] by simp
chaieb@29687
  2421
      finally have "?lhs $n = ?rhs$n" using en 
chaieb@29687
  2422
	by (simp add: fps_sin_def fps_uminus_def ring_simps power_Suc)}
chaieb@29687
  2423
    then have "?lhs $ n = ?rhs $ n" 
chaieb@29687
  2424
      by (cases "even n", simp_all add: fps_deriv_def fps_sin_def 
chaieb@29687
  2425
	fps_cos_def fps_uminus_def) }
chaieb@29687
  2426
  then show ?thesis by (auto simp add: fps_eq_iff)
chaieb@29687
  2427
qed
chaieb@29687
  2428
chaieb@29687
  2429
lemma fps_sin_cos_sum_of_squares:
chaieb@29687
  2430
  "fps_cos c ^ 2 + fps_sin c ^ 2 = 1" (is "?lhs = 1")
chaieb@29687
  2431
proof-
chaieb@29687
  2432
  have "fps_deriv ?lhs = 0"
chaieb@29687
  2433
    apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc)
chaieb@29687
  2434
    by (simp add: fps_power_def ring_simps fps_const_neg[symmetric] del: fps_const_neg)
chaieb@29687
  2435
  then have "?lhs = fps_const (?lhs $ 0)"
chaieb@29687
  2436
    unfolding fps_deriv_eq_0_iff .
chaieb@29687
  2437
  also have "\<dots> = 1"
chaieb@29687
  2438
    by (auto simp add: fps_eq_iff fps_power_def nat_number fps_mult_nth fps_cos_def fps_sin_def)
chaieb@29687
  2439
  finally show ?thesis .
chaieb@29687
  2440
qed
chaieb@29687
  2441
chaieb@29687
  2442
definition "fps_tan c = fps_sin c / fps_cos c"
chaieb@29687
  2443
chaieb@29687
  2444
lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)"
chaieb@29687
  2445
proof-
chaieb@29687
  2446
  have th0: "fps_cos c $ 0 \<noteq> 0" by (simp add: fps_cos_def)
chaieb@29687
  2447
  show ?thesis 
chaieb@29687
  2448
    using fps_sin_cos_sum_of_squares[of c]
chaieb@29687
  2449
    apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv add: fps_const_neg[symmetric] ring_simps power2_eq_square del: fps_const_neg)
chaieb@29687
  2450
    unfolding right_distrib[symmetric]
chaieb@29687
  2451
    by simp
chaieb@29687
  2452
qed
chaieb@29687
  2453
end